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Solution of the Cauchy problem for a time-dependent Schrödinger equation

J. Math. Phys. 49, 072102 (2008); doi:10.1063/1.2938698

Published 9 July 2008

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Maria Meiler,1 Ricardo Cordero–Soto,2 and Sergei K. Suslov2
1Department of Mathematics, Munich University of Technology, Boltzmannstrasse 3, D-85747 Garching, Munich, Germany
2Department of Mathematics and Statistics, Arizona State University, Tempe, Arizona 85287, USA
We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schrödinger equation with certain time-dependent Hamiltonian operator of a modified oscillator. The dynamical SU(1,1) symmetry of the harmonic oscillator wave functions, Bargmann's functions for the discrete positive series of the irreducible representations of this group, the Fourier integral of a weighted product of the Meixner–Pollaczek polynomials, a Hankel-type integral transform, and the hyperspherical harmonics are utilized in order to derive the corresponding Green function. It is then generalized to a case of the forced modified oscillator. The propagators for two models of the relativistic oscillator are also found. An expansion formula of a plane wave in terms of the hyperspherical harmonics and solution of certain infinite system of ordinary differential equations are derived as by-products. ©2008 American Institute of Physics
History: Received 5 December 2007; accepted 11 May 2008; published 9 July 2008
Permalink: http://link.aip.org/link/?JMAPAQ/49/072102/1
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KEYWORDS and PACS

Keywords
PACS
  • 03.65.Ge
    Solutions of wave equations: bound states in quantum mechanics
  • 02.30.Hq
    Ordinary differential equations
  • 02.30.Uu
    Integral transforms
  • 02.30.Nw
    Fourier analysis
  • YEAR: 2008

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ISSN:
0022-2488 (print)   1089-7658 (online)
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