Exactly solvable Schrödinger operators

J. Dereziński
Department of Mathematical Methods in Physics,
Warsaw University,
Hoża 74, 00-682 Warszawa, Poland,
email jan.derezinski@fuw.edu.pl

M. Wrochna
RTG “Mathematical Structures in Modern Quantum Physics”,
Institute of Mathematics, University of Göttingen,
Bunsenstr. 3-5, D-37073 Göttingen, Germany
email wrochna@uni-math.gwdg.de
March 6, 2015

Contents
1 Introduction 2

2 Second order homogeneous differential equations 4

2.1 The Bose invariant of a second order differential equation . . . . . . . . . . . . . . . . . . 5
2.2 Schwarz derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 The Liouville transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Natanzon’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Schrödinger operators reducible to the hypergeometric equation 7

3.1 Hypergeometric equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 The Riemann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 Trigonometric Pöschl-Teller potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.4 Hyperbolic Pöschl-Teller potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.5 Scarf potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.6 More about Pöschl-Teller and Scarf potentials . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.7 Manning-Rosen potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.8 Eckart potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.9 Rosen-Morse potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.10 More about Manning-Rosen, Eckart and Rosen-Morse potentials . . . . . . . . . . . . . . 14

4 Schrödinger operators reducible to the rescaled confluent equation 14

4.1 Symmetries of the rescaled confluent equation . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Hydrogen atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3 Rotationally symmetric harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.4 Morse potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1
5 Schrödinger equations reducible to the translated harmonic oscillator 17
5.1 Translated harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.2 Special potential I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.3 Special potential II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Abstract
We systematically describe and classify 1-dimensional Schrödinger equations that can be solved
in terms of hypergeometric type functions. Beside the well-known families, we explicitly describe 2
new classes of exactly solvable Schrödinger equations that can be reduced to the Hermite equation.

1 Introduction
Exactly solvable 1-dimensional Schrödinger equations play an important role in quantum physics. The
best known are the harmonic oscillator and the radial equation for the hydrogen atom, which are covered
in every course of quantum mechanics. A number of other examples were discovered in the 30’s of the
last century. They include (known also under other names) trigonometric and hyperbolic Pöschl-Teller
potentials [PT], the Scarf potential [Sca], the Eckart potential [Eck], the Manning-Rosen potential [MR]
and the Rosen-Morse potential. All these examples can be reduced to the hypergeometric equation (see
eg. [WW]). One should also mention the Morse potential that leads to the confluent equation. Problems
involving these potentials are often used in classes on quantum mechanics, see eg. the well-known problem
book of Flügge [Flü]. A number of techniques have been developed to study their properties, such as the
factorization method [HI] and the closely related method of the superpotential (see for instance [CKS]
and [Cot]).
In later years J. N. Ginocchio [Gin] discovered that these examples can be generalized to a larger
class of potentials equivalent to the hypergeometric or confluent equation. Later, this class was extended
by G. A. Natanzon [Nat] and further generalized by R. Milson [Mil]. These classes, besides the hitherto
known potentials, are not very practical in applications, since they are not given by explicit expressions.
There exist even more general classes of potentials that can be called exactly solvable, found by
A. Khare and U. Sukhatme [KS]. They are, however, expressed in exotic special functions.
The literature on exactly solvable Schrödinger equations is very large. The subject is in fact very
useful for applications, especially in quantum physics. We also believe that it is quite beautiful, which in
the existing literature is perhaps not so easy to see.
In our paper we would like to systematically describe basic classes of exactly solvable Schrödinger
equations. Most of the material of our paper is scattered in the literature, notably in [Nat], [Mil] and
[CKS] (see also [Bos]). Our treatment is, however, somewhat more systematic than what we could find in
the literature. For instance, we distinguish between the complex classification and the real classification.
It also seems that for the first time we explicitly describe 2 new classes of exactly solvable Schrödinger
equations, which can be reduced to the Hermite equation (see Subsects 5.2 and 5.3).
By a (stationary 1-dimensional) Schrödinger equation we will mean an equation of the form

−∂r2 + V (r) − E φ(r) = 0.

(1.1)

This equation can be interpreted as the eigenvalue problem for the operator

H := −∂r2 + V (r). (1.2)

An operator of the form (1.2) will be called a (1-dimensional) Schrödinger operator. V (r) will be called
a potential and the parameter E an energy.
(1.2) can be interpreted as an operator in a number of ways. If V is a holomorphic function of the
complex variable on some open Ω ⊂ C, then r can be interpreted as a complex variable and (1.2) can be

2
viewed as an operator on holomorphic functions. In this case it is natural to allow E to be a complex
parameter. The corresponding eigenvalue equation (1.1) will be then called the Schrödinger equation of
the complex variable. Let us note that, complex affine transformations preserve the class of Schrödinger
equations of the complex variable. (By a complex affine transformations we mean r 7→ ar + b, where
a 6= 0, b are complex constants).
One can also interpret r as a real variable in some open I ⊂ R. The operator (1.2) is then viewed as
acting on functions on I. Understood in this way, (1.1) will be called the Schrödinger equation of the real
variable. Of special interest is then the case of real potential. Clearly, real affine transformations preserve
the class of real Schrödinger equations of the real variable.
Our paper is organized as follows. First, we will briefly discuss some general facts related to 2nd
order linear differential equations. In particular, we will describe basic ingredients of the so-called Bose-
Natanzon method, which permits to obtain a class of Schrödinger equations equivalent to a chosen linear
equation. Note that it applies to more general situations than those described in the literature ([Nat],
[Mil]). We will formulate a criterion that determines when the Bose-Natanzon method can be used.
Next, we will focus on hypergeometric type equations. Recall that equations of the form

