SUM RULES FOR JACOBI MATRICES

AND THEIR APPLICATIONS TO SPECTRAL THEORY

ROWAN KILLIP
1
AND BARRY SIMON
2
Abstract. We discuss the proof of and systematic application of Cases
sum rules for Jacobi matrices. Of special interest is a linear combination
of two of his sum rules which has strictly positive terms. Among our
results are a complete classication of the spectral measures of all Jacobi
matrices J for which J J0 is HilbertSchmidt, and a proof of Nevais
conjecture that the Szeg o condition holds if J J0 is trace class.
1. Introduction
In this paper, we will look at the spectral theory of Jacobi matrices, that
is, innite tridiagonal matrices,
J =
_
_
_
_
_
b
1
a
1
0 0
a
1
b
2
a
2
0
0 a
2
b
3
a
3

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
_
_
_
_
_
(1.1)
with a
j
> 0 and b
j
R. We suppose that the entries of J are bounded,
that is, sup
n
[a
n
[ + sup
n
[b
n
[ < so that J denes a bounded self-adjoint
operator on
2
(Z
+
) =
2
(1, 2, . . . ). Let
j
be the obvious vector in
2
(Z
+
),
that is, with components
jn
which are 1 if n = j and 0 if n ,= j.
The spectral measure we associate to J is the one given by the spectral
theorem for the vector
1
. That is, the measure dened by
m

(E)
1
, (J E)
1

1
) =
_
d(x)
x E
. (1.2)
There is a one-one correspondence between bounded Jacobi matrices and
unit measures whose support is both compact and contains an innite num-
ber of points. As we have described, one goes fromJ to by the spectral the-
orem. One way to nd J, given , is via orthogonal polynomials. Applying
Date: December 3, 2001.
1
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, U.S.A.
E-mail: killip@math.ias.edu. Supported in part by NSF grant DMS-9729992.
2
Department of Mathematics 25337, California Institute of Technology, Pasadena,
CA 91125, U.S.A. E-mail: bsimon@caltech.edu. Supported in part by NSF grant DMS-
9707661.
1
2 R. KILLIP AND B. SIMON
the GramSchmidt process to x
n

n=0
, one gets orthonormal polynomials
P
n
(x) =
n
x
n
+ with
n
> 0 and
_
P
n
(x)P
m
(x) d(x) =
nm
. (1.3)
These polynomials obey a three term recurrence:
xP
n
(x) = a
n+1
P
n+1
(x) +b
n+1
P
n
(x) +a
n
P
n1
(x), (1.4)
where a
n
, b
n
are the Jacobi matrix coecients of the Jacobi matrix with
spectral measure (and P
1
0).
The more usual convention in the orthogonal polynomial literature is
to start numbering of a
n
and b
n
with n = 0 and then to have (1.4)
with (a
n
, b
n
, a
n1
) instead of (a
n+1
, b
n+1
, a
n
). We made our choice to start
numbering of J at n = 1 so that we could have z
n
for the free Jost function
(well known in the physics literature with z = e
ik
) and yet arrange for the
Jost function to be regular at z = 0. (Cases Jost function in [6, 7] has a
pole since where we use u
0
below, he uses u
1
because his numbering starts
at n = 0.) There is, in any event, a notational conundrum which we solved
in a way that we hope wont oend too many.
An alternate way of recovering J from is the continued fraction expan-
sion for the function m

(z) near innity,

m

(E) =
1
E +b
1
+
a
2
1
E +b
2
+
(1.5)
Both methods for nding J essentially go back to Stieltjes monumental
paper [57]. Three-term recurrence relations appeared earlier in the work
of Chebyshev and Markov but, of course, Stieltjes was the rst to consider
general measures in this context. While [57] does not have the continued
fraction expansion given in (1.5), Stieltjes did discuss (1.5) elsewhere. Wall
[62] calls (1.5) a J-fraction and the fractions used in [57], he calls S-fractions.
This has been discussed in many places, for example, [24, 56].
That every J corresponds to a spectral measure is known in the orthogo-
nal polynomial literature as Favards theorem (after Favard [15]). As noted,
it is a consequence for bounded J of Hilberts spectral theorem for bounded
operators. This appears already in the HellingerToeplitz encyclopedic ar-
ticle [26]. Even for the general unbounded case, Stones book [58] noted this
consequence before Favards work.
Given the one-one correspondence between s and Js, it is natural to ask
how properties of one are reected in the other. One is especially interested
in Js close to the free matrix, J
0
with a
n
= 1 and b
n
= 0, that is,
J
0
=
_
_
_
_
0 1 0 0 . . .
1 0 1 0 . . .
0 1 0 1 . . .
0 0 1 0 . . .
_
_
_
_
(1.6)
SUM RULES FOR JACOBI MATRICES 3
In the orthogonal polynomial literature, the free Jacobi matrix is taken
as
1
2
of our J
0
since then the associated orthogonal polynomials are precisely
Chebyshev polynomials (of the second kind). As a result, the spectral mea-
sure of our J
0
is supported by [2, 2] and the natural parametrization is
E = 2 cos .
Here is one of our main results:
Theorem 1. Let J be a Jacobi matrix and the corresponding spectral
measure. The operator J J
0
is HilbertSchmidt, that is,
2

n
(a
n
1)
2
+

b
2
n
< (1.7)
if and only if has the following four properties:
(0) (BlumenthalWeyl Criterion) The support of is [2, 2] E
+
j

N+
j=1

j

N
j=1
where N

are each zero, nite, or innite, and E

+
1
> E
+
2
>
> 2 and E

1
< E

2
< < 2 and if N

is innite, then
lim
j
E

j
= 2.
(1) (Quasi-Szego Condition) Let
ac
(E) = f(E) dE where
ac
is the
Lebesgue absolutely continuous component of . Then
_
2
2
log[f(E)]
_
4 E
2
dE > . (1.8)
(2) (LiebThirring Bound)
N+

j=1
(E
+
j
2)
3/2
+
N

j=1
(E

j
+ 2)
3/2
< (1.9)
(3) (Normalization)
_
d(E) = 1
Remarks. 1. Condition (0) is just a quantitative way of writing that the
essential spectrum of J is the same as that of J
0
, viz. [2, 2], consistent with
the compactness of J J
0
. This is, of course, Weyls invariance theorem
[63, 45]. Earlier, Blumenthal [5] proved something close to this in spirit for
the case of orthogonal polynomials.
2. Equation (1.9) is a Jacobi analog of a celebrated bound of Lieb and
Thirring [37, 38] for Schr odinger operators. That it holds if J J
0
is
HilbertSchmidt has also been recently proven by HundertmarkSimon [27],
although we do not use the
3
2
-bound of [27] below. We essentially reprove
(1.9) if (1.7) holds.
3. We call (1.8) the quasi-Szeg o condition to distinguish it from the Szeg o
condition,
_
2
2
log[f(E)](4 E
2
)
1/2
dE > . (1.10)
This is stronger than (1.8) although the dierence only matters if f vanishes
extremely rapidly at 2. For example, like exp((2[E[)

) with
1
2
<
3
2
.
Such behavior actually occurs for certain Pollaczek polynomials [8].
4 R. KILLIP AND B. SIMON
4. It will often be useful to have a single sequence e
1
(J), e
2
(J), . . . ob-
tained from the numbers [E

j
2[ by reordering so e
1
(J) e
2
(J) 0.
By property (1), for any J with J J
0
HilbertSchmidt, the essential
support of the a.c. spectrum is [2, 2]. That is,
ac
gives positive weight to
any subset of [2, 2] with positive measure. This follows from (1.8) because
f cannot vanish on any such set. This observation is the Jacobi matrix ana-
logue of recent results which show that (continuous and discrete) Schr odinger
operators with potentials V L
2
, p 2, or [V (x)[ (1 +x
2
)
/2
, > 1/2,
have a.c. spectrum. (It is known that the a.c. spectrum can disappear once
p > 2 or 1/2.) Research in this direction began with Kiselev [29] and
culminated in the work of ChristKiselev [11], Remling [47], DeiftKillip [13],
and Killip [28]. Especially relevant here is the work of DeiftKillip who used
sum rules for nite range perturbations to obtain an a priori estimate. Our
work diers from theirs (and the follow-up papers of MolchanovNovitskii
Vainberg [40] and LaptevNabokoSafronov [36]) in two critical ways: we
deal with the half-line sum rules so the eigenvalues are the ones for the prob-
lem of interest and we show that the sum rules still hold in the limit. These
developments are particularly important for the converse direction (i.e., if
obeys (03) then J J
0
is HilbertSchmidt).
In Theorem 1, the only restriction on the singular part of on [2, 2] is
in terms of its total mass. Given any singular measure
sing
supported on
[2, 2] with total mass less than one, there is a Jacobi matrix J obeying
(1.7) for which this is the singular part of the spectral measure. Similarly,
the only restriction on the norming constants, that is, the values of (E

j
),
is that their sum must be less than one.
In the related setting of Schr odinger operators on R, Denisov [14] has
constructed an L
2
potential which gives rise to embedded singular continu-
ous spectrum. In this vein see also Kiselev [30]. We realized that the key
to Denisovs result was a sum rule, not the particular method he used to
construct his potentials. We decided to focus rst on the discrete case where
one avoids certain technicalities, but are turning to the continuum case.
While (1.8) is the natural condition when J J
0
is HilbertSchmidt, we
have a one-directional result for the Szeg o condition. We prove the following
conjecture of Nevai [43]:
Theorem 2. If J J
0
is in trace class, that is,

n
[a
n
1[ +

n
[b
n
[ < , (1.11)
then the Szeg o condition (1.10) holds.
Remark. Nevai [42] and GeronimoVan Assche [22] prove the Szego con-
dition holds under the slightly stronger hypothesis

n
(log n)[a
n
1[ +

n
(log n)[b
n
[ < .
We will also prove
SUM RULES FOR JACOBI MATRICES 5
Theorem 3. If J J
0
is compact and
(i)

j
[E
+
j
2[
1/2
+

j
[E

j
+ 2[
1/2
< (1.12)
(ii) liminf
N
a
1
. . . a
N
> 0
then (1.10) holds.
We will prove Theorem 2 from Theorem 3 by using a
1
2
power Lieb
Thirring inequality, as proven by HundertmarkSimon [27].
For the special case where has no mass outside [2, 2] (i.e., N
+
=
N

= 0), there are over seventy years of results related to Theorem 1 with
important contributions by Szeg o [59, 60], Shohat [49], Geronomius [23],
Krein [33], and Kolmogorov [32]. Their results are summarized by Nevai
[43] as:
Theorem 4 (Previously Known). Suppose is a probability measure sup-
ported on [2, 2]. The Szeg o condition (1.10) holds if and only if
(i) J J
0
is HilbertSchmidt.
(ii)

(a
n
1) and

b
n
are (conditionally) convergent.
Of course, the major dierence between this result and Theorem 1 is
that we can handle bound states (i.e., eigenvalues outside [2, 2]) and the
methods of Szego, Shohat, and Geronimus seem unable to. Indeed, as we will
see below, the condition of no eigenvalues is very restrictive. A second issue
is that we focus on the previously unstudied (or lightly studied; e.g., it is
mentioned in [39]) condition which we have called the quasi-Szeg o condition
(1.8), which is strictly weaker than the Szeg o condition (1.10). Third, related
to the rst point, we do not have any requirement for conditional convergence
of

N
n=1
(a
n
1) or

N
n=1
b
n
.
The Szego condition, though, has other uses (see Szego [60], Akhiezer [2]),
so it is a natural object independently of the issue of studying the spectral
condition.
We emphasize that the assumption that has no pure points outside
[2, 2] is extremely strong. Indeed, while the Szeg o condition plus this
assumption implies (i) and (ii) above, to deduce the Szeg o condition requires
only a very small part of (ii). We will prove the following that includes
Theorem 4.
Theorem 4

. If (J) [2, 2] and

(i) limsup
N

N
n=1
log(a
n
) > ,
then the Szeg o condition holds. If (J) [2, 2] and either (i) or the Szego
condition holds, then
(ii)

n=1
(a
n
1)
2
+

n=1
b
2
n
<
(iii) lim
N

N
n=1
log(a
n
) exists (and is nite)
(iv) lim
N

N
n=1
b
n
exists (and is nite).
In particular, if (J) [2, 2], then (i) implies (ii)(iv).
6 R. KILLIP AND B. SIMON
In Nevai [41], it is stated and proven (see pg. 124) that

n=1
[a
n
1[ <
implies the Szego condition, but it turns out that his method of proof only
requires our condition (i). Nevai informs us that he believes his result was
probably known to Geronimus.
The key to our proofs is a family of sum rules stated by Case in [7]. Case
was motivated by Flaschkas calculation of the rst integrals for the Toda
lattice for nite [16] and doubly innite Jacobi matrices [17]. Cases method
of proof is partly patterned after that of Flaschka in [17].
To state these rules, it is natural to change variables from E to z via
E = z +
1
z
. (1.13)
We choose the solution of (1.13) with [z[ < 1, namely
z =
1
2
_
E
_
E
2
4

, (1.14)
where we take the branch of

with

> 0 for > 0. In this way, E z
is the conformal map of C[2, 2] to D z [ [z[ < 1, which takes
to 0 and (in the limit) 2 to 1. The points E [2, 2] are mapped to
z = e
i
where E = 2 cos .
The conformal map suggests replacing m

by
M

(z) = m

_
E(z)
_
= m

_
z + z
1
_
=
_
z d(x)
1 xz +z
2
. (1.15)
We have introduced a minus sign so that ImM

(z) > 0 when Imz > 0.

Note that ImE > 0 m

(E) > 0 but E z maps the upper half-plane to

the lower half-disk.
If obeys the BlumenthalWeyl criterion, M

is meromorphic on D with
poles at the points (

j
)
1
where
[
j
[ > 1 and E

j
=

j
+ (

j
)
1
. (1.16)
As with E

j
, we renumber

j
to a single sequence [
1
[ [
2
[ 1.
By general principles, M

has boundary values a.e. on the circle,

M

(e
i
) = lim
r1
M

(re
i
) (1.17)
with M

(e
i
) = M

(e
i
) and ImM

(e
i
) 0 for (0, ).
From the integral representation (1.2),
Imm

(E + i0) =
d
ac
dE
(1.18)
so using dE = 2 sin d = (4 E
2
)
1/2
d, the quasi-Szego condition
(1.10) becomes
4
_

0
log[ImM

(e
i
)] sin
2
d >
SUM RULES FOR JACOBI MATRICES 7
and the Szeg o condition (1.8) is
_

0
log[ImM

(e
i
)] d > .
Moreover, we have by (1.18) that
2

_

0
Im[M

(e
i
)] sin d =
ac
(2, 2) 1. (1.19)
With these notational preliminaries out of the way, we can state Cases
sum rules. For future reference, we give them names:
C
0
:
1
4
_

log
_
sin
ImM(e
i
)
_
d =

j
log[
j
[

j
log[a
j
[ (1.20)
and for n = 1, 2, . . . ,
C
n
:

1
2
_

log
_
sin
ImM(e
i
)
_
cos(n) d +
1
n

j
(
n
j

n
j
)
=
2
n
Tr
_
T
n
_
1
2
J
_
T
n
_
1
2
J
0
_
_
(1.21)
where T
n
is the n
th
Chebyshev polynomial (of the rst kind).
We note that Case did not have the compact form of the right side of
(1.21), but he used implicitly dened polynomials which he didnt recognize
as Chebyshev polynomials (though he did give explicit formulae for small
n). Moreover, his arguments are formal. In an earlier paper, he indicates
that the conditions he needs are
[a
n
1[ +[b
n
[ C(1 + n
2
)
1
(1.22)
but he also claims this implies N
+
< , N

< , and, as Chihara [9]

noted, this is false. We believe that Cases implicit methods could be made
to work if

n[[a
n
1[ +[b
n
[] < rather than (1.22). In any event, we will
provide explicit proofs of the sum rulesindeed, from two points of view.
One of our primary observations is the power of a certain combination of
the Case sum rules, C
0
+
1
2
C
2
. It says
P
2
:
1
2
_

log
_
sin
ImM()
_
sin
2
d +

j
[F(E
+
j
) + F(E

j
)]
=
1
4

j
b
2
j
+
1
2

j
G(a
j
)
(1.23)
where G(a) = a
2
1 log[a[
2
and F(E) =
1
4
[
2

2
log[[
4
], with
given by E = +
1
, [[ > 1 (c.f. (1.16)).
8 R. KILLIP AND B. SIMON
As with the other sum rules, the terms on the left-hand side are purely
spectralthey can be easily found from ; those on the right depend in a
simple way on the coecients of J.
The signicance of (1.23) lies in the fact that each of its terms is nonneg-
ative. It is not dicult to see (see the end of Section 3) that F(E) 0 for
E R [2, 2] and that G(a) 0 for a (0, ). To see that the integral is
also nonnegative, we employ Jensens inequality. Notice that y log(y)
is convex and
2

