Rational extension and Jacobi-type Xm solutions of a

CRM-ICMAT Workshop on Exceptional Orthogonal Polynomials
and exact Solutions in Mathematical Physics
Segovia (Spain), September 7–12, 2014.
Rational extension and Jacobi-type Xm solutions
of a quantum nonlinear oscillator
Axel Schulze-Halberg 1
Barnana Roy 2
1. Department of Mathematics and Actuarial Science and Department of Physics,
Indiana University Northwest, 3400 Broadway, Gary IN 46408, USA.
2. Physics & Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108,
India.
Abstract
The purpose of this work is to construct a rational extension to the
quantum model of a nonlinear oscillator, commonly referred to as the
“Mathews-Lakshmanan oscillator” [2]. The latter oscillator system originates from the classical Lagrangian
L
(λ + 1) x2
x˙ 2
−
,
2 λ x2 + 2
2 λ x2 + 2
=
(1)
for a real-valued parameter λ < 0. The Lagrangian (1) resembles a system
under the presence of a position-dependent mass m and potential energy
V , given by
m =
1
V =
λ x2 + 1
(λ + 1) x2
.
2 λ x2 + 2
A quantum model associated with the Lagrangian (1) that was recently
studied in [1] (see also references therein), is governed by the following
boundary-value problem of Dirichlet-type
1
(1 − |λ|) x2
(1 − |λ| x2 ) Ψ00 − |λ| x Ψ0 + 2 E −
Ψ = 0, x ∈ 0, |λ|− 2(2)
2
1 − |λ| x
1
Ψ (0) = Ψ |λ|− 2 = 0,
(3)
where E is a real spectral parameter. The problem (2), (3) admits an
infinite discrete spectrum and an associated orthogonal set of solutions
located in a weighted L2 -space. This can be explained by noticing that
the above problem is equivalent to the well-known trigonometric Scarf
system, to which it is related by means of a coordinate change and a
reparametrization. Since the trigonometric Scarf system allows for a rational extension in terms of Jacobi-type Xm EOPs [3], one can expect that
the problem (2), (3) can be rationally extended as indicated in diagram
1 below. In particular, we establish a mapping E that provides a rational
extension of the nonlinear oscillator model through the relationship
E
=
P2 ◦ E ◦ P1 ,
(4)
where E stands for the rational extension associated with the trigonometric Scarf model, while P1 and P2 denote coordinate changes from the
nonlinear oscillator to the trigonometric Scarf system and vice versa, respectively. By evaluating (4) explicitly, we find that the rationally extended counterpart to the initial problem (2), (3) contains an additional
term in the potential that can be expressed in terms of Jacobi-type Xm
EOPs. We omit to display the explicit form of the extended potential
due to the length of the expressions that are involved. We construct the
infinite discrete spectrum of the extended model in closed form, as well
as the corresponding set of solutions that are normalizable in the same
weighted L2 -space as their counterparts for the initial problem. We obtain
from our construction (4) the explicit results
En
=
1
1
|λ| (2 m − 2 n − 1)2 + (4 n − 4 m + 3)
2
2
Ψn
=
1 λ+2
1
x (2 − 2 x2 |λ|) 4 + 4 | λ |
( 1 | λ+2 |, 1 ,m)
Pn 2 λ 2
2 |λ| x2 − 1 ,
1 λ+2 ,− 1
−1−
(
2| λ |
2)
Pm
(2 |λ| x2 − 1)
where P and P stand for a Jacobi polynomial and a Jacobi type Xm EOP,
respectively, while n, m are a nonnegative integers that satisfy n ≥ m.
Moreover, it turns out that normalizability of the solutions as well as
the existence of the discrete spectrum depends on the parameter λ. In
particular, we observe that λ must be chosen within a particular interval
that depends on the parameter m.
Extension of the NLO
E
NLO
-
NLO
extended
6
Coordinate
change
Coordinate
change
P1
P2
Extension of the TS
?
TS
E
-
TS
extended
Diagram 1: Link between rational extensions of the nonlinear oscillator
(NLO) and the trigonometric Scarf (TS) systems.
References
[1] J.F. Cari˜
nena, M.F. Ra˜
nada and M. Santander: A quantum exactly solvable nonlinear oscillator with quasi-harmonic behaviour. Ann. Phys. 322
(2007) 434-459.
[2] P.M. Mathews and M. Lakshmanan: On a unique nonlinear oscillator.
Quart. Appl. Math. 32 (1974) 215-218.
[3] C. Quesne: Exceptional orthogonal polynomials and new exactly solvable
potentials in quantum mechanics. J. Phys. Conf. Ser. 380 (2012) 012016

Download Report