2025/09/01 by Masahiro Kaminaga, Kaminaga, Masahiro
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Random Matrices and Applications #Spectral Theory in Mathematical Physics #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.2509.01225
25 pages, no figure
openalex publication_date 2025/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29 · arxiv created 2026/07/31 · arxiv updated 2026/08/03
We study an elliptic transmission problem associated with a Stark operator and a δ interaction on a compact Lipschitz interface in \mathbb Rd. The background differential expression contains the unbounded coefficient -Fx1, and the interface strength is an arbitrary real function α∈ L^∞(Σ). We introduce a transmission class with piecewise H1 regularity near the interface and an L2 action away from it. This setting gives Dirichlet traces in H1/2(Σ) and weak normal derivatives in H-1/2(Σ) without assuming smoothness of Σ. We prove that the transmission conditions define a self-adjoint realization of the formal operator HF,0+αδΣ. We also obtain a boundary resolvent formula in terms of the free Stark resolvent and a bounded operator from H-1/2(Σ) to H1/2(Σ). The formula implies that the resolvent difference is compact on L2(\mathbb Rd). Consequently, if F≠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.