0
sin
2
d = 1 so
1
2
_

log
_
sin()
ImM(e
i
)
_
sin
2
d =
1
2
2

_

0
log
_
ImM
sin
_
sin
2
() d

1
2
log
_
2

_

0
(ImM) sin() d
_
=
1
2
log[
ac
(2, 2)] 0 (1.24)
by (1.19).
The hard work in this paper will be to extend the sum rule to equalities
or inequalities in fairly general settings. Indeed, we will prove the following:
Theorem 5. If J is a Jacobi matrix for which the right-hand side of (1.23)
is nite, then the left-hand side is also nite and LHS RHS.
Theorem 6. If is a probability measure that obeys the BlumenthalWeyl
criterion and the left-hand side of (1.23) is nite, then the right-hand side
of (1.23) is also nite and LHS RHS.
In other words, the P
2
sum rule always holds although both sides may be
innite. We will see (see Proposition 3.4) that G(a) has a zero only at a = 1
where G(a) = 2(a1)
2
+O((a1)
3
) so the RHS of (1.23) is nite if and only
if

b
2
n
+

(a
n
1)
2
< , that is, J is HilbertSchmidt. On the other hand,
we will see (see Proposition 3.5) that F(E
j
) = ([E
j
[2)
3/2
+O(([E
j
[2)
2
) so
the LHS of (1.23) is nite if and only if the quasi-Szeg o condition (1.8) and
LiebThirring bound (1.9) hold. Thus, Theorems 5 and 6 imply Theorem 1.
The major tool in proving the Case sum rules is a function that arises in
essentially four distinct guises:
(1) The perturbation determinant dened as
L(z; J) = det
_
(J z z
1
)(J
0
z z
1
)
1

. (1.25)
(2) The Jost function, u
0
(z; J) dened for suitable z and J. The Jost
solution is the unique solution of
a
n
u
n+1
+b
n
u
n
+a
n1
u
n1
= (z +z
1
)u
n
(1.26)
n 1 with a
0
1 which obeys
lim
n
z
n
u
n
= 1. (1.27)
The Jost function is u
0
(z; J) = u
0
.
SUM RULES FOR JACOBI MATRICES 9
(3) Ratio asymptotics of the orthogonal polynomials P
n
,
lim
n
P
n
(z + z
1
)z
n
. (1.28)
(4) The Szego function, normally only dened when N
+
= N

= 0:
D(z) = exp
_
1
4
_
log

sin()f(2 cos )

e
i
+z
e
i
z
d
_
(1.29)
where d = f(E)dE +d
sing
.
These functions are not all equal, but they are closely related. L(z; J) is
dened for [z[ < 1 by the trace class theory of determinants [25, 53] so long
as J J
0
is trace class. We will see in that case it has a continuation to
z [ [z[ 1, z ,= 1 and, when J J
0
is nite rank, it is a polynomial.
The Jost function is related by L by
u
0
(z; J) =
_

1
a
j
_
1
L(z; J). (1.30)
Indeed, we will dene all u
n
by formulae analogous to (1.30) and show that
they obey (1.26)/(1.27). The Jost solution is normally constructed using
existence theory for the dierence equation (1.26). We show directly that
the limit in (1.28) is u
0
(J, z)/(1 z
2
). Finally, the connection of D(z) to
u
0
(z) is
D(z) = (2)
1/2
(1 z
2
) u
0
(z; J)
1
. (1.31)
Connected to this formula, we will prove that
[u
0
(e
i
)[
2
=
sin
ImM

()
, (1.32)
from which (1.31) will follow easily when JJ
0
is nice enough. The result for
general trace class JJ
0
is obviously new since it requires Nevais conjecture
to even dene D in that generality. It will require the analytic tools of this
paper.
In going from the formal sum rules to our general results like Theorems 4
and 5, we will use three technical tools:
(1) That the map
_

log(
sin
ImM
) sin
2
d and the similar map with
sin
2
d replaced by d is weakly lower semicontinuous. As we will
see, these maps are essentially the negatives of entropies and this will
be a known upper semicontinuity of an entropy.
(2) Rather than prove the sum rules in one step, we will have a way to
prove them one site at a time, which yields inequalities that go in the
opposite direction from the semicontinuity in (1).
(3) A detailed analysis of how eigenvalues change as a truncation is re-
moved.
In Section 2, we discuss the construction and properties of the pertur-
bation determinant and the Jost function. In Section 3, we give a proof
of the Case sum rules for nice enough J J
0
in the spirit of Flaschkas
10 R. KILLIP AND B. SIMON
[16] and Cases [7] papers, and in Section 4, a second proof implementing
tool (2) above. Section 5 discusses the Szego and quasi-Szeg o integrals as
entropies and the associated semicontinuity, and Section 6 implements tool
(3). Theorem 5 is proven in Section 7, and Theorem 6 in Section 8.
Section 9 discusses the C
0
sum rule and proves Nevais conjecture.
The proof of Nevais conjecture itself will be quite simplethe C
0
sum
rule and semicontinuity of the entropy will provide an inequality that shows
the Szego integral is nite. We will have to work quite a bit harder to show
that the sum rule holds in this case, that is, that the inequality we get is
actually an equality.
In Section 10, we turn to another aspect that the sum rules expose: the
fact that a dearth of bound states forces a.c. spectrum. For Schr odinger
operators, there are many V s which lead to ( + V ) = [0, ). This
always happens, for example, if V (x) 0 and lim
|x|
V (x) = 0. But for
discrete Schr odinger operators, that is, Jacobi matrices with a
n
1, this
phenomenon is not widespread because (J
0
) has two sides. Making b
n
0
to prevent eigenvalues in (, 2) just forces them in (2, )! We will prove
two somewhat surprising results (the e
n
(J) are dened in Remark 6 after
Theorem 1).
Theorem 7. If J is a Jacobi matrix with a
n
1 and

n
[e
n
(J)[
1/2
< ,
then
ac
(J) = [2, 2].
Theorem 8. Let W be a two-sided Jacobi matrix with a
n
1 and no
eigenvalues. Then b
n
= 0, that is, W = W
0
, the free Jacobi matrix.
We emphasize that Theorem 8 does not presuppose any reectionless
condition.
Acknowledgments. We thank F. Gesztesy, N. Makarov, P. Nevai,
M.B. Ruskai, and V. Totik for useful discussions. R.K. would like to thank
T. Tombrello for the hospitality of Caltech where this work was initiated.
2. Perturbation Determinants and the Jost Function
In this section we introduce the perturbation determinant
L(z; J) = det
__
J E(z)
__
J
0
E(z)
_
1

; E(z) = z +z
1
and describe its analytic properties. This leads naturally to a discussion
of the Jost function commencing with the introduction of the Jost solution
(2.63). The section ends with some remarks on the asymptotics of orthogonal
polynomials. We begin, however, with notation, the basic properties of
J
0
, and a brief review of determinants for trace class and HilbertSchmidt
operators. The analysis of L begins in earnest with Theorem 2.5.
Throughout, J represents a matrix of the form (1.1) thought of as an
operator on
2
(Z
+
). The special case a
n
1, b
n
0 is denoted by J
0
and
SUM RULES FOR JACOBI MATRICES 11
J = J J
0
constitutes the perturbation. If J is nite rank (i.e., for large
n, a
n
= 1 and b
n
= 0), we say that J is nite range.
It is natural to approximate the true perturbation by one of nite rank.
We dene J
n
as the semi-innite matrix,
J
n
=
_
_
_
_
_
_
_
_
_
_
b
1
a
1
0
a
1
b
2
a
2
. . . . . . . . .
. . . b
n1
a
n1
a
n1
b
n
1
1 0 1
1 0 . . .
_
_
_
_
_
_
_
_
_
_
(2.1)
that is, J
n
has b
m
= 0 for m > n and a
m
= 1 for m > n 1. Notice that
J
n
J
0
has rank at most n.
We write the nn matrix obtained by taking the rst n rows and columns
of J (or of J
n
) as J
n;F
. The n n matrix formed from J
0
will be called
J
0;n;F
.
A dierent class of associated objects will be the semi-innite matrices
J
(n)
obtained from J by dropping the rst n rows and columns of J, that
is,
J
(n)
=
_
_
_
_
b
n+1
a
n+1
0 . . .
a
n+1
b
n+2
a
n+2
. . .
0 a
n+2
b
n+3
. . .
. . . . . . . . . . . .
_
_
_
_
(2.2)
As the next preliminary, we need some elementary facts about J
0
, the
free Jacobi matrix. Fix z with [z[ < 1. Look for solutions of
u
n+1
+ u
n1
= (z +z
1
)u
n
, n 2 (2.3)
as sequences without any a priori conditions at innity or n = 1. The
solutions of (2.3) are linear combinations of the two obvious solutions u

given by
u

n
(z) = z
n
. (2.4)
Note that u
+
is
2
at innity since [z[ < 1. The linear combination that
obeys
u
2
= (z +z
1
)u
1
as required by the matrix ending at zero is (unique up to a constant)
u
(0)
n
(z) = z
n
z
n
. (2.5)
Noting that the Wronskian of u
(0)
and u
+
is z
1
z, we see that (J
0

E(z))
1
has the matrix elements (z
1
z)
1
u
(0)
min(n,m)
(z)u
+
max(n,m)
(z) either
by a direct calculation or standard Greens function formula. We have thus
proven that
(J
0
E(z))
1
nm
= (z
1
z)
1
[z
|mn|
z
m+n
] (2.6)
12 R. KILLIP AND B. SIMON
=
min(m,n)1

j=0
z
1+|mn|+2j
(2.7)
where the second comes from (z
1
z)(z
1n
+z
3n
+ +z
n1
) = z
n
z
n
by telescoping. (2.7) has two implications we will need later:
[z[ 1 [(J
0
E(z)
1
nm
[ min(n, m)[z[
1+|mn|
(2.8)
and that while the operator (J
0
E(z))
1
becomes singular as [z[ 1, the
matrix elements do not; indeed, they are polynomials in z.
We need an additional fact about J
0
:
Proposition 2.1. The characteristic polynomial of J
0;n;F
is
det(E(z) J
0,n;F
) =
(z
n1
z
n+1
)
(z
1
z)
= U
n
_
1
2
E(z)
_
(2.9)
where U
n
(cos ) = sin[(n + 1)]/ sin() is the Chebyshev polynomial of the
second kind. In particular,
lim
n
det[E(z) J
0;n+j;F
]
det[E(z) J
0;n;F
]
= z
j
. (2.10)
Proof. Let
g
n
(z) = det(E(z) J
0;n;F
). (2.11)
By expanding in minors
g
n+2
(z) = (z + z
1
)g
n+1
(z) g
n
(z).
Given that g
1
= z +z
1
and g
0
= 1, we obtain the rst equality of (2.9) by
induction. The second equality and (2.10) then follow easily.
In Section 4, we will need
Proposition 2.2. Let T
m
be the Chebyshev polynomial (of the rst kind):
T
m
(cos ) = cos(m). (2.12)
Then
Tr
_
T
m
_
1
2
J
0,n;F
_
=
_
n m = 2(n + 1); Z

1
2

1
2
(1)
m
otherwise.
(2.13)
In particular, for m xed, once n >
1
2
m 1 the trace is independent of n.
Proof. As noted above, the characteristic polynomial of J
0,n;F
is U
n
(E/2).
That is, det[2 cos() J
0;n;F
] = sin[(n + 1)]/ sin[]. This implies that the
eigenvalues of J
0;n;F
are given by
E
(k)
n
= 2 cos
_
k
n + 1
_
k = 1, . . . , n. (2.14)
SUM RULES FOR JACOBI MATRICES 13
So by (2.12), T
m
_
1
2
E
(k)
n
_
= cos
_
km
n+1
_
. Thus,
Tr
_
T
m
_
1
2
J
0;n;F
_
=
n

k=1
cos
_
km
n + 1
_
=
1
2

1
2
(1)
m
+
1
2
n+1

k=n
exp
_
ikm
n + 1
_
.
The nal sum is 2n + 2 if m is a multiple of 2(n + 1) and 0 if it is not.
As a nal preliminary, we discuss Hilbert space determinants [25, 52, 53].
Let I
p
denote the Schatten classes of operators with norm |A|
p
= Tr([A[
p
)
as described for example, in [53]. In particular, I
1
and I
2
are the trace class
and HilbertSchmidt operators, respectively.
For each A I
1
, one can dene a complex-valued function det(1+A) (see
[25, 53, 52]), so that
[det(1 +A)[ exp(|A|
1
) (2.15)
and A det(1 +A) is continuous; indeed [53, pg. 48],
[det(1 + A) det(1 +B)[ |AB|
1
exp(|A|
1
+|B|
1
+ 1). (2.16)
We will also use the following properties:
A, B I
1
det(1 +A) det(1 + B) = det(1 +A+ B + AB) (2.17)
AB, BA I
1
det(1 +AB) = det(1 +BA) (2.18)
(1 + A) is invertible if and only if det(1 +A) ,= 0 (2.19)
z A(z) analytic det(1 +A(z)) analytic (2.20)
If A is nite rank and P is a nite-dimensional self-adjoint projection,
PAP = A det(1 +A) = det
PH
(1
PH
+PAP), (2.21)
where det
PH
is the standard nite-dimensional determinant.
For A I
2
, (1 + A)e
A
1 I
1
, so one denes (see [53, pp. 106108])
det
2
(1 +A) = det((1 +A)e
A
). (2.22)
Then
[det
2
(1 +A)[ exp(|A|
2
2
) (2.23)
[det
2
(1 + A) det
2
(1 + B)[ |A B|
2
exp((|A|
2
+|B|
2
+ 1)
2
) (2.24)
and, if A I
1
,
det
2
(1 +A) = det(1 + A)e
Tr(A)
(2.25)
or
det(1 +A) = det
2
(1 +A)e
Tr(A)
. (2.26)
To estimate the I
p
norms of operators we use
Lemma 2.3. If A is a matrix and | |
p
the Schatten I
p
norm [53], then
14 R. KILLIP AND B. SIMON
(i)
|A|
2
2
=

n,m
[a
nm
[
2
(2.27)
(ii)
|A|
1

n,m
[a
nm
[ (2.28)
(iii) For any j and p,

n
[a
n,n+j
[
p
|A|
p
p
(2.29)
Proof. (i) is standard. (ii) follows from the triangle inequality for | |
1
and
the fact that a matrix which a single nonzero matrix element, , has trace
norm [[. (iii) follows from a result of Simon [53, 51] that
|A|
p
p
= sup
_

n
[
n
, A
n
)[
p

n
,
n
orthonormal sets
_
.

The following factorization will often be useful. Dene

c
n
= max([a
n1
1[, [b
n
[, [a
n
1[)
which is the maximum matrix element in the n
th
row and n
th
column. Let
C be the diagonal matrix with matrix elements c
n
. Dene U by
J = C
1/2
UC
1/2
. (2.30)
Then U is a tridiagonal matrix with matrix elements bounded by 1 so
|U| 3. (2.31)
One use of (2.30) is the following:
Theorem 2.4. Let c
n
= max([a
n1
1[, [b
n
[, [a
n
1[). For any p [1, ),
1
3
_

n
[c
n
[
p
_
1/p
|J|
p
3
_

n
[c
n
[
p
_
1/p
. (2.32)
Proof. The right side is immediate from (2.30) and Holders inequality for
trace ideals [53]. The leftmost inequality follows from (2.29) and
_

n
[c
n
[
p
_
1/p

n
[b
n
[
p
_
1/p
+ 2
_

n
[a
n
[
p
_
1/p
.

With these preliminaries out of the way, we can begin discussing the
perturbation determinant L. For any J with J I
1
(by (2.32) this is
equivalent to

[a
n
1[ +

[b
n
[ < ), we dene
L(z; J) = det
__
J E(z)
_ _
J
0
E(z)
_
1

(2.33)
SUM RULES FOR JACOBI MATRICES 15
for all [z[ < 1. Since
(J E)(J
0
E)
1
= 1 + J(J
0
E)
1
, (2.34)
the determinant in (2.33) is of the form 1 +A with A I
1
.
Theorem 2.5. Suppose J I
1
.
(i) L(z; J) is analytic in D z [ [z[ < 1.
(ii) L(z; J) has a zero in D only at points z
j
where E(z
j
) is an eigenvalue
of J, and it has zeros at all such points. All zeros are simple.
(iii) If J is nite range, then L(z; J) is a polynomial and so has an ana-
lytic continuation to all of C.
Proof. (i) follows from (2.20).
(ii) If E
0
= E(z
0
) is not an eigenvalue of J, then E
0
/ (J) since E : D
C[2, 2] and
ess
(J) = [2, 2]. Thus, (J E
0
)/(J
0
E
0
) has an inverse
(namely, (J
0
E
0
)/(J E
0
)), and so by (2.19), L(z; J) ,= 0. If E
0
is an
eigenvalue, (J E
0
)/(J
0
E
0
) is not invertible, so by (2.19), L(z
0
; J) = 0.
Finally, if E(z
0
) is an eigenvalue, eigenvalues of J are simple by a Wronskian
argument. That L has a simple zero under these circumstances comes from
the following.
If P is the projection onto the eigenvector at E
0
= E(z
0
), then (J
E)
1
(1 P) has a removable singularity at E
0
. Dene
C(E) = (J E)
1
(1 P) +P (2.35)
so
(J E)C(E) = 1 P + (E
0
E)P. (2.36)
Dene
D(E) (J
0
E)C(E) (2.37)
= JC(E) + (J E)C(E)
= 1 P + (E
0
E)P JC(E)
= 1 + trace class.
Moreover,
D(E)[(J E)/(J
0
E)] = (J
0
E)[1 P + (E
0
E)P](J
0
E)
1
= 1 + (J
0
E)[P + (E
0
E)P](J E)
1
.
Thus by (2.17) rst and then (2.18),
det(D(E(z)))L(z; J) = det(1 + (J
0
E)[P + (E
0
E)P](J
0
E)
1
)
= det(1 P + (E
0
E)P)
= E
0
E(z),
where we used (2.21) in the last step. Since L(z; J) has a zero at z
0
and
E
0
E(z) = (z z
0
)[1
1
zz
0
] has a simple zero, L(z; J) has a simple zero.
16 R. KILLIP AND B. SIMON
(iii) Suppose J has range N, that is, N = maxn [ [b
n
[ +[a
n1
1[ > 0
and let P
(N)
be the projection onto the span of
j

