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Short-range potential and a model of operator extension theory for resonators with semitransparent boundary. (English. Russian original) Zbl 0952.35110
Math. Notes 65, No. 5, 590-597 (1999); translation from Mat. Zametki 65, No. 5, 703-711 (1999).
Let \(\Omega ^{\text{in}}\) be a domain in \(\mathbb{R}^{3}\) (or \(\mathbb{R}^{2}\)) with a smooth boundary \(\Gamma \), let \(\Omega ^{\text{ex}}:=\mathbb{R}^{3}\setminus \overline{ \Omega ^{\text{in}}}\) (or \(\Omega ^{\text{ex}}:=\mathbb{R}^{2}\setminus \overline{\Omega ^{\text{in}}}\)). Let \(\Delta _{0}^{\text{ex}}\), \(\Delta _{0}^{\text{ex}}\) denote the Laplace operators in \(\Omega ^{\text{in}}\), \(\Omega ^{\text{ex}}\), respectively, defined on the sets of functions vanishing on \(\Gamma \). The operator \(-\Delta _{0}:=- \overline{( \Delta _{0}^{\text{in}}\oplus \Delta _{0}^{\text{ex}}) }\) is symmetric and has deficiency indices \(\left( \infty ,\infty \right) \). There are considered its simplest selfadjoint extensions that are determined by the finite-dimensional constraints between the limit values of both the functions and their normal derivatives on both sides of the boundary. The precise meaning of that is given in Theorem 1. Theorem 2 says that the resolvent R(\(\lambda \)) of a fixed extension with an appropriate domain is the limit, as \(\varepsilon \rightarrow 0\), of the resolvents R\(_{\varepsilon }\)(\( \lambda \)) of the disturbances H\(_{\varepsilon }\) of the Laplace operator with domain H\(^{2}\), the Sobolev space. The limit is taken in the Banach space of bounded linear operators acting from \(L_{2}\) into the Sobolev space \(H^{1}\). All this is interpreted in terms of the propagation of an electronic wave in quantum wave guide as well as in those of the transport properties of two-barrier structures.
MSC:
35Q40 PDEs in connection with quantum mechanics
81V10 Electromagnetic interaction; quantum electrodynamics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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