Isoperimetric problems for singular interactions in the plane

Pavel Exner Czech Academy of Sciences, Czech Republic

We consider isoperimetric problems motivated by the theory of leaky quantum wires. Two examples are discussed. In the first one 2D point interaction of the same coupling are placed at vertices of an equilateral N-polygon and we ask about the configuration which maximized the ground-state energy of the corresponding Hamiltonian. It is shown a regular polygon is a sharp local maximizer, the question about its global uniqueness is reduced to an interesting geometrical problem. In the second example we prove similar results for the operator given formally as −Δ − αδ(x − Γ) supported by closed loops Γ of a fixed length in the plane.