N
j=1
. As P
(N)
J = J,
J(J
0
z)
1
= P
(N)
P
(N)
J(J
0
z)
1
.
By (2.18),
L(z; J) = det(1 + P
(N)
J(J
0
z)
1
P
(N)
).
Thus by (2.7), L(z; J) is a polynomial if J is nite range.
Remarks. 1. By this argument, if J has range n, L(z; J) is the determinant
of an n n matrix whose ij element is a polynomial of degree i + j + 1.
That implies that we have shown L(z; J) is a polynomial of degree at most
2n(n + 1)/2 + n = (n + 1)
2
. We will show later it is actually a polynomial
of degree at most 2n 1.
2. The same idea shows that if

n
[(a
n
+ 1)
n
[ + [b
n

n
[ < for some
> 1, then C
1/2
(J
0
z)
1
C
1/2
is trace class for [z[ < , and thus L(z; J)
has an analytic continuation to z [ [z[ < .
We are now interested in showing that L(z; J), dened initially only on
D, can be continued to D or part of D. Our goal is to show:
(i) If

n=1
n[[a
n
1[ + [b
n
[] < , (2.38)
then L(z; J) can be continued to all of

D, that is, extends to a function
continuous on

D and analytic in D.
(ii) For the general trace class situation, L(z; J) has a continuation to

D1, 1.
(iii) As x real approaches 1, [L(x; J)[ is bounded by expo(1)/(1[x[).
We could interpolate between (i) and (iii) and obtain more information
about cases where (2.38) has n replaced by n

with 0 < < 1 or even log n

(as is done in [42, 22]), but using the theory of Nevanlinna functions and
(iii), we will be able to handle the general trace class case (in Section 9), so
we forgo these intermediate results.
Lemma 2.6. Let C be diagonal positive trace class matrix. For [z[ < 1,
dene
A(z) = C
1/2
(J
0
E(z))
1
C
1/2
. (2.39)
Then, as a HilbertSchmidt operator-valued function, A(z) extends contin-
uously to

D 1, 1. If

n
nc
n
< , (2.40)
it has a HilbertSchmidt continuation to

D.
Proof. Let A
nm
(z) be the matrix elements of A(z). It follows from [z[ < 1
and (2.6)/(2.8) that
[A
nm
(z)[ 2c
1/2
n
c
1/2
m
[z 1[
1
[z + 1[
1
(2.41)
SUM RULES FOR JACOBI MATRICES 17
[A
nm
(z)[ min(m, n)c
1/2
n
c
1/2
m
(2.42)
and each A
n,m
(z) has a continuous extension to

D. It follows from (2.41),
the dominated convergence theorem, and

n,m
(c
1/2
n
c
1/2
m
)
2
=
_

n
c
n
_
2
that so long as z stays away from1, A
mn
(z)
n,m
is continuous in
2
((1, )
(1, )) so A(z) is HilbertSchmidt and continuous on

D1, 1. Moreover,
(2.42) and

n,m
_
min(m, n)c
1/2
n
c
1/2
m

mn
mnc
n
c
m
=
_

n
nc
n
_
2
imply that A(z) is HilbertSchmidt on

D if (2.40) holds.
Remark. When (2.40) holdsindeed, when

c
n
< (2.43)
for any > 0we believe that one can show A(z) has trace class boundary
values onD1, 1 but we will not provide all the details since the Hilbert
Schmidt result suces. To see this trace class result, we note that ImA(z) =
(A(z) A

(z))/2i has a rank 1 boundary value as z e

i
; explicitly,
ImA(e
i
)
mn
= c
1/2
n
c
1/2
m
(sinm)(sinn)
(sin)
. (2.44)
Thus, ImA(e
i
) is trace class and is Holder continuous in the trace norm if
(2.43) holds. Now Re A(e
i
) is the Hilbert transform of a H older continuous
trace class operator-valued function and so trace class. This is because
when a function is H older continuous, its Hilbert transform is given by a
convergent integral, hence limit of Riemann sums. Because of potential
singularities at 1, the details will be involved.
Lemma 2.7. Let J be trace class. Then
t(z) = Tr((J)(J
0
E(z))
1
) (2.45)
has a continuation to

D1, 1. If (2.38) holds, t(z) can be continued to

D.
Remark. We are only claiming t(z) can be continued to D, not that it
equals the trace of (J)(J
0
E(z))
1
since J(J
0
E(z))
1
is not even a
bounded operator for z D!
Proof. t(z) = t
1
(z) + t
2
(z) +t
3
(z) where
t
1
(z) =

b
n
(J
0
E(z))
1
nn
t
2
(z) =

(a
n
1)(J
0
E(z))
1
n+1,n
18 R. KILLIP AND B. SIMON
t
3
(z) =

(a
n
1)(J
0
E(z))
1
n,n+1
.
Since, by (2.6), (2.8),
[(J
0
E(z))
1
nm
[ 2[z 1[
1
[z + 1[
1
[(J
0
E(z))
1
nm
[ min(n, m),
the result is immediate.
Theorem 2.8. If J is trace class, L(z; J) can be extended to a continuous
function on

D1, 1 with
[L(z; J)[ exp
_
c
_
|J|
1
+ |J|
2
1

[z 1[
2
[z + 1[
2
_
(2.46)
for a universal constant, c. If (2.38) holds, L(z; J) can be extended to all of

D with
[L(z; J)[ exp
_
c
_
1 +

n=1
n
_
[a
n
1[ +[b
n
[

_
2
_
(2.47)
for a universal constant, c.
Proof. This follows immediately from (2.22), (2.23), (2.25), and the last two
lemmas and their proofs.
While we cannot control |C
1/2
(J
0
E(z))
1
C
1/2
|
1
for arbitrary z with
[z[ 1, we can at the crucial points 1 if we approach along the real axis,
because of positivity conditions.
Lemma 2.9. Let C be a positive diagonal trace class operator. Then
lim
|x|1
x real
(1 [x[)|C
1/2
(J
0
E(x))
1
C
1/2
|
1
= 0. (2.48)
Proof. For x < 0, E(x) < 2, and J
0
E(x) > 0, while for x > 0, E(x) > 2,
so J
0
E(x) < 0. It follows that
|C
1/2
(J
0
E(x))
1
C
1/2
|
1
= [Tr(C
1/2
(J
0
E(x))
1
C
1/2
)[

n
c
n
[(J
0
E(x))
1
nn
[. (2.49)
By (2.6),
(1 [x[)[(J
0
E(x))
1
nn
[ 1
and by (2.7) for each xed n,
lim
|x|1
x real
(1 [x[)[(J
0
E(x))
1
nn
[ = 0.
Thus (2.49) and the dominated convergence theorem proves (2.48).
Theorem 2.10.
limsup
|x|1
x real
(1 [x[) log[L(x; J)[ 0 (2.50)
SUM RULES FOR JACOBI MATRICES 19
Proof. Use (2.30) and (2.18) to write
L(x; J) = det(1 +UC
1/2
(J
0
E(x))
1
C
1/2
)
and then (2.15) and (2.31) to obtain
log[L(x; J)[ |UC
1/2
(J
0
E(x))
1
C
1/2
|
1
3|C
1/2
(J
0
E(x))
1
C
1/2
|
1
.
The result now follows from the lemma.
Next, we want to nd the Taylor coecients for L(z; J) at z = 0, which
we will need in the next section.
Lemma 2.11. For each xed h > 0 and [z[ small,
log
_
1
h
E(z)
_
=

n=1
2
n
_
T
n
(0) T
n
(
1
2
h)

z
n
(2.51)
where T
n
(x) is the n
th
Chebyshev polynomial of the rst kind: T
n
(cos ) =
cos(n). In particular, T
2n+1
(0) = 0 and T
2n
(0) = (1)
n
.
Proof. Consider the following generating function:
g(x, z)

n=1
T
n
(x)
z
n
n
=
1
2
log[1 2xz +z
2
]. (2.52)
The lemma now follows from
log
_
1
2x
z + z
1
_
= 2[g(0, z) g(x, z)] =

2
n
_
T
n
(0) T
n
(x)

z
n
by choosing x = h/2. The generation function is well known (Abramowitz
and Stegun [1, Formula 22.9.8] or Szeg o [60, Equation 4.7.25]) and easily
proved: for R and [z[ < 1,
g
z
(cos , z) =
1
z

n=1
cos(n)z
n
=
1
2z

n=1
_
_
ze
i
_
n
+
_
ze
i
_
n
_
=
cos() + z
z
2
2z cos + 1
=
1
2

z
log[1 2xz +z
2
]
at x = cos . Integrating this equation from z = 0 proves (2.52) for x
[1, 1] and [z[ < 1. For more general x one need only consider C and
require [z[ < exp[ Re [.
20 R. KILLIP AND B. SIMON
Lemma 2.12. Let A and B be two self-adjoint mm matrices. Then
log det
__
A E(z)
_ _
B E(z)
_
1

n=0
c
n
(A, B)z
n
(2.53)
where
c
n
(A, B) =
2
n
Tr
_
T
n
_
1
2
A
_
T
n
_
1
2
B
_
. (2.54)
Proof. Let
1
, . . . ,
m
be the eigenvalues of A and
1
, . . . ,
m
the eigenvalues
of B. Then
det
_
AE(z)
B E(z)
_
=
m

j=1
_

j
E(z)

j
E(z)
_
log det
_
AE(z)
B E(z)
_
=
m

j=1
log[1
j
/E(z)] log[1
j
/E(z)]
so (2.53)/(2.54) follow from the preceding lemma.
Theorem 2.13. If J is trace class, then for each n, T
n
(J/2) T
n
(J
0
/2)
is trace class. Moreover, near z = 0,
log[L(z; J)] =

n=1
c
n
(J)z
n
(2.55)
where
c
n
(J) =
2
n
Tr
_
T
n
_
1
2
J
_
T
n
_
1
2
J
0
_
. (2.56)
In particular,
c
1
(J) = Tr(J J
0
) =

m=1
b
m
(2.57)
c
2
(J) =
1
2
Tr(J
2
J
2
0
) =
1
2

m=1
[b
2
m
+ 2(a
2
m
1)]. (2.58)
Proof. To prove T
n
(J/2) T
n
(J
0
/2) is trace class, we need only show that
J
m
J
m
0
=

m1
j=1
J
j
J J
m1j
is trace class, and thats obvious! Let

J
n;F
be J
n;F
extended to
2
(Z
+
) by setting it equal to the zero matrix on

2
(j n). Let

J
0,n
be J
0
with a
n+1
set equal to zero. Then

J
n;F
(

J
0,n
E)
1
J(J
0
E)
1
in trace norm, which means that
det
_
J
n;F
E(z)
J
0,n;F
E(z)
_
L(z; J). (2.59)
This convergence is uniform on a small circle about z = 0, so the Taylor
series coecients converge. Thus (2.53)/(2.54) imply (2.55)/(2.56).
SUM RULES FOR JACOBI MATRICES 21
Next, we look at relations of L(z; J) to certain critical functions beginning
with the Jost function. As a preliminary, we note (recall J
(n)
is dened in
(2.2)),
Proposition 2.14. Let J be trace class. Then for each z

D1, 1,
lim
n
L(z; J
(n)
) = 1 (2.60)
uniformly on compact subsets of

D1, 1. If (2.38) holds, (2.60) holds
uniformly in z for all z in

D.
Proof. Use (2.16) and (2.24) with B = 1 and the fact that |J
(n)
|
1
0 in
the estimates above.
Next, we note what is essentially the expansion of det(J E(z)) in minors
in the rst row:
Proposition 2.15. Let J be trace class and z

D1, 1. Then
L(z; J) = (E(z) b
1
)zL(z; J
(1)
) a
2
1
z
2
L(z; J
(2)
). (2.61)
Proof. Denote (J
(k)
)
n;F
by J
(k)
n;F
, that is, the n n matrix formed by rows
and columns k + 1, . . . , k +n of J. Then expanding in minors,
det(E J
n;F
) = (E b
1
) det(E J
(1)
n1;F
) a
2
1
det(E J
(2)
n2;F
). (2.62)
Divide by det(E J
0;n;F
) and take n using (2.59). (2.61) follows if
one notes
det(E J
0;nj;F
)
det(E J
0;n;F
)
z
j
by (2.10).
We now dene for z

D1, 1 and n = 1, . . . , ,
u
n
(z; J) =
_

j=n
a
j
_
1
z
n
L(z; J
(n)
) (2.63)
u
0
(z; J) =
_

j=1
a
j
_
1
L(z; J). (2.64)
u
n
is called the Jost solution and u
0
the Jost function. The innite product
of the as converges to a nonzero value since a
j
> 0 and

j
[a
j
1[ < .
We have:
Theorem 2.16. The Jost solution, u
n
(z; J), obeys
a
n1
u
n1
+ (b
n
E(z))u
n
+ a
n
u
n+1
= 0, n = 1, 2, . . . (2.65)
where a
0
1. Moreover,
lim
n
z
n
u
n
(z; J) = 1. (2.66)
22 R. KILLIP AND B. SIMON
Proof. (2.61) for J replaced by J
(n)
reads
L(z; J
(n)
) = (E(z) b
n+1
)zL(z; J
(n+1)
) a
2
n+1
z
2
L(z; J
(n+2)
),
from which (2.65) follows by multiplying by z
n
(

j=n+1
a
j
)
1
. Since lim
n

j=n
a
j
=
1, (2.66) is just a rewrite of (2.60).
Remarks. 1. If (2.6) holds, one can dene u
n
for z = 1.
2. By Wronskian methods, (2.65)/(2.66) uniquely determine u
n
(z; J).
Theorem 2.16 lets us improve Theorem 2.5(iii) with an explicit estimate
on the degree of L(z; J).
Theorem 2.17. Let J have range n, that is, a
j
= 1 if j n, b
j
= 0 if
j > n. Then u
0
(z; J) and so L(z; J) is a polynomial in z of degree at most
2n 1. If b
n
,= 0, then L(z; J) has degree exactly 2n 1. If b
n
= 0 but
a
n1
,= 1, then L(z; J) has degree 2n 2.
Proof. The dierence equation (2.65) can be rewritten as
_
u
n1
u
n
_
=
_
(E b
n
)/a
n1
a
n
/a
n1
1 0
__
u
n
u
n+1
_
=
1
za
n1
A
n
(z)
_
u
n
u
n+1
_
, (2.67)
where
A
n
(z) =
_
z
2
+ 1 b
n
z a
n
z
a
n1
z 0
_
. (2.68)
If J has range n, J
(n)
= J
0
and a
n
= 1. Thus by (2.63), u

(z; J) = z

if
n. Therefore by (2.67),
_
u
0
u
1
_
= (a
1
a
n1
)
1
z
n
A
1
(z) A
n
(z)
_
z
n
z
n+1
_
= (a
1
a
n1
)
1
A
1
(z) A
n
(z)
_
1
z
_
= (a
1
a
n1
)
1
A
1
(z) A
n1
(z)
_
1 b
n
z
a
n1
z
_
. (2.69)
Since A
j
(z) is a quadratic, (2.69) implies u
0
is a polynomial of degree at
most 2(n 1) + 1 = 2n 1. The top left component of A
j
contains z
2
while everything else is of lower order. Proceeding inductively, the top left
component of A
1
(z) A
n1
(z) is z
2n2
+O(z
2n3
). Thus if b
n
,= 0,
u
0
= (a
1
. . . a
n1
)
1
b
n
z
2n1
+ O(z
2n2
),
proving u
0
has degree 2n 1. If b
n
= 0, then
A
n1
(z)
_
1
a
n1
z
_
=
_
1 + (1 a
2
n1
)z
2
b
n1
z
a
n2
z
_
so inductively, one sees that
u
0
= (a
1
. . . a
n1
)
1
(1 a
2
n1
)z
2n2
+ O(z
2n3
)
SUM RULES FOR JACOBI MATRICES 23
and u
0
has degree 2n 2.
Remark. Since the degree of u is the number of its zeros (counting multi-
plicities), this can be viewed as a discrete analog of the Regge [46]Zworski
[64] resonance counting theorem.
Recall the denitions (1.2) and (1.15) of the m-function which we will
denote for now by M(z; J) = (E(z) J)
1
11
.
Theorem 2.18. If J I
1
then for [z[ < 1 with L(z; J) ,= 0, we have
M(z; J) =
zL(z; J
(1)
)
L(z; J)
(2.70)
=
u
1
(z; J)
u
0
(z; J)
. (2.71)
Proof. (2.71) follows from (2.70) and (2.63)/(2.64). (2.70) is essentially
Cramers rule. Explicitly,
M(z; J) = lim
n
(E(z) J
n;F
)
1
11
= lim
n
det(E J
(1)
n1;F
)
det(E J
n;F
)
= lim
n
w
n
x
m
y
n
where (by (2.59) and (2.10))
w
n
=
det(E J
(1)
n1;F
)
det(E J
0;n1;F
)
L(z; J
(1)
)
x
n
=
det(E J
0;n;F
)
det(E J
n;F
)
L(z; J)
1
y
n
=
det(E J
0;n1;F
)
det(E J
0;n;F
)
z.

