Bound States in Bent Soft Waveguides

Abstract

The aim of this paper is to show that a two-dimensional Schrödinger operator with the potential in the form of a `ditch' of a fixed profile can have a geometrically induced discrete spectrum; this happens if such a potential channel has a single or multiple bends being straight outside a compact. Moreover, under stronger geometric restrictions the claim remains true in the presence of a potential bias at one of the channel `banks'.

Revised version, to appear in J. Spectral Theory; 29 pages, four figures, some parts of the argument made more detailed