arXiv subjects

Matěj Tušek

Publications and source records attributed to Matěj Tušek.

15 recordsLinked to original sources

arXiv2026

Continuum limit of discretized matrix-valued Fourier multipliers

Building upon a recent result by H. Cornean, H. Garde, and A. Jensen concerning continuum limits of discrete Dirac operators, we extend the analysis to a wide class of block operator matrices. This class includes, among others, the bilayer graphene Hamiltonian. Our main goal is to find norm estimates for the difference between the resolvents of continuous operators and their discrete counterparts embedded in the continuum in a specific way. While some discretization schemes lead directly to convergence in the generalized norm resolvent sense as the mesh parameter tends to zero, others require the addition of a suitable correction term to ensure the convergence.

math-ph↗

arXiv2025

Non-self-adjoint Dirac operators on graphs

In this paper we introduce and study generally non-self-adjoint realizations of the Dirac operator on an arbitrary finite metric graph. Employing the robust boundary triple framework, we derive, in particular, a variant of the Birman Schwinger principle for its eigenvalues, and with an example of a star shaped graph we show that the point spectrum may exhibit diverse behaviour. Subsequently, we find sufficient and necessary conditions on transmission conditions at the graph's vertices under which the Dirac operator on the graph is symmetric with respect to the parity, the time reversal, or the charge conjugation transformation.

math-ph↗

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

On two-dimensional Dirac operators with $δ$-shell interactions supported on unbounded curves with straight ends

In this paper we study the self-adjointness and spectral properties of two-dimensional Dirac operators with electrostatic, Lorentz scalar, and anomalous magnetic $δ$-shell interactions with constant weights that are supported on a smooth unbounded curve that is straight outside a compact set and whose ends are rays that are not parallel to each other. For all possible combinations of interaction strengths we describe the self-adjoint realizations and compute their essential spectra. Moreover, we prove in different situations the existence of geometrically induced discrete eigenvalues.

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

Two-dimensional Dirac operators with general $δ$-shell interactions supported on a straight line

In this paper the two-dimensional Dirac operator with a general hermitian $δ$-shell interaction supported on a straight line is introduced as a self-adjoint operator and its spectral properties are investigated in detail. In particular, it is demonstrated that the singularly continuous spectrum is always empty and that by switching a certain $δ$-shell interaction on, it is possible to generate an eigenvalue in the gap of the spectrum of the free operator or to partially or even fully close the gap. This suggests that the studied operators may serve as interesting continuum toy-models for Dirac materials. Finally, approximations by Dirac operators with regular potentials are presented.

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↗

arXiv2021

Spectral transition for Dirac operators with electrostatic $δ$-shell potentials supported on the straight line

In this note the two dimensional Dirac operator $A_η$ with an electrostatic $δ$-shell interaction of strength $η\in\mathbb R$ supported on a straight line is studied. We observe a spectral transition in the sense that for the critical interaction strengths $η=\pm 2$ the continuous spectrum of $A_η$ inside the spectral gap of the free Dirac operator $A_0$ collapses abruptly to a single point.

math.SP↗

arXiv2021

General $δ$-shell interactions for the two-dimensional Dirac operator: self-adjointness and approximation

In this work we consider the two-dimensional Dirac operator with general local singular interactions supported on a closed curve. A systematic study of the interaction is performed by decomposing it into a linear combination of four elementary interactions: electrostatic, Lorentz scalar, magnetic, and a fourth one which can be absorbed by using unitary transformations. We address the self-adjointness and the spectral description of the underlying Dirac operator, and moreover we describe its approximation by Dirac operators with regular potentials.

math.AP↗

arXiv2020

Spectral analysis of the multi-dimensional diffusion operator with random jumps from the boundary

We develop a Hilbert-space approach to the diffusion process of the Brownian motion in a bounded domain with random jumps from the boundary introduced by Ben-Ari and Pinsky in 2007. The generator of the process is introduced by a diffusion elliptic differential operator in the space of square-integrable functions, subject to non-self-adjoint and non-local boundary conditions expressed through a probability measure on the domain. We obtain an expression for the difference between the resolvent of the operator and that of its Dirichlet realization. We prove that the numerical range is the whole complex plane, despite the fact that the spectrum is purely discrete and is contained in a half-plane. Furthermore, for the class of absolutely continuous probability measures with square-integrable densities we characterise the adjoint operator and prove that the system of root vectors is complete. Finally, under certain assumptions on the densities, we obtain enclosures for the non-real spectrum and find a sufficient condition for the non-zero eigenvalue with the smallest real part to be real. The latter supports the conjecture of Ben-Ari and Pinsky that this eigenvalue is always real.

math.SP↗

arXiv2020

Approximation of one-dimensional relativistic point interactions by regular potentials revised

We show that the one-dimensional Dirac operator with quite general point interaction may be approximated in the norm resolvent sense by the Dirac operator with a scaled regular potential of the form $1/\varepsilon~h(x/\varepsilon)\otimes B$, where $B$ is a suitable $2\times 2$ matrix. Moreover, we prove that the limit does not depend on the particular choice of $h$ as long as it integrates to a constant value.

math-ph↗

arXiv2018

Location of hot spots in thin curved strips

The maxima and minima of Neumann eigenfunctions of thin tubular neighbourhoods of curves on surfaces are located in terms of the maxima and minima of Neumann eigenfunctions of the underlying curves. In particular, the hot spots conjecture for a new large class of domains (possibly non-convex and non-Euclidean) is proved.

math.AP↗

arXiv2018

A geometric Iwatsuka type effect in quantum layers

We study motion of a charged particle confined to Dirichlet layer of a fixed width placed into a homogeneous magnetic field. If the layer is planar and the field is perpendicular to it the spectrum consists of infinitely degenerate eigenvalues. We consider translationally invariant geometric perturbations and derive several sufficient conditions under which a magnetic transport is possible, that is, the spectrum, in its entirety or a part of it, becomes absolutely continuous.

math-ph↗

arXiv2018

Qualitative analysis of magnetic waveguides for two-dimensional Dirac fermions

We focus on the confinement of two-dimensional Dirac fermions within the waveguides created by realistic magnetic fields. Understanding of their band structure is of our main concern. We provide easily applicable criteria, mostly depending only on the asymptotic behavior of the magnetic field, that can guarantee existence or absence of the energy bands and provide valuable insight into the systems where analytical solution is impossible. The general results are employed in specific systems where the waveguide is created by the magnetic field of a set of electric wires or magnetized strips.

cond-mat.mes-hall↗

arXiv2012

Indirect Coulomb Energy for Two-Dimensional Atoms

In this manuscript we provide a family of lower bounds on the indirect Coulomb energy for atomic and molecular systems in two dimensions in terms of a functional of the single particle density with gradient correction terms.

math-ph↗