HEPHY-PUB 800/05

May 2005

RELATIVISTIC HARMONIC OSCILLATOR

arXiv:hep-ph/0501268v2 20 Sep 2005

LI Zhi-Feng‡
Department of Physics, Chongqing University, 400044 Chongqing, China

LIU Jin-Jin
Department of Modern Physics, University of Science and Technology of China, Hefei, China

Wolfgang LUCHA∗
Institute for High Energy Physics, Austrian Academy of Sciences,
Nikolsdorfergasse 18, A-1050 Vienna, Austria

MA Wen-Gan
Department of Modern Physics, University of Science and Technology of China, Hefei, China

Franz F. SCHÖBERL†
Institute for Theoretical Physics, University of Vienna,
Boltzmanngasse 5, A-1090 Vienna, Austria

Abstract

We study the semirelativistic Hamiltonian operator composed of the relativistic kinetic energy
and a static harmonic-oscillator potential in three spatial dimensions and construct, for bound
states with vanishing orbital angular momentum, its eigenfunctions in “compact form,” i.e., as
power series, with expansion coefficients determined by an explicitly given recurrence relation.
The corresponding eigenvalues are fixed by the requirement of normalizability of the solutions.

PACS numbers: 03.65.Pm, 03.65.Ge

‡
Present address: Institute for Theoretical Physics, University of Vienna, Boltzmanngasse
5, A-1090 Vienna, Austria
∗
E-mail address: wolfgang.lucha@oeaw.ac.at
†
E-mail address: franz.schoeberl@univie.ac.at
 1

1 Introduction: relativistic harmonic-oscillator problem

The simplest and perhaps most straightforward generalization of the Schrödinger operators of
standard nonrelativistic quantum theory towards the inclusion of relativistic kinematics leads
to Hamiltonians H that involve the relativistic kinetic energy, or relativistically covariant form
of the free energy, of a particle of mass m and momentum p, given by the square-root operator
q
T (p) ≡ p2 + m2 , p ≡ |p| ,

and a coordinate-dependent static interaction potential V (x). In the one-body case, they read
q
H= p2 + m2 + V (x) . (1)

The eigenvalue equation of this Hamiltonian is usually called the “spinless Salpeter equation.”
It may be regarded as a well-defined approximation to the Bethe–Salpeter formalism [1] for the
description of bound states within relativistic quantum field theories, obtained when assuming
that all bound-state constituents interact instantaneously and propagate like free particles [2].
Among others, it yields semirelativistic descriptions of hadrons as bound states of quarks [3,4].
In general, the above semirelativistic Hamiltonian H is, unfortunately, a nonlocal operator:
either the relativistic kinetic energy, T (p), in configuration space or, in general, the interaction
potential in momentum space is a nonlocal operator. Because of the nonlocality it is somewhat
difficult to obtain rigorous analytical statements about the solutions of its eigenvalue equation.
Thus sophisticated methods have been developed to extract information about these solutions;
for details and comparisons of the various approaches, consult, for example, the reviews [5–10].
Analytical or, at least, semianalytical1 expressions for both upper and lower bounds on the
eigenvalues of some self-adjoint operator may be found by combining the minimum–maximum
principle [11–13] and appropriate operator inequalities [14–18]. The outcome of this procedure
is sometimes called the “spectral comparison theorem.” In Sec. 3 below, we will use this kind of
bounds in order to estimate the accuracy of our findings for the eigenvalues of the operator (1).
Accordingly, we recall in Appendix A the proof of the “translation” of some inequality satisfied
by two operators into the corresponding relations between their discrete eigenvalues, by briefly
sketching all basic assumptions and the line of argument. A very systematic path for obtaining
such operator inequalities is provided by rather simple geometrical considerations summarized
under the notion “envelope theory” [19–24]. These envelope techniques may be generalized to
systems composed of arbitrary numbers of relativistically moving interacting particles [25–27].
For particular potentials V, semianalytical lower bounds on the ground-state energy eigenvalue
of the semirelativistic Hamiltonian (1), and hence on the entire spectrum of H, can be found by
the appropriate generalization of the local-energy theorem [13,28–30] to our case of relativistic
kinematics [23,31], or by applying the optimized (Beckner–Brascamp–Lieb) version of Young’s
inequality for convolutions to some integral formulation of the spinless Salpeter equation [32].
Purely numerical solutions of the spinless Salpeter equation may be computed in numerous
ways. The semirelativistic Hamiltonian H can be approximated by some effective Hamiltonian
which is of nonrelativistic shape but uses parameters that depend on expectation values of the
momenta [33,34]. Upper bounds of, in principle, arbitrarily high precision on the eigenvalues of
a self-adjoint operator can be found [18,35–39] with the help of the Rayleigh–Ritz (variational)
1
We regard a bound on some eigenvalue of a given self-adjoint operator as semianalytical if it can be derived
by the — in general, numerical — optimization of an analytically known expression over a single real variable.
 2

