A magnetic version of the Smilansky–Solomyak model. (8th November 2017) - Barseghyan, Diana; Exner, Pavel
Abstract: We analyze spectral properties of two mutually related families of magnetic Schrödinger operators, H S m ( A ) = ( i ∇ + A ) 2 + ω 2 y 2 + λ y δ ( x ) and H ( A ) = ( i ∇ + A ) 2 + ω 2 y 2 + λ y 2 V ( x y ) in L 2 ( R 2 ), with the parameters ω > 0 and λ < 0, where A is a vector potential corresponding to a homogeneous magnetic field perpendicular to the plane and V is a regular nonnegative and compactly supported potential. We show that the spectral properties of the operators depend crucially on the one-dimensional Schrödinger operators L = − d 2 d x 2 + ω 2 + λ δ ( x ) and L ( V ) = − d 2 d x 2 + ω 2 + λ V ( x ), respectively. Depending on whether the operators L and L ( V ) are positive or not, the spectrum of H S m ( A ) and H ( V ) exhibits a sharp transition.