A comprehensive review of the discrete quantum mechanics with the pure imaginary shifts
and the real shifts is presented in parallel with the corresponding results in the ordinary …

We prove that every rational extension of the quantum harmonic oscillator that is exactly
solvable by polynomials is monodromy free, and therefore can be obtained by applying a …

Infinite families of multi-indexed orthogonal polynomials are discovered as the solutions of
exactly solvable one-dimensional quantum mechanical systems. The simplest examples, the …

We adapt the notion of the Darboux transformation to the context of polynomial Sturm–
Liouville problems. As an application, we characterize the recently described X m Laguerre …

A simple derivation is presented of the four families of infinitely many shape-invariant
Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials. The …

It was recently conjectured that every system of exceptional orthogonal polynomials is
related to a classical orthogonal polynomial system by a sequence of Darboux …

It has been recently discovered that exceptional families of Sturm–Liouville orthogonal
polynomials exist, that generalize in some sense the classical polynomials of Hermite …

Combining recent results on rational solutions of the Riccati–Schrödinger equations for
shape invariant potentials to the finite difference Bäcklund algorithm and specific symmetries …

In this paper we state and prove some properties of the zeros of exceptional Jacobi and
Laguerre polynomials. Generically, the zeros of exceptional polynomials fall into two …

The existence of a novel enlarged shape invariance property valid for some rational
extensions of shape-invariant conventional potentials, first pointed out in the case of the …