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Abstract: We discuss a new completely integrable case of the time-dependentSchroedinger equation in $R^n$ with variable coefficients for a modifiedoscillator, which is dual with respect to the time inversion to a model of thequantum oscillator recently considered by Meiler, Cordero-Soto, and Suslov. Asecond pair of dual Hamiltonians is also found in the momentum representation.Our examples show that in mathematical physics and quantum mechanics a changein the direction of time may require a total change of the system dynamics inorder to return the system back to its original quantum state. Particularsolutions of the corresponding Schroedinger equations are also obtained. AHamiltonian structure of the classical integrable problem and its quantizationare also discussed.

Author: Ricardo Cordero-Soto, Sergei K. Suslov


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