An infinite family of superintegrable deformations of the Coulomb potential

Abstract

We introduce a new family of Hamiltonians with a deformed Kepler- Coulomb potential dependent on an indexing parameter k. We show that this family is superintegrable for all rational k and compute the classical trajectories and quantum wave functions. We show that this system is related, via coupling constant metamorphosis, to a family of superintegrable deformations of the harmonic oscillator given by Tremblay, Turbiner and Winternitz. In doing so, we prove that all Hamiltonians with an oscillator term are related by coupling constant metamorphosis to systems with a Kepler-Coulomb term, both on Euclidean space. We also look at the effect of the transformation on the integrals of the motion, the classical trajectories and the wave functions and give the transformed integrals explicitly for the classical system. Comment: 11 pages, typos corrected and references updated.

Full text

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arXiv:1003.5230v2 [math-ph] 11 May 2010
An infinite family of superintegrable deformations
of the Coulomb potential
Sarah Post1, Pavel Winternitz2
Centre de recherches math´ ematiques, C.P. 6128 succ. Centre-Ville, Montreal
(QC) H3C 3J7, Canada1
Centre de recherches math´ ematiques and D´ epartement de math´ ematiques et de
statistique, C.P. 6128 succ. Centre-Ville, Montreal (QC) H3C 3J7, Canada2
E-mail: post@CRM.UMontreal.CA, wintern@CRM.UMontreal.CA
Abstract.
Coulomb potential dependent on an indexing parameter k. We show that this
family is superintegrable for all rational k and compute the classical trajectories
and quantum wave functions. We show that this system is related, via coupling
constant metamorphosis, to a family of superintegrable deformations of the
harmonic oscillator given by Tremblay, Turbiner and Winternitz. In doing so,
we prove that all Hamiltonians with an oscillator term are related by coupling
constant metamorphosis to systems with a Kepler-Coulomb term, both on
Euclidean space. We also look at the effect of the transformation on the integrals
of the motion, the classical trajectories and the wave functions and give the
transformed integrals explicitly for the classical system.
We introduce a new family of Hamiltonians with a deformed Kepler-
PACS numbers: 03.65.FD, 02.30.K,11.30.Na
1. Introduction
The purpose of this article is to introduce an infinite family of classical and quantum
systems with the Hamiltonian
αk2
4r2cos2(k
VDC
k
= −Q
r+
2φ)+
βk2
4r2sin2(k
2φ)
(1)
HDC
k
= p2
1+ p2
2+ VDC
k
,HDC
k
= −∆ + VDC
k, ? p is the linear momentum and
k
(2)
where (r,φ) are polar coordinates with 0 ≤ φ ≤2π
∆ is the Laplacian on 2 dimensional Euclidean space. The superscript denotes that it
is a deformed Coulomb potential and the subscript shows the dependence on k. This
system was given for k = 1 in [1] and k = 2 in [2].
We shall show that the classical system is superintegrable for all rational values
of k in that it allows two independent integrals of motion, besides the Hamiltonian.
Both are polynomial in the momentum, one of second order and the other a higher
order polynomial. We show that all bounded classical trajectories of these systems are
closed and the motion is periodic. For the quantum system, we show that Schr¨ odinger
equation is exactly solvable and the energy levels are essentially the same as those
of the Coulomb system.We will also show that these systems are related to a

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An infinite family of superintegrable deformations of the Coulomb potential2
family of superintegrable deformations of the harmonic oscillator via coupling constant
metamorphosis.
Superintegrable systems can be classified by the degree of the highest order
integral of the motion, excluding the Hamiltonian. Superintegrable systems of first-
order are directly related to Lie groups of point transformations while superintegrable
systems of second-order are characterized by separability in multiple coordinate
systems. Both types are considered to be well understood [3, 4, 5, 6, 7, 8]. The
classification of higher-order superintegrable systems remains an open problem and
has been a subject of much recent activity [9, 10, 11, 12, 13, 14, 15, 16, 2, 17].
Most relevant to this paper, is the recent discovery of a family of superintegrable
deformations of the harmonic oscillator [18] indexed by a parameter k, referred to
by others in subsequent articles as the TTW system.
a classical or quantum Hamiltonian with potential V = ω2ρ2+ αk2ρ−2sec2(kθ) +
βk2ρ−2csc2(kθ). This system has ignited much recent work on its conjectured
superintegrability for certain values of k. Specifically, for integer k, it was conjectured
to have an independent integral of the motion of order 2k, in addition to the second-
order integral defining separation of variables.
The original authors proved exact-solvability for all k and superintegrability for
k = 1,2,3,4. Later they demonstrated the periodicity of bounded trajectories in
the classical case for rational k [19], supporting the conjecture. In recent papers,
the classical superintegrability was proven for rational k [20] and the quantum
superintegrability was proven for odd k [21].
of the superintegrability of the quantum system for rational k was given in [22].
An important tool in the analysis of superintegrable systems is the coupling
constant metamorphosis, also referred to as the St¨ ackel transform which maps one
Hamiltonian to another [23, 24]. This transform is particularly useful in the study
of integrable and superintegrable systems because it carries an associated mapping of
the integrals of the motion. In this paper, we will see that the new Hamiltonian given
above is St¨ ackel equivalent to the TTW system and so the integrals of the motion,
trajectories and wave functions will be intimately linked.
In Sections 2 and 3, we find the classical trajectories and then the quantum
wave functions. In Section 4 we discuss the St¨ ackel transform and prove that systems
with oscillator terms in the potential are St¨ ackel equivalent to systems with Kepler-
Coulomb terms and describe the effect of the St¨ ackel transform on the trajectories
or wave functions and on the integrals of the motion. In Section 5, we apply these
theorems to determine higher-order integrals of the motion for the classical system.
It can be defined both as
Most recently, a constructive proof
2. The Classical Trajectories
We consider the classical trajectories of the system HDC
that the bounded trajectories are closed and the motion is periodic. We follow the
procedure in [25] and separate the action as S = S1(r) + S2(φ) − Et. The Hamilton-
Jacobi equation separates as
− A = r2(∂S1
?
4cos2(k
k
given by (1-2) and prove
∂r)2− Qr − Er2
(∂S2
∂φ)2+
(3)
−A = −
αk2
2φ)+
βk2
4sin2(k
2φ)
?
(4)

