A Transmission Framework for Stark Operators with $\hat{I}´$ Interactions on a Compact Lipschitz Interface

Abstract

We study an elliptic transmission problem associated with a Stark operator and a $\hat{I}´$ interaction on a compact Lipschitz interface in ${\mathbb{R}}^d$. The background differential expression contains the unbounded coefficient $-F{x}_1$, and the interface strength is an arbitrary real function $\hat{I}\pm \in L^{\infty }(\hat{I}\pounds )$. We introduce a transmission class with piecewise $H^1$ regularity near the interface and an $L^2$ action away from it. This setting gives Dirichlet traces in $H^{1/2}(\hat{I}\pounds )$ and weak normal derivatives in $H^{-1/2}(\hat{I}\pounds )$ without assuming smoothness of $\hat{I}\pounds$. We prove that the transmission conditions define a self-adjoint realization of the formal operator $H_{F,0}+\hat{I}\pm \hat{I}{´}_{\hat{I}}\pounds$. We also obtain a boundary resolvent formula in terms of the free Stark resolvent and a bounded operator from $H^{-1/2}(\hat{I}\pounds )$ to $H^{1/2}(\hat{I}\pounds )$. The formula implies that the resolvent difference is compact on $L^2({\mathbb{R}}^d)$. Consequently, if $F\ne 0$, the essential spectrum of the interacting operator is $\mathbb{R}$. The result supplies a direct boundary reduction for an interface problem whose background potential is not bounded and is not translation invariant.

25 pages, no figure