Discrete Quantum Mechanics

Abstract

A comprehensive review of the discrete quantum mechanics with the pure imaginary shifts and the real shifts is presented in parallel with the corresponding results in the ordinary quantum mechanics. The main subjects to be covered are the factorised Hamiltonians, the general structure of the solution spaces of the Schroedinger equation (Crum's theorem and its modification), the shape invariance, the exact solvability in the Schroedinger picture as well as in the Heisenberg picture, the creation/annihilation operators and the dynamical symmetry algebras, the unified theory of exact and quasi-exact solvability based on the sinusoidal coordinates, the infinite families of new orthogonal (the exceptional) polynomials. Two new infinite families of orthogonal polynomials, the X_\ell Meixner-Pollaczek and the X_\ell Meixner polynomials are reported.

61 pages, 1 figure. Comments and references added

References in corpus (10)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• 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
• 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
• On factorization of q-difference equation for continuous q-Hermite polynomials

Cited by in corpus (32)

• Asymptotic behaviour of zeros of exceptional Jacobi and Laguerre polynomials
• Orthogonal Polynomials from Hermitian Matrices II
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials
• Moments of random matrices and hypergeometric orthogonal polynomials
• Prepotential approach to solvable rational extensions of Harmonic Oscillator and Morse potentials
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : III
• Casoratian Identities for the Wilson and Askey-Wilson Polynomials
• DARBOUX partners of pseudoscalar Dirac potentials associated with exceptional orthogonal polynomials
• Equivalences of the Multi-Indexed Orthogonal Polynomials
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : II
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials IV : closure relations and creation/annihilation operators
• Non-Hermitian invisibility in tight-binding lattices
• Exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice
• Exactly Solvable Discrete Quantum Mechanical Systems and Multi-indexed Orthogonal Polynomials of the Continuous Hahn and Meixner-Pollaczek Types
• Solvable Discrete Quantum Mechanics: q-Orthogonal Polynomials with |q|=1 and Quantum Dilogarithm
• Generalized Rayleigh and Jacobi processes and exceptional orthogonal polynomials
• Perturbations around the zeros of classical orthogonal polynomials
• New Determinant Expressions of the Multi-indexed Orthogonal Polynomials in Discrete Quantum Mechanics
• Type II Multi-indexed Little $q$-Jacobi and Little $q$-Laguerre Polynomials
• Algebraic localization-delocalization phase transition in moving potential wells on a lattice
• Dual Polynomials of the Multi-Indexed ($q$-)Racah Orthogonal Polynomials
• Non-polynomial extensions of solvable potentials a la Abraham-Moses
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials V : Racah and $q$-Racah types
• Discrete orthogonality relations for multi-indexed Laguerre and Jacobi polynomials
• Casoratian Identities for the Discrete Orthogonal Polynomials in Discrete Quantum Mechanics with Real Shifts
• New Finite Type Multi-Indexed Orthogonal Polynomials Obtained From State-Adding Darboux Transformations
• Another Type of Forward and Backward Shift Relations for Orthogonal Polynomials in the Askey Scheme
• Multi-indexed Orthogonal Polynomials of a Discrete Variable and Exactly Solvable Birth and Death Processes
• Discrete Orthogonality Relations for the Multi-Indexed Orthogonal Polynomials in Discrete Quantum Mechanics with Pure Imaginary Shifts
• Scalar field stochastic dynamics in de Sitter spacetime from exact solutions of quantum deficient oscillators
• Some Difference Relations for Orthogonal Polynomials of a Continuous Variable in the Askey Scheme
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials VI : Meixner-Pollaczek and continuous Hahn types