26 research outputs found
Exceptional orthogonal polynomials and generalized Schur polynomials
We show that the exceptional orthogonal polynomials can be viewed as
confluent limits of the generalized Schur polynomials introduced by Sergeev and
Veselov.Comment: arXiv admin note: text overlap with arXiv:1309.375
Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials
We construct rational extensions of the Darboux-P\"oschl-Teller and isotonic
potentials via two-step confluent Darboux transformations. The former are
strictly isospectral to the initial potential, whereas the latter are only
quasi-isospectral. Both are associated to new families of orthogonal
polynomials, which, in the first case, depend on a continuous parameter. We
also prove that these extended potentials possess an enlarged shape invariance
property
Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials
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 finite number of state-deleting Darboux transformations
on the harmonic oscillator. Equivalently, every exceptional orthogonal
polynomial system of Hermite type can be obtained by applying a Darboux-Crum
transformation to the classical Hermite polynomials. Exceptional Hermite
polynomial systems only exist for even codimension 2m, and they are indexed by
the partitions \lambda of m. We provide explicit expressions for their
corresponding orthogonality weights and differential operators and a separate
proof of their completeness. Exceptional Hermite polynomials satisfy a 2l+3
recurrence relation where l is the length of the partition \lambda. Explicit
expressions for such recurrence relations are given.Comment: 25 pages, typed in AMSTe
New rational extensions of solvable potentials with finite bound state spectrum
Using the disconjugacy properties of the Schr\"odinger equation, it is
possible to develop a new type of generalized SUSY QM partnership which allows
to generate new solvable rational extensions for translationally shape
invariant potentials having a finite bound state spectrum. For this we prolong
the dispersion relation relating the energy to the quantum number out of the
physical domain until a disconjugacy sector. The prolonged excited states
Riccati-Schr\"odinger (RS) functions are used to build Darboux-B\"acklund
transforms which give regular isospectral extensions of the initial potential.
We give the spectra of these extensions in terms of new orthogonal polynomials
and study their shape invariance properties
Solvable rational extensions of the Morse and Kepler-Coulomb potentials
We show that it is possible to generate an infinite set of solvable rational
extensions from every exceptional first category translationally shape
invariant potential. This is made by using Darboux-B\"acklund transformations
based on unphysical regular Riccati-Schr\"odinger functions which are obtained
from specific symmetries associated to the considered family of potentials