Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials

Abstract

Infinite families of multi-indexed orthogonal polynomials are discovered as the solutions of exactly solvable one-dimensional quantum mechanical systems. The simplest examples, the one-indexed orthogonal polynomials, are the infinite families of the exceptional Laguerre and Jacobi polynomials of type I and II constructed by the present authors. The totality of the integer indices of the new polynomials are finite and they correspond to the degrees of the `virtual state wavefunctions' which are `deleted' by the generalisation of Crum-Adler theorem. Each polynomial has another integer n which counts the nodes.

7 pages, 1 figure. Comments and references added. Typo corrected(4,5 lines below eq.(5)). To appear in Phys.Lett.B

References in corpus (15)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
• Two-step Darboux transformations and exceptional Laguerre polynomials
• Solvable rational extensions of the isotonic oscillator
• Exceptional orthogonal polynomials and the Darboux transformation
• Orthogonal Polynomials from Hermitian Matrices
• Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states
• Exact solution in the Heisenberg picture and annihilation-creation operators
• Dirac(-Pauli), Fokker-Planck equations and exceptional Laguerre polynomials
• The Exceptional (X_{\ell}) (q)-Racah Polynomials
• Exceptional Askey-Wilson type polynomials through Darboux-Crum transformations
• Dual Christoffel transformations
• q-oscillator from the q-Hermite Polynomial
• Quasi Exactly Solvable Difference Equations
• A new family of shape invariantly deformed Darboux-Pöschl-Teller potentials with continuous \ell

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