@article{6f029f9cd9f542689b4a48d624a4c2af,

title = "Solution of the Cauchy problem for a time-dependent Schr{\"o}dinger equation",

abstract = "We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schr{\"o}dinger 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.",

author = "Maria Meiler and Ricardo Cordero-Soto and Sergei Suslov",

note = "Funding Information: This paper is written as a part of the summer 2007 program on analysis of the Mathematical and Theoretical Biology Institute (MTBI) at Arizona State University. The MTBI/SUMS undergraduate research program is supported by the National Science Foundation (Grant No. DMS–0502349), the National Security Agency (Grant No. DOD–H982300710096), the Sloan Foundation, and the Arizona State University. The authors are grateful to Professor Carlos Castillo-Ch{\'a}vez for support and Ref. . We thank Professor Dick Askey, Professor Slim Ibrahim, Professor Erik Koelink, Professor Hank Kuiper, Professor Svetlana Roudenko, and Professor Andreas Ruffing for valuable comments. One of us (Maria Meiler) thanks MTBI/SUMS and the Department of Mathematics and Statistics at Arizona State University for hospitality. The authors are grateful to the referee for valuable suggestions. We dedicate this paper to the 90th birthday of Professor Ya. A. Smorodinskiĭ.",

year = "2008",

doi = "10.1063/1.2938698",

language = "English (US)",

volume = "49",

journal = "Journal of Mathematical Physics",

issn = "0022-2488",

publisher = "American Institute of Physics Publising LLC",

number = "7",

}