2022-08-08T10:47:04Zhttps://soar-ir.repo.nii.ac.jp/oaioai:soar-ir.repo.nii.ac.jp:000117402021-09-02T05:46:43Z1169:1170Discrete quantum mechanicsOdake, SatoruSasaki, RyuA 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 factorized Hamiltonians, the general structure of the solution spaces of the Schrodinger equation (Crum's theorem and its modification), the shape invariance, the exact solvability in the Schrodinger 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, and the infinite families of new orthogonal (the exceptional) polynomials. Two new infinite families of orthogonal polynomials, the X-l Meixner-Pollaczek and the X-l Meixner polynomials, are reported.ArticleJOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL. 44(35):353001 (2011)journal articleIOP PUBLISHING LTD2011-09application/pdfJOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL35443530011751-8113AA12185372https://soar-ir.repo.nii.ac.jp/record/11740/files/Discrete_quantum_mechanics.pdfeng10.1088/1751-8113/44/35/353001https://doi.org/10.1088/1751-8113/44/35/353001CopyrightÂ© 2011 IOP Publishing Ltd / This is an author-created, un-copyedited version of an article accepted for publication in J. Phys. A-Math. Theor. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher-authenticated version is available online at DOI: 10.1088/1751-8113/44/35/353001 .