Theorem 2.18 allows us to link [u

0
[ and [L[ on [z[ = 1 to Im(M) there:
Theorem 2.19. Let J be trace class. Then for all ,= 0, , the boundary
value lim
r1
M(re
i
; J) M(e
i
; J) exists. Moreover,
[u
0
(e
i
; J)[
2
ImM(e
i
; J) = sin . (2.72)
Equivalently,
[L(e
i
; J)[
2
ImM(e
i
; J) =
_

j=1
a
2
j
_
sin . (2.73)
24 R. KILLIP AND B. SIMON
Proof. By (2.61), (2.73) is equivalent to (2.72). If [z[ = 1, then E( z) =
E(z) since z = z
1
. Thus, u
n
(z; J) and u
n
( z; J) solve the same dierence
equation. Since z
n
u
n
(z; J) 1 and a
n
1, we have that
a
n
[u
n
( z; J)u
n+1
(z; J) u
n
(z; J)u
n+1
( z; J)] z z
1
.
Since the Wronskian of two solutions is constant, if z = e
i
,
a
n
[u
n
(e
i
; J)u
n+1
(e
i
; J) u
n
(e
i
; J)u
n+1
(e
i
; J)] = 2i sin.
Since a
0
= 1 and u
n
( z; J) = u
n
(z; J), we have that
Im[ u
0
(e
i
; J) u
1
(e
i
; J)] = sin. (2.74)
(2.74) implies that u
0
(e
i
; J) ,= 0 if ,= 0, , so by (2.71), M(z; J) extends
to

D1, 1. Since u
1
(e
i
; J) = u
0
(e
i
; J)M(e
i
; J) (by (2.71)), (2.74) is
the same as (2.72).
If J has no eigenvalues in R[2, 2] and (2.38) holds so u
0
(z; J) has a
continuation to

D, then
u
0
(z; J) = exp
_
1
2
_
2
0
e
i
+z
e
i
z
log[u
0
(e
i
; J)[ d
_
(2.75)
= exp
_

1
4
_
2
0
e
i
+z
e
i
z
log
_
[ImM(e
i
; J)[
[sin[
_
d
_
(2.76)
= exp
_

1
4
_
2
0
e
i
+z
e
i
z
log
_
f(2 cos )
[sin[
_
d
_
(2.77)
= (2)
1/2
(1 z
2
) D(z)
1
(2.78)
where D is the Szego function dened by (1.29) and f(E) =
dac
dE
. In the
above, (2.75) is the PoissonJensen formula [48]. It holds because under
(2.38), u
0
is bounded on

D and by (2.74), and the fact that u
1
is bounded,
log(u
0
) at worst has a logarithmic singularity at 1. (2.76) follows from
(2.72) and (2.77) from (1.18). To obtain (2.78) we use
1
2
(1 z
2
) = exp
_
1
2
_
e
i
z
e
i
+ z
log
_
2 sin
2

_
d
_
which is the PoissonJensen formula for
1
2
(1 z
2
)
2
if we note that [(1
e
2i
)
2
[ = 4 sin
2
.
As a nal remark on perturbation theory and Jost functions, we note how
easy they make Szego asymptotics for the polynomials:
Theorem 2.20. Let J be a Jacobi matrix with J trace class. Let P
n
(E)
be an orthonormal polynomial associated to J. Then for [z[ < 1,
lim
n
z
n
P
n
(z + z
1
) =
u
0
(z; J)
(1 z
2
)
(2.79)
with convergence uniform on compact subsets of D.
SUM RULES FOR JACOBI MATRICES 25
Remarks. 1. By looking at (2.79) near z = 0, one gets results on the
asymptotics of the leading coecients of P
n
(E), that is, a
n,nj
in P
n
(E) =

n
k=0
a
n,k
E
k
; see Szego [60].
2. Alternatively, if Q
n
are the monic polynomials,
lim
n
z
n
Q
n
(z +z
1
) =
L(z; J)
(1 z
2
)
. (2.80)
Proof. This is essentially (2.59). For let
Q
n
(E) = det(E J
n;F
). (2.81)
Expanding in minors in the last rows shows
Q
n
(E) = (E b
n
)Q
n1
(E) a
2
n1
Q
n2
(E) (2.82)
with Q
0
(E) = 1 and Q
1
(E) = Eb
1
. It follows Q
n
(E) is the monic orthog-
onal polynomial of degree n (this is well known; see, e.g. [3]). Multiplying
(2.81) by (a
1
, . . . , a
n1
)
1
, we see that
P
n
(E) = (a
1
. . . a
n
)
1
Q
n
(E) (2.83)
obeys (1.4) and so are the orthonormal polynomials. It follows then from
(2.59) and (2.9) that
L(z; J) = lim
n
z
1
z
z
(n+1)
Q
n
(z) = lim
n
(1 z
2
)z
n
Q
n
(z)
which implies (2.80) and, given (2.83) and lim
n
(a
1
, . . . , a
n
)
1
exists, also
(2.79).
3. The Sum Rule: First Proof
Following Flaschka [17] and Case [6, 7], the Case sum rules follow from
the construction of L(z; J), the expansion (2.55) of log[L(z; J)] at z = 0, the
formula (2.73) for [L(e
i
; J)[, and the following standard result:
Proposition 3.1. Let f(z) be analytic in a neighborhood of

D, let z
1
, . . . , z
m
be the zeros of f in D and suppose f(0) ,= 0. Then
log[f(0)[ =
1
2
_
2
0
log[f(e
i
)[ d +
m

j=1
log[z
j
[ (3.1)
and for n = 1, 2, . . . ,
Re(
n
) =
1

_
2
0
log[f(e
i
)[ cos(n) d Re
_
m

j=1
z
n
j
z
n
j
n
_
(3.2)
where
log
_
f(z)
f(0)
_
= 1 +

n=1

n
z
n
(3.3)
for [z[ small.
26 R. KILLIP AND B. SIMON
Remarks. 1. Of course, (3.1) is Jensens formula. (3.2) can be viewed as a
derivative of the PoissonJensen formula, but the proof is so easy we give
it.
2. In our applications, f(z) = f( z) so
n
are real and the zeros are real
or come in conjugate pairs. Therefore, Re can be dropped from both sides
of (3.2) and the dropped from z
i
.
Proof. Dene the Blaschke product,
B(z) =
m

j=1
[z
j
[
z
j
z
j
z
1 z z
j
for which we have
log[B(z)] =
m

j=1
log[z
j
[ + log
__
1
z
z
j
__
log(1 z z
j
)
=
m

j=1
log[z
j
[

n=1
z
n
m

j=1
z
n
j
z
n
j
n
. (3.4)
By a limiting argument, we can suppose f has no zeros on D. Then
f(z)/B(z) is nonvanishing in a neighborhood of

D, so g(z) log[f(z)/B(z)]
is analytic there and by (3.3)/(3.4), its Taylor series
g(z) =

n=0
c
n
z
n
has coecients
c
0
= log[f(0)] +
m

j=1
log[z
j
[
c
n
=
n
+
m

j=1
[z
n
j
z
n
j
]
n
.
Substituting d =
dz
iz
and cos(n) =
1
2
(z
n
+ z
n
) in the Cauchy integral
formula,
1
2i
_
2
0
g(z)
dz
z
n+1
=
_
c
n
if n 0
0 if n 1,
we get integral relations whose real part is (3.1) and (3.2).
While this suces for the basic sum rule for nite range J, which is the
starting point of our analysis, we note three extensions:
(1) If f(z) is meromorphic in a neighborhood of

D with zeros z
1
, . . . , z
m
and poles p
1
, . . . , p
k
, then (3.1) and (3.2) remain true so long as one makes
SUM RULES FOR JACOBI MATRICES 27
the changes:
m

j=1
log[z
j
[
m

j=1
log[z
j
[
k

j=1
log[p
j
[ (3.5)
m

j=1
z
n
j
z
n
j
n

m

j=1
z
n
j
z
n
j
n

k

j=1
p
n
j
p
n
j
n
(3.6)
for we write f(z) = f
1
(z)/

k
j=1
(z p
j
) and apply Proposition 3.1 to f
1
and
to

k
j=1
(z p
j
). We will use this extension in the next section.
(2) If f has continuous boundary values on D, we know its zeros in D
obey

j=1
(1 [z
j
[) < (so the Blaschke product converges) and we have
some control on log[f(re
i
)[ as r 1, one can prove (3.1)(3.2) by a limit-
ing argument. We could use this to extend the proof of Cases inequalities
to the situation

n[[a
n
1[ + [b
n
[] < . We rst use a Bargmann bound
(see [10, 19, 20, 27]) to see there are only nitely many zeros for L and
(2.73) to see the only place log[L[ can be singular is at 1. The argument in
Section 9 that sup
r
_
[log

[L(re
i
)[]
2
d < lets us control such potential
singularities. Since Section 9 will have a proof in the more general case of
trace class J, we do not provide the details. But we would like to empha-
size that proving the sum rules in generality Case claims in [6, 7] requires
overcoming technical issues he never addresses.
(3) The nal (one might say ultimate) form of (3.1)/(3.2) applies when
f is a Nevanlinna function, that is, f is analytic in D and
sup
0<r<1
1
2
_
2
0
log
+
[f(re
i
)[ d < , (3.7)
where log
+
(x) = max(log(x), 0). If f is Nevanlinna, then ([48, pg. 311];
essentially one uses (3.1) for f(z/r) with r < 1),

j=1
(1 [z
j
[) < (3.8)
and ([48, pg. 310]) the Blaschke product converges. Moreover (see [48,
pp. 247, 346]), there is a nite real measure d
(f)
on D so
log[f(re
i
)[ d d
(f)
() (3.9)
weakly, and for Lebesgue a.e. ,
lim
r1
log[f(re
i
)[ = log[f(e
i
)[ (3.10)
and
d
(f)
() = log[f(e
i
)[ d + d
(f)
s
() (3.11)
where d
(f)
s
() is singular with respect to Lebesgue measure d on D.
d
(f)
s
() is called the singular inner component.
28 R. KILLIP AND B. SIMON
By using (3.1)/(3.2) for f(z/r) with r 1 and (3.9), we immediately have:
Theorem 3.2. Let f be a Nevanlinna function D and let z
j

N
j=1
(N =
1, 2, . . . , or ) be its zeros. Suppose f(0) ,= 0. Let log[f(e
i
)[ be the
a.e. boundary values of f and d
(f)
s
() the singular inner component. Then
log[f(0)[ =
1
2
_
2
0
log[f(e
i
)[ d +
1
2
_
2
0
d
(f)
s
() +
N

j=1
log[z
j
[ (3.12)
and for n = 1, 2, . . . ,
Re(
n
) =
1

_
2
0
log[f(e
i
)[ cos(n) d
+
1

_
2
0
cos(n) d
(f)
s
() Re
_
N

j=1
z
n
j
z
n
j
n
_ (3.13)
where
n
is given by (3.3).
We will use this form in Section 9.
Now suppose that J has nite range, and apply Proposition 3.1 to
L(z; J). Its zeros in D are exactly the image under E z of the (sim-
ple) eigenvalues of J outside [2, 2] (Theorem 2.5(ii)). The expansion of
log[L(z; J)] at z = 0 is given by Theorem 2.13 and log[L(e
i
; J)[ is given by
(2.73). We have thus proven:
Theorem 3.3 (Cases Sum Rules: Finite Rank Case). Let J be nite rank.
Then, with [
1
(J)[ [
2
(J)[ > 1 dened so
j
+
1
j
are eigenvalues
of J outside [2, 2], we have
C
0
:
1
4
_
2
0
log
_
sin
ImM(e
i
)
_
d =

j
log[
j
[

n=1
log(a
n
) (3.14)
C
n
:
1
2
_
2
0
log
_
sin
ImM(e
i
)
_
cos(n) d (3.15)
=
1
n

j
(
n
j

n
j
) +
2
n
Tr
_
T
n
_
1
2
J
_
T
n
_
1
2
J
0
_
_
In particular,
P
2
:
1
2
_
2
0
log
_
sin
ImM
_
sin
2
d +

j
F(e
j
) =
1
4

n
b
2
n
+
1
2

n
G(a
n
),
(3.16)
where
G(a) = a
2
1 log(a
2
) (3.17)
and
F(e) =
1
4
(
2

2
log[[
4
); e = +
1
, [[ > 1. (3.18)
SUM RULES FOR JACOBI MATRICES 29
Remarks. 1. Actually, when J is nite rank all eigenvalues must lie outside
[2, 2] it is easily checked that the corresponding dierence equation has
no (nonzero) square summable solutions. While eigenvalues may occur at
2 or 2 when J I
1
, there are none in (2, 2). This follows from the fact
that lim
r1
M(re
i
; J) exists for (0, ) (see Theorem 2.19) or alternately
from the fact that one can construct two independent solutions u
n
(e
i
, J)
whose linear combinations are all non-L
2
.
2. In (2.73),
log[L[ =
1
2
log

sin
ImM

n=1
log a
n
,
the

n=1
log a
n
term is constant and contributes only to C
0
since
_
2
0
cos(n) d =
0.
3. As noted, P
2
is C
0
+
1
2
C
2
.
4. We have looked at the combinations of sum rules that give sin
4

and sin
6
hoping for another miracle like the one below that for sin
2
, the
function G and F that result are positive. But we have not found anything
but a mess of complicated terms that are not in general positive.
P
2
is especially useful because of the properties of G and F:
Proposition 3.4. The function G(a) = a
2
1 2 log(a) for a (0, ) is
nonnegative and vanishes only at a = 1. For a 1 small,
G(a) = 2(a 1)
2
+O((a 1)
3
). (3.19)
Proof. By direct calculations, G(1) = G

(1) = 0 and
G

(a) =
2(1 +a
2
)
a
2
2
so G(a) (a 1)
2
(since G(1) = G

(1) = 0). (3.19) follows from G

(1) =
4.
Proposition 3.5. The function F(e) given by (3.18) for [e[ > 2 is strictly
positive, and for [e[ 2 small,
F(e) =
2
3
([e[ 2)
3/2
+ O(([e[ 2)
2
). (3.20)
In addition,
F(e)
2
3
(e
2
4)
3/2
. (3.21)
Proof. Let R() =
1
4
(
2

2
log[[
4
) for 1 and compute
R

() =
1
2
_
+
3

_
=
1
2
_
+ 1

_
2
1

( 1)
2
.
Thus
R

() = 2( 2)
2
+ O(( 1)
3
)
and since 1, ( + 1)/ 2 and
1
1 so
R

() 2( 1)
2
.
30 R. KILLIP AND B. SIMON
Since R(1) = 0, we have
R()
2
3
( 1)
3
(3.22)
and
R() =
2
3
( 1)
3
+ O(( 1)
4
). (3.23)
Because F(e) = F(e), we can suppose e > 2 so > 1. As =
1
2
[e +

e
2
4 ], 1 = (e 2)
1/2
+O(e 2) and (3.23) implies (3.20). Lastly,
( 1)
1
=
_
e
2
4 ,
so (3.22) implies (3.21).
4. The Sum Rule: Second Proof
In this section, we will provide a second proof of the sum rules that never
mentions a perturbation determinant or a Jost function explicitly. We do
this not only because it is nice to have another proof, but because this proof
works in a situation where we a priori know the m-function is analytic in
a neighborhood of

D and the other proof does not apply. And this is a
situation we will meet in proving Theorem 6. On the other hand, while we
could prove Theorems 1, 2, 3, 5, 6 without Jost functions, we denitely need
them in our proof in Section 9 of the C
0
-sum rule for the trace class case.
The second proof of the sum rules is based on the continued fraction
expansion of m (1.5). Explicitly, we need,
M(z; J)
1
= (z + z
1
) +b
1
+a
2
1
M(z; J
(1)
) (4.1)
which one obtains either from the Weyl solution method of looking at M
(see [24, 56]) or by writing M as a limit of ratio of determinants
M(z; J) = lim
n
det(E(z) J
(1)
n1;F
)
det(E(z) J
n;F
)
(4.2)
and expanding the denominator in minors in the rst row. For any j, (4.1)
holds for z D. Suppose though that we know M has a meromorphic
continuation to a neighborhood of

D and consider (4.1) with z = e
i
:
M(e
i
; J)
1
= 2 cos + b
1
+a
2
1
M(e
i
; J
(1)
). (4.3)
Taking imaginary parts of both sides,
ImM(e
i
; J)
[M(e
i
; J)[
2
= a
2
1
ImM(e
i
; J
(1)
) (4.4)
or, letting
g(z; J) =
M(z; J)
z
(Note: because
M(z; J) = (z + z
1
J)
1
11
= z(1 + z
2
zJ)
1
11
= z +O(z
2
) (4.5)
SUM RULES FOR JACOBI MATRICES 31
near zero, g is analytic in D), we have
1
2
_
log
_
ImM(e
i
; J)
sin
_
log
_
ImM(e
i
; J
1
)
sin
__
= log a
1
+ log[g(e
i
; J)[.
(4.6)
To see where this is heading,
Theorem 4.1. Suppose M(z; J) is meromorphic in a neighborhood of

D.
Then J and J
(1)
have nitely many eigenvalues outside [2, 2] and if
C
0
(J) =
1
4
_
2
0
log
_
sin
ImM(e
i
; J)
_
d
N

j=1
log[
j
(J)[ (4.7)
(with
j
as in Theorem 3.3), then
C
0
(J) = log(a
1
) + C
0
(J
(1)
). (4.8)
In particular, if J is nite rank, then the C
0
sum rule holds:
C
0
(J) =

n=1
log(a
n
). (4.9)
Proof. The eigenvalues, E
j
, of J outside [2, 2] are precisely the poles of
M(E; J) and so the poles of M(z; J) under E
j
= z
j
+ z
1
j
. By (4.1), the
poles of M(z; J
(1)
) are exactly the zeros of M(z; J). Thus
j
(J)
1
are the
poles of M(z; J) and
j
(J
(1)
)
1
are its zeros. Since g(0; J) = 1 by (4.5),
(3.1)/(3.5) becomes
1
2
_
log([g(e
i
, J)[ d =

j
log([
j
(J)[) +

j
log([
j
(J
(1)
)[).
(4.6) and this formula implies (4.8). By (4.3), if M(z; J) is meromorphic in
a neighborhood of

