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Set-valued mappings specified by regularization of the Schrödinger equation with degeneration. (Russian, English) Zbl 1210.81029
Zh. Vychisl. Mat. Mat. Fiz. 46, No. 4, 683-699 (2006); translation in Comput. Math. Math. Phys. 46, No. 4, 651-665 (2006).
Summary: The Cauchy problem for the Schrödinger equation with an operator degenerating on a half-line and a family of regularized Cauchy problems with uniformly elliptic operators, whose solutions approximate the solution to the degenerate problem, are considered. A set-valued mapping is investigated that takes a bounded operator to a set of partial limits of values of its quadratic form on solutions of the regularized problems when the regularization parameter tends to zero. The dynamics of quantum states are determined by applying an averaging procedure to the set-valued mapping.

MSC:
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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