Abstract
In this talk, we present recent new results in the field of quantum waveguides
- Schrödinger operators in curved regions - with 'soft' walls. We investigate the properties of their spectrum in two dimensions in the setting of the generalized bookcover shape, that is, Schrödinger operator with the potential in the form of a ditch consisting of a finite curved part and straight asymptotes which are parallel or almost parallel pointing in the same direction. We show how the eigenvalues accumulate when the angle between the asymptotes tends to zero. In case of parallel asymptotes the existence of a discrete spectrum depends on the ditch profile. We prove that it is absent in the weak-coupling case, on the other hand, it exists provided the transverse potential is strong enough. We also present a numerical example in which the critical strength can be assessed.