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Spectra of self-adjoint extensions and applications to solvable Schrödinger operators. (English) Zbl 1163.81007
The paper aims at providing a self-contained presentation of the theory of self-adjoint extensions by means of so-called boundary triples. The departure point is the Krein resolvent formula and its link with the technique of boundary triples. An abstract machinery of boundary value problems is tailored to give an insight into some models of modern mathematical physics: those involving singular perturbations, point perturbations, hybrid spaces and quantum graphs. Specifically, new results are presented on properties of spectra of equilateral quantum graphs and arrays of quantum dots, where the complete dimension reduction is performed. The study of isolated eigenvalues of self-adjoint extensions generalizes previously known results to the case of operators with infinite deficiency indices.

MSC:
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics
47B25 Linear symmetric and selfadjoint operators (unbounded)
47A10 Spectrum, resolvent
46N50 Applications of functional analysis in quantum physics
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