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Journal of Physics A Mathematical and General

New classes of analytic solutions of the three-state problem

Artur Mikhailovich Ishkhanyan

Published Jul 7, 2000• 8 citations
Publisher

Abstract

A rich multiparametric class of solutions of the time-dependent Schrödinger equations, including many integrable cases known at present of the three-state problem, is derived. In addition, the class contains an infinite number of new multiparametric integrable subfamilies. Examples of such four-parametric new subfamilies of asymmetric, in general, amplitude and frequency modulation functions permitting solution of the three-state problem in terms of the Clausen function are presented.

DOI
10.1088/0305-4470/33/28/308

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A rich multiparametric class of solutions of the time-dependent Schrödinger equations, including many integrable cases known at present of the three-state problem, is derived. In addition, the class contains an infinite number of new multiparametric integrable subfamilies. Examples of such four-parametric new subfamilies of asymmetric, in general, amplitude and frequency modulation functions permitting solution of the three-state problem in terms of the Clausen function are presented.
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We solve the Schrodinger equation analytically for systems whose potentials have a certain time-dependence (which is monotonously decreasing) and general coordinate-dependences.Only a few time-dependent systems have been reported to be analytically solved whose potentials are constant, linear, and quadratic functions of coordinate with arbitrary time-dependences.From a different perspective, we focus on the time-dependent systems whose potentials are monotonously decreasing functions of time with arbitrary coordinate dependences.Time-dependent potential of any coordinate-dependence can be handled analytically by transforming it to a time-independent potential of known solutions if its time-dependence is monotonously decreasing.We do this by a unitary transformation of the wavefunction and variable transformations to change the Schrodinger equation to be time-independent in new variables.These variables are then determined by solving a set of simple differential equations.This way we are able to find and to obtain analytical solutions for time-dependent potentials which we mention above.