technique as immediate consequence of the minimum–maximum theorem [11–13]. The spinless

Salpeter equation may also be converted into an equivalent matrix eigenvalue problem [40–45].
A particular rôle for H is played by the spherically symmetric harmonic-oscillator potential
V (x) = a r 2 , r ≡ |x| , a>0;
this potential defines the “relativistic harmonic-oscillator problem,” posed by the Hamiltonian
q
H= p2 + m2 + a r 2 . (2)
The eigenvalue equation of H, for eigenstates |ψi, H |ψi = E |ψi, involves only one parameter:
upon factorizing off some overall energy scale a1/3 by performing the canonical transformation
r
r → 1/3 , p → a1/3 p
a
and rescaling both mass m and eigenvalue E according to m = a1/3 µ and E = a1/3 ε, it reads
q 
p2 + µ2 + r 2 |ψi = ε |ψi .

In momentum-space representation this eigenvalue equation reduces to a Schrödinger problem

[−∆p + V (p)] ψ(p) = ε ψ(p) , (3)
with an interaction potential reminiscent of the square root of the relativistic kinetic energy:
q
V (p) ≡ p2 + µ 2 .
We would like to take advantage of this facet of the relativistic harmonic-oscillator problem
in order to derive for its bound-state eigenfunctions, in this analysis, a compact expression and
to move thereby beyond only numerically calculated exact solutions without having to rely on
perturbation theory, maybe paving the way for the eventual construction of analytic solutions.

2 Analytical solutions for bound-state eigenfunctions

We exploit the fact that, in contrast to the general case, for a harmonic-oscillator potential (as,
with due care, for any potential of the form V ∝ r 2n , n = 1, 2, 3, . . .) the eigenvalue equation of
the Hamiltonian H is an ordinary differential equation, parametrized by µ and its eigenvalue ε.
Focusing on the ground state or purely radial excitations, let us introduce, for eigenstates
with vanishing relative orbital angular momentum ℓ, the reduced radial wave function y(p) by
√
y(p) ≡ 4 π p ψ(p) (ℓ = 0) ,
which is, of course, subject to the normalization condition
Z∞
dp |y(p)|2 = 1 .
0

In accordance with Eq. (3) the reduced radial wave function satisfies a reduced radial equation:
d2 y
(p) = [V (p) − ε] y(p) . (4)
dp2
 3

From its definition, this reduced radial wave function y(p) has to vanish at the origin: y(0) = 0.
Moreover, the analysis of the normalizable solutions of the eigenvalue equation (4) reveals that
y(p) behaves like p for small p, that is, for p ≪ 1; hence, its derivative with respect to p at the
point p = 0 is a nonvanishing constant, which may be absorbed into the overall normalization:
dy
(0) = 1 .
dp

We construct all solutions of Eq. (4) in form of Taylor-series expansions by using the ansatz
∞
X pn
y(p) = cn
n=0 n!

