Resolvents of self-adjoint extensions with mixed boundary conditions. (English) Zbl 1143.47017

Let \(H\) be a symmetric operator with equal deficiency numbers, \((\mathcal H,\Gamma_1,\Gamma_2)\) be its space of boundary values (boundary triplet). An abstract version of the Krein resolvent formula gives an expression for the resolvent of any selfadjoint extension \(\widetilde H\) of \(H\) in terms of a selfadjoint linear relation in \(\mathcal H\) corresponding to \(\widetilde H\).
The author considers the description of \(\widetilde H\) by the boundary conditions of the form \(A\Gamma_1u=B\Gamma_2u\), where \(A,B\) are bounded linear operators, \(AB^*=BA^*\) (earlier, mostly some special “canonical” forms of such abstract boundary conditions were used). For such a description, an explicit form of the resolvent formula is obtained, enabling a spectral analysis of selfadjoint extensions. As examples, the author considers point interactions in one dimension, Schrödinger operators on spaces consisting of pieces with different dimensions, and the Laplacian on a half-plane.


47B25 Linear symmetric and selfadjoint operators (unbounded)
35P05 General topics in linear spectral theory for PDEs
47E05 General theory of ordinary differential operators
47N50 Applications of operator theory in the physical sciences
47F05 General theory of partial differential operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
Full Text: DOI arXiv


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