Geometrically Induced Spectrum of the Schr?dinger Operator

Abstract

New and recent results concerning the spectrum of the operator with a boundary condition on a given curve in the Ÿ(x,y) - plane are presented. The geometry of the curve influences the nature of the spectrum, and the situations discussed are (i) there is a spectral interval [-c^ 2 /4, ) , (ii) there are discrete eigenvalues in[- ,-c ^2 /4) . The aim is to present material which avoids some of the technicalities in the literature.