σ(z)∂z2 + τ (z)∂z + η f (z) = 0,

(1.3)

where σ(z), τ (z) are polynomials with

degσ ≤ 2, degτ ≤ 1,

and η is a number, are sometimes called hypergeometric type equations (see A.F. Nikiforov and V.B. Uvarov
[NU]). Solutions of (1.3) are called hypergeometric type functions. Hypergeometric type functions are very
well understood and include the hypergeomeric function, the confluent function, Hermite, Laguerre and
Jacobi polynomials, etc. Traditionally, one classifies hypergeometric type equations into several distinct
classes invariant under complex affine transformations. In each class one chooses a simple representative,
to which the whole class can be reduced. Such representatives are the hypergeometric equation, the
confluent equation, etc.
In our analysis, we will concentrate on 3 specific equations:
(1) the hypergeometric equation

z(1 − z)∂z2 + (c − (a + b + 1)z)∂z − ab f (z) = 0;

(1.4)

(2) the rescaled confluent equation

z∂z2 + (c − γz)∂z − a f (z) = 0.

(1.5)

(3) the translated harmonic oscillator

−∂z2 + θ2 z 2 + ρz + λ f (z) = 0.


(1.6)

Note that all hypergeometric type equations with a nonzero σ are related with one of the above
equations by elementary operations
• multiplication of f (z) by a function, multiplication of the equation by a function,
• a change of variables not depending on the parameters a, b, c (respectively a, c, γ or θ, ρ, λ).
(1) If σ is 2nd order and has 2 distinct roots, an affine transformation reduces (1.3) to (1.4).
(2) If σ is 2nd order and has only one root, then a transformation involving z 7→ z −1 reduces (1.3) to
(1.5) with γ 6= 0.

3
(3) If σ is 1st order, then an affine transformation reduces (1.3) to (1.5).
(4) If σ is 0th order and nonzero, then (1.3) is equivalent to (1.6) with θ 6= 0.
Obviously, (1.4) and (1.5) are themselves equations of the hypergeometric type. (1.6) is not, but can
be reduced to a hypergeometric type equation. This reduction is depends on whether θ 6= 0 or θ = 0.
The former case leads to a hypergeometric type equation with a constant σ, as mentioned above. In the
case θ = 0, (1.6) is the Airy equation, which can be reduced to a special case of the hypergeometric type
equation by a transformation involving z 7→ z 3 .
Equations (1.4), (1.5) and (1.6) are very well understood and have well known solutions. We will
describe classes of solvable potentials reducible to one of the equations (1.4), (1.5) or (1.6). We will
describe both the complex and the real classification of such potentials.
It is natural to consider first a classification of exactly solvable Schrödinger equations of the complex
variable. Note that all solvable potentials that we consider are holomorphic on the complex plane, apart
from some isolated singularities. Obviously, we can always use a complex affine transformation to put the
equation in a convenient form. One can also move a complex constant from the potential to the energy.
Consider a family of exactly solvable holomorphic potentials. Suppose that for a selected subfamily of
parameters the potential is real if restricted to an open, possibly infinite interval I ⊂ C. By elementary
properties of holomorphic functions, if we extend the interval I, the potential is still real until we hit
a singularity. A real affine transformation can be used to put the equation in a convenient form. In
particular, we can always assume that I is a subset of the real line. We can also move a real constant
from the potential to the energy.
The above discussion motivates the following definition: We will say that an open interval I ⊂ R is
a natural real domain for a certain family of potentials if it ends either at −∞, +∞ or at a singularity
of the potential, and no singularities lie inside I. When describing the real classification, we will always
specify a natural real domain of the potential.
Clearly, if I is a natural real domain, then the operator H = −∂r2 + V (r) is hermitian on Cc∞ (I). It is
then natural to ask about self-adjoint extensions of H. We will not discuss this question here. We plan
to consider it in later papers.
In Equations (1.4), (1.5) and (1.6) we have 3 arbitrary parameters. Therefore, in all the solvable cases
that we will describe, the potential depends on 2 parameters, since the third parameter is responsible for
the energy E.
Acknowledgement The research of J. D. was supported in part by the grant N N201 270135 of the
Polish Ministry of Science and Higher Education.

2 Second order homogeneous differential equations

The main object of our paper are ordinary homogeneous 2nd order linear differential equations, that is
equations of the form
a(r)∂r2 + b(r)∂r + c(r) φ(r) = 0.


(2.7)
It will be convenient to treat (2.7) as the problem of finding the kernel of the operator

A(r, ∂r ) := a(r)∂r2 + b(r)∂r + c(r). (2.8)

We will then say that the equation (2.7) is given by the operator (2.8).
We will treat r either as a complex or a real variable. In the complex case we will usually assume
that the coefficients are analytic.
In this section we describe some general facts related to the theory of equations of the form (2.7) and
their reduction to the Schrödinger equation.

4
2.1 The Bose invariant of a second order differential equation
Suppose that we are given a second order differential equation (2.7). Then we can always eliminate the
first order term as follows. We divide from the left by a(r), set
Z r 
b(t)
h(r) := exp dt .
2a(t)
We check that
a(r)−1 h(r)A(r, ∂r )h(r)−1 = ∂r2 + I(r),
where, using the notation a0 := d
dr a(r),

4ac − 2ab0 + 2ba0 − b2

I= . (2.9)
4a2
Clearly, φ̌(r) := h(r)φ(r) solves the equation
∂r2 + I(r) φ̌(r) = 0.


