Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBqj3BWbIqubWexLMBb50ujbqegm0B % 1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqr % Ffpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0F % irpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaa % GcbaWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgaiuaacqWF % pepucqWFtepvaaa!46A4! $$ \mathcal{P}\mathcal{

Abstract

We develop a systematic approach to construct novel completely solvable rational potentials. Second-order supersymmetric quantum mechanics dictates the latter to be isospectral to some well-studied quantum systems. PT symmetry may facilitate reconciling our approach to the requirement that the rationally-extended potentials be singularity free. Some examples are shown.

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

“…After noting [3] that the first extended potentials enlarged the class of known translationally shape invariant potentials in supersymmetric quantum mechanics (SUSYQM) [4,5,6], it appeared convenient to use a SUSYQM technique to construct some additional examples of EOP and potentials [7,8], in agreement with previous works on algebraic deformations of shape invariant potentials [9,10].…”

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].…”

“…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].…”

“…First, we will review the construction of rationally-extended Morse potentials by using a standard SUSYQM approach, similar to that previously employed for some other potentials [14,15,28,30].…”

P T ${\mathcal {PT}}$ Symmetry in Natanzon-class Potentials

“…The investigations were extended to some Natanzon-class potentials too [27][28][29] and even to potentials solved in terms of exceptional orthogonal polynomials [30]. It was found that some potentials have strictly real energy spectrum, while others support complex conjugate energy eigenvalues too.…”