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An infinite family of superintegrable deformations of the Coulomb potential3
and for S1and S2, we have
S1(r) =
?
1
r
?
Er2+ Qr − Adr,S2(φ) =
??
A −
αk2
4cos2(k
2φ)−
βk2
4sin2(k
2φ).
The trajectories will then satisfy
∂S
∂E=∂S1
∂E− t = δ1,
∂S
∂A=∂S1
∂A+∂S2
∂A= δ2. (5)
It remains to integrate these equations under the condition that the motion be
bounded,
Er2
i+ Qri− A = 0,
Also, we have the requirement
0 ≤ u1≤ cos2(k
These conditions give restrictions on the choices of parameters. We summarize these
in Table 1 below.
0 ≤ r1≤ r ≤ r2,i = 1,2.(6)
2φ) ≤ u2≤ 1,
−Au2
i+ (A − β(k
2)2+ α(k
2)2)ui− α(k
2)2= 0.(7)
Table 1. Parameter restrictions for bounded trajectories
RestrictionEffect
D1≡ Q2+ 4AE > 0
Q > 00 < r2
A > 00 < r1and
u1< cos2(k
E < 0r1< r < r2
Restriction
4(β + α))2−αβk4
A −k2
β > 0
Effect
uireal
u2> 0
u1> 0
rirealD2≡ (A −k2
4
> 0
4|β − α| > 0
2φ) < u2
α > 0u2< 1
Under the restrictions given in the table above, the solutions for (5) are
(t + δ1) +2√−E
0 =−2Er − Q
√D1
+ sin
?4(−E)3/2
Qa
?
Er2+ Qr − A
?
(8)
0 = −δ2+
1
2√Aarcsin(2A − Qr
r√D1
) +
1
2k√Aarcsin(2Asin2(k
2φ) − A +k2
√D2
4(β − α)
) (9)
and hence r has the period of
for rational k = c/d with c,d integer. Rewriting (9),
Qπ
2(−E)3/2in t. These trajectories are also periodic in φ
0 = −2
and using the Chebyshev polynomials defined as,
√
Acδ2+c + d
2
π−carccos(2A − Qr
r√D1
)−darccos(2Asin2(k
2φ) − A +k2
√D2
4(β − α)
)(10)
Tn(x) = cos(narccos(x)),Un(x) =sin((n + 1)arccos(x))
sinarccosx
(11)
we obtain,
0 = −Tc(2A − Qr
r√D1
) + cos(C)Td
?
2Asin2(k
2φ) − A +k2
√D2
4(β − α)
?
(12)
+sin(C)Ud−1
?
2Asin2(k
2φ) − A +k2
√D2
4(β − α)
??
?
?
?1 −
?
2Asin2(k
2φ) − A +k2
√D2
4(β − α)
?2

