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arXiv · 2509.01225

A Transmission Framework for Stark Operators with $\delta$ Interactions on a Compact Lipschitz Interface

Masahiro Kaminaga
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Abstract

We study an elliptic transmission problem associated with a Stark operator and a $\delta$ interaction on a compact Lipschitz interface in $\mathbb R^d$. The background differential expression contains the unbounded coefficient $-Fx_1$, and the interface strength is an arbitrary real function $\alpha\in L^\infty(\Sigma)$. 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 $H^{1/2}(\Sigma)$ and weak normal derivatives in $H^{-1/2}(\Sigma)$ without assuming smoothness of $\Sigma$. We prove that the transmission conditions define a self-adjoint realization of the formal operator $H_{F,0}+\alpha\delta_\Sigma$. We also obtain a boundary resolvent formula in terms of the free Stark resolvent and a bounded operator from $H^{-1/2}(\Sigma)$ to $H^{1/2}(\Sigma)$. The formula implies that the resolvent difference is compact on $L^2(\mathbb R^d)$. Consequently, if $F\ne0$, 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.

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Masahiro Kaminaga. 2025-09-01. A Transmission Framework for Stark Operators with $\delta$ Interactions on a Compact Lipschitz Interface. https://arxiv.org/abs/2509.01225

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