(2.10)
(2.10) will be called the canonical form of the equation (2.7). (2.9) will be called the Bose invariant
of (2.7) [Bos]. (The name “Bose invariant” is used eg. by Milson [Mil], however the object itself was
clearly known before Bose).
It is easy to check that for any functions f, g the equation
f (r)A(r, ∂r )g(r)φ(r) = 0 (2.11)
has the same Bose invariant as (2.7). Conversely, if an equation has the same Bose invariant as (2.7), it
equals (2.11) for some f, g.

2.2 Schwarz derivative

The Schwarz derivative of r 7→ r(y) appears naturally in the context of 2nd order ordinary differential
equations and was known to XIX century mathematicians. It was mentioned already by Kummer in 1836
[Kum]. The universally adopted name Schwarz derivative was introduced by A. Cayley, who derived its
basic properties in 1880 [Cay]. For more historical comments together with numerous applications we
refer the reader to a book by Osgood [Osg].
The Schwarz derivative is defined as
 00 0 2
1 r00 (y)

r (y)
{r, y} = − . (2.12)
r0 (y) 2 r0 (y)
Here are some basic properties of the Schwarz derivative:
Proposition 2.1 1. Let y 7→ s(y), s 7→ r(s) be two functions. Then
{r, y} = {r, s} s0 (y)2 + {s, y} . (2.13)

2. Let r 7→ y(r) be the inverse of y 7→ r(y). Then

{y, r} = −[r0 (y)]−2 {r, y} . (2.14)
ay+b
3. {r, y} ≡ 0 ⇔ r(y) = cy+d , ad − bc 6= 0
We will need the following fact:
Proposition 2.2 Suppose that z 0 = z p (1 − z)q . Then
1 2p−2
(1 − z)2q−2 (p2 − 2p)(1 − z)2 + (q 2 − 2q)z 2 − 2pqz(1 − z) .


{z, r} = z (2.15)
2

5
2.3 The Liouville transformation
Consider a 2nd order equation in the canonical form, that is
(∂r2 + I(r))φ(r) = 0. (2.16)
Let us make the transformation r = r(y) in this equation. We obtain
1 r00 (y)
∂y2 φ(r(y)) − ∂y φ(r(y)) + I(r(y))φ(r(y)) = 0 (2.17)
[r0 (y)]2 [r0 (y)]3
The resulting equation we transform again to its canonical form:
(∂y2 + J(y))ψ(y) = 0, (2.18)
where
1
ψ(y) = φ(r(y)), (2.19)
[r0 (y)]2
1
J(y) = [r0 (y)]2 I(r(y)) +
{r, y} , (2.20)
2
The above procedure is called the Liouville transformation of (2.16) by the change of variables r = r(y)
[Lio]. One can check using (2.17), that the composition of two Liouville transformations, first by the
change of variables r = r(y), then y = y(z), is the Liouville transformation by the change of variables
r = r(y(z)).

2.4 Natanzon’s problem

Let A(a1 , . . . , ak ; z, ∂z ) be a 2nd order differential operator depending on k parameters, such that its
Bose invariant can be written as
I(b1 , . . . , bn ; z) = b1 I1 (z) + . . . + bn In (z) (2.21)
for some (at least one) n ≤ k linearly independent functions I1 , . . . , In (independent of a1 , . . . , ak ), and
some n numbers bi (a1 , . . . , ak ). Then one can solve the following problem (which we will call Natanzon’s
problem for A(z, ∂z )):

Find all potentials V (r) such that the 1-dimensional stationary Schrödinger equation
−∂r2 + V (r) − E φ(E, r) = 0


(2.22)
can be transformed to the equation given by A(a1 , . . . , ak ; z, ∂z ) for some a1 (E), . . . , ak (E). We allow the
following operations:
(1) multiplication of both sides of the equation by some f (E, r);
(2) substitution of φ̌(E, r) := g(E, r)φ(E, r) for some g(E, r);
(3) change of coordinates r 7→ z independent of E.

This is the problem solved by Natanzon in the case of the hypergeometric equation and by Milson in the
general case of hypergeometric type equations. In the following, we recall the construction they used, in
a slightly generalized form.
Let us consider an arbitrary equation of the form (2.22). Obviously, its Bose invariant equals E −V (r).
Clearly, the transformations (1) and (2) allow us to transform A(a1 , . . . , ak ; z, ∂z ) to its canonical form
−∂z2 − I(b1 , . . . , bn ; z). (2.23)

6
Thus (2.22) can be transformed to A(a1 , . . . , ak ; z, ∂z ) if

2
1
I(b1 , . . . , bn ; z) = (r0 (z)) (E − V r(z) + {r, z}, (2.24)
2
that is, if the two Bose invariants are related by a Liouville transformation. Using (2.21) we rewrite
(2.24) as
2
1
b1 (E)I1 (z) + . . . + bn (E)In (z) = (r0 (z)) (E − V r(z) + {r, z}. (2.25)
2
By assumption, bi depend on E and Ii (z) are independent of E. Thus, the dependence of bi on E is
linear. Therefore, by transforming linearly (b1 , . . . , bn ) into (b̃1 , . . . , b̃n−1 , E), we can assume that

I(b1 , . . . , bn ; z) = b̃1 I˜1 (z) + . . . + b̃n−1 I˜n−1 (z) + E I˜n (z),

for some functions I˜i (z), which are linear combinations of Ii (z). Then we can write

2
1
E I˜n (z) + b̃1 I˜1 (z) + . . . + b̃n−1 I˜n−1 (z) = (r0 (z)) (E − V r(z) + {r, z}. (2.26)
2
Therefore, Natanzon’s problem is solved by the following pair of equations
 
−2 1
V b̃1 , . . . , b̃n−1 , r(z) := (r0 (z)) b̃1 I˜1 (z) + . . . + b̃n−1 I˜n−1 (z) − {r, z} ,


2
0 2 ˜
(r (z)) = In (z). (2.27)

We can rewrite (2.27) in an equivalent way as

  1
2
(z 0 (r)) b̃1 I˜1 (z(r)) + . . . + b̃n−1 I˜n−1 (z(r)) + {z, r},


V b̃1 , . . . , b̃n−1 , r :=
2
−2
(z 0 (r)) = I˜n (z(r)). (2.28)

If desired, we can renormalize E by substracting a constant.

