Geodesic
Time reversal for modified oscillators
Teoretičeskaâ i matematičeskaâ fizika, Tome 162 (2010) no. 3, pp. 345-380 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider a new completely integrable case of the time-dependent Schrödinger equation in $\mathbb R^n$ with variable coefficients for a modified oscillator that is dual (with respect to time reversal) to a model of the quantum oscillator. We find a second pair of dual Hamiltonians in the momentum representation. The examples considered show that in mathematical physics and quantum mechanics, a change in the time direction may require a total change of the system dynamics to return the system to its original quantum state. We obtain particular solutions of the corresponding nonlinear Schrödinger equations. We also consider a Hamiltonian structure of the classical integrable problem and its quantization.
Keywords: Cauchy initial value problem, Schrödinger equation with variable coefficients, Green's function, propagator, time reversal, hyperspherical harmonic, nonlinear Schrödinger equation. 
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R. Cordero-Soto; S. K. Suslov. Time reversal for modified oscillators. Teoretičeskaâ i matematičeskaâ fizika, Tome 162 (2010) no. 3, pp. 345-380. http://geodesic.mathdoc.fr/item/TMF_2010_162_3_a2/

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