Unified Theory of Annihilation-Creation Operators for Solvable (`Discrete') Quantum Mechanics

Abstract

The annihilation-creation operators $a^{(\pm )}$ are defined as the positive/negative frequency parts of the exact Heisenberg operator solution for the `sinusoidal coordinate'. Thus $a^{(\pm )}$ are hermitian conjugate to each other and the relative weights of various terms in them are solely determined by the energy spectrum. This unified method applies to most of the solvable quantum mechanics of single degree of freedom including those belonging to the `discrete' quantum mechanics.

43 pages, no figures, LaTeX2e, with amsmath, amssymb

References in corpus (5)

• The structure relation for Askey-Wilson polynomials
• An Algebraic Construction of Generalized Coherent States for Shape-Invariant Potentials
• Equilibrium Positions, Shape Invariance and Askey-Wilson Polynomials
• Calogero-Sutherland-Moser Systems, Ruijsenaars-Schneider-van Diejen Systems and Orthogonal Polynomials
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Cited by in corpus (52)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials
• Orthogonal Polynomials from Hermitian Matrices
• Krein-Adler transformations for shape-invariant potentials and pseudo virtual states
• Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials
• Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states
• Infinitely many shape invariant potentials and cubic identities of the Laguerre and Jacobi polynomials
• Exact solution in the Heisenberg picture and annihilation-creation operators
• Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum Transformations
• Modification of Crum's Theorem for `Discrete' Quantum Mechanics
• Dirac(-Pauli), Fokker-Planck equations and exceptional Laguerre polynomials
• Discrete Quantum Mechanics
• Multi-indexed (q-)Racah Polynomials
• Extensions of solvable potentials with finitely many discrete eigenstates
• Exactly Solvable Birth and Death Processes
• Exceptional Askey-Wilson type polynomials through Darboux-Crum transformations
• Orthogonal Polynomials from Hermitian Matrices II
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials
• Dual Christoffel transformations
• q-oscillator from the q-Hermite Polynomial
• Spectrum generating algebras for position-dependent mass oscillator Schrodinger equations
• Casoratian Identities for the Wilson and Askey-Wilson Polynomials
• Quasi Exactly Solvable Difference Equations
• Prepotential approach to exact and quasi-exact solvabilities
• Unified theory of exactly and quasi-exactly solvable `Discrete' quantum mechanics: I. Formalism
• Multi-indexed Meixner and Little $q$-Jacobi (Laguerre) Polynomials
• Simple unified derivation and solution of Coulomb, Eckart and Rosen-Morse potentials in prepotential approach
• Quasi-Linear Algebras and Integrability (the Heisenberg Picture)
• Global Solutions of Certain Second-Order Differential Equations with a High Degree of Apparent Singularity
• New Quasi Exactly Solvable Difference Equation
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials IV : closure relations and creation/annihilation operators
• Exactly Solvable Discrete Quantum Mechanical Systems and Multi-indexed Orthogonal Polynomials of the Continuous Hahn and Meixner-Pollaczek Types
• Exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice
• Solvable Discrete Quantum Mechanics: q-Orthogonal Polynomials with |q|=1 and Quantum Dilogarithm
• Perturbations around the zeros of classical orthogonal polynomials
• Bethe Ansatz Solutions to Quasi Exactly Solvable Difference Equations
• Exactly and quasi-exactly solvable `discrete' quantum mechanics
• New Determinant Expressions of the Multi-indexed Orthogonal Polynomials in Discrete Quantum Mechanics
• Exact Heisenberg operator solutions for multi-particle quantum mechanics
• Discrete orthogonality relations for multi-indexed Laguerre and Jacobi polynomials
• 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
• New Finite Type Multi-Indexed Orthogonal Polynomials Obtained From State-Adding Darboux Transformations
• Exactly Solvable Quantum Mechanics
• Interpolation of SUSY quantum mechanics
• Another Type of Forward and Backward Shift Relations for Orthogonal Polynomials in the Askey Scheme
• Spectral intertwining relations in exactly solvable quantum-mechanical systems
• A relativistic model of the $N$-dimensional singular oscillator
• 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
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials VI : Meixner-Pollaczek and continuous Hahn types