Note that in practical applications the value of the above method depends on whether it is possible
to invert the relation r 7→ z expressing it in terms of standard functions. In the next sections, we
will describe such solutions of the Natanzon problem for the three classes of equations described in the
introduction.

3 Schrödinger operators reducible to the hypergeometric equa-

tion
3.1 Hypergeometric equation
The hypergeometric equation is given by the operator

F(a, b; c; z, ∂z ) := z(1 − z)∂z2 + (c − (a + b + 1)z)∂z − ab, (3.29)

where a, b, c are arbitrary complex parameters. Its Bose invariant equals

(1 − a2 − b2 + 2ab)z 2 + 2(−2ab + ac + bc − c)z + 2c − c2

2 I1 (a, b; c; z) = .
4z 2 (1 − z)2

7
 1
It satisfies condition (2.21) for n = 3. The functions Ii (z), i = 1, 2, 3 can be taken to be 4z2 (1−z)
1 1 0 2 ˜ 2p 2q
4z(1−z)2 and 4z(1−z) . It is natural to demand that (r (z)) = I3 (z) is a function of the form z (1 − z) .
It is easy to see that this gives the following possibilities for I˜3 (z):
1 1 1
(1) , (2) 2 , (3) ,
4z(1 − z) 4z (1 − z) 4z(1 − z)2
1 1 1
(4) 2 2
, (5) 2 , (6) . (3.30)
4z (1 − z) 4z 4(1 − z)2
We will see that each of these ansatzes leads to exactly solvable potentials considered in the literature.
Note also that the formula for the Schwarz derivative (2.15) will be handy.

3.2 The Riemann equation

It is well known that it is useful to consider the hypergeometric equation as a special case of the so-called
Riemann equation. The Riemann equation is defined as the class of equations on the Riemann sphere C̄
having 3 regular singular points. The following theorem summarizes the basic theory of these equations:
Theorem 3.1 (1) Suppose that we are given a 2nd order differential equation on the Riemann sphere
having 3 singular points z1 , z2 , z3 , all of them regular singular points with the following indices
z1 : ρ1 , ρ̃1 ,
z2 : ρ2 , ρ̃2 ,
z3 : ρ3 , ρ̃3 .
Then the following condition is satisfied:
ρ1 + ρ̃1 + ρ2 + ρ̃2 + ρ3 + ρ̃3 = 1. (3.31)
Such an equation, normalized to have coefficient 1 at the 2nd derivative, is always equal to
 
z1 z2 z3
P  ρ1 ρ2 ρ3 z, ∂z  φ(z) = 0, (3.32)
ρ̃1 ρ̃2 ρ̃3
where  
z1 z2 z3  
ρ1 +ρ̃1 −1 ρ2 +ρ̃2 −1 ρ3 +ρ̃3 −1
P  ρ1 ρ2 ρ3 z, ∂z  := ∂z2 − z−z1 + z−z2 + z−z3 ∂z
ρ̃1 ρ̃2 ρ̃3
ρ1 ρ̃1 (z1 −z2 )(z1 −z3 ) ρ2 ρ̃2 (z2 −z3 )(z2 −z1 ) ρ3 ρ̃3 (z3 −z1 )(z3 −z2 )
+ (z−z 2
1 ) (z−z2 )(z−z3 )
+ (z−z2 )2 (z−z3 )(z−z1 ) + (z−z3 )2 (z−z1 )(z−z2 ) .

(2) Let z 7→ w = h(z) = az+bcz+d . (Transformations of this form are called homographies or Möbius
transformations). We can always assume that ad − bc = 1. Then
   
h(z1 ) h(z2 ) h(z3 ) z1 z2 z3
P  ρ1 ρ2 ρ3 w, ∂w  = (cz + d)4 P  ρ1 ρ2 ρ3 z, ∂z  ,
ρ̃1 ρ̃2 ρ̃3 ρ̃1 ρ̃2 ρ̃3
(3)
 
z1 z2 z3
(z − z1 )−λ (z − z2 )λ P  ρ1 ρ2 ρ3 z, ∂z  (z − z1 )λ (z − z2 )−λ
ρ̃1 ρ̃2 ρ̃3
 
z1 z2 z3
= P  ρ1 − λ ρ2 + λ ρ3 z, ∂z  .
ρ̃1 − λ ρ̃2 + λ ρ̃3

8
 Clearly, in all above formulas one of zi can equal ∞, with an obvious meaning of various expressions.
For convenience we give the expression for the Riemann operator with z3 = ∞:
 
z1 z2 ∞
P  ρ1 ρ2 ρ3 z, ∂z 
ρ̃1 ρ̃2 ρ̃3
 
2 ρ1 + ρ̃1 − 1 ρ2 + ρ̃2 − 1
= ∂z − + ∂z
z − z1 z − z2
ρ1 ρ̃1 (z1 − z2 ) ρ2 ρ̃2 (z2 − z1 ) ρ3 ρ̃3
+ 2
+ 2
+ (3.33)
(z − z1 ) (z − z2 ) (z − z2 ) (z − z1 ) (z − z1 )(z − z2 )

The hypergeometric equation is a special case of the Riemann equation, since

 
0 1 ∞
F(a, b; c; z, ∂z ) = z(1 − z)P  0 0 a z, ∂z 
1−c c−a−b b

= z(1 − z)∂z2 + (c − (a + b + 1)z)∂z − ab.

