Schroedinger operators with singular interactions: a model of tunneling resonances

Details Schroedinger operators with singular interactions: a model of tunneling resonances Schroedinger operators with singular interactions: a model of tunneling resonances

2003-12-22

arxiv.org/abs/math-ph/0312055v1

J. Phys. A37 (2004), 8255-8277

Physics

Mathematical Physics

LaTeX2e, 34 pages

Scientific paper

10.1088/0305-4470/37/34/005

We discuss a generalized Schr\"odinger operator in $L^2(\mathbb{R}^d), d=2,3$, with an attractive singular interaction supported by a $(d-1)$-dimensional hyperplane and a finite family of points. It can be regarded as a model of a leaky quantum wire and a family of quantum dots if $d=2$, or surface waves in presence of a finite number of impurities if $d=3$. We analyze the discrete spectrum, and furthermore, we show that the resonance problem in this setting can be explicitly solved; by Birman-Schwinger method it is cast into a form similar to the Friedrichs model.

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