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Solution of the Cauchy problem for a time-dependent Schrödinger equation. (English) Zbl 1152.81557

Summary: 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 \(\mathrm{SU}(1,1)\) 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.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q40 PDEs in connection with quantum mechanics
35J10 Schrödinger operator, Schrödinger equation
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C55 Spherical harmonics
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