with the expansion coefficients

dn y
cn ≡ n (0) , n = 0, 1, 2, . . . .
dp

The solution of the eigenvalue equation (4) is then clearly tantamount to the determination of
the expansion coefficients cn . The first three of these expansion coefficients are known trivially:

c0 = y(0) = 0 ,
dy
c1 = (0) = 1 ,
dp
d2 y
c2 = (0) = [(V − ε) y](0) = 0 .
dp2

[For the sake of notational simplicity, we suppress, in accordance with our above remark, in the
following that normalization factor of y(p) which guarantees its unity norm and assume y(p) to
be normalized such that the value of the first nonvanishing expansion coefficient is one: c1 = 1.]
Upon insertion of the eigenvalue equation (4) followed by the application of Leibniz’s theorem,
the nontrivial expansion coefficients cn , n ≥ 3, may be shown to satisfy the recurrence relation:
" #
dn+2 y dn d2 y dn [(V − ε) y]
cn+2 = (0) = n (0) = (0)
dpn+2 dp dp2 dpn
n
!" #
X n dk (V − ε) dn−k y
= (0)
k=0
k dpk dpn−k
n
!
X n dk V
= (µ − ε) cn + (0) cn−k
k=1
k dpk
n
!
X n
= (µ − ε) cn + µ1−k dk cn−k , n = 1, 2, 3, . . . . (5)
k=1
k

Here, for the last step, we abbreviated the k-th derivative of the potential V by a coefficient dk :
k k
√
d V d x2 + 1 p
dk ≡ µk−1 k (0) = k
(0) , x≡ , µ>0, k = 0, 1, 2, . . . .
dp dx µ
 4

In the case µ = m = 0, the solutions involve Airy’s function Ai(z); cf., e. g., Refs. [18,23,33,34].
According
√ to the above definition, we have d0 = 1. Furthermore, by inspection of the function
f (x) = x2 + 1, it is easy to convince oneself that all odd derivatives of f (x) vanish at x = 0,
d2k+1 = 0 for all k = 0, 1, 2, . . . , (6)
whereas all even derivatives of f (x) at x = 0 necessarily satisfy the (simple) recurrence relation
d2k+2 = (1 − 4 k 2 ) d2k , k = 0, 1, 2, . . . .
By induction, the solution of this recurrence relation for the nonvanishing coefficients d2k reads
" #2
k−1 (2 k − 2)!
d2k = (−1) (2 k − 1) k−1 , k = 1, 2, 3, . . . . (7)
2 (k − 1)!
Taking into account the observation (6), we obtain c3 = µ−ε and, for the coefficients cn , n ≥ 4,
[n/2] !
X n
cn+2 = (µ − ε) cn + µ1−2k d2k cn−2k , n = 2, 3, 4, . . . ,
k=1
2k
where 

n
  
 for n even, n = 2, 4, 6, . . . ,
n 2
≡
2 

n−1
for n odd, n = 3, 5, 7, . . . .
2
Thus the recurrence relation (5) for all expansion coefficients cn decomposes into one involving
only the even coefficients c2n , n = 2, 3, 4, . . . , and one involving only the odd coefficients c2n+1 ,
n = 2, 3, 4, . . . ; recalling c0 = 0 and c2 = 0, we conclude that all the even coefficients cn vanish:
c2n = 0 for all n = 0, 1, 2, . . . .
With the result (7), the recurrence relation for the (nonvanishing) odd coefficients finally reads
n
!
X 2n+1
c2n+3 = (µ − ε) c2n+1 + µ1−2k d2k c2n−2k+1
k=1
2k
n
! " #2
X 2n+1 (2 k − 2)!
1−2k k−1
= (µ − ε) c2n+1 + c2n−2k+1 µ (−1) (2 k − 1) k−1 ,
k=1
2k 2 (k − 1)!
n = 1, 2, 3, . . . . (8)
In summary, upon constructing the relevant expansion coefficients according to this recurrence
relation the analytical expressions for all reduced radial wave functions (of ℓ = 0 bound states)
y(p) of the relativistic harmonic-oscillator problem (2) are given by the power-series expansion
∞
X p2n+1
y(p) = c2n+1 . (9)
n=0 (2 n + 1)!
The various solutions y(p) of the eigenvalue equation (3) are characterized or discriminated
by different sets of expansion coefficients cn . By construction, apart from the first coefficient c1 ,
all expansion coefficients for a given solution depend on the corresponding energy eigenvalue ε.
Evaluating the recurrence relation (8) for just the first term, the series (9) explicitly starts with
( " # )
p2 3 p4
y(p) = p 1 + (µ − ε) + (µ − ε)2 + + ... .
3! µ 5!
 5