D, so is M(z; J
(1)
). So we can iterate (4.8). The free M
function is
M(z; J
0
) = z (4.10)
(e.g., by (2.7) with m = n = 1), so C
0
(J
0
) = 0 and thus, if J is nite rank,
the remainder is zero after nitely many steps.
To get the higher-order sum rules, we need to compute the power series
for log(g(z; J)) about z = 0. For low-order, we can do this by hand. Indeed,
by (4.1) and (4.5) for J
(1)
,
g(z; J) = (z[(z + z
1
) b
1
a
2
1
z + O(z
2
)])
1
= (1 b
1
z (a
2
1
1)z
2
+ O(z
3
))
1
= 1 +b
1
z + ((a
2
1
1) + b
2
1
)z
2
+O(z
3
)
so since log(1 + w) = w
1
2
w
2
+ O(w
3
),
log(g(z; J)) = b
1
z + (
1
2
b
2
1
+a
2
1
1)z
2
+ O(z
3
). (4.11)
32 R. KILLIP AND B. SIMON
Therefore, by mimicking the proof of Theorem 4.1, but using (3.2)/(3.6) in
place of (3.1)/(3.5), we have
Theorem 4.2. Suppose M(z; J) is meromorphic in a neighborhood of

D.
Let
C
n
(J) =
1
2
_
2
0
log
_
sin
ImM(e
i
)
_
cos(n) +
1
n
_

j
(J)
n

j
(J)
n
_
.
(4.12)
Then
C
1
(J) = b
1
+C
1
(J
(1)
) (4.13)
C
2
(J) = [
1
2
b
2
1
+ (a
2
1
1) +C
2
(J
(2)
)]. (4.14)
If
P
2
(J) =
1
2
_
2
0
log
_
sin
ImM(e
i
)
_
sin
2
d +

j
F(e
j
(J)) (4.15)
with F given by (3.18), then writing G(a) = a
2
a 2 log(a) as in (3.17)
P
2
(J) =
1
4
b
2
1
+
1
2
G(a
1
) +P
2
(J
(1)
). (4.16)
In particular, if J is nite rank, we have the sum rules C
1
, C
2
, P
2
of
(3.15)/(3.16).
To go to order larger than two, we need to systematically expand log(g(z; J))
as follows: We begin by noting that by (4.2) (Cramers rule),
g(z; J) = lim
n
g
n
(z; J) (4.17)
where
g
n
(z; J) =
z
1
det(z +z
1
J
(1)
n1;F
)
det(z +z
1
J
n;F
)
(4.18)
=
1
1 +z
2
det(1 E(z)
1
J
(1)
n1;F
)
det(1 E(z)
1
J
n;F
)
(4.19)
where we used z(E(z)) = 1 + z
2
and the fact that because the numerator
has a matrix of order one less than the denominator, we get an extra factor
of E(z). We now use Lemma 2.11, writing F
j
(x) for
2
j
[T
j
(0) T
j
(x/2)],
log g
n
(z; J) = log(1 +z
2
) +

j=1
z
j
_
Tr
_
F
j
(J
(1)
n1;F
)
_
Tr
_
F
j
(J
n;F
)
_
_
(4.20)
= log(1 +z
2
)

j=1
z
2j
j
(1)
j
SUM RULES FOR JACOBI MATRICES 33
+

j=1
2z
j
j
_
Tr
_
T
j
_
1
2
J
n;F
_
_
Tr
_
T
j
_
1
2
J
(1)
n;F
_
__
(4.21)
where we picked up the rst sum because J
n;F
has dimension one greater
than J
(1)
n1;F
so the T
j
(0) terms in F
j
(J
n;F
) and J
(1)
n1;F
contribute dierently.
Notice

j=1
z
2j
j
(1)
j
= log(1 + z
2
)
so the rst two terms cancel! Since g
n
(z; J) converges to g(z; J) in a neigh-
borhood of z = 0, its Taylor coecients converge. Thus
Proposition 4.3. For each j,

j
(J, J
(1)
) = lim
n
_
Tr
_
T
j
_
1
2
J
n;F
_
_
Tr
_
T
j
_
1
2
J
(1)
n1;F
_
__
(4.22)
exists, and for z small,
log g(z; J) =

j=1
2z
j
j

j
(J, J
(1)
). (4.23)
Remark. Since
(J
(1)
)
mm
= (J

)
m+1 m+1
if m , the dierence of traces on the right side of (4.22) is constant for
n > j, so one need not take the limit.
Plugging this into the machine that gives Theorem 4.1 and Theorem 4.2,
we obtain
Theorem 4.4. Let C
n
(J) be given by (4.12) and by (4.22). Then
C
n
(J) =
2
n

n
(J, J
(1)
) +C
n
(J
(1)
). (4.24)
In particular, if J is nite rank, we have the sum rule C
n
of (3.15).
Proof. The only remaining point is why if J is nite rank, we have recovered
the same sum rule as in (3.15). Iterating (4.24) when J has rank m gives
C
n
(J) =
2
n
m

j=1

n
(J
(j1)
, J
(j)
)
= lim

2
n
_
Tr
_
T
n
_
1
2
J
;F
_
T
n
_
1
2
J
0,m;F
_
__
(4.25)
while (3.15) reads
C
n
(J) =
2
n
Tr
_
T
n
_
J
2
_
T
n
_
J
0
2
__
= lim

2
n
_
Tr
_
J
;F
2
_
Tr
_
J
0,;F
2
__
. (4.26)
That (4.25) and (4.26) are the same is a consequence of Proposition 2.2.
34 R. KILLIP AND B. SIMON
5. Entropy and Lower Semicontinuity
of the Szeg o and Quasi-Szeg o Terms
In the sum rules C
0
and P
2
of most interest to us, there appear two terms
involving integrals of logarithms:
Z(J) =
1
4
_
2
0
log
_
sin
ImM(e
i
, J)
_
d (5.1)
and
Q(J) =
1
2
_
2
0
log
_
sin
ImM(e
i
, J)
_
sin
2
d. (5.2)
One should think of M as related to the original spectral measure on
(J) [2, 2] as
ImM(e
i
) =
d
ac
dE
(2 cos ) (5.3)
in which case, (5.1), (5.2) can be rewritten
Z(J) =
1
2
_
2
2
log
_

4 E
2
2 d
ac
/dE
_
dE

4 E
2
(5.4)
and
Q(J) =
1
4
_
2
2
log
_

4 E
2
2 d
ac
/dE
_
_
4 E
2
dE. (5.5)
Our main result in this section is to view Z and Q as functions of and to
prove if
n
weakly, then Z(
n
) (resp. Q(
n
)) obeys
Z() liminf Z(
n
); Q() liminf Q(
n
), (5.6)
that is, that Z and Q are weakly lower semicontinuous. This will let us
prove sum rule-type inequalities in great generality.
The basic idea of the proof will be to write variational principles for Z and
J as suprema of weakly continuous functions. Indeed, as Totik has pointed
out to us, Szeg os theorem (as extended to the general, not only a.c., case
[2, 18]) gives what is essentially Z(J) by a variational principle; explicitly,
exp
_
1

_
2
0
log
_
d
ac
d
_
d
_
= inf
P
_
1
2
_

[P(e
i
)[
2
d()
_
(5.7)
where P runs through all polynomials with P(0) = 1, which can be used
to prove the semicontinuity we need for Z. It is an interesting question of
what is the relation between (5.7) and the variational principle (5.16) below.
It also would be interesting to know if there is an analog of (5.7) to prove
semicontinuity of Q.
We will deduce the semicontinuity by providing a variational principle.
We originally found the variational principle based on the theory of Legendre
transforms, then realized that the result was reminiscent of the inverse Gibbs
variation principle for entropy (see [55, pg. 271] for historical remarks; the
principle was rst written down by LanfordRobinson [35]) and then realized
that the quantities of interest to us arent merely reminiscent of entropy, they
SUM RULES FOR JACOBI MATRICES 35
are exactly relative entropies where is the second variable rather than the
rst one that is usually varied. We have located the upper semicontinuity
of the relative entropy in the second variable in the literature (see, e.g.,
[12, 34, 44]), but not in the generality we need it, so especially since the
proof is easy, we provide it below. We use the notation
log

(x) = max(log(x), 0). (5.8)

Denition. Let , be nite Borel measures on a compact Hausdor space,
X. We dene the entropy of relative to , S( [ ), by
S( [ ) =
_
if is not -ac

_
log(
d
d
)d if is -ac.
(5.9)
Remarks. 1. Since log

(x) = log
+
(x
1
) x
1
and
_ _
d
d
_
1
d =
__
x

d
d
,= 0
__
(X) < ,
the integral in (5.9) can only diverge to , not to +.
2. If d = f d, then
S( [ ) =
_
f log(f) d, (5.10)
the more usual formula for entropy.
Lemma 5.1. Let be a probability measure. Then
S( [ ) log (X). (5.11)
In particular, if is also a probability measure,
S( [ ) 0. (5.12)
Equality holds in (5.12) if and only if = .
Proof. If is not -ac, (5.11)/(5.12) is trivial, so suppose = f d and let
d =
{x|f(x)=0}
d (5.13)
so and are mutually ac. Then,
S( [ ) =
_
log
_
d
d
_
d
log
__ _
d
d
_
d
_
(5.14)
= log (X)
log (X) (5.15)
where we used Jensens inequality for the concave function log(x). For
equality to hold in (5.12), we need equality in (5.15) (which says = ) and
in (5.14), which says, since log is strictly convex, that d/d is a constant.
When (X) = (X) = 1, this says = .
36 R. KILLIP AND B. SIMON
Theorem 5.2. For all , ,
S( [ ) = inf
__
F(x) d
_
(1 + log F) d(x)
_
(5.16)
where the inf is over all real-valued continuous functions F with
min
xX
F(x) > 0.
Proof. Let us use the notation
G(F, , ) =
_
F(x) d
_
(1 + log F) d(x)
for any nonnegative function F with F L
1
(d) and log F L
1
(d).
Suppose rst that is -ac with d = f d and F is positive and contin-
uous. Let A = x [ f(x) ,= 0 and dene by (5.13). As log(a) is concave,
log(a) a 1 so for a, b > 0,
ab
1
1 + log(ab
1
) = 1 + log(a) log b. (5.17)
Thus for x A,
F(x)f(x)
1
1 + log F(x) log f(x).
Integrating with d and using
_
F(x) d
_
F(x) d =
_
A
F(x)f(x)
1
d,
we have that
_
F(x) d
_
(1 + log F(x)) d(x) +S( [ )
or
S( [ ) G(F, , ). (5.18)
To get equality in (5.16), take F = f so
_
d and
_
F d cancel. Of course,
f may not be continuous or strictly positive, so we need an approximation
argument. Given N, , let
f
N,
(x) =
_

_
N if f(x) N
f(x) if f(x) N
if f(x) .
Let f
,N,
(x) be continuous functions with f
,N,
N so that as ,
f
,N,
f
N,
in L
1
(X, d+ d). For N > 1, ff
1
N,
1 +f, so we have

_
log(f
N,
) d = (X)
_
log(f
1
N,
)
d
(X)
(X) log
__
ff
1
N,
d
(X)
_
(X) log
_
1 +
(X)
(X)
_
<
SUM RULES FOR JACOBI MATRICES 37
and thus, since log(f
N,
) increases as 0, f
N,=0
lim
0
f
N,
has
log f
N,=0
L
1
(d) and the integrals converge. It follows that as
and then 0,
G(f
,N,
, , ) G(f
N,
, , ) G(f
N,=0
, , ).
Now take N . By monotonicity,
_
log f
N,=0
d converges to
_
log f d
which may be innite. And
_
f
N,=0
d
_
d 0 so G(f
N,=0
, , ) S( [
), and we have proven (5.18).
Next, suppose is not -ac. Thus, there is a Borel subset A X with
(A) > 0 and (A) = 0. By regularity of measures, we can nd K A
compact and for any , U

open so K A U

and
(K) > 0 (U

) < . (5.19)
By Urysohns lemma, nd F

continuous with
1 F

(x)
1
all x, F

1
on K, F

1 on XU

. (5.20)
Then
_
F

d (XU

) +
1
(U

) (X) + 1
while
_
(1 + log F

) d log(
1
)(K)
so
G(F

, , ) (X) + 1 log(
1
)(K)
as 0, proving the right side of (5.16) is .
As an inmum of continuous functions is upper semicontinuous, we have
Corollary 5.3. S( [ ) is jointly weakly upper semicontinuous in and ,
that is, if
n
w
and
n
w
, then
S( [ ) limsup
n
S(
n
[
n
).
Remarks. 1. In our applications,
n
will be xed.
2. This proof can handle functions other than log. If
_
log((d/d)
1
) d
is replaced by
_
G((d/d)) d where G is an arbitrary increasing concave
function with lim
y0
G(y) = , there is a variational principle where 1 +
log F in (5.18) is replaced by H(F(x)) with H(y) = inf
x
(xy g(x)).
To apply this to Z and Q, we note
Proposition 5.4. (a) Let
d
0
(E) =
1
2
_
4 E
2
dE. (5.21)
Then
Q(J) =
1
2
S(
0
[
J
). (5.22)
38 R. KILLIP AND B. SIMON
(b) Let
d
1
(E) =
1

dE

4 E
2
. (5.23)
Then
Z(J) =
1
2
log(2)
1
2
S(
1
[
J
). (5.24)
Remarks. 1. Both
0
and
1
are probability measures, as is easily checked
by setting E = 2 cos .
2. d
0
is the spectral measure for J
0
. For M(z; J
0
) = z and thus
ImM(e
i
; J
0
) = sin so m(E; J
0
) =
1
2

4 E
2
and
1

ImmdE = d
0
.
3. d
1
is the spectral measure for the whole-line free Jacobi matrix and
also for the half-line matrix with b
n
= 0, a
1
=

2, a
2
= a
3
= = 1. An
easy way to see this is to note that after E = 2 cos , d
1
() =
1

d and so
the orthogonal polynomials are precisely the normalized scaled Chebyshev
polynomials of the rst kind that have the given values of a
j
.
Proof. (a) Follows immediately from (5.16) if we note that
d
0
d
=
d
0
dE
_
d
ac
dE
=

4 E
2
2 d
ac
/dE
.
(b) As above,
d
1
d
= 2(4 E
2
)
1

4 E
2
2 d
ac
/dE
.
Thus
Z(J) = c
1
2
S(
1
[
J
)
where
c =
1
2
_
2
2
log
_
2
4 E
2
_
_
4 E
2
dE
=
1
4
_
2
0
log[2 sin
2
] d
=
1
2
log(2) +
1
2
_
2
0
log[sin[ d
=
1
2
log(2) +
1
2
_
2
0
log

1 e
i
2

d
=
1
2
log(2) + log
_
1
2
_
=
1
2
log(2)
where we used Jensens formula for f(z) =
1
2
(1 z
2
) to do the integral.
Remark. As a check on our arithmetic, consider the Jacobi matrix

J with
a
1
=

2 and all other as and bs the same as for J
0
so d

J
is d
1
. The sum
rule, C
0
, for this case says that
Z(

J) = log(

2) =
1
2
log 2
SUM RULES FOR JACOBI MATRICES 39
since there are no eigenvalues and a
1
=

2. But
1
=
J
, so S(
1
[
J
) =
0. This shows once again that c =
1
2
log 2 (actually, it is essentially the
calculation we diddone the long way around!).
Given this proposition, Lemma 5.1, and Corollary 5.3, we have
Theorem 5.5. For any Jacobi matrix,
Q(J) 0 (5.25)
and
Z(J)
1
2
log(2). (5.26)
If
Jn

J
weakly, then
Z(J) liminf Z(J
n
). (5.27)
and
Q(J) liminf Q(J
n
) (5.28)
We will call (5.27) and (5.28) lower semicontinuity of Z and Q.
6. Fun and Games with Eigenvalues
Recall that J
n
denotes the Jacobi matrix with truncated perturbation, as
given by (2.1). In trying to get sum rules, we will approximate J by J
n
and
need to estimate eigenvalues of J
n
in terms of eigenvalues of J. Throughout
this section, X denotes a continuous function on R with X(x) = X(x),
X(x) = 0 if [x[ 2, and X is monotone increasing in [2, ). Our goal is to
prove:
Theorem 6.1. For any J and all n, we have N

(J
n
) N

(J) + 1 and
(i) E

1
(J
n
) E

1
(J) + 1
(ii) E

k+1
(J
n
) E

k
(J)
In particular, for any function X of the type described above,
N

(Jn)

j=1
X(E

j
(J
n
)) X(E

1
(J) + 1) +
N

(J)

j=1
X(E

j
(J)). (6.1)
Theorem 6.2. If J J
0
is compact, then
lim
n
N

(Jn)

j=1
X(E

j
(J
n
)) =
N

(J)

j=1
X(E

j
(J)). (6.2)
This quantity may be innite.
Proof of Theorem 6.1. To prove these results, we pass fromJ to J
n
in several
intermediate steps.
(1) We pass from J to J
n;F
.
(2) We pass from J to J
n;F
d
n,n
J

n;F
where d
n,n
is the matrix with
1 in the n, n place and zero elsewhere.
(3) We take a direct sum of J

n;F
and J
0
d
1,1
.
40 R. KILLIP AND B. SIMON
(4) We pass from this direct sum to J
n
.
Step 1: J
n;F
is just a restriction of J (to
2
(1, . . . , n)). The min-max
principle [45] implies that under restrictions, the most positive eigenvalues
become less positive and the most negative, less negative. It follows that
N

(J
n;F
) N

(J) (6.3)
E

j
(J
n;F
) E

j
(J). (6.4)
Step 2: To study E
+
j
, we add d
n,n
, and to study E

j
, we subtract d
n,n
.
The added operator d
n,n
has two critical properties: It is rank one and its
norm is one. From the norm condition, we see
[E

1
(J

n;F
) E

1
(J
n;F
)[ 1 (6.5)
so
E
+
1
(J
+
n;F
) E
+
1
(J
n;F
) + 1
E
+
1
(J) + 1. (6.6)
(Note (6.5) and (6.6) hold for all indices j, not just j = 1, but we only need
j = 1.) Because d
n,n
is rank 1, and positive, we have
E
+
m+1
(J
n;F
) E
+
m+1
(J
+
n;F
) E
+
m
(J
n;F
)
and so, by (6.4),
E
+
m+1
(J
+
n;F
) E
+
m
(J) (6.7)
and thus also
N

(J

n;F
) N

(J) + 1. (6.8)
Step 3: Take the direct sum of J

n;F
and J
0
d
11
. This should be interpreted
as a matrix with entries
_
J

n;F
(J
0
d
11
)

k,
=
_

_
(J

n;F
)
k,
k, n
(J
0
d
11
)
kn,n
k, > n
0 otherwise.
Since J
0
d
11
has no eigenvalues, (6.7) and (6.8) still hold.
Step 4: Go from the direct sum to J
n
. In the + case, we add the 2 2
matrix in sites n, n + 1:
dJ
+
=
_
1 1
1 1
_
and, in the case,
dJ

=
_
1 1
1 1
_
dJ
+
is negative, so it moves eigenvalues down, while dJ

is positive. Thus
E
+
m+1
(J
n
) E
+
m+1
(J
+
n;F
) E
+
m
(J)
SUM RULES FOR JACOBI MATRICES 41
and
N

(J
n
) N

(J

n;F
) N

(J) + 1.

