# Relations between geometry and the principal eigenvalue in some point-interaction models

Seminário do GFM

IIIUL, B1-01

2012-09-06 15:00
2012-09-06 16:00
2012-09-06
15:00
..
16:00

Pavel Exner (Doppler Institute for Mathematical Physics and Applied Mathematics, Praga, República Checa)

In this talk I am going to present several results on relations between the principal eigenvalue of various "atractive" point-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 Omega–in R^d (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 Rd (d = 1, 2, 3) and on quantum graphs as well as their "continuous" analogues.

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 Rd (d = 1, 2, 3) and on quantum graphs as well as their "continuous" analogues.