In this paper, we analyze a free quantum particle in a straight Dirichlet waveguide which has at its axis two Dirichlet barriers of lengths separated by a window of length 2a. It is known that if the barriers are semi-infinite, i.e., we have two adjacent waveguides coupled laterally through the boundary window, the system has for any a > 0 a finite number of eigenvalues below the essential spectrum threshold. Here, we demonstrate that for large but finite the system has resonances which converge to the said eigenvalues as , and derive the leading term in the corresponding asymptotic expansion.

We investigate a class of generalized Schrdinger operators in L²(³) with a singular interaction supported by a smooth curve . We find a strong-coupling asymptotic expansion of the discrete spectrum in the case when is a loop or an infinite bent curve which is asymptotically straight. It is given in terms of an auxiliary one-dimensional Schrdinger operator with a potential determined by the curvature of . In the same way, we obtain asymptotics of spectral bands for a periodic curve. In particular, the spectrum is shown to have open gaps in this case if is not a straight line and the singular interaction is strong enough.

We discuss discuss spectral and scattering properties of a particle confined to a straight Dirichlet tube in R''with a family of point interactions.

Given a 3-holed sphere decomposition of an orientable closed surface, it is shown that each orientation preserving homeomorphism of the surface is isotopic to a composition AB where A is a product of positive Dehn twists and B is a product of negative Dehn twists on the decomposition curves.

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