Every Riemann equation can be brought to the form of the hypergeometric equation by applying (1) and
(2) of Theorem 3.1.
From Theorem 3.1 we also see that symmetries of the hypergeometric equation are better visible if
we replace a, b, c with α, β, µ:

α=c−1 β = a + b − c, µ = a − b;
1+α+β−µ
a= 2 , b = 1+α+β+µ
2 , c = 1 + α.

In fact, the new parameters coincide with the differences of the indices of the points 0, 1, ∞:

α = ρ1 − ρ̃1 , β = ρ2 − ρ̃2 , µ = ρ3 − ρ̃3 .

In the parameters α, β, µ, the Bose invariant of the hypergeometric equation has a more symmetric form:

(1 − α2 )(1 − z) + (1 − β 2 )z + (µ2 − 1)(1 − z)z

2 I1 (α, β, µ; z) = .
4z 2 (1 − z)2

We can summarize the relation between the hypergeometric equation and its canonical form by
β
  α 1
α 1 1
−β−1
−z 2 + 2 −1 (1 − z) 2 + 2 −1 F α+β+µ+1
2 , α+β−µ+1
2 ; 1 + α; z, ∂ − −
z z 2 2 (1 − z) 2 2

∞
 
0 1
β µ
= −P  α2 + 12 2 + 1
2
1
2 − 2 z, ∂z 
β µ
− α2 + 12 − 2 + 12 − 2 − 21

1 1 1
= −∂z2 + α2 − 1 + β2 − 1 − µ2 − 1




. (3.34)
4z 2 (1 − z) 4z(1 − z) 2 4z(1 − z)

It will be natural to introduce the parameters

1 2 1
κ := (α − β 2 ), δ := (α2 + β 2 ). (3.35)
2 2

9
In some cases the parameter κ will be replaced by
i 2
τ := (α − β 2 ) = iκ. (3.36)
2
We will describe two complex classes of exactly solvable potentials depending on two complex param-
eters. Within each complex class there will be three real classes of exactly solvable potentials depending
on two real parameters.

3.3 Trigonometric Pöschl-Teller potential

In this subsection we consider Ansatz (1) of (3.30). We set
r 1 1
z = sin2 , which solves z 0 = z 2 (1 − z) 2 . (3.37)
2
This leads to the Schrödinger equation

µ2
 
2 tPT
−∂r + Vδ,κ (r) − φ(r) = 0, (3.38)
4

where
   
tPT 2 1 1 2 1 1
Vδ,κ (r) := α − 2 r
+ β − r (3.39)
4 4 sin 2 4 4 cos2 2
 
1 1 cos r
= δ− +κ 2 .
4 sin2 r sin r

This potential was proposed and solved by G. Pöschl and E. Teller [PT]. It is usually called the Pöschl-
Teller potential, sometimes also the Pöschl-Teller potential of the first kind or the trigonometric Scarf
potential.
A natural real domain for this potential is ]0, π[. If κ, δ are real, then the potential is real on this
domain.
Explicitly, the reduction of (3.38) to the hypergeometric equation is derived as follows:
α 1 β 1
  α 1 β 1
−z 2 + 4 (1 − z) 2 + 4 F α+β+µ+1
2 , α+β−µ+1
2 ; 1 + α; z, ∂z z − 2 − 4 (1 − z)− 2 − 4

∞
 
0 1
α 1 β 1 µ
= −z(1 − z)P 
2 + 4 2 + 4 2 z, ∂z 
β
− 2 + 41
α
− 2 + 41 − µ2

   
1 1
= −z(1 − z) + ∂z2 − ∂z
2z 2(1 − z)
µ2
   
1 1 1 1
+ α2 − + β2 − − (3.40)
4 4z 4 4(1 − z) 4
 1  1  1  1 µ2
= −∂r2 + α2 − + β 2
− r − .
4 4 sin2 2r 4 4 cos2 2 4

10
3.4 Hyperbolic Pöschl-Teller potential
We continue with (1) of (3.30). We set
r 1 1
z = − sinh2 , which solves z 0 = −(−z) 2 (1 − z) 2 . (3.41)
2
This leads to the Schrödinger equation
µ2
 
−∂r2 + Vδ,κ
hPT
(r) + φ(r) = 0, (3.42)
4
where
   
hPT 2 1 1 2 1 1
Vδ,κ (r) := α − − β − (3.43)
4 4 sinh2 2r 4 4 cosh2 r
2
 
1 1 cosh r
= δ− 2 +κ .
4 sinh r sinh2 r
This potential was also proposed and solved by G. Pöschl and E. Teller [PT]. In the literature it is known
as hyperbolic, generalized Pöschl-Teller potential, or the Pöschl-Teller potential of the second kind.
A natural real domain for this potential is ]0, ∞[. If κ, δ are real, then the potential is real on this
domain.
To see that (3.42) can be solved in terms of the hypergeometric equation, we first repeat the compu-
tations leading to (3.40), and then set z = − sinh2 2r .

3.5 Scarf potential

We continue with (1) of (3.30). We set
1 r r 1 1
z= − i cosh sinh , which solves z 0 = (−z) 2 (1 − z) 2 . (3.44)
2 2 2
This leads to the Schrödinger equation
µ2
 
2 S
−∂r + Vδ,τ (r) + φ(r) = 0, (3.45)
4
where
 
S 1 1 sinh r
Vδ,τ (r) := − δ − −τ .
4 cosh2 r cosh2 r
This potential was proposed and solved by F. Scarf [Sca]. In the literature it is often called the hyperbolic
Scarf potential.
A natural real domain for this potential is ] − ∞, ∞[. If δ, τ are real, then the potential is real on this
domain.
To see that (3.45) can be solved in terms of the hypergeometric equation, we first repeat the compu-
tations leading to (3.40), and then set z = 21 − i cosh 2r sinh 2r .

