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F. Trèves (1980)
Introduction to Pseudodifferential and Fourier Integral Operators
Differential Equations, Vol. 39, No. 1, 2003, pp. 73–82. Translated from Differentsial'nye Uravneniya, Vol. 39, No. 1, 2003, pp. 70–77. Original Russian Text Copyright c 2003 by Korovina. PARTIAL DIFFERENTIAL EQUATIONS Construction of a Self-Adjoint Extension of the Schr odinger Operator with a Potential Concentrated on a Pencil of Planes: II M. V. Korovina Moscow State University, Moscow, Russia Received May 15, 2001 In the present paper, we continue the research [1] and consider Schr odinger equations with a potential concentrated on a pencil of planes whose intersection has a nonzero dimension. In other words, the problem is to construct self-adjoint extensions of the corresponding operator, describe them in terms of boundary conditions posed on each plane, and analyze the semiboundedness of these extensions. Such a model arises, for example, in the analysis of many-particle problems in quantum mechanics. We consider local boundary conditions, i.e., conditions of Skornyakov{Ter- Martirosyan type [2], and some generalizations, since they provide pair interaction of particles and hence are most natural from the physical viewpoint. We shall show that there are no self- adjoint extensions corresponding to Skornyakov{Ter-Martirosyan conditions in the entire L (R ); therefore, just as in [3], we have to choose
Differential Equations – Springer Journals
Published: Oct 5, 2004
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