Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials

We extend and generalize the construction of Sturm-Liouville problems for a family of Hamiltonians constrained to fulfill a third-order shape-invariance condition and focusing on the "-2x/3" hierarchy of solutions to the fourth Painlev\'e transcendent. Such a construction has been previously addressed in the literature for some particular cases but we realize it here in the most general case. The corresponding potential in the Hamiltonian operator is a rationally extended oscillator defined in terms of the conventional Okamoto polynomials, from which we identify three different zero-modes constructed in terms of the generalized Okamoto polynomials. The third-order ladder operators of the system reveal that the complete set of eigenfunctions is decomposed as a union of three disjoint sequences of solutions, generated from a set of three-term recurrence relations. We also identify a link between the eigenfunctions of the Hamiltonian operator and a special family of exceptional Hermite polynomial.

Numerical study of the SWKB condition of novel classes of exactly solvable systems

The supersymmetric WKB (SWKB) condition is supposed to be exact for all known exactly solvable quantum mechanical systems with the shape invariance. Recently, it was claimed that the SWKB condition was not exact for the extended radial oscillator, whose eigenfunctions consisted of the exceptional orthogonal polynomial, even the system possesses the shape invariance. In this paper, we examine the SWKB condition for the two novel classes of exactly solvable systems: one has the multi-indexed Laguerre and Jacobi polynomials as the main parts of the eigenfunctions, and the other has the Krein–Adler Hermite, Laguerre and Jacobi polynomials. For all of them, one can always remove the [Formula: see text]-dependency from the condition, and it is satisfied with a certain degree of accuracy.

Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of ℏ

We show that the method developed by Gangopadhyaya, Mallow, and their coworkers to deal with (translational) shape invariant potentials in supersymmetric quantum mechanics and consisting in replacing the shape invariance condition, which is a difference-differential equation, which, by an infinite set of partial differential equations, can be generalized to deformed shape invariant potentials in deformed supersymmetric quantum mechanics. The extended method is illustrated by several examples, corresponding both to ℏ-independent superpotentials and to a superpotential explicitly depending on ℏ.

RATIONALLY EXTENDED SHAPE INVARIANT POTENTIALS IN ARBITRARY D DIMENSIONS ASSOCIATED WITH EXCEPTIONAL Xm POLYNOMIALS

Rationally extended shape invariant potentials in arbitrary D-dimensions are obtained by using point canonical transformation (PCT) method. The bound-state solutions of these exactly solvable potentials can be written in terms of <em>X_m</em> Laguerre or <em>X_m</em> Jacobi exceptional orthogonal polynomials. These potentials are isospectral to their usual counterparts and possess translationally shape invariance property.

ALGEBRAIC DESCRIPTION OF SHAPE INVARIANCE REVISITED

<span>We revisit the algebraic description of shape invariance method in one-dimensional quantum mechanics. In this note we focus on four particular examples: the Kepler problem in flat space, the Kepler problem in spherical space, the Kepler problem in hyperbolic space, and the Rosen-Morse potential problem. Following the prescription given by Gangopadhyaya et al., we introduce new nonlinear algebraic systems and solve the bound-state problems by means of representation theory.</span>