Existence of the transfer matrix for a class of nonlocal potentials in two dimensions

Abstract

Evanescent waves are waves that decay or grow exponentially in regions of the space void of interaction. In potential scattering defined by the Schr\"odinger equation, $(-\nabla^2+v)\psi=k^2\psi$ for a local potential $v$, they arise in dimensions greater than one and are present regardless of the details of $v$. The approximation in which one ignores the contributions of the evanescent waves to the scattering process corresponds to replacing $v$ with a certain energy-dependent nonlocal potential $\hat{\mathscr{V}}_k$. We present a dynamical formulation of the stationary scattering for $\hat{\mathscr{V}}_k$ in two dimensions, where the scattering data are related to the dynamics of a quantum system having a non-self-adjoint, unbounded, and nonstationary Hamiltonian operator. The evolution operator for this system determines a two-dimensional analog of the transfer matrix of stationary scattering in one dimension which contains the information about the scattering properties of the potential. Under rather general conditions on $v$, we establish the strong convergence of the Dyson series expansion of the evolution operator and prove the existence of the transfer matrix for $\hat{\mathscr{V}}_k$ as a densely-defined operator acting in $\mathbb{C}^2\otimes L^2(-k,k)$.