3 Explicit bound-state eigenfunctions and energy levels

The central result of our present investigation of the “relativistic harmonic-oscillator problem”
defined by the Hamiltonian H of Eq. (2) is an analytical expression for the reduced radial parts
y(p) of all eigenfunctions ψ(p) of H for vanishing angular momentum in form of a Taylor series:
√ ∞
X p2n+1
y(p) ≡ 4 π p ψ(p) = c2n+1 ,
n=0 (2 n + 1)!
where the corresponding expansion coefficients c2n+1 , n = 0, 1, 2, . . . , are either fixed to c1 = 1,
c3 = µ−ε or, for c2n+1 , n ≥ 2, determined by the recurrence relation (8). For a given solution of
this eigenvalue problem these expansion coefficients and thus the resulting reduced radial wave
function y(p) depend on the parameter µ ≡ m/a1/3 and the corresponding energy eigenvalue ε.
Taking into account the necessary requirement of normalizability of Hilbert-space eigenstates,
the inversion of the latter relation may be exploited to determine the energy eigenvalues ε from
the knowledge of the dependence of ε on the coefficients cn , as derived in the preceding section.
In order to fulfil its normalization condition, any solution y(p) must vanish in the limit p → ∞:

lim y(p) = 0 .
p→∞
(10)

Fixing the energy eigenvalue ε of a chosen bound state in this manner and using this particular
value of the parameter ε in the expansion (9) then yields the corresponding wave function y(p).
Our principal concern is beyond doubt the semianalytical approach developed in Sec. 2 and
summarized above. Nevertheless, it might be instructive to construct explicitly a few examples
of solutions in numerical or graphical form. These results can be compared with the outcome of
some straightforward (but merely numerical) integration of the Schrödinger equation (4). This
will provide a useful check of the correctness of our solutions and justify the present formalism.
In actual computations, the infinite series (9) has to be truncated, for practical purposes, to
a reasonably large but definitely finite number N of terms considered in this expansion of y(p):
N
X p2n+1
y(p) ≃ c2n+1 . (11)
n=0 (2 n + 1)!

In this case, the wave function y(p) will approach zero, as required by the constraint (10), not
at infinity but already for a finite value, say p,
b of the momentum p — before it starts to diverge.
This technique gives the energy eigenvalues ε with a precision determined by one’s choice of N:
ε = ε(N). Likewise, the momentum boundary pb will also change with N: pb = p(N). b For a given
value of µ, which quantifies the relative importance of particle mass m and harmonic-oscillator
coupling strength a, every wave function resulting from this truncation procedure involves two
dimensionless parameters: the relevant energy eigenvalue ε(N) of the Hamiltonian (2), and the
characteristic momentum p(N). b Their values will be determined simultaneously, in accordance
with the above requirement on y(p) [to approach zero at p(N)]b by an appropriate fit procedure.
In other words, the value of p(N),
b in particular, cannot be varied freely; it is fixed for chosen N.
Let us illustrate this procedure for both ground state and first radial excitation, i. e., for the
two bound states defined by vanishing orbital angular momentum and radial quantum number
nr = 0, 1, resp. Figure 1 shows the corresponding reduced radial wave functions y(p) for µ = 30
as obtained by inspecting the functional form of y(p) resulting from different choices of ε and p, b
if taking into account 45 terms in their Taylor series (9), that is, if choosing N = 44 in Eq. (11).
Table 1 summarizes the relevant numerical parameter values emerging from such construction.
 6

y
1.75

1.5

1.25

0.75

0.5

0.25

p
2 4 6 8 10
(a)
y

0.5

p
3 6 9 12
-0.5

-1

(b)
√
Figure 1: Radial eigenfunctions in momentum space y(p) ≡ 4 π p ψ(p) of the ground state (a)
and the first radial√
excitation (b) of the relativistic harmonic-oscillator problem, defined by the
Hamiltonian H = p2 + µ2 +r 2 , with µ = 30, and for 45 terms in the Taylor expansion of y(p).

