In this note we investigate the asymptotic behavior of the s-numbers of the resolvent difference of two generalized self-adjoint, maximal dissipative or maximal accumulative Robin Laplacians on a bounded domain Ω with smooth boundary ∂Ω. For this we apply the recently introduced abstract notion of quasi boundary triples and Weyl functions from extension theory of symmetric operators together with Krein type resolvent formulae and well-known eigenvalue asymptotics of the Laplace-Beltrami operator on ∂Ω. It is shown that the resolvent difference of two generalized Robin Laplacians belongs to the Schattenvon Neumann class of any order p for whichMoreover, we also give a simple sufficient condition for the resolvent difference of two generalized Robin Laplacians to belong to a Schatten-von Neumann class of arbitrary small order. Our results extend and complement classical theorems due to M.Š. Birman on Schatten-von Neumann properties of the resolvent differences of Dirichlet, Neumann and self-adjoint Robin Laplacians.

We consider symmetric operators of the form S := A ⊗ I T + I H ⊗ T where A is symmetric and T = T * is (in general) unbounded. Such operators naturally arise in problems of simulating point contacts to reservoirs. We construct a boundary triplet ΠS for S * preserving the tensor structure. The corresponding γ-field and Weyl function are expressed by means of the γ-field and Weyl function corresponding to the boundary triplet ΠA for A * and the spectral measure of T . Applications to 1-D Schrödinger and Dirac operators are given. A model of electron transport through a quantum dot assisted by cavity photons is proposed. In this model the boundary operator is chosen to be the well-known Jaynes-Cumming operator which is regarded as the Hamiltonian of the quantum dot.Mathematics Subject Classification: 47A80, 47B25, 81Q05, 81Q37

The model of zero width slits based on the operator’s extension theory in an indefinite metric space for the wave scattering by a resonator with a narrow slit is constructed.

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The asymptotics (in the width of windows) of eigenvalues and bands for two-dimensional waveguides and three-dimensional layers coupled through small windows is obtained. The technique is matching of asymptotic expansions of the solutions of boundary value problems.