Higher-Order Susy, Exactly Solvable Potentials, and Exceptional Orthogonal Polynomials

Abstract

Exactly solvable rationally-extended radial oscillator potentials, whose wavefunctions can be expressed in terms of Laguerre-type exceptional orthogonal polynomials, are constructed in the framework of kth-order supersymmetric quantum mechanics, with special emphasis on k = 2. It is shown that for µ = 1, 2, and 3, there exist exactly µ distinct potentials of µth type and associated families of exceptional orthogonal polynomials, where µ denotes the degree of the polynomial g µ arising in the denominator of the potentials.Running head: Higher-order SUSY

Extensions of a class of similarity solutions of Fokker-Planck equation with time-dependent coefficients and fixed/moving boundaries

“…In the case ν = 1, (22) reduces to (19) with µ > 0 and σ = 1 as expected. The new system can be considered as a kind of deformed system based on the parameter ν.…”

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

“…3 and 7. Recently, the X m EOP (and associated potentials) were generalized to multiindexed families X m 1 ,m 2 ,...,m k , obtained through the use of multi-step Darboux alge-braic transformations [23], the Crum-Adler mechanism [24], higher-order SUSYQM [25] or multi-step Darboux-Backlund transformations [26].…”

“…25, in particular, the procedure for constructing rationally-extended radial oscillator potentials and associated Laguerre EOP in higher-order SUSYQM was considered with special emphasis on second order. The number of distinct potentials and EOP families corresponding to a given degree µ of the polynomial arising in the potential denominator (which is some definite function of m 1 , m 2 , .…”