On the spectral theory of Schrodinger and Dirac operators with point interactions and quantum graphs

Abstract: Modern concepts of extension theory of symmetric operators, including concepts of boundary triples, corresponding Weyl functions, and boundary operators will be discussed. Applications to Schrodinger and Dirac operators with point interactions, as well as to quantum graphs, will be demonstrated. It turns out that certain spectral properties of each of these operators (deficiency indices, selfadjointness, discreteness, lower semiboundedness, etc) strictly correlate to that of a special difference operator. In the first two cases the corresponding difference operator is generated by a special Jacobi matrix. This matrix appears as a boundary operator of the corresponding Schrodinger and Dirac realization in an appropriate boundary triple. In the case of quantum graphs a similar role is played by a certain weighted discrete Laplacian on the underlying discrete graph, which also appears as a boundary operator.

mathematical physics

Audience: researchers in the discipline

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