The exact results can be easily computed with the aid of a (standard) integration technique
designed for solving the Schrödinger equation numerically [46]. For our two examples the exact
wave functions y(p) prove to be practically indistinguishable, at least by the eye, from the ones
extracted from the series (11) with N = 44. This explains why we refrain from plotting also the
former in Fig. 1. For the momentum range depicted in Fig. 1, that is, for 0 ≤ p ≤ p,b the relative
−8
differences of the areas under corresponding curves are (of the order) 10 ; more precisely, they
are given by 2×10−8 for nr = 0, the ground state, and 3×10−8 for nr = 1, the first excited level.
 7

Table 1: Dimensionless (by scaling) energy eigenvalues ε(N) and characteristic√momenta p(N) b
of the relativistic harmonic-oscillator problem posed by the Hamiltonian H = p + µ +r 2 as
2 2

well as the classical turning points p of the corresponding nonrelativistic motion for the ground
state and the first radial excitation (identified by their radial quantum number nr = 0, 1), with
mass-vs.-coupling strength parameter µ = 30, and N = 44 or N = 14 in our Taylor series (11).

N nr ε(N) − µ p(N)
b p

44 0 0.38627 10 4.82
1 0.89864 12 7.36

14 0 0.38854 8.5 4.82

From a straightforward consideration of the “classical turning points” of the corresponding

(and well understood) nonrelativistic harmonic-oscillator problem defined by the Hamiltonian

p2
HNR = µ + + r2 ,
2µ

the maximum “classical” momenta, p, are found, in terms of the radial quantum number nr , as
q
2
p = (4 nr + 3) 2 µ , nr = 0, 1, 2, . . . .

By inspecting Table 1 we note with satisfaction that for both energy levels under consideration
the numerical values of the suitable pb turn out to be far beyond their classical counterparts p.
In order to get, at least, some vague idea of the dependence of our findings on the amount of
truncation represented by N < ∞, we inspect the ground-state wave function y(p) constructed
again for µ = 30 but by truncating the expansion (9) to the rather modest number of 15 terms,
which means to set N = 14 in Eq. (11). Figure 2 confronts this approximate wave function y(p)
with its exact behaviour for the ground state. While for small and intermediate momenta there
is still perfect agreement with the exact result [46], we observe a clearly discernible discrepancy
between approximate and exact curve for large momenta. Table 1 tells us that even for N = 14
our crucial momentum pb is still comfortably above the corresponding classical turning point, p.
Moreover, comparing the cases N = 14 and N = 44, we learn that the value of pb increases with
increasing number N. Of course, the naive expectation would be that pb behaves like pb → ∞ for
N → ∞, that is, when removing the truncation and restoring the full series expansion for y(p).
The minimum number of terms to be taken into account in the Taylor-series expansions (9)
required in order to achieve some given precision of one’s results will depend, of course, on both
the bound state under study and the desired accuracy. From our above remarks we feel entitled
to conclude that a reasonable (and manageable) number N ≃ 40 produces satisfactory results.
To our knowledge, at present the best semianalytical upper and lower bounds to the energy
eigenvalues of the relativistic harmonic-oscillator problem are provided simultaneously by the
combination of minimum–maximum principle with operator inequalities [18] and the envelope
theory [19,21]: at least for the relativistic harmonic oscillator the envelope bounds [19,21] may
be shown [23] to be quantitatively equivalent to the bounds derived in Appendix A of Ref. [18].
 8

y
1.75

1.5

1.25

0.75

0.5

0.25

p
2 4 6 8

Figure 2: Momentum-space wave function y(p) of Fig. 1(a) resulting from consideration of only
15 terms in its polynomial approximation (11) [full line], in comparison with the corresponding
exact ground-state wave function of the relativistic harmonic-oscillator operator [dashed line].