Proof of Theorem 6.2. We have, since J J

0
is compact, that
|J
n
J| sup
mn+1
[b
m
[ + 2 sup
mn
[a
m
[ 0.
Thus
[E

j
(J
n
) E

j
(J)[ |J
n
J| 0. (6.9)
If

N

(J)
j=1
X(E

j
(J)) = , then, by (6.9), for all xed m,
liminf
N

(Jn)

j=1
X(E

j
(J
n
)) liminf
m

j=1
X(E

j
(J
n
))
=
m

j=1
X(E

j
(J))
so taking m to innity, (6.2) results.
If the sum is nite, (6.9), dominated convergence and (6.1) imply (6.2).

7. Jacobi Data Dominate Spectral Data in P

2
Our goal in this section is to prove Theorem 5. Explicitly, for a Jacobi
matrix, J, let
D
2
(J) =
1
4

j=1
b
2
j
+
1
2

j=1
G(a
j
) (7.1)
with G = a
2
1 2 log(a) as in (3.17). For a probability measure, on R,
dene
P
2
() =
1
2
_

log
_
sin
ImM

(e
i
)
_
sin
2
d +

j
F(E
j
) (7.2)
where E
j
are the mass points of outside [2, 2] and F is given by (3.18).
Recall that ImM

(e
i
) d
ac
/dE at E = 2 cos . We will let
J
be the
measure associated with J by the spectral theorem and J

the Jacobi matrix

associated to .
Then Theorem 5 says
Theorem 7.1. If J J
0
is HilbertSchmidt so that D
2
(J) < , then
P
2
(
J
) < and
P
2
(
J
) D
2
(J). (7.3)
42 R. KILLIP AND B. SIMON
Proof. Let J
n
be a truncation of J given by (2.1). Then D
2
(J
n
) is monotone
increasing with limit D
2
(J). This is nite because JJ
0
is HilbertSchmidt.
By the denition (5.5),
P
2
() = Q(J) +

j
F(E
j
)
=
1
2
S(
0
, ) +

j
F(E
j
) (7.4)
by (5.22). Since Q 0 (5.25) and F > 0 (Proposition 3.5), (7.4) is a sum of
positive terms. Moreover, by Theorem 6.2,

j
F(E
j
(J
n
))

j
F(E
j
(J))
even if the right side is innite, and by (5.28), Q(J) limsup Q(J
n
). It
follows that
P
2
(J

) limsup
_
Q(J
n
) +

F
_
E
j
(J
n
)
_
_
= limsup[D
2
(J
n
)] (by Theorem 3.3)
= D
2
(J).
Thus P
2
(
J
) < and (7.3) holds.
The result in this section is essentially a quantitative version of the main
result in DeiftKillip [13].
8. Spectral Data Dominate Jacobi Data in P
2
Our goal in this section is to prove the following, which is essentially
Theorem 6:
Theorem 8.1. If is a probability measure with P
2
() < , then
D
2
(J

) P
2
() (8.1)
and so J

is HilbertSchmidt.
The idea of the proof is to start with a case where we have the sum rule
and then pass to successively more general cases where we can prove an
inequality of the form (8.1). There will be three steps:
(1) Prove the inequality in the case M

is meromorphic in a neighborhood
of

D.
(2) Prove the inequality in the case
0
where is a positive real
number and
0
is the free Jacobi measure (5.18).
(3) Prove the inequality in the case P
2
() < .
Proposition 8.2. Let J be a Jacobi matrix for which M

has a meromorphic
continuation to a neighborhood of

D. Then
D
2
(J) P
2
(J). (8.2)
SUM RULES FOR JACOBI MATRICES 43
Proof. By Theorem 4.2,
P
2
(J) =
1
4
b
2
1
+
1
2
G(a
1
) +P
2
(J
(1)
).
so iterating,
P
2
(J) =
1
4
m

j=1
b
2
j
+
1
2
m

j=1
G(a
j
) +P
2
(J
(m)
)

1
4
m

j=1
b
2
j
+
1
2
m

j=0
G(a
j
)
since P
2
(J
(m)
) 0. Now G 0, so we can take m and obtain
(8.2).
Remark. If M

has a meromorphic continuation into z [ [z[ < for some

> 1, then by a theorem of Geronimo [21],

[a
n
1[
n
+ [b
n
[
n
< for
all < , so the sum rule also follows from the methods of Section 3. We
prefer to avoid the use of Geronimos theorem.
Given any J and associated M-function M(z; J), there is a natural ap-
proximating family of M-functions meromorphic in a neighborhood of

D.
Lemma 8.3. Let M

be the M-function of a probability measure obeying

the BlumenthalWeyl condition, and dene
M
(r)
(z) = r
1
M

(rz) (8.3)
for 0 < r < 1. Then, there is a set of probability measures
(r)
so that
M
(r)
= M

(r) .
Proof. Return to the E variable. Since M
(r)
(z) is meromorphic in a neigh-
borhood of

D with ImM
(r)
(z) > 0 if Imz > 0,
m
(r)
(E) = M
(r)
(z(E))
(where z(E) + z(E)
1
= E with [z[ < 1) is meromorphic on C[2, 2] and
Herglotz. It follows that it is the Borel transform of a measure
(r)
of total
weight lim
E
Em
(r)
(E) = lim
z0
z
1
M
(r)

(z) = 1.
Proposition 8.4. Let be a probability measure obeying the Blumenthal
Weyl condition and

0
(8.4)
where
0
is the free Jacobi measure (the measure with M

0
(z) = z) and
> 0. Then
D
2
(J

) P
2
(). (8.5)
Proof. We claim that
limsup
r1
_
log[ImM

(r) (e
i
)[ d
_
log[ImM

(e
i
)[ d. (8.6)
44 R. KILLIP AND B. SIMON
Accepting (8.6) for the moment, let us complete the proof. The eigenvalues
of
(r)
correspond to s of the form

k
(J

(r) ) =

k
(J)
r
for those k with [
k
(J)[ < r. Thus

f(E

k
(J

(r) )) is monotone increasing

to

f(E

k
(J

)), so (8.6) shows that

P
2
() limsup P
2
(
(r)
). (8.7)
Moreover, M

(r) (z) M

(z) uniformly on compact subsets of D which

means that the continued fraction parameters for m
(r)
(E), which are the
Jacobi coecients, must converge. Thus for any N,
1
4
N

j=1
b
2
j
+
1
2
N1

j=1
G(a
j
) = lim
r1
1
4
N

j=1
_
b
(r)
j
_
2
+
1
2
N1

j=1
G(a
(r)
j
)
liminf D
2
(J

(r) )
liminf P
2
(
(r)
) (by Proposition 8.2)
P
2
() (by (8.7))
so (8.5) follows by taking N .
Thus, we need only prove (8.6). Since M

(r) () = r
1
M

(re
i
) M

(e
i
)
for a.e. , Fatous lemma implies that
liminf
r1
_
log
+
[ImM

(r) ()[ d
_
log
+
[ImM

()[ d. (8.8)
On the other hand, (8.4) implies [ImM

(z)[ [Imz[, so [ImM

(r) (z)[
[Imz[. Thus uniformly in r,
[ImM

(r) (e
i
)[ [sin[. (8.9)
Thus
log

[ImM

(r) (e
i
)[ log log[sin[,
so, by the dominated convergence theorem,
lim
_
log

([ImM

(r) [) d =
_
log

([ImM

()[) d.
This, together with (8.8) and log(x) = log
+
(x) + log

(x) implies (8.6).

Remark. Semicontinuity of the entropy and (8.9) actually imply one has
equality for the limit in (8.6) rather than inequality for the limsup.
Proof of Theorem 8.1. For each (0, 1), let

= (1 )+
0
. Since

obeys (8.4) and the BlumenthalWeyl criterion,

D
2
(J

) P
2
(

). (8.10)
SUM RULES FOR JACOBI MATRICES 45
Let M

and note that

ImM

(e
i
) = (1 ) ImM(e
i
) + sin
so
log[ImM

(e
i
)[ = log(1 ) + log

ImM(e
i
) +

1
sin

.
We see that up to the convergent log(1) factor, log[ImM

(e
i
)[ is monotone
in , so by the monotone convergence theorem,
P
2
() = lim
0
P
2
(

) (8.11)
(the eigenvalue terms are constant in , since the point masses of

have
the same positions as those of !).
On the other hand, since

weakly, as in the last proof,

D
2
(J

) liminf D
2
(J

). (8.12)
(8.10)(8.12) imply (8.1).
9. Consequences of the C
0
Sum Rule
In this section, we will study the C
0
sum rule and, in particular, we
will prove Nevais conjecture (Theorem 2) and several results showing that
control of the eigenvalues can have strong consequences for J and
J
, specif-
ically Theorems 4

and 7. While Nevais conjecture will be easy, the more

complex results will involve some machinery, so we provide this overview:
(1) By employing semicontinuity of the Szeg o term, we easily get a C
0
-
inequality that implies Theorems 2 and 7 and the part of Theorem 4

that says J J
0
is HilbertSchmidt.
(2) We prove Theorem 4.1 under great generality when there are no
eigenvalues and use that to prove a semicontinuity in the other direc-
tion, and thereby show that the Szeg o condition implies a C
0
-equality
when there are no eigenvalues, including conditional convergence of

n
(a
n
1).
(3) We use the existence of a C
0
-equality to prove a C
1
-equality, and
thereby conditional convergence of

n
b
n
.
(4) Returning to the trace class case, we prove that the perturbation
determinant is a Nevanlinna function with no singular inner part,
and thereby prove a sum rule in the Nevai conjecture situation.
Theorem 9.1 ( Theorem 3). Let J be a Jacobi matrix with
ess
(J)
[2, 2] and

k
e
k
(J)
1/2
< , (9.1)
limsup
N
N

j=1
log(a
j
) > . (9.2)
46 R. KILLIP AND B. SIMON
Then
(i)
ess
(J) = [2, 2].
(ii) The Szeg o condition holds, that is,
Z(J) <
with Z given by (5.1).
(iii)
ac
(J) = [2, 2]; indeed, the essential support of
ac
is [2, 2].
Remarks. 1. We emphasize (9.2) says > , not < , that is, it is a
condition which prevents the a
n
s from being too small (on average).
2. We will see below that (9.1) and (9.2) also imply [a
j
1[ 0 and
[b
j
[ 0 and that at least inequality holds for the C
0
sum rule:
Z(J)

k
log[
k
(J)[ limsup
N
N

j=1
log(a
j
) (9.3)
holds.
Proof. Pick N
1
, N
2
, . . . (tending to ) so that
inf

_
N

j=1
log(a
j
)
_
> (9.4)
and let J
N

be given by (2.1). By Theorem 3.3,

Z(J
N

)
N

j=1
log(a
j
) +

log([
k
(J
N

)[)
inf

j=1
log(a
j
) +

log([
k
(J)[) + 2 log([
1
(J)[ + 2) (9.5)
where in (9.5) we used Theorem 6.1 and the fact that the

solving to
e
1
(J) + 3 =

+

1
has

(
1
(J)) + 2 (i.e.,

+

1
1 +
1
+
1
1
).
For later purposes, we note that if [b
n
(J)[ + [a
n
(J) 1[ 0, Theorem 6.2
implies we can drop the last term in the limit.
Now use (5.27) and (9.5) to see that
Z(J) liminf Z(J
N

) < .
This proves (ii). But (ii) implies
dac
dE
> 0 a.e. on E [2, 2], that
is, [2, 2] is the essential support of
ac
. That proves (iii). (i) is then
immediate.
Proof of Theorem 2 (Nevais Conjecture). We need only check that J J
0
trace class implies (9.1) and (9.2). The niteness of (9.1) follows from a
bound of HundertmarkSimon [27],

[ [e
k
(J)[ [e
k
(J) + 4[ ]
1/2

n
[b
n
[ + 2[a
n
1[
SUM RULES FOR JACOBI MATRICES 47
where e
k
(J) = [E

[ 2 so [e[ [e + 4[ = (E

)
2
4.
Condition (9.2) is immediate for, as is well-known, a
j
> 0 and

([a
j
[
1) < implies

a
j
is absolutely convergent, that is,

[log(a
j
)[ < .
Corollary 9.2 ( Theorem 7). A discrete half-line Schr odinger operator
(i.e., a
n
1) with
ess
(J) [2, 2] and

e
n
(J)
1/2
< has
ac
= [2, 2].
This is, of course, a special case of Theorem 9.1 but a striking one dis-
cussed further in Section 10. In particular, if a
n
1 and b
n
= n

w
n
where
<
1
2
and w
n
are identically distributed independent random variables with
distribution g() d with g L

and supp(g) bounded, then it is known

that [2, 2] is dense pure point spectrum (see Simon [54]). It follows that
J must also have innitely many eigenvalues outside [2, 2], indeed, enough
that

e
n
(J)
1/2
= .
Next, we deduce some additional aspects of Theorem 4

:
Corollary 9.3. If
ess
(J) [2, 2] and (9.1), (9.2) hold, then J J
0
I
2
,
that is,

b
2
n
+

(a
n
1)
2
< . (9.6)
Proof. By Theorem 6, (9.6) holds if

k
e
k
(J)
3/2
< , and Q(J) (given
by (5.21)) is nite. By (9.1) and e
k
(J)
3/2
e
1
(J)e
k
(J)
1/2
, we have that

e
k
(J)
3/2
< . Moreover, Z(J) < (i.e., Theorem 9.1) implies Q(J) <
. For, in any event,
_
ImM d < implies
_
2
0
log

_
sin
ImM
_
sin
2
() < and
_
2
0
log

_
sin
ImM
_
d < .
Thus
Z(J) <
_
2
0
log
+
_
sin
ImM
_
d <

_
2
0
log
+
_
sin
ImM
_
sin
2
<
Q(J) < .

What remains to be shown of Theorem 4

is the existence of the condi-

tional sums. We will start with

(a
n
1). Because

(a
n
1)
2
< ,
it is easy to see that

(a
n
1) is conditionally convergent if and only if

log(a
n
) is conditionally convergent. By (9.5) and the fact that J J
0
is
compact, we have:
Proposition 9.4. If (9.2) holds and (J) [2, 2], that is, no eigenvalues
outside [2, 2], then
Z(J) limsup
_
N

j=1
log(a
j
)
_
. (9.7)
48 R. KILLIP AND B. SIMON
We are heading towards a proof that
Z(J) liminf
_
N

j=1
log(a
j
)
_
(9.8)
from which it follows that the limit exists and equals Z(J).
Lemma 9.5. If (J) [2, 2], then log[z
1
M(z; J)] lies in every H
p
(D)
space for p < . In particular, z
1
M(z; J) is a Nevanlinna function with
no singular inner part.
Proof. In D(1, 0), we can dene Arg M(z; J) (, ) and Arg z
(, ) since ImM(z; J)/ Imz > 0. Thus g(z; J) = z
1
M(z; J) in the same
region has argument in (, ). But Arg g is single-valued and continuous
across (1, 0) since M has no poles and precisely one zero at z = 0. Thus
Arg g L

. It follows by Rieszs theorem on conjugate functions ([48,

pg. 351]) that log(g) H
p
(D) for any p < . Since it lies in H
1
, g is
Nevanlinna. Since for p > 1, any H
p
function, F, has boundary values
F(re
i
) F(e
i
) in L
p
, log(g) has no singular part in its boundary value.