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An infinite family of superintegrable deformations of the Coulomb potential
where C = (−2√Apδ2+ (c + d)/2π). Under the given restrictions on the parameters,
the implicit function for r = r(φ) given by (12) is well defined and periodic, with
period τ =2
kπ in φ.
We note that these trajectories and their determining equations bear a striking
resemblance to those obtained for the TTW system [19]. In fact, the implicit function
determining r = r(φ) is identical under a change of variables and parameters. We
shall see in a later section why this is so.
4
3. Eigenfunctions for the Quantum System
In this section, we solve for the wave functions of the quantum system HDC
(1-2) and show that the system is exactly solvable. That is, its energy values can be
calculated algebraically and the eigenfunctions can be realized as polynomials modulo
a gauge function [26, 27].
We assume a solution Ψ = R(r)S(φ) and separate the equation HDC
k
given by
k
Ψ = EΨ as
?
?
−∂2
r−1
r∂r+A
r2−Q
r− E
?
R(r) = 0.(13)
−∂2
φ+
αk2
4cos2(kφ
2)+
βk2
4sin2(kφ
2)− A
?
S(φ) = 0.(14)
We look for solutions which can be written as gauge transformations of a
polynomial, a characteristic of exact solvability. In order to obtain such a form of
solutions, we require that α and β be greater that −1/4 and rewrite α = a(a−1),β =
b(b − 1).
If we take a gauge transformation with gauge G1= r
radial equation (13) will have polynomial solutions if we restrict to quantized values of
the energy E = −Q2(2n+1+2√A)−2. If we make a gauge transformation with G2=
cos(k
solutions if we restrict to quantized values of the parameter A = k2(2m + a + b)2/4.
A set of solutions for the Schr¨ odinger equation is given by Jacobi polynomial
multiplied by Laguerre polynomials
Ψ = G1G2L2√A
n
√Ae2r√−E, the transformed
2φ)asin(k
2φ)bthen the transformed angular equation (14) will have polynomial
?
2r√−E
?
Pa−1
m
2,b−1
2
(−cos(kφ)) (15)
with energy
E =
−Q2
(2(n + km) + 1 + ka + kb)2.(16)
For a given rational k = c/d the energy levels are indexed by an integer N = dn+cm
and their degeneracy is D = [dN/c] + 1. This coincides with the degeneracy of an
anisotropic oscillator with frequency ratio c/d.
These wave functions are in agreement with the solutions for the k = 1 case
previously analyzed [?].The quantum system is indeed exactly solvable and we
recover the requirements A > 0,E < 0 with a slight relaxing in the restrictions on the
parameters α,β which are both required to be greater that −1/4 instead of positive as
in the classical case. Here we see again the relation between these eigenfunctions and
those of the TTW system which differ by the same change of variables and parameters
as the classical trajectories. In the next section we shall see why this is the case as we
prove some theorems which are directly relevant to our systems.

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An infinite family of superintegrable deformations of the Coulomb potential5
4.
coordinates with a harmonic oscillator term in the potential
Coupling Constant Metamorphosis of Hamiltonians separable in polar
Consider a classical Hamiltonian H =ˆ H −˜EU, whereˆ H includes the kinetic energy
and part of the potential independent of the coupling constant (−˜E). We can then
write the Hamilton-Jacobi equation H = E and solve for˜E to obtain a new Hamilton-
Jacobi equation˜ H ≡ U−1(ˆ H − E) =˜E. For H and˜ H the role of coupling constant
and the energy are interchanged. The quantum version follows by direct analogy.
Such a transform is called coupling constant metamorphosis or St¨ ackel transform.
A remarkable characteristic of coupling constant metamorphosis is that there is an
associated mapping of the integrals of the motion to the new system. Such mapping
was given first for classical integrable Hamiltonians and later extended to quantum
systems with second-order integrals [23, 24]. More recently, the quantum transform
was extended to higher order constants of the motion [28].
The theorems which define the transforms are given below with their proofs.
Theorem 1 Given a classical Hamiltonian H =ˆ H −˜EU, whereˆ H is independent
of the arbitrary parameter˜E, with an integral of the motion L(˜E). If we define the
St¨ ackel transform of H and L as˜ H ≡ U−1(ˆ H−E) and˜L ≡ L(˜H) then˜L is an integral
of the motion for˜ H.
To prove this, we use an identity for Poisson brackets given by
{F(pi,qj,f(pi,qj)),G} = {F(pi,qj,τ),G}|τ=f(pi,qj)+∂F(pi,qj,τ)
∂τ
|τ=f(pi,qj){f(pi,qj),G}
where pi,qjare the conjugate position and momenta and τ is a parameter. With this,
we compute,
{˜ H,˜L} = {1
= {1
= − {U,L}1
and since H|˜ E=˜
There is an associated theorem for quantum systems though we must make a
further assumption about the form of the integral of motion in order to get a well
defined integral.
Theorem 2 Given a quantum Hamiltonian H =ˆH −˜EU, whereˆH is independent of
the arbitrary parameter˜E, with an integral of the motion L =?[n
L as˜H = U−1(ˆH − E) and˜L =?[n
parity of N, with respect to dµ then˜H will be self-adjoint and˜L will have the same
parity as L with respect to the metric Udµ.
U(H +˜EU − E),L|˜ E=˜
U(H +˜EU − E),L}|˜ E=˜
H}
H+ ∂˜ EL(˜E)|˜ E=˜
H{˜ H,˜ H}
U2(H − E)|˜ E=˜
H
H= E, we see that {˜ H,˜L} = 0 and we have proved the theorem.
2]
j=0KN−2j˜Ej, where
Kihave degree i as differential operators. If we define the St¨ ackel transform of H and
2]
j=0KN−2j˜Hj, then [˜H,˜L] = 0.
Furthermore, if H is self-adjoint and L is self or skew adjoint, depending on the
The proof of this theorem uses the fact that the K′
that for all integer j, we have
is do not depend on˜E to show
[L,H] = [
[N
?
2]
j=0
KN−2j˜Ej,ˆH +˜EU] = 0 ⇐⇒ [KN−2j,ˆH] + [KN−2j+2,U] = 0,(17)