Optimization of the Lowest Eigenvalue for Leaky Star Graphs

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schrödinger operator with an attractive -interaction of a xed strength the support of which is a star graph with nitely many edges of an equal length L 2 (0;1]. Under the constraint of xed number of the edges and xed length of them, we prove that the lowest eigenvalue is maximized by the fully symmetric star graph. The proof relies on the Birman-Schwinger principle, properties of the Macdonald function, and on a geometric inequality for polygons circumscribed into the unit circle.