However, a discussion, in full generality, of all implications of such operator inequalities for the
eigenvalues of the operators considered appears clearly off the mainstream of our presentation.
Therefore, as already promised in the Introduction, the general relationship is demonstrated in
Appendix A. The bounds we need here are derived from this general theorem by specializing to
the case of the relativistic harmonic-oscillator problem posed by the Hamiltonian operator (2);
all operator inequalities required by this procedure may be generated by, e. g., envelope theory.
For a bound state of vanishing relative orbital angular momentum ℓ, that is, for a purely radial
excitation, identified by the radial quantum number nr = 0, 1, 2, . . . (identical to the number of
nodes of the corresponding wave function), the bounds on the dimensionless eigenvalue ε read
s  s 
P2 P2
min  µ2 + 2L + r 2  ≤ ε ≤ min  µ2 + U + r2 ,
r>0 r r>0 r2

where, in three spatial dimensions, our envelope-theory upper-bound parameter PU is given by

3
PU = 2 nr + , nr = 0, 1, 2, . . . ,
2
while the lower-bound parameter PL required for our envelope bounds is related to the zeros z0
of Airy’s function Ai(z) [47] (−z0 = 2.338107, 4.087949, 5.520560, 6.786708, 7.944134, . . .) by
 3/2
−z0
PL = 2 , Ai(z0 ) = 0 .
3
Resulting values of PL for the lowest-lying ℓ = 0 bound states are listed in Table 2 [9,19,21–23].
 9

Table 2: Numerical values of the parameter PL used by envelope theory for the lower bounds on
the energy levels of the relativistic harmonic-oscillator problem in three spatial dimensions, for
the lowest-lying ℓ = 0 bound states identified by their radial quantum number nr = 0, 1, 2, . . . .

nr PL

0 1.3760835
1 3.1813129
2 4.9925543
3 6.8051369
4 8.6182269

Table 3 compares, for the lowest four ℓ = 0 energy levels (identified by their radial quantum
number nr = 0, 1, 2, 3) of our Hamiltonian (2), the approximate eigenvalues, ε(N), obtained by
the present approach by a truncation of the power series (11) to N = 14 or N = 44 terms, with
the corresponding semianalytical upper (εU ) and lower (εL ) bounds mentioned above as well as
with the (numerically exact) eigenvalues εnum , computed by a method developed for the purely
numerical solution of (nonrelativistic) Schrödinger equations [46]. For N = 14, the polynomial
in ε resulting from the suitably adapted boundary condition (10) has only two real roots at all.
Moreover, both of these approximate values are still above our (semianalytical) upper bounds.
In contrast to this rather crude approximation, for N = 44 the eigenvalues ε(N) of already the
three lowest energy levels fit perfectly to the ranges spanned by the semianalytical bounds. For
the ground state, in particular, ε(44) reproduces the exact result at least to five decimal places.

Table 3: “Compact-origin” eigenvalues ε(N), their upper (εU ) and lower (εL) bounds, and their
exact values εnum for the lowest ℓ = 0 states of the Hamiltonian H/a1/3 in Eq. (2), with µ = 30.

Radial Excitation nr 0 1 2 3

εU − µ 0.38668 0.90032 1.41179 1.92111

εL − µ 0.35478 0.81862 1.28222 1.74440
εnum − µ 0.38627 0.89857 1.40768 1.91366
ε(N = 14) − µ 0.38854 0.93616 — —
ε(N = 44) − µ 0.38627 0.89864 1.41032 1.94319

4 Summary and Conclusions

Our compact result for the reduced ℓ = 0 eigenfunctions of the Hamiltonian (2) is given by the
power series (9), with expansion coefficients c1 = 1, c3 = µ−ε, and c2n+1 , n ≥ 2, determined by
the recurrence relation (8). Both the numerical determination of all energy eigenvalues and the
explicit construction of the corresponding eigenfunctions of the relativistic harmonic-oscillator
 10

problem is then achieved by forcing our solutions to satisfy, in addition, the constraint imposed
by the requirement of normalizability of bound-state wave functions. Comparing these explicit
solutions with the outcomes of purely numerical integration procedures reveals that at least for
the lowest-lying energy levels our semianalytical approach reproduces, already for a truncation
of the Taylor series (9) to a moderate number of expansion terms, the exact solutions with high
accuracy. Of course, if one is interested only in numerical solutions of the problem under study,
their straightforward computation with the help of some integration algorithm should produce
the desired result more easily than their extraction from our Taylor series by means of Eq. (10).

Acknowledgements
One of us (F. F. S.) would like to thank the University of Science and Technology of China of
the Chinese Academy of Sciences, and the Department of Physics of the Chongqing University
for hospitality during his stay in China, during which part of the present work has been done.

