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Solution of the Cauchy problem for a timedependent Schrödinger equation
 J. Math. Phys
"... Abstract. We construct an explicit solution of the Cauchy initial value problem for the ndimensional Schrödinger equation with certain timedependent Hamiltonian operator of a modified oscillator. The dynamical SU (1, 1) symmetry of the harmonic oscillator wave functions, Bargmann’s functions for t ..."
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Abstract. We construct an explicit solution of the Cauchy initial value problem for the ndimensional Schrödinger equation with certain timedependent 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 Hankeltype 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 a byproduct. 1.
The Cauchy problem for a forced harmonic oscillator
 Rev. Mex. Fis. E
"... Abstract. We construct an explicit solution of the Cauchy initial value problem for the onedimensional Schrödinger equation with timedependent Hamiltonian operator for the forced harmonic oscillator. The corresponding Green function (propagator) is derived with the help of the generalized Fourier t ..."
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Cited by 3 (1 self)
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Abstract. We construct an explicit solution of the Cauchy initial value problem for the onedimensional Schrödinger equation with timedependent Hamiltonian operator for the forced harmonic oscillator. The corresponding Green function (propagator) is derived with the help of the generalized Fourier transform and a relation with representations of the Heisenberg–Weyl group N (3) in a certain special case first and then is extended to the general case. A three parameter extension of the classical Fourier integral is discussed as a byproduct. In addition, we also solve an initial value problem to a similar diffusiontype equation. 1.
The time inversion for modified oscillators
"... Abstract. We discuss a new completely integrable case of the timedependent Schrödinger equation with variable coefficients for a modified oscillator, which is dual with respect to the time inversion to a model of the quantum oscillator recently considered by Meiler, CorderoSoto, and Suslov. A seco ..."
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Abstract. We discuss a new completely integrable case of the timedependent Schrödinger equation with variable coefficients for a modified oscillator, which is dual with respect to the time inversion to a model of the quantum oscillator recently considered by Meiler, CorderoSoto, and Suslov. A second pair of dual Hamiltonians is also found. Our examples show that in mathematical physics and quantum mechanics a change in the direction of time may require a total change of the system dynamics in order to return the system back to its original quantum state. Particular solutions of the corresponding nonlinear Schrödinger equations are also obtained. 1.