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.
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Sylwia Kondej, David Krejčiřík, Spectral Analysis of a Quantum System with a Double Line Singular Interaction. Publ. Res. Inst. Math. Sci. 49 (2013), no. 4, pp. 831–859DOI 10.4171/PRIMS/121