An extension of Bochner’s problem: Exceptional invariant subspaces

Abstract

We prove an extension of Bochner's classical result that characterizes the classical polynomial families as eigenfunctions of a second-order differential operator with polynomial coefficients. The extended result involves considering differential operators with rational coefficients and the requirement is that they have a numerable sequence of polynomial eigenfunctions p 1 , p 2 , . . . of all degrees except for degree zero. The main theorem of the paper provides a characterization of all such differential operators. The existence of such differential operators and polynomial sequences is based on the concept of exceptional polynomial subspaces, and the converse part of the main theorem rests on the classification of codimension one exceptional subspaces under projective transformations, which is performed in this paper.

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“…(14) and (15)) belong to a class of exceptional orthogonal polynomials, which has been the topic of some recent mathematical study [14]. Our SSUSY approach provides us with a convenient way of constructing and generalizing such polynomials.…”

“…Against this background, the recent introduction of two new classes of exceptional orthogonal polynomials [14] and their occurrence in the bound-state wavefunctions of two novel rational potentials isospectral to some well-known conventional ones [15] have led us to re-examine the construction of such pairs of partner potentials. These examples have suggested us an alternative approach to the usual one, which consists in searching for a reduction of the initial Schrödinger equation general solution to some elementary function [16].…”

Revisiting (Quasi-)Exactly Solvable Rational Extensions of the Morse Potential

“…After the introduction of the first families of exceptional orthogonal polynomials (EOP) in the context of Sturm-Liouville theory [11,12], the realization of their usefulness in constructing new SI extensions of ES potentials in quantum mechanics [13,14,15], and the rapid developments that followed in this area [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30], it soon appeared that only some of the well-known SI potentials led to rational extensions connected with EOP. In this category, one finds the radial oscillator [13,15,16,17,18,22,23,24], the Scarf I (also called trigonometric Pöschl-Teller or Pöschl-Teller I) [13,15,16,17,22,24], and the generalized Pöschl-Teller (also termed hyperbolic Pöschl-Teller or Pöschl-Teller II) [14,16,17].…”

RATIONALLY-EXTENDED RADIAL OSCILLATORS AND LAGUERRE EXCEPTIONAL ORTHOGONAL POLYNOMIALS IN kTH-ORDER SUSYQM

“…. , in the context of Sturm-Liouville theory [1,2] and the realization of their usefulness in building new exactly solvable rational extensions of known quantum potentials [3], a lot of work has been devoted to generalizing these families and the associated exactly solvable potentials, as well as to providing several different (but equivalent) approaches to the problem.…”