We consider Schrdinger operators in L2(R3) with a singular interaction supported by a finite curve . We present a proper definition of the operators and study their properties, in particular, we show that the discrete spectrum can be empty if is short enough. If it is not the case, we investigate properties of the eigenvalues in the situation when the curve has a hiatus of length 2. We derive an asymptotic expansion with the leading term which a multiple of ln.

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We investigate the operator -\Delta-lpha\delta (x-\Gamma) in -\Delta-lpha\delta (x-\Gamma) , where -\Delta-lpha\delta (x-\Gamma) is a smooth surface which is either compact or periodic and satisfies suitable regularity requirements. We find an asymptotic expansion for the lower part of the spectrum as -\Delta-lpha\delta (x-\Gamma) which involves a``two-dimensional''comparison operator determined by the geometry of the surface -\Delta-lpha\delta (x-\Gamma) . In the compact case the asymptotics concerns negative eigenvalues, in the periodic case Floquet eigenvalues. We also give a bandwidth estimate in the case when a periodic -\Delta-lpha\delta (x-\Gamma) decomposes into compact connected components. Finally, we comment on analogous systems of lower dimension and other aspects of the problem.