We consider a Hamiltonian with N point interactions in {\bb R}^ d,\: d= 2, 3, all with the same coupling constant, placed at vertices of an equilateral polygon {\cal P} _N. It is shown that the ground-state energy is locally maximized by a regular polygon. The question whether the maximum is global is reduced to an interesting geometric problem.

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

We consider a pair of straight adjacent quantum waveguides of constant, and in general different widths. These waveguides are coupled laterally by a pair of windows in the common boundary, not necessarily of the same length, at a distance 2l. The Hamiltonian is the respective Dirichlet Laplacian. We analyze the asymptotic behavior of the discrete spectrum as the window distance tends to infinity for the generic case, i.e., for eigenvalues of the corresponding one-window problems separated from the threshold.