arXiv subjects

Marco Vogel

Publications and source records attributed to Marco Vogel.

5 recordsLinked to original sources

arXiv2026

On the discrete spectrum of Dirac operators with Lorentz-scalar $\delta$-shell interactions supported on unbounded curves

We consider the massive Dirac operator (with positive mass) in the plane with an attractive Lorentz-scalar $\delta$-shell interaction of strength $\tau\in(-\infty,0)\setminus\{-2\}$ supported on a $C^\infty$-smooth curve $\Sigma\subset\mathbb{R}^2$ being a local deformation of the broken line. This singular interaction is defined by imposing a suitable transmission condition on the curve $\Sigma$ in the operator domain. Such a Dirac operator is self-adjoint and has a gap in the essential spectrum, whose size is explicit and depends on the mass and the interaction strength. We show that the number of discrete eigenvalues in the gap is finite. Under the assumption that one of the domains bounded by $\Sigma$ is convex, we prove that the corresponding Dirac operator has a non-empty discrete spectrum, provided that $\tau$ is either sufficiently small or sufficiently large in absolute value. The result holds for any perturbation of a broken line of any opening angle as described above, and this discrete spectrum is induced by the geometry, since for the same type of a singular interaction supported on the straight line the discrete spectrum is empty.

math.SP↗

arXiv2025

Laplacian eigenvalues for large negative Robin parameters on domains with outward peaks

We study the asymptotic behavior of individual eigenvalues of the Laplacian in domains with outward peaks for large negative Robin parameters. A large class of cross-sections is allowed, and the resulting asymptotic expansions reflect both the sharpness of the peak and the geometric shape of its cross-section. The results are an extension of previous works dealing with peaks whose cross-sections are balls.

math.AP↗

arXiv2023

Asymptotics of Robin eigenvalues on sharp infinite cones

Let $ω\subset\mathbb{R}^n$ be a bounded domain with Lipschitz boundary. For $\varepsilon>0$ and $n\in\mathbb{N}$ consider the infinite cone $Ω_{\varepsilon}:=\big\{(x_1,x')\in (0,\infty)\times\mathbb{R}^n: x'\in\varepsilon x_1ω\big\}\subset\mathbb{R}^{n+1}$ and the operator $Q_{\varepsilon}^α$ acting as the Laplacian $u\mapsto-Δu$ on $Ω_{\varepsilon}$ with the Robin boundary condition $\partial_νu=αu$ at $\partialΩ_\varepsilon$, where $\partial_ν$ is the outward normal derivative and $α>0$. We look at the dependence of the eigenvalues of $Q_\varepsilon^α$ on the parameter $\varepsilon$: this problem was previously addressed for $n=1$ only (in that case, the only admissible $ω$ are finite intervals). In the present work we consider arbitrary dimensions $n\ge2$ and arbitrarily shaped "cross-sections" $ω$ and look at the spectral asymptotics as $\varepsilon$ becomes small, i.e. as the cone becomes "sharp" and collapses to a half-line. It turns out that the main term of the asymptotics of individual eigenvalues is determined by the single geometric quantity $N_ω:=\dfrac{\mathrm{Vol}_{n-1} \partialω}{\mathrm{Vol}_n ω}$. More precisely, for any fixed $j\in \mathbb{N}$ and $α>0$ the $j$th eigenvalue $E_j(Q^α_\varepsilon)$ of $Q^α_\varepsilon$ exists for all sufficiently small $\varepsilon>0$ and satisfies $E_j(Q^α_\varepsilon)=-\dfrac{N_ω^2\,α^2}{(2j+n-2)^2\,\varepsilon^2}+O\left(\dfrac{1}{\varepsilon}\right)$ as $\varepsilon\to 0^+$. The paper also covers some aspects of Sobolev spaces on infinite cones, which can be of independent interest.

math.SP↗

arXiv2023

Asymptotics of Robin eigenvalues for non-isotropic peaks

Let $Ω\subset \mathbb{R}^3$ be an open set such that \begin{align*} &Ω\cap (-δ,δ)^3=\left\{(x_1,x_2,x_3)\in \mathbb{R}^2\times(0,δ): \, \left(\frac{x_1}{x_3^p},\frac{x_2}{x_3^q}\right)\in(-1,1)^2\right\}\subset\mathbb{R}^{3}, \\ &Ω\setminus [-δ,δ]^3 \text{ is