2024/10/14 by Lukáš Heriban, Heriban, Lukáš, Markus Holzmann +7
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2410.10448
openalex publication_date 2024/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we introduce and study a family of self-adjoint realizations of the Laplacian in L2(ℝ2) with a new type of transmission conditions along a closed bi-Lipschitz curve Σ. These conditions incorporate jumps in the Dirichlet traces both of the functions in the operator domains and of their Wirtinger derivatives and are non-local. Constructing a convenient generalized boundary triple, they may be parametrized by all compact hermitian operators in L2(Σ;ℂ2). Whereas for all choices of parameters the essential spectrum is stable and equal to [0, +∞), the discrete spectrum exhibits diverse behaviour. While in many cases it is finite, we will describe also a class of parameters for which the discrete spectrum is infinite and accumulates at -∞. The latter class contains a non-local version of the oblique transmission conditions. Finally, we will connect the current model to its relativistic counterpart studied recently in [L. Heriban, M. Tušek: Non-local relativistic δ-shell interactions].