Shape Invariant Potentials in "Discrete Quantum Mechanics"
Satoru Odake,Ryu Sasaki +1 more
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TLDR
Shape invariance is an important ingredient of many exactly solvable quantum mechanics and several examples of shape invariant ''discrete quantum mechanical systems'' are introduced and discussed in some detail as mentioned in this paper.Abstract:
Shape invariance is an important ingredient of many exactly solvable quantum mechanics. Several examples of shape invariant ``discrete quantum mechanical systems" are introduced and discussed in some detail. They arise in the problem of describing the equilibrium positions of Ruijsenaars-Schneider type systems, which are "discrete" counterparts of Calogero and Sutherland systems, the celebrated exactly solvable multi-particle dynamics. Deformed Hermite and Laguerre polynomials are the typical examples of the eigenfunctions of the above shape invariant discrete quantum mechanical systems.read more
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Infinitely many shape invariant potentials and new orthogonal polynomials
Satoru Odake,Ryu Sasaki +1 more
TL;DR: In this paper, the shape invariant potentials obtained by deforming the radial oscillator and the trigonometric/hyperbolic Poschl-Teller potentials in terms of their degree l polynomial eigen functions are presented.
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Exactly solvable quantum mechanics and infinite families of multi-indexed orthogonal polynomials
Satoru Odake,Ryu Sasaki +1 more
TL;DR: In this paper, infinite families of multi-indexed orthogonal polynomials are discovered as the solutions of exactly solvable one-dimensional quantum mechanical systems, where each polynomial has another integer n which counts the nodes and the totality of the integer indices correspond to the degrees of the virtual state wavefunctions which are deleted by the generalisation of Crum-Adler theorem.
Journal ArticleDOI
Unified theory of annihilation-creation operators for solvable ('discrete') quantum mechanics
Satoru Odake,Ryu Sasaki +1 more
TL;DR: The annihilation-creation operators a (±) are defined as the positive/negative frequency parts of the exact Heisenberg operator solution for thesinusoidal coordinate as discussed by the authors, and the relative weights of various terms in them are solely determined by the energy spectrum.
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Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey–Wilson polynomials
Satoru Odake,Ryu Sasaki +1 more
TL;DR: In this article, a set of exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomorphisms are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical Hamiltonians.
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Orthogonal Polynomials from Hermitian Matrices
Satoru Odake,Ryu Sasaki +1 more
TL;DR: In this article, a unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of Hermitian matrices of finite or infinite dimensions.
References
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Orthogonal Polynomials
TL;DR: In this paper, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed and orthogonality on the unit circle is not discussed.
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Supersymmetry and quantum mechanics
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The factorization method
Leopold Infeld,T. E. Hull +1 more
TL;DR: The first-order differential-difference factorization method as mentioned in this paper is an operational procedure which enables us to answer, in a direct manner, questions about eigenvalue problems which are of importance to physicists.
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The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
Roelof Koekoek,René F. Swarttouw +1 more
TL;DR: The Askey-scheme of hypergeometric orthogonal polynomials was introduced in this paper, where the q-analogues of the polynomial classes in the Askey scheme are given.
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Solution of the One‐Dimensional N‐Body Problems with Quadratic and/or Inversely Quadratic Pair Potentials
TL;DR: In this paper, the quantum-mechanical problems of N 1-dimensional equal particles of mass m interacting pairwise via quadratic (harmonical) and/or inverse (centrifugal) potentials is solved.
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