99-279 G.F. Dell'Antonio, R. Figari, A. Teta

Schr\"{o}dinger Equation with Moving Point Interactions in Three Dimensions. (36K, LaTex) Jul 27, 99

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Abstract. We consider the motion of a non relativistic quantum particle in $R^{3}$ subject to $n$ point interactions which are moving on given smooth trajectories. Due to the singular character of the time-dependent interaction, the corresponding Schr\"{o}dinger equation does not have solutions in a strong sense and, moreover, standard perturbation techniques cannot be used. Here we prove that, for smooth initial data, there is a unique weak solution by reducing the problem to the solution of a Volterra integral equation involving only the time variable. It is also shown that the evolution operator uniquely extends to a unitary operator in $L^{2}(R^{3})$.

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