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

• URL: http://arxiv.org/abs/2207.10054v1
• Date: Wed, 20 Jul 2022 17:34:05 GMT
• Title: Existence of the transfer matrix for a class of nonlocal potentials in two dimensions
• Authors: Farhang Loran and Ali Mostafazadeh
• Abstract summary: 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, $(-nabla2+v)psi=k2psi$ for a local potential $v$, they arise in dimensions greater than one and are present regardless of the details of $v$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\mathbb{C}^2\otimes L^2(-k,k)$.

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