arXiv subjects

Lukáš Heriban

Publications and source records attributed to Lukáš Heriban.

3 recordsLinked to original sources

arXiv2025

Two-dimensional Schrödinger operators with non-local singular potentials

In this paper we introduce and study a family of self-adjoint realizations of the Laplacian in $L^2(\mathbb{R}^2)$ 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 $L^2(Σ;\mathbb{C}^2)$. Whereas for all choices of parameters the essential spectrum is stable and equal to $[0, +\infty)$, 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 $-\infty$. 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].

math.SP↗

arXiv2023

Non-local relativistic $δ$-shell interactions

In this paper, new self-adjoint realizations of the Dirac operator in dimension two and three are introduced. It is shown that they may be associated with the formal expression $\mathcal{D}_0+|Fδ_Σ\rangle\langle Gδ_Σ|$, where $\mathcal{D}_0$ is the free Dirac operator, $F$ and $G$ are matrix valued coefficients, and $δ_Σ$ stands for the single layer distribution supported on a hypersurface $Σ$, and that they can be understood as limits of the Dirac operators with scaled non-local potentials. Furthermore, their spectral properties are analysed.

math-ph↗

arXiv2022

Non-self-adjoint relativistic point interaction in one dimension

The one-dimensional Dirac operator with a singular interaction term which is formally given by $A\otimes|δ_0\rangle\langleδ_0|$, where $A$ is an arbitrary $2\times 2$ matrix and $δ_0$ stands for the Dirac distribution, is introduced as a closed not necessarily self-adjoint operator. We study its spectral properties, find its non-relativistic limit and also address the question of regular approximations. In particular, we show that, contrary to the case of local approximations, for non-local approximating potentials, coupling constants are not renormalized in the limit.

math-ph↗