3.6 More about Pöschl-Teller and Scarf potentials

Both kinds of Pöschl-Teller potentials and the Scarf potential are real cases of the same complex case.
To see this, consider eg. the hyperbolic Pöschl-Teller potential as a function of the complex parameter
r. It is holomorphic away from singularities at iπn, n ∈ Z.

11
 hPT
For real δ, κ, Vδ,κ (r) is real on iR and R + iπn. On each halfline ]0, ∞[+iπn and ] − ∞, 0[+iπn we
obtain the hyperbolic Pöschl-Teller potential. On each interval ]iπn, iπ(n+1)[ we obtain the trigonometric
Pöschl-Teller potential.
hPT
For real δ, iκ,, Vδ,κ (r) is real on R + iπ(n + 12 ). On each of these lines we obtain the Scarf potential.
Above, we used Ansatz (1) to derive Scarf and both kinds of Pöschl-Teller potentials. Alternatively,
one can use Ansatzes (2) or (3). To see this it is enough to consider Ansatz (3). In fact, we first repeat
computations analogous to (3.40):

∞
 
0 1
β µ
−z(1 − z)2 P  α2 + 14 2
1
2 + 4 z, ∂z 
− α2 + 41 − β2 µ
− 2 + 14

 1 
1 
= −z(1 − z)2 ∂z2 + − ∂z
2z 1−z
2
   
1 1 − z β 1 (1 − z)
+ α2 − + − µ2 − .
4 4z 4 4 4

We set
r 1
z = tgh2 , which solves z 0 = z 2 (1 − z), (3.46)
2
obtaining the hyperbolic Pöschl-Teller potential,
r 1
z = −tg2 , which solves z 0 = −(−z) 2 (1 − z), (3.47)
2
obtaining the trigonometric Pöschl-Teller potential, or
r 1
z = ctgh2 , which solves z 0 = z 2 (1 − z), (3.48)
2
obtaining the Scarf potential.

3.7 Manning-Rosen potential

Let us consider Ansatz (3) of (3.30). We set
1
z= , which solves z 0 = 2z(z − 1). (3.49)
1 + e2r
This leads to the Schrödinger equation

−∂r2 + Vκ,µ
MR


(r) + δ φ(r) = 0, (3.50)

where
µ2
 
MR sinh r 1 1
Vκ,µ (r) := −κ − − .
cosh r 4 4 cosh2 r
This potential was proposed and solved by M. F. Manning and N. Rosen [MR]. In the literature it is also
called the Woods-Saxon potential [SW] (for instance in [Flü]), also the hyperbolic Rosen-Morse potential.
A natural real domain for this potential is ] − ∞, ∞[. The potential is real if r ∈] − ∞, ∞[ and κ, β 2
are real.

12
 Here is a derivation of (3.50) from the hypergeometric equation:
α β α β
−4z 1+ 2 (1 − z)1+ 2 F( α+β+µ+1
2 , α+β−µ+1
2 ; 1 + α; z, ∂z )z − 2 (1 − z)− 2

∞
 
0 1
β
= −4z 2 (z − 1)2 P  α
2 2 µ + 12 z, ∂z 
− α2 − β2 −µ + 12

 1 
1 
= −4z 2 (1 − z)2 ∂z2 + − ∂z + α2 (1 − z) + β 2 z − (µ2 − 1)z(1 − z) (3.51)
z 1−z

e2r − 1 e2r
= −∂r2 + δ + κ 2r
− (µ2 − 1) . (3.52)
1+e (1 + e2r )2

3.8 Eckart potential

We still consider Ansatz (3) of (3.30).
We set
1
z= , which solves z 0 = 2z(1 − z), (3.53)
1 − e−2r
and use (3.35). This leads to the Schrödinger equation

−∂r2 + Vκ,µ
E


(r) + δ φ(r) = 0, (3.54)

where
µ2
 
E cosh r 1 1
Vκ,µ (y) := −κ + − .
sinh r 4 4 sinh2 r
This potential was proposed and solved by C. Eckart [Eck]. In the literature (for instance [Flü]) it is
also called the Hulthen potential [Hul], sometimes also the generalized Morse potential, because of its
similarity to the Morse potential, see Subsect. 4.4.
A natural real domain for this potential is ]0, ∞[. If κ, β 2 are real, then the potential is real on this
domain.
To derive the Eckart potential, we first repeat the computations leading to (3.51), and then set
z = 1−e1−2r .

3.9 Rosen-Morse potential

Once again, we consider Ansatz (3) of (3.30).
We set
1
z= , which solves z 0 = 2iz(1 − z). (3.55)
1 − e2ir
This leads to the Schrödinger equation

−∂r2 + Vτ,µ
RM


(r) − δ φ(r) = 0, (3.56)

where
µ2
 
RM cos r 1 1
Vτ,µ (y) := τ + −.
sin r sin2 r
4 4
This potential is known as the Rosen-Morse potential, also the trigonometric Rosen-Morse potential
(altough this name is widely used in the literature, we were unable to explain decisively its origin).

13
 A natural real domain for this potential is ]0, π[. If τ, µ2 are real, then the potential is real on this
domain.
To derive the Rosen-Morse potential, we first repeat the computations leading to (3.51), then set
z = 1−e12ir .

