Lower bound on the spectrum of the two-dimensional Schrödinger operator with a $\delta$-perturbation on a curve

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.