A Combination of minimum–maximum principle with

operator inequality: “spectral comparison theorem”
It is a simple exercise to relate discrete eigenvalues of two operators satisfying some inequality.
Consider for some operator A, with domain D(A), its eigenvalue equation A |αk i = ak |αk i,
k = 0, 1, 2, . . . , for its set of eigenstates {|αk i, k = 0, 1, 2, . . .}, corresponding to its eigenvalues

hαk | A |αk i
ak ≡ , k = 0, 1, 2, . . . ,
hαk |αk i

and likewise for some operator B, with domain D(B), its eigenvalue equation B |βk i = bk |βk i,
k = 0, 1, 2, . . . , for its set of eigenstates {|βk i, k = 0, 1, 2, . . .}, corresponding to its eigenvalues

hβk | B |βk i
bk ≡ , k = 0, 1, 2, . . . .
hβk |βk i

Assume that both these operators A and B are self-adjoint: A† = A, B † = B. This implies that
all their eigenvalues are real: a∗k = ak , b∗k = bk , k = 0, 1, 2, . . . . Let these eigenvalues be ordered
according to a0 ≤ a1 ≤ a2 ≤ · · · , b0 ≤ b1 ≤ b2 ≤ · · · . Consider only the discrete eigenvalues ak
of A below the onset of the essential spectrum of the operator A. Assume that the operator A is
bounded from below. Assume that the operators A and B are related by an operator inequality
of the form A ≤ B, which implies that B too is bounded from below. In order to derive, for any
k = 0, 1, 2, . . . , the relationship between ak and bk , focus on some arbitrary (k+1)-dimensional
subspace Dk+1 of the domain D(A) of A: Dk+1 ⊆ D(A). Employing the appropriate form of the
minimum–maximum principle, the operator inequality A ≤ B translates into an upper bound
on the eigenvalue ak of A which involves all expectation values of B within this subspace Dk+1 :
hψ| A |ψi hψ| B |ψi
ak ≤ sup ≤ sup for all k = 0, 1, 2, . . . . (12)
|ψi∈Dk+1 hψ|ψi |ψi∈Dk+1 hψ|ψi

Now, in order to relate the supremum over the expectation values of B to the eigenvalues bk
of B, consider a particular subspace Dk+1 , namely, that space that is spanned by the first k+1
 11

eigenvectors of the operator B, that is, by precisely those eigenvectors of B that correspond to
the first k +1 eigenvalues b0 , b1 , . . . , bk of B: Dk+1 ⊆ D(B) ⊆ D(A). Then, clearly, every |ψi in
Dk+1 is a linear combination of the eigenstates {|βi i, i = 0, 1, . . . , k} of B, with coefficients ci :
k
X
|ψi = ci |βi i for all |ψi ∈ Dk+1 .
i=0

For any subspace Dk+1 , k = 0, 1, 2, . . . , use of this expansion of |ψi yields for its norm squared
k
X
hψ|ψi = |ci |2 hβi |βi i for all |ψi ∈ Dk+1
i=0

and, with this and bi ≤ bk for all i = 0, 1, . . . , k, an upper bound on all expectation values of B:
k
X k
X
2
hψ| B |ψi = |ci | bi hβi |βi i ≤ bk |ci |2 hβi |βi i = bk hψ|ψi for all |ψi ∈ Dk+1 ,
i=0 i=0

which means
hψ| B |ψi
≤ bk for all |ψi ∈ Dk+1 ,
hψ|ψi
hψ| B |ψi
= bk for |ψi = |βk i ∈ Dk+1 .
hψ|ψi
Therefore the supremum of all expectation values of B over Dk+1 is just the eigenvalue bk of B:

hψ| B |ψi
sup = bk for all k = 0, 1, 2, . . . .
|ψi∈Dk+1 hψ|ψi

Thus, inserting this identity in the chain of inequalities (12) proves that corresponding discrete
eigenvalues ak , bk of semibounded self-adjoint operators A, B that satisfy A ≤ B are related by

ak ≤ bk for all k = 0, 1, 2, . . . .

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