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.

We consider a simple model which is a caricature of a crystal interacting with a radiation field. The model has two bands of continuous spectrum and the particle can pass from the upper one to the lower by radiating a photon, the coupling between the excited and deexcited states being of a Friedrichs type. Under suitable regularity and analyticity assumptions we find the continued resolvent and show that for weak enough coupling it has a curve-type singularity in the lower halfplane which is a deformation of the upper-band spectral cut. We then find a formula for the decay amplitude and show that for a fixed energy it is approximately exponential at intermediate times, while the tail has a power-like behaviour.

No abstract uploaded!