Spectral isoperimetric inequality for the ${\delta }^{\prime }$-interaction on a contour

Abstract

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schrödinger operator with an attractive ${\delta }^{\prime }$-interaction of a fixed strength, the support of which is a $C^2$-smooth contour. Under the constraint of a fixed length of the contour, we prove that the lowest eigenvalue is maximized by the circle. The proof relies on the min-max principle and the method of parallel coordinates.

contribution to the proceedings of the 3rd workshop "Mathematical Challenges of Zero-Range Physics: rigorous results and open problems"