Geodesic
Lower bound on the~spectrum of the~two-dimensional Schr\"odinger operator with a~$\delta$-perturbation on a~curve
Teoretičeskaâ i matematičeskaâ fizika, Tome 162 (2010) no. 3, pp. 397-407

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We consider the two-dimensional Schrödinger operator with a $\delta$-potential supported by curve. For the cases of infinite and closed finite smooth curves, we obtain lower bounds on the spectrum of the considered operator that are expressed explicitly in terms of the interaction strength and a parameter characterizing the curve geometry. We estimate the bottom of the spectrum for a piecewise smooth curve using parameters characterizing the geometry of the separate pieces. As applications of the obtained results, we consider curves with a finite number of cusps and general “leaky” quantum graph as the support of the $\delta$-potential.
Keywords: Schrödinger operator, singular potential, spectral estimate, Birman–Schwinger transformation. 
@article{TMF_2010_162_3_a4,
     author = {I. S. Lobanov and V. Yu. Lotoreichik and I. Yu. Popov},
     title = {Lower bound on the~spectrum of the~two-dimensional {Schr\"odinger} operator with a~$\delta$-perturbation on a~curve},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {397--407},
     publisher = {mathdoc},
     volume = {162},
     number = {3},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2010_162_3_a4/}
}
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I. S. Lobanov; V. Yu. Lotoreichik; I. Yu. Popov. Lower bound on the~spectrum of the~two-dimensional Schr\"odinger operator with a~$\delta$-perturbation on a~curve. Teoretičeskaâ i matematičeskaâ fizika, Tome 162 (2010) no. 3, pp. 397-407. http://geodesic.mathdoc.fr/item/TMF_2010_162_3_a4/