3.10 More about Manning-Rosen, Eckart and Rosen-Morse potentials

The Manning-Rosen, Eckart and Rosen-Morse potentials are all real cases of the same complex case. To
see this, consider eg. the Eckart potential as a function of the complex parameter r. It is holomorphic
away from singularities at iπn, n ∈ Z.
For real κ, µ2 , Vκ,µ
E
(r) is real on the lines R + iπn 1
2 . On each line R + i(m + 2 )π, m ∈ Z, we obtain
the Manning-Rosen potential. On each halfline ]0, ∞[+iπm and ] − ∞, 0[+iπm, we obtain the Eckart
potential.
For real iκ, µ2 , Vκ,µ
E
(r) is real on iR. On each interval ]iπm, iπ(m + 1)[ we obtain the Rosen-Morse
potential.
Ansatzes (5) and (6) lead to the same classes of exactly solvable potentials as Ansatz (4). To see this
it is enough to consider Ansatz (5). In fact, we first repeat computations analogous to (3.51):

α+1 β α+1 β
−4z − 2 (1 − z)2+ 2 F( α+β+µ+1
2 , α+β+−µ+1
2 ; 1 + α; z, ∂z )z − 2 (1 − z)− 2

∞
 
0 1
β µ
= −4(1 − z)2 P  α + 12 2 2 z, ∂z 
−α + 12 − β2 − µ2

 
2 1
= −4(1 − z) − ∂z∂z2
1−z
1−z 1 1−z
+(α2 − 1) 2 + β 2 − µ2 .
z z z
We set
z = 1 − e2r , which solves z 0 = −2(1 − z), (3.57)
obtaining the Eckart potential,

z = 1 + e2r , which solves z 0 = 2(1 − z), (3.58)

obtaining the Manning-Rosen potential,

z = 1 − ei2r , which solves z 0 = −2i(1 − z), (3.59)

obtaining the Rosen-Morse potential,

4 Schrödinger operators reducible to the rescaled confluent equa-

tion
One of basic exactly solvable equations is the confluent equation, given by the operator

F(a; c; z, ∂z ) := z∂z2 + (c − z)∂z − a. (4.60)

14
It is convenient to consider (4.60) in parallel with the equation given by the operator

F(a, b; −; z, ∂z ) := z 2 ∂z2 + (−1 + (1 + a + b)z)∂z + ab. (4.61)

The equation given by (4.61) is sometimes called the 2 F0 equation. Note that

z a F(a, b; −; z, ∂z )z −a = wF(a; 1 + a − b; w, ∂w ), w = −z −1 , z = −w−1 .

Hence the 2 F0 equation is equivalent to the confluent equation. The relationship between the parameters
is
c = 1 + a − b, b = 1 + a − c.
Another exactly solvable equation that we will consider in this section is sometimes called the 0 F1
equation. It is given by
F(c; z, ∂z ) := z∂z2 + c∂z − 1, (4.62)
and is equivalent to the Bessel equation.
Clearly, the confluent, 2 F0 and 0 F1 equations belong to the class of hypergeometric type equations.
The basic equation of this section will be (1.5), given by

F(a; c; γ; z; ∂z ) := z∂z2 + (c − γz)∂z − a. (4.63)

It will be called the rescaled confluent equation. Note that in the case γ = 1 (4.63) coincides with the
confluent equation. If γ 6= 0, (4.63) can be reduced to the confluent equation by rescaling (and hence
also to the 2 F0 equation). If γ = 0 and a 6= 0, by rescaling (4.63) reduces to the 0 F1 equation (4.62). If
γ = a = 0, (4.63) is a trivial equation.
The Bose invariant of the rescaled confluent equation equals
−γ 2 z 2 + 2(cγ − 2a)z + 2c − c2
1 I1 (a, b; γ; z) = .
4z 2

4.1 Symmetries of the rescaled confluent equation

Let us first describe symmetries of the rescaled confluent equation.

z c−1 F(a; c; γ; z, ∂z )z 1−c = F(a − cγ + γ; 2 − c; γ; z, ∂z ); (4.64)

e−γz F(a; c; γ; z, ∂z )eγz = −F(c − a; c; γ; w, ∂w ), z = −w; (4.65)

Besides, the scaling acts as follows:

F(a; c; γ; z, ∂z ) = γF(a/γ; c; 1; w, ∂w ), w = γz.

It is convenient to introduce new parameters α, ν:

γ+αγ−ν
a= 2 , c = 1 + α;

α = c − 1 = a − b, ν = cγ − 2a = 1 − a − b.
In the new parameters the Bose invariant of the rescaled confluent equation has a more symmetric form:
γ2 ν 1 − α2
1 I1 (α, ν, γ; z) = − + + .
4 2z 4z 2
Thus the starting point for the further analysis will be the equation
γ2  α2
 
ν 1 1
−∂z2 + − + − φ(z) = 0. (4.66)
4 2z 4 4 z2

15
 We will describe 3 classes of Schrödinger operators solved using the confluent equation corresponding
 2
to three obvious choices for r0 (z) :

1 1 1
(1) , (2) , (3) 2 . (4.67)
4 4z 4z

4.2 Hydrogen atom

We consider (1) of (4.67). We set
z = 2r. (4.68)
This leads to the Schrödinger equation

−∂r2 + Vα,ν (r) + γ 2 φ(r) = 0,

(4.69)

with the potential

α2
 
ν 1 1
Vα,ν (r) := − + − .
r 4 4 r2
(4.69) is the radial part of the Schrödinger equation for the hydrogen atom.
]0, ∞[ is a natural real domain for this equation. If ν, α2 are real, then so is Vα,ν on ]0, ∞[.
The derivation of (4.69) from (4.66) is immediate:

γ2  α2
 
2 ν 1 1
4 −∂z + − + −
4 2z 4 4 z2
2
ν  α 1 1

= −∂r2 + γ 2 − + − .
r 4 4 r2

4.3 Rotationally symmetric harmonic oscillator

We consider (2) of (4.67). We set
√
z = r2 , which solves z 0 = 2 z. (4.70)

