A new class of solvable models in quantum mechanics describing point interactions on the line

Abstract

The authors provide a detailed analysis of properties of the Schrodinger operator in L2(R) which formally can be written as -d2/dx2+ Sigma y in gamma nu y delta '(.-y) where delta ' is the derivative of Dirac's delta -function and Y contained in/implied by R is discrete. This model allows for an explicit calculation of spectral properties. Special emphasis is given to the periodic case Y=Z, nu = nu j, j in Z where the spectrum and the density of states are explicitly computed. Also the spectrum for a half-crystal is given. The authors study in detail spectral consequences when various defects and impurities are added to the periodic case.