Pavel Exner: Relations between geometry and the principal eigenvalue in some contact-interaction models

In this talk I am going to present several results on relations between the principal eigenvalue of various "atractive" contact-interaction Hamiltonians and the underlying geometry. The first question concerns polymer rings, that is, point interactions on a loop with an upper bound to distance to the neighbours; it is shown that the ground-state energy is minimized by a regular polygon. Then we will consider a point interaction in a region in the Euclidean space of dimension d=2,3, and derive a condition under which the eigenvalue increases as the interaction site moves. Finally, we will discuss behaviour of the principle eigenvalue w.r.t. increasing distances between the interaction sites both in Euclidean space of dimension d=1,2,3, and on quantum graphs as well as their "continuous" analogues.