This leads to the Schrödinger equation

−∂r2 + Vα,γ (r) − 2ν φ(r) = 0,

(4.71)

with the potential  

2 2 1 2 1
Vα,γ (r) := γ r + α − .
4 r2
]0, ∞[ is a natural real domain. For real γ 2 , α2 , the potential is real on ]0, ∞[. By scaling, the only
different real cases are γ 2 = 1, 0, −1.
For γ 2 = 1 the equation is the radial part of the rotationally symmetric harmonic oscillator. For
2
γ = 0 the equation is the radial part of the Helmholtz equation.
Here is an explicit derivation of (4.71) from (4.66):

γ2  α2
 
1 ν 1 1 1
4z 1− 4 −∂z2 + − + − 2
z4
4 2z 4 4 z
 1 1
= −4z∂z2 − 2∂z + γ 2 z − 2ν + α2 −
4 z
2 2 2

2 1 1
= −∂r + γ r − 2ν + α − .
4 r2

16
4.4 Morse potential
We consider (3) of (4.67). We set

z = e−r , which solves z 0 = −z. (4.72)

This leads to the Schrödinger equation

α2
 
2
−∂r + Vν,γ (r) + φ(r) = 0, (4.73)
4

where
γ 2 −2r ν −r
Vν,γ (r) = e − e .
4 2
] − ∞, ∞[ is a natural real domain. For real ν, γ 2 , the potential is real on ] − ∞, ∞[. By translation,
the only different real cases are γ 2 = 1, 0, −1.
Here is an explicit derivation of (4.73) from (4.66):

γ2  α2
 
1 ν 1 1 1
z 2− 2 −∂z2 + − + − 2
z2
4 2z 4 4 z
2 2
γ ν α
= −z 2 ∂z2 − z∂z + z 2 − z +
4 2 4
2 γ 2 −2r ν −r α2
= −∂r + e − e + .
4 2 4

5 Schrödinger equations reducible to the translated harmonic

oscillator
The last family of exactly solvable Schrödinger equations that we describe is (1.6), given by

−∂z2 + θ2 z 2 + ρz + λ. (5.74)

We will call (5.74) the translated harmonic oscillator equation. It is already in the canonical form.
If θ2 6= 0, then (5.74) is just the translation of the Schrödinger equation for usual harmonic oscillator.
It is then equivalent to the Hermite equation, given by

G(a, y, ∂y ) := (∂y2 − 2y∂y − 2a).

In fact, we have
y2 y2
θe− 2 G(a, y, ∂y )e 2

= θ −∂y2 + y 2 + 2a − 1

ρ2
= −∂z2 + θ2 z 2 + ρz + + 2a − 1,
2θ2
√
where y = θ z + 2θρ2 .

For θ = 0, ρ 6= 0, (5.74) is equivalent to the Airy equation

(∂y2 + y)ψ(y) = 0,

17
which in turn is equivalent to a special case of the 0 F1 equation:

−∂z2 + ρz + λ
2
ρ 3 −∂y2 + y


=
 
2 1 2
= (ρ3) 3 w 3 F ; w, ∂w ,
3
1 2
where we set y = ρ 3 z + ρ− 3 λ, and w = 3−2 y 3 .
We will describe 3 classes of Schrödinger operators that can be reduced to (5.74) corresponding to
 2
three obvious choices for r0 (z) :
(1) 1, (2) z, (3) z 2 . (5.75)

5.1 Translated harmonic oscillator

We consider Ansatz (1) of (5.75), which corresponds to the most obvious choice of the energy, that is −λ.
We rename the variable z = r. This leads to the Schrödinger equation

−∂r2 + Vθ,ρ (r) + λ φ(r) = 0,

(5.76)

where

Vθ,ρ (r) = θ2 r2 + ρr.

5.2 Special potential I

We can choose the energy to be −ρ, that is Ansatz (2) of (5.75). This corresponds to the substitution
  23
3r 1
z= , which solves z 0 = z − 2 ,
2

and leads to the Schrödinger equation

−∂r2 + Vθ,λ (r) + ρ φ(r) = 0,

(5.77)

where
  23   23
3r 2 5 1
Vθ,λ (r) = θ2 +λ − .
2 3r 36 r2

In fact,
1
1
z 4 −1 −∂z2 + θ2 z 2 + ρz + λ z − 4
 
1 1 λ 1 1 1
= − ∂z2 + 2 ∂z + θ2 z + ρ + − + 2
z 2z z 4 4 z3
  32   23    2
2 2 3r 2 1 1 2
= −∂r + θ +ρ+λ − + 2 .
2 3r 4 4 3r

Note the following intriguing feature of the above potential: the coefficient at r−2 is fixed and one
cannot change it by rescaling the variable r.

18
5.3 Special potential II
We can choose the energy to be −θ2 , that is Ansatz (3) of (5.75). This corresponds to the substitution
1
z = (2r) 2 , which solves z 0 = z −1 ,

and leads to the Schrödinger equation

−∂r2 + Vρ,λ (r) + θ2 φ(r) = 0,

(5.78)

where
ρ λ 3 1
Vρ,λ (r) = 1 + − .
(2r) 2 2r 16 r2

In fact,
1
1
z 2 −2 −∂z2 + θ2 z 2 + ρz + λ z − 2
 
1 2 1 2 ρ λ 1 1 1
= − ∂ + ∂z + θ + + − +
z2 z z3 z z2 2 22 z 4
 
ρ λ 1 1 1
= −∂r2 + θ2 + 1 + − + .
(2r) 2 2r 2 22 22 r2

Again, the coefficient at r−2 is fixed and one cannot change it by rescaling the variable r.

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20