Lower bound on the spectrum of the Schrödinger operator in the plane with delta-potential supported by a curve

Details Lower bound on the spectrum of the Schrödinger operator in the plane with delta-potential supported by a curve Lower bound on the spectrum of the Schrödinger operator in the plane with delta-potential supported by a curve

2009-07-09

arxiv.org/abs/0907.1543v1

Theor. Math. Phys. 162 No. 3 (2010) 397--407

Mathematics

Spectral Theory

Scientific paper

We consider the Schr\"odinger operator in the plane with delta-potential supported by a curve. For the cases of an infinite curve and a finite loop we give estimates on the lower bound of the spectrum expressed explicitly through the strength of the interaction and a parameter which characterizes geometry of the curve. Going further we cut the curve into finite number of pieces and estimate the bottom of the spectrum using the parameters for the pieces. As an application of the elaborated theory we consider a curve with a finite number of cusps and "leaky" quantum graphs.

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