Proposition 9.6. Let (J) [2, 2]. Suppose Z(J) < . Let C
0
, C
n
be
given by (4.10) and (4.15) (where the (J) terms are absent). Then the
step-by-step sum rules, (4.8), (4.13), (4.14), (4.24) hold. In particular,
Z(J) = log(a
1
) + Z(J
(1)
) (9.9)
C
1
(J) = b
1
+ C
1
(J
(1)
). (9.10)
Proof. (4.4) and therefore (4.1) hold. Thus, we only need apply Theorem 3.2
to g, noting that we have just proven that g has no singular inner part.
Theorem 9.7. If J is such that Z(J) < and (J) [2, 2], then
(i) lim
N

N
j=1
log(a
j
) exists.
(ii) The limit in (i) is Z(J).
(iii)
lim
n
Z(J
(n)
) = 0 (= Z(J
0
)) (9.11)
Proof. By (9.9),
Z(J) +
n

j=1
log(a
j
) = Z(J
(n)
). (9.12)
Since J J
0

2
,
J
(n)
J
0
weakly, and so, by (5.27), liminf Z(J
(n)
) 0,
or by (9.12),
liminf
_
n

j=1
log(a
j
)
_
Z(J). (9.13)
SUM RULES FOR JACOBI MATRICES 49
But (9.7) says
limsup
_
n

j=1
log(a
j
)
_
Z(J).
Thus the limit exists and equals Z(J), proving (i) and (ii). Moreover, by
(9.12), (i) and (ii) imply (iii).
If Z( ) had a positive integrand, (9.11) would immediately imply C
1
(J
(n)
)
0 as n , in which case, iterating (9.10) would imply that

n
j=1
b
j
is con-
ditionally convergent. Z( ) does not have a positive integrand but a theme
is that concavity often lets us treat it as if it does. Our goal is to use (9.11)
and the related lim
n
Q(J
(n)
) = 0 (which follows from Theorem 5) to still
prove that C
1
(J
(n)
) 0. We begin with
Lemma 9.8. Let d be a probability measure and suppose f
n
0,
_
f
n
d
1, and
lim
n
_
log(f
n
) d = 0. (9.14)
Then
_
[log(f
n
)[ d +
_
[f
n
1[ d 0. (9.15)
Proof. Let
H(y) = log(y) 1 + y. (9.16)
Then
(i) H(y) 0 for all y.
(ii) inf
|y1|
H(y) > 0
(iii) H(y)
1
2
y if y > 8.
(i) is concavity of log(y), (ii) is strict concavity, and (iii) holds because
log y 1 +
1
2
y is monotone on (2, ) and > 0 at y = 8 since log(8) is
slightly more than 2.
Since
_
(f
n
1) d 0, (9.14) and (i) implies that
_
f
n
(x) d(x) 1 (9.17)
and
lim
n
_
H(f
n
(x)) d(x) 0. (9.18)
Since H 0, (ii) and the above imply f
n
1 in measure:
(x [ [f
n
(x) 1[ > ) 0. (9.19)
By (i), (iii) and (9.18),
_
fn(x)>8
[f
n
(x)[ d 0. (9.20)
50 R. KILLIP AND B. SIMON
Now (9.19)/(9.20) imply that
_
[f
n
(x) 1[ d(x) 0
and this together with (9.18) implies
_
[log(f
n
)[ d = 0.
Proposition 9.9. Suppose Z(J) < and (J) [2, 2]. Then
lim
n
_

log
_
sin
ImM(e
i
, J
(n)
)
_

d = 0. (9.21)
Proof. By (9.11), the result is true if [ [ is dropped. Thus it suces to show
lim
n
_

log

_
sin
ImM(e
i
, J
(n)
)
_
d = 0
or equivalently,
lim
n
_

log
+
_
ImM(e
i
, J
(n)
)
sin
_
d = 0. (9.22)
Now, let d
0
() =
1

sin
2
d and f
n
() = (sin)
1
ImM(e
i
, J
(n)
). By
(1.19),
_

f
n
() d
0
() 1 (9.23)
and by Theorem 5 (and Corollary 9.3, which implies |J
(n)
J
0
|
2
2
0),
_
log(f
n
()) d
0
() 0
so, by Lemma 9.8, we control [log[ and so log
+
, that is,
lim
n
_

log
+
_
ImM(e
i
, J
(n)
)
sin
_
sin
2
d = 0. (9.24)
Thus, to prove (9.22), we need only prove
lim
0
limsup
n
_
||<
or
||<
log
+
_
ImM(e
i
, J
(n)
)
sin
_
d = 0. (9.25)
To do this, use
log
+
_
a
b
_
log
+
(a) + log

(b) = 2 log
+
(a
1/2
) + log

(b)
2a
1/2
+ log

(b)
with a = sin ImM(e
i
, J
(n)
) and b = sin
2
. The contribution of log

(b)
in (9.25) is integrable and n-independent, and so goes to zero as 0. The
contribution of the 2a
1/2
term is, by the Schwartz inequality, bounded by
(4)
1/2
_
4
_

f
n
() d
0
()
_
1/2
also goes to zero as 0. Thus (9.25) is proven.
SUM RULES FOR JACOBI MATRICES 51
The following concludes the proofs of Theorems 4 and 4

.
Theorem 9.10. If Z(J) < and (J) [2, 2], then
lim
N
N

j=1
b
j
exists and equals
1
2
_
2
0
log
_
sin
ImM(e
i
)
_
cos() d. (9.26)
Proof. By Proposition 9.9, C
1
(J
(n)
) 0 and, by (9.10),
C
1
(J) =
n

j=1
b
j
+ C
1
(J
(n)
).

As a nal topic in this section, we return to the general trace class case
where we want to prove that the C
0
(and other) sum rules hold; that is, we
want to improve the inequality (9.5) to an equality. The key will be to show
that in this case, the perturbation determinant is a Nevanlinna function
with vanishing inner singular part.
Proposition 9.11. Let J J
0
be trace class. Then, the perturbation deter-
minant L(z; J) is in Nevanlinna class.
Proof. By (2.19), if J
n
is given by (2.1), then
L(z; J
n
) L(z; J) (9.27)
uniformly on compact subsets of D. Thus
sup
0<r<1
_
2
0
log
+
[L(re
i
; J)[
d
2
sup
n
sup
0<r<1
_
2
0
log
+
[L(re
i
; J
n
)[
d
2
= sup
n
_
2
0
log
+
[L(e
i
; J
n
)[
d
2
(9.28)
where (9.28) follows from the monotonicity of the integral in r (see [48,
pg. 336]) and the fact that L(z; J
n
) is a polynomial.
In (9.28), write log
+
[L[ = log[L[ + log

[L[. By Jensens formula, (3.1),

and L(0; J) = 1,
_
2
0
log[L(e
i
; J
n
)[
d
2
=
Nn

j=1
log[
j
(J
n
)[
and this is uniformly bounded in n by the
1
2
LiebThirring inequality of
HundertmarkSimon [27], together with Theorem 6.1. On the other hand,
by (2.69),
2 log

[L(e
i
; J
n
)[ = log

_
n1

j=1
a
2
j
sin
ImM(e
i
; J
n
)
_
52 R. KILLIP AND B. SIMON
2
n1

j=1
log

(a
j
) + log(sin ) + log
+
_
ImM(e
i
; J
n
)
_
(9.29)
since log

[ab/c[ = [log(a) + log(b) log(c)]

log

(a) +log

(b) +log
+
(c).
The rst term in (9.29) is -independent and uniformly bounded in n
since

j=1
[a
j
1[ < . The second term is integrable. For the nal term,
we note that log
+
(y) y so by (1.19), the integral over is uniformly
bounded.
Remark. Our proof that L is Nevanlinna used

k
(e
k
(J))
1/2
< as input.
If we could nd a proof that did not use this a priori, we would have, as a
consequence, a new proof that

k
e
k
(J)
1/2
< since

[1
k
(J)
1
] <
is a general property of Nevanlinna functions.
Proposition 9.12. If J I
2
, the singular inner part of L(z; J), if any, is
a positive point mass at z = 1 and/or at z = 1.
Proof. By Theorem 2.8, L(z; J) is continuous on

D1, 1 and by (2.73),
it is nonvanishing on e
i
[ ,= 0, . It follows that on any closed interval,
I (0, ) (, 2), log[L(re
i
, J)[ d converges to an absolutely continuous
measure, so the support of the singular inner part is 1.
Returning to (9.29) and using log
+
(x) = 2 log
+
(x
1/2
) 2x
1/2
, we see that
log

[L[ lies in L
2
, that is,
sup
n
sup
0<r<1
_
[log

[L(re
i
, J
n
)[]
2
d
2
<
and this implies log

[L(re
i
, J)[ has an a.c. measure as its boundary value.
Thus 1 can only be positive pure points.
Remark. We will shortly prove L has no singular inner part. However, we
can ask a closely related question. If [

log(a
n
)[ < and

e
k
(J)
1/2
<
so Z(J) < , does the sum rule always hold or is there potentially a positive
singular part in some suitable object?
Theorem 2.10 will be the key to proving that L(x; J) has no pure point
singular part. The issue is whether the Blaschke product can mask the
polar singularity, since, if not, (2.46) says there is no polar singularity in L
which combines the singular inner part, outer factor, and Blaschke product.
Experts that we have consulted tell us that the idea that Blaschke products
cannot mask poles goes back to Littlewood and is known to experts, although
our approach in the next lemma seems to be a new and interesting way of
discussing this:
Lemma 9.13. Let f(z) be a Nevanlinna function on D. Then for any

0
D,
lim
r1
[log(1 r)
1
]
1
_
r
0
log[f(ye
i
0
)[ dy = 2
s
(
0
). (9.30)
SUM RULES FOR JACOBI MATRICES 53
Proof. Let B be a Blaschke product for f. Then ([48, pg. 346]),
log[f(z)[ = log[B(z)[ +
_

P(z, ) d() (9.31)

where P is the Poisson kernel
P(re
i
, ) =
(1 r
2
)
(1 + r
2
2r cos( ))
and d() is the boundary value of log[f(re
i
)[ d, that is, outer plus singular
inner piece. By an elementary estimate,
sup
r,,
(1 r)P(re
i
, ) <
and
lim
r1
(1 r)P(re
i
, ) =
_
0 ,=
2 =
and thus
(1 r)
_

P(re
i
, ) d() 2
s
().
This means (9.30) is equivalent to
lim
r1
[log(1 r)
1
]
1
_
r
0
log[B(ye
i
0
)[ dy = 0 (9.32)
for any Blaschke product. Without loss, we can take
0
= 0 in (9.32). Now
let
b

(z) =
[[

z
1 z
so
B(z) =

z
i
b
z
i
(z)
and note that for 0 < x < 1 and any D,
[b

(x)[ [b
||
(x)[,
so
log[B(x)[

z
i
log[b
|z
i
|
(z)[. (9.33)
Thus, also without loss, we can suppose all the zeros z
i
lie on (0, 1).
If (0, 1), a straightforward calculation (or Maple!) shows
_
1
0
log[b

(x)[ dx = log
_
1

_
+
1
2

log
_
1
1
_
. (9.34)
We claim that for a universal constant C and r >
3
4
, >
1
2
,

_
r
0
log[b

(x)[ dx C(1 ) log(1 r)

1
. (9.35)
54 R. KILLIP AND B. SIMON
Accepting (9.35) for the moment, by (9.33), we have for r >
3
4
,

_
r
0
log[B(x)[ dx
n

j=1
(
j
) + C
_

j=n+1
(1
j
)
_
log(1 r)
1
where () is the right side of (9.34). Dividing by log(1 r)
1
and using
() < , we see
limsup
_
log(1 r)
1
_

_
r
0
log[B(x)[ dx
__
C

j=n+1
(1
j
).
Taking n , we see that the limsup is 0. Since log[B(x)[ > 0, the limit
is 0 as required by (9.32). Thus, the proof is reduced to establishing (9.35).
Note rst that if 1 > >
1
2
,
1

(1 + ) < 4. Moreover, if g() = log(

1

),
then g

() =
1

2
< 0, so g() (1 ). Thus, if () is the right side of
(9.34) and >
1
2
, then
() 1 + 4(1 ) log
_
1
1
_
. (9.36)
Suppose now
1 (1 r)
2
. (9.37)
Then

_
r
0
log[b

(x)[ dx () (1 )
_
1 + 4 log
_
1
1
__
(1 )
_
1 + 8 log
_
1
1 r
__
(9.38)
by (9.37).
On the other hand, suppose
(1 ) (1 r)
2
. (9.39)
By an elementary estimate (see, e.g., [48, pg. 310]),
[1 b

(x)[
2
1 x
(1 ). (9.40)
If (9.39) holds and x < r, then, by (9.40),
[1 b

(x)[
2(1 r)
2
1 x
2(1 r) <
1
2
(9.41)
since r is supposed larger than
3
4
. If u (
1
2
, 1), then
log u =
_
1
u
dy
y
2(1 u)
so if (9.41) holds,
log(b

(x)) 2(1 b

(x))
4(1 )
1 x
SUM RULES FOR JACOBI MATRICES 55
and so
_
r
0
log(b

(x)) dx 4(1 ) log(1 r)

1
. (9.42)
We have thus proven (9.38) if (9.37) holds and (9.42) if (9.39) holds. To-
gether this proves (9.35).
Theorem 9.14. Let J J
0
be trace class. Then the Nevanlinna function
L(z; J) has a vanishing singular inner component and all the sum rules
C
0
, C
1
, . . . hold with no singular term.
Proof. By Proposition 9.12, the only possible singular parts are positive
points at 1. By (9.30) and the estimate (2.50), these point masses are
absent. Thus the singular part vanishes and the sum rules hold by Theo-
rem 3.2.
10. Whole-Line Schr odinger Operators
With No Bound States
Our goal in this section is to prove Theorem 8 that the only whole-line
Schr odinger operator with (W) [2, 2] is W
0
, the free operator. We do
this here because it illustrates two themes: that absence of bound states is
a strong assertion and that sum rules can be very powerful tools.
Given two sequences of real numbers a
n

n=
, b
n

n=
with a
n
> 0,
we will denote by W the operator on
2
(Z) dened by
(Wu)
n
= a
n1
u
n1
+ b
n
u
n
+ a
n+1
u
n+1
. (10.1)
W
0
is the operator with a
n
1, b
n
0. The result we will prove is:
Theorem 10.1. Let W be a whole-line operator with a
n
1 and (W)
[2, 2]. Then W = W
0
, that is, b
n
0.
The proof works if
limsup
n
m
_
m

j=n
log(a
j
)
_
0.
The strategy of the proof will be to establish analogs of the C
0
and C
2
sum
rules. Unlike the half-line case, the integrand inside the Szeg o-like integral
will be nonnegative. The C
0
sum rule will then imply this integrand is zero
and the C
2
sum will therefore yield

n
b
2
n
= 0. As a preliminary, we note:
Proposition 10.2. If a
n
1 and (W) [2, 2], then

n
b
2
n
< . In
particular, b
n
0 as [n[ .
Proof. Let J be a Jacobi matrix obtained by restricting to 1, 2, . . . . By
the min-max principle [45], (J) [2, 2]. By Corollary 9.3,

n=1
b
2
n
< .
Similarly, by restricting to 0, 1, 2, . . . , , we obtain

0
n=
b
2
n
< .
56 R. KILLIP AND B. SIMON
Let W
(n)
for n = 1, 2, . . . be the operator with
_

_
a
(n)
j
1
b
(n)
j
= b
j
if [j[ n
b
(n)
j
= 0 if [j[ > n.
(10.2)
Then, Proposition 10.2 and the proofs of Theorems 6.1 and 6.2 immediately
imply:
Theorem 10.3. If a
n
1 and (W) [2, 2], then W
(n)
has at most four
eigenvalues in R[2, 2] (up to two in each of (, 2) and (2, )) and
for j = 1, . . . , 4,
[e
j
(W
(n)
)[ 0 (10.3)
as n .
Note: As in the Jacobi case, e
j
(W) is a relabeling of [E

j
(W)[ 2 in de-
creasing order.
To get the sum rules, we need to study whole-line perturbation deter-
minants. We will use the same notation as for the half-line, allowing the
context to distinguish the two cases. So, let W = W W
0
be trace class
and dene
L(z; W) = det((W E(z)(W
0
E(z))
1
) (10.4)
= det(1 +W(W
0
E(z))
1
) (10.5)
where as usual, E(z) = z + z
1
.
The calculation of the perturbation series for L is algebraic and so imme-
diately extends to imply:
Proposition 10.4. If W is trace class, for each n, T
n
(W/2) T
n
(W
0
/2)
is trace class. Moreover, for [z[ small,
log[L(z; W)] =

n=1
c
n
(W)z
n
(10.6)
where c
n
(W) is
c
n
(W) =
2
n
Tr
_
T
n
_
1
2
W
_
T
n
_
1
2
W
0
_
_
. (10.7)
In particular,
c
2
(W) =
1
2

m=1
b
2
m
+ 2(a
2
m
1). (10.8)
The free resolvent, (W
0
E(z))
1
, has matrix elements that we can com-
pute as we did to get (2.9),
(W
0
E(z))
1
nm
= (z
1
z)
1
z
|mn|
which has poles at z = 1. We immediately get
SUM RULES FOR JACOBI MATRICES 57
Proposition 10.5. If W is nite rank, L(z; J) is a rational function on
C with possible singularities only at z = 1.
Remarks. 1. If W has b
0
= 1, all other elements zero, then
L(z; W) = 1 (z
1
z)
1
=
(1 z z
2
)
(1 z
2
)
has poles at 1, so poles can occur.
2. The rank one operator
R(z) = (z
1
z)
1
z
m+n
is such that if W = C
1/2
UC
1/2
with C nite rank, then
C
1/2
[(W
0
E(z)
1
R(z)]C
1/2
is entire. Using this, one can see L(z; J) has a pole of order at most 1 when
W is nite rank. We will see this below in another way.
If z

D1, 1, we can dene a Jost solution u
+
n
(z; W) so that (2.65)
holds for all n Z and
lim
n
z
n
u
+
n
(z; W) = 1. (10.9)
Moreover, if W has nite rank, u
+
n
is a polynomial in z for each n 0.
Moreover, for n < 0, z
n
u
+
n
is a polynomial in z by using (2.67).
Similarly, we can construct u

n
solving (2.65) for all n Z with
lim
n
z
n
u

n
(z; W) = 1.
As above, if W is nite rank, u

n
is a polynomial in z if n 0 and for
n > 0, z
n
u

n
is a polynomial.
Proposition 10.6. Let W be trace class. Then for z

D1, 1 and all
n Z,
L(z; W) = (z
1
z)
1
_

j=
a
j
_
a
n
[u
+
n
(z; W)u

n+1
(z; W) u

n
(z; W)u
+
n+1
(z; W)].
(10.10)
Proof. Both sides of (10.10) are continuous in W, so we need only prove the
result when W is nite rank. Moreover, by constancy of the Wronskian,
the right side of (10.10) is independent of n so we need only prove (10.10)
when [z[ < 1 and n is very negativeso negative it is to the left of the
support of W, that is, choose R so a
n
= 1, b
n
= 0 if n < R, and we will
prove that (10.10) holds for n < R.
For n < R, z
n
and z
n
are two solutions of (2.65) so in that region we
have
u
+
n
=

z
n
+

z
n
. (10.11)
58 R. KILLIP AND B. SIMON
Taking the Wronskian of u
+
n
given by (10.11) and u

n
= z
n
at some point
n < R, we see
RHS of (10.10) =

j=
a
j
_
. (10.12)
Let W
n
, W
0;n
be the Jacobi matrices on
2
(n+ 1, n+2, . . . , ) obtained by
truncation. On the one hand, as with the proof of (2.74), for [z[ < 1,
L(z; W) = lim
n
det((W
n
E(z))(W
n;0
E(z))
1
) (10.13)
and on the other hand, for n < R, by (2.64),
RHS of (10.13) =
_

n
a
n
_
z
n
u
+
n
(z; W).
Thus
L(z; W) =
_

n
a
n
_
lim
n
z
n
u
+
n
(z; W)
=
_

n
a
n
_
lim
n
(

z
2n
)
=
_

n
a
n
_

(10.14)
since [z[ < 1 and n . Comparing (10.12) and (10.14) yields (10.10).

Note: In (10.11),

use for left since they are related to scattering

from the left.
Corollary 10.7. If W is nite rank, then (1 z
2
)L(z; W) is a polynomial
and, in particular, L(z; W) is a rational function.
Proof. By (10.10), this is equivalent to z(u
+
0
u

1
u

0
u
+
1
) being a polynomial.
But u
+
0
, zu

1
, u

0
, and u
+
1
are all polynomials.
Let W be nite rank. Since L is meromorphic in a neighborhood of

D
and analytic in D, Proposition 3.1 immediately implies the following sum
rule:
Theorem 10.8. If W is nite rank, then
C
0
:
1
2
_
2
0
log[L(e
i
; W)[ d =
N(W)

j=1
log[
j
(W)[ (10.15)
C
n
:
1

_
2
0
log[L(e
i
; W)[ cos(n) d
=
1
n
N(W)

j=1
[
n
j

n
j
]
2
n
Tr
_
T
n
_
1
2
W
_
T
n
_
1
2
W
0
_
_
(10.16)
SUM RULES FOR JACOBI MATRICES 59
for n 1.
The nal element of our proof is an inequality for L(e
i
; W) that depends
on what a physicist would call conservation of probability.
Proposition 10.9. Let W be trace class. Then for all ,= 0, ,
[L(e
i
; W)[

j=
a
j
. (10.17)
Proof. As above, we can suppose that W is nite range. Choose R so that
all nonzero matrix elements of W have indices lying within (R, R). By
(10.12), (10.17) is equivalent to
[

[ 1 (10.18)
where

is given by (10.11).
Since u
+
n
(z; W) is real for z real, we have
u
+
n
( z; W) = u
+
n
(z; W).
Thus for z = e
i
, ,= 0, , and n < R,
u
+
n
(e
i
; W) =

(e
i
)e
in
+

(e
i
)e
in
u
+
n
(e
i
; W) =

(e
i
) e
in
+

(e
i
) e
+in
.
Computing the Wronskian of the left-hand sides for n > R, where u
+
n
= z
n
and then the Wronskian of the right-hand sides for n < R, we nd
i(sin) = i(sin)[[

[
2
[

[
2
]
or, since ,= 0, ,
[

[
2
= 1 +[

[
2
(10.19)
from which (10.18) is obvious.
Remark. In terms of the transmission and reection coecients of scattering
theory [61],

= 1/t,

= r/t, (10.19) is [r[

2
+[t[
2
= 1 and (10.18) is [t[ 1.
We are now ready for
Proof of Theorem 10.1. Let W
(n)
be given by (10.2). Then, by (10.3) and
C
0
(10.15),
lim
n
1
2
_
2
0
log[L(e
i
; W
(n)
)[ d = 0. (10.20)
Since a
n
1, (10.17) implies log[L(e
i
; W
(n)
)[ 0, and so (10.20) implies
lim
n
1
2
_
2
0
cos(2) log[L(e
i
; W
(n)
)[ d = 0. (10.21)
By (10.8), a
m
1, C
2
, and (10.3), we see
lim
n

|j|<n
b
2
j
= 0
which implies b 0.
60 R. KILLIP AND B. SIMON
Finally, a remark on why this result holds that could provide a second
proof (without sum rules) if one worked out some messy details. Here is a
part of the idea:
Proposition 10.10. Let b
n
be a bounded sequence and W the associated
whole-line Schr odinger operator (with a
n
1). Let
A() =

b
n
e
|n|
. (10.22)
If
limsup
0
A() > 0, (10.23)
W has spectrum in (2, ), and if
liminf
0
A() < 0, (10.24)
then W has spectrum in (, 2).
Proof. Let (10.23) hold and set

(n) = e
|n|/2
. Then
(W
0

)(n) =
_
2 cosh(

2
) if n ,= 0
[2 cosh(

2
) 2 sinh(

2
)]

(n) if n = 0.
It follows that
(

, W

)(n) = 2 cosh
_

2
_
|

|
2
+ A() 2 sinh
_

2
_
= 2|

|
2
+ 2
_
cosh
_

2
_
1
_
|

|
2
+A() 2 sinh
_

2
_
.
(10.25)
Now, sinh(/2) 0 as 0 and since |

|
2
= O(
1
) and cosh(/2)1 =
O(
2
), 2[cosh(/2) 1]|

|
2
0 as 0. If there is a sequence with
limA(
n
) > 0, for n large, (
n
, (W 2)
n
) > 0 which implies there is
spectrum in (2, ).
If (10.24) holds, use

(n) = (1)
n
e
|n|/2
and a similar calculation to
(
n
, (W + 2)
n
) < 0.
This proof is essentially a variant of the weak coupling theory of Simon
[50]. Those ideas immediately show that if

n[b
n
[ < (10.26)
and

b
n
= 0 (so Proposition(10.10) does not apply), then W has eigenval-
ues in both (2, ) and (, 2) unless b 0. This reproves Theorem 10.1
when (10.25) holds by providing explicit eigenvalues outside [2, 2]. It is
likely using these ideas as extended in [4, 31], one can provide an alternate
proof of Theorem 10.1. In any event, the result is illuminated.
SUM RULES FOR JACOBI MATRICES 61
References
[1] M. Abramowitz, I.A. Stegun, Handbook of mathematical functions with formulas,
graphs, and mathematical tables, For sale by the Superintendent of Documents,
U.S. Government Printing Oce, Washington, D.C., 1964.
[2] N.I. Akhiezer, Theory of Approximation, Dover Publications, New York, 1956.
[3] Ju.M. Berezanski, Expansions in eigenfunctions of selfadjoint operators, Translated
from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. Trans-
lations of Mathematical Monographs, American Mathematical Society, Providence,
R.I., 1968.
[4] R. Blankenbecler, M.L. Goldberger, and B. Simon, The bound state of weakly coupled
long range one-dimensional quantum Hamiltonians, Ann. Phys. 108 (1977), 6978.
[5] O. Blumenthal, Ueber die Entwicklung einer willk urlichen Funktion nach den Nen-
nern des Kettenbruches f ur

() d
z
, Ph.D. dissertation, Gottingen 1898.
[6] K.M. Case, Orthogonal polynomials from the viewpoint of scattering theory, J. Math.
Phys. 15 (1974), 21662174.
[7] K.M. Case, Orthogonal polynomials. II, J. Math. Phys. 16 (1975), 14351440.
[8] T.S. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and its
Applications Vol. 13, Gordon and Breach, New York-London-Paris, 1978.
[9] T.S. Chihara, Orthogonal polynomials whose distribution functions have nite point
spectra, SIAM J. Math. Anal. 11 (1980), 358364.
[10] T.S. Chihara and P. Nevai, Orthogonal polynomials and measures with nitely many
point masses, J. Approx. Theory 35 (1982), 370380.
[11] M. Christ and A. Kiselev, Absolutely continuous spectrum for one-dimensional
Schr odinger operators with slowly decaying potentials: Some optimal results, J.
Amer. Math. Soc. 11 (1998), 771797.
[12] J.E. Cohen, J.H.B. Kemperman, and Gh. Zbaganu, Comparisons of Stochastic Ma-
trices. With Applications in Information Theory, Statistics, Economics, and Popu-
lation Sciences, Birkhauser Boston, Boston, MA, 1998.
[13] P. Deift and R. Killip, On the absolutely continuous spectrum of one-dimensional
Schr odinger operators with square summable potentials, Comm. Math. Phys. 203
(1999), 341347.
[14] S.A. Denisov, On the coexistence of absolutely continuous and singular continuous
components of the spectral measure for some Sturm-Liouville operators with square
summable potentials, preprint.
[15] J. Favard, Sur les polynomes de Tchebiche, C.R. Acad. Sci. Paris 200 (1935),
20522053.
[16] H. Flaschka, The Toda lattice. I. Existence of integrals, Phys. Rev. B (3) 9 (1974),
19241925.
[17] H. Flaschka, On the Toda lattice. II. Inverse-scattering solution, Progr. Theoret.
Phys. 51 (1974), 703716.
[18] J.B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.
[19] J.S. Geronimo, An upper bound on the number of eigenvalues of an innite dimen-
sional Jacobi matrix, J. Math. Phys. 23 (1982), 917921.
[20] J.S. Geronimo, On the spectra of innite-dimensional Jacobi matrices, J. Approx.
Theory 53 (1988), 251265.
[21] J.S. Geronimo, Scattering theory, orthogonal polynomials, and q-series, SIAM J.
Math. Anal. 25 (1994), 392419.
[22] J.S. Geronimo and W. Van Assche, Orthogonal polynomials with asymptotically pe-
riodic recurrence coecients, J. Approx. Theory 46 (1986), 251283.
[23] L.Ya. Geronimus, Orthogonal polynomials: Estimates, Asymptotic Formulas, and
Series of Polynomials Orthogonal on the Unit Circle and on an Interval, Consultants
Bureau, New York, 1961.
62 R. KILLIP AND B. SIMON
[24] F. Gesztesy and B. Simon, m-functions and inverse spectral analysis for nite and
semi-innite Jacobi matrices, J. Anal. Math. 73 (1997), 267297.
[25] I.C. Gohberg and M.G. Krein, Introduction to the theory of linear nonselfadjoint op-
erators, Translations of Mathematical Monographs, Vol. 18, American Mathematical
Society, Providence, R.I., 1969.
[26] E. Hellinger and O. Toeplitz, Integralgleichungen und Gleichungen mit unendlichvie-
len Unbekannten, Encyklopadie der Mathematischen Wissenschaften II C13 (1928),
13351616.
[27] D. Hundertmark and B. Simon, Lieb-Thirring inequalities for Jacobi matrices,
preprint.
[28] R. Killip, Perturbations of one-dimensional Schr odinger operators preserving the
absolutely continuous spectrum, preprint.
[29] A. Kiselev, Absolutely continuous spectrum of one-dimensional Schr odinger opera-
tors and Jacobi matrices with slowly decreasing potentials, Comm. Math. Phys. 179
(1996), 377400.
[30] A. Kiselev, Imbedded Singular Continuous Spectrum for Schr odinger Operators, pre-
pint.
[31] M. Klaus, On the bound state of Schr odinger operators in one dimension, Ann.
Physics 108 (1977), 288300.
[32] A.N. Kolmogoro, Stationary sequences in Hilberts space (Russian), Bolletin
Moskovskogo Gosudarstvenogo Universiteta. Matematika 2 (1941).
[33] M.G. Krein, On a generalization of some investigations of G. Szeg o, V. Smirno
and A. Kolmogoro, C.R. (Doklady) Acad. Sci. USSR (N.S.) 46 (1945), 9194.
[34] S. Kullback and R.A. Leibler, On information and suciency, Ann. Math. Statistics
22 (1951), 7986.
[35] O.E. Lanford and D.W. Robinson, Statistical mechanics of quantum spin systems.
III, Comm. Math. Phys. 9 (1968), 327338.
[36] A. Laptev, S. Naboko, and O. Safronov, Absolutely continuous spectrum of Jacobi
matrices, preprint.
[37] E.H. Lieb and W. Thirring, Bound for the kinetic energy of fermions which proves
the stability of matter, Phys. Rev. Lett. 35 (1975), 687689. Errata 35 (1975), 1116.
[38] E.H. Lieb and W. Thirring, Inequalities for the moments of the eigenvalues of
the Schr odinger Hamiltonian and their relation to Sobolev inequalities, in Stud-
ies in Mathematical Physics. Essays in Honor of Valentine Bargmann, pp. 269303,
Princeton University Press, Princeton, NJ, 1976.
[39] A. Mate, P. Nevai, and V. Totik, Szegos extremum problem on the unit circle, Ann.
of Math. 134 (1991), 433453.
[40] S. Molchanov, M. Novitskii, and B. Vainberg, First KdV integrals and absolutely
continuous spectrum for 1-D Schr odinger operator, Comm. Math. Phys. 216 (2001),
195213.
[41] P. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, 185
pp.
[42] P. Nevai, Orthogonal polynomials dened by a recurrence relation, Trans. Amer.
Math. Soc. 250 (1979), 369384.
[43] P. Nevai, Orthogonal polynomials, recurrences, Jacobi matrices, and measures, in
Progress in Approximation Theory (Tampa, FL, 1990), pp. 79104, Springer Ser.
Comput. Math., 19, Springer, New York, 1992.
[44] M. Ohya and D. Petz, Quantum Entropy and Its Use, Texts and Monographs in
Physics, Springer-Verlag, Berlin, 1993.
[45] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV. Analysis of
Operators, Academic Press, New York, 1978.
[46] T. Regge, Analytic properties of the scattering matrix, Nuovo Cimento (10) 8 (1958),
671679.
SUM RULES FOR JACOBI MATRICES 63
[47] C. Remling, The absolutely continuous spectrum of one-dimensional Schr odinger
operators with decaying potentials, Comm. Math. Phys. 193 (1998), 151170.
[48] W. Rudin, Real and Complex Analysis, 3rd edition, Mc-Graw Hill, New York, 1987.
[49] J.A. Shohat, Theorie Generale des Polinomes Orthogonaux de Tchebichef, Memorial
des Sciences Mathematiques, Vol. 66, pp. 169, Paris, 1934.
[50] B. Simon, The bound state of weakly coupled Schr

dinger operators in one and two

dimensions, Ann. Phys. 97 (1976), 279288.
[51] B. Simon, Analysis with weak trace ideals and the number of bound states of
Schr odinger operators, Trans. Amer. Math. Soc. 224 (1976), 367380.
[52] B. Simon, Notes on innite determinants of Hilbert space operators, Adv. Math. 24
(1977), 244273.
[53] B. Simon, Trace Ideals and Their Applications, London Mathematical Society Lec-
ture Note Series, 35, Cambridge University Press, Cambridge-New York, 1979.
[54] B. Simon, Some Jacobi matrices with decaying potential and dense point spectrum,
Comm. Math. Phys. 87 (1982), 253258.
[55] B. Simon, The Statistical Mechanics of Lattice Gases. Vol. I, Princeton Series in
Physics, Princeton University Press, Princeton, NJ, 1993.
[56] B. Simon, The classical moment problem as a self-adjoint nite dierence operator,
Adv. Math. 137 (1998), 82203.
[57] T.J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse 8
(1894), J, 1122; 9 (1894), A, 147; Oeuvres 2, 402566. Also published in Memoires
presentes par divers savants `a lAcademie des sciences de lInstitut National de
France, vol. 33, 1196.
[58] M.H. Stone, Linear Transformations in Hilbert Spaces and Their Applications to
Analysis, Amer. Math. Soc. Colloq. Pub. vol. 15, New York, 1932.
[59] G. Szego, Beitr age zue Theorie der Toeplitzschen Formen, II, Math. Z. 9 (1921),
167190.
[60] G. Szego, Orthogonal Polynomials, 4th edition. American Mathematical Society,
Colloquium Publications, Vol. XXIII. American Mathematical Society, Providence,
R.I., 1975.
[61] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, American
Mathematical Society, Providence, R.I., 1999.
[62] H.S. Wall, Analytic Theory of Continued Fractions, AMS Chelsea Publ., American
Mathematical Society, Providence, R.I., 1948.
[63] H. Weyl,

Uber beschr ankte quadratische Formen, deren Dierenz vollstetig ist, Rend.
Circ. Mat. Palermo 27 (1909), 373392.
[64] M. Zworski, Distribution of poles for scattering on the real line, J. Funct. Anal. 73
(1987), 277296.