Rational extension and Jacobi-type {\boldmath{$X_m$}} solutions of a quantum nonlinear oscillator

Abstract: We construct a rational extension of a recently studied nonlinear quantum oscillator model. Our extended model is shown to retain exact solvability, admitting a discrete spectrum and corresponding closed-form solutions that are expressed through Jacobi-type $X_m$ exceptional orthogonal polynomials.

• The paper introduces a rational extension of a nonlinear quantum oscillator, achieving closed-form spectral solutions.
• The methodology leverages point transformations and exceptional Jacobi-type Xₘ polynomials to maintain exact solvability.
• The work integrates position-dependent mass effects, offering a framework for generalizing quantum mechanical models.

Rational Extension and Jacobi-Type $X_m$ Solutions of a Quantum Nonlinear Oscillator

Introduction

The paper presents a rational extension of a nonlinear quantum oscillator model that admits exact solvability. By extending the quantum oscillator with exceptional orthogonal polynomials of the Jacobi-type $X_m$, the authors retain the discrete spectrum characteristic and express solutions in closed form. This extension builds upon several existing approaches found in quantum mechanics, such as supersymmetric quantum mechanics, Darboux transformations, and prepotential methods.

Nonlinear Oscillator Model

Originally introduced in the literature for its intriguing properties, the underlined nonlinear oscillator model includes a nonlinearity in the potential and a term allowing the interpretation of a position-dependent mass, both of which are conducive to harmonic forms of solutions. The governing equation of this model includes a potential and enforces Dirichlet-type boundary conditions. The potential is characterized as:

$V = \frac{(1-|\lambda|)x^2}{1-|\lambda|x^2}$

with exact solvability demonstrated through closed-form spectral values and solutions expressible in terms of special functions such as Legendre functions.

Rational Extension

For extending the model, the authors utilize a connection with the trigonometric Scarf potential via a point transformation. The key innovation here is applying known results from the rational extension of the Scarf system to the nonlinear oscillator. The extended model yields solutions in terms of the $X_m$ Jacobi-type exceptional polynomials:

$$R(u) = -2m(a-b-m+1)-(a-b-m+1)\left[a+b+(a-b+1)\sin(u)\right]\frac{P_{m-1}^{(-a,b)}[\sin(u)]}{P_{m}^{(-a-1,b-1)}[\sin(u)]}+\ldots$$

The construction maintains the discrete spectrum through appropriate transformations and parameter redefinitions.

Hamiltonian and Position-Dependent Mass

The proposed system views the Hamiltonian as dependent on a position-variable mass, thus differing from traditional Schrödinger representations. The quantum version is defined in a weighted Hilbert space, distinguishing it from typical $L^2$ spaces.

Spectral Analysis and Solutions

The method ensures that the extended system's spectral properties align with canonical expectations by using $X_m$ exceptional orthogonal polynomials, yielding:

$$E_n = \frac{|\lambda|}{2}(2n+1)^2 + 2n + \frac{3}{2}$$

which demonstrates the method's efficacy in maintaining or enhancing pre-existing spectral structures.

Practical Considerations and Applications

The paper's contribution lies in the generalizability of the approach beyond Schrödinger-type equations, impacting other nonlinear potential and oscillator problems in quantum mechanics. Furthermore, by offering exact solutions through an innovative application of point transformations and use of exceptional orthogonal polynomials, it provides a model for addressing similar challenges in related quantum mechanical problems.

Conclusion

This research contributes significantly to the field of quantum mechanics by constructing a rational extension of a nonlinear quantum oscillator. It retains exact solvability while extending the solution space through $X_m$ Jacobi-type polynomials. The presented framework can serve as a basis for further exploration in extending quantum mechanical systems using exceptional polynomials beyond standard implementations.