Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Abstract

The self-adjoint Schrödinger operator A δ,α with a δ-interaction of constant strength α supported on a compact smooth hypersurface C is viewed as a self-adjoint extension of a natural underlying symmetric operator S in L 2 (R n ). The aim of this note is to construct a boundary triple for S * and a self-adjoint parameter Θ δ,α in the boundary space L 2 (C) such that A δ,α corresponds to the boundary condition induced by Θ δ,α . As a consequence the well-developed theory of boundary triples and their Weyl functions can be applied. This leads, in particular, to a Krein-type resolvent formula and a description of the spectrum of A δ,α in terms of the Weyl function and Θ δ,α .

Resonance asymptotics for a pair quantum waveguides with common semitransparent perforated wall

“…The conditions of such type appear if one considers singular potential supported on hypersurface. These potentials are intensively investigated during last two decades (see, e.g., [16][17][18][19][20][21][22]). We construct asymptotics in window size, so one of the most important parameters is a -we will consider it as half of the window size.…”

On formal asymptotic expansion of resonance for quantum waveguide with perforated semitransparent barrier

“…The conditions of such type appear if one considers singular potential supported on hypersurface. These potentials have been intensively investigated during last two decades (see, e.g., [21][22][23][24][25]).…”

Construction of spectral decomposition for non-self-adjoint friedrichs model operator

“…The classical definition of the Weyl function is considered in [7] with the help of the boundary triple for the Schr dinger operator, and in [8][9][10] Weyl theory was investigated for a self-directed Schr dinger operator. The establishment of conditions for the existence of the Weyl function and the resolvent structure was investigated without involving the operator spectral decomposition.…”