Publications of the Research Institute for Mathematical Sciences


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Volume 49, Issue 4, 2013, pp. 831–859
DOI: 10.4171/PRIMS/121

Spectral Analysis of a Quantum System with a Double Line Singular Interaction

Sylwia Kondej[1] and David Krejčiřík[2]

(1) Institute of Physics, University of Zielona Góra, ul. Szafrana 4a, 65-246, Zielona Gora, Poland
(2) Department of Mathematics, Czech Technical University in Prague, Trojanova 13, 120 00, Prague 2, Czechia

We consider a non-relativistic quantum particle interacting with a singular potential supported by two parallel straight lines in the plane. We locate the essential spectrum under the hypothesis that the interaction asymptotically approaches a constant value, and nd conditions which guarantee either the existence of discrete eigenvalues or Hardy-type inequalities. For a class of our models admitting mirror symmetry, we also establish the existence of embedded eigenvalues and show that they turn into resonances after introducing a small perturbation.

Keywords: Schrödinger operator, singular perturbation, spectral analysis, Hardy inequality, resonance

Kondej Sylwia, Krejčiřík David: Spectral Analysis of a Quantum System with a Double Line Singular Interaction. Publ. Res. Inst. Math. Sci. 49 (2013), 831-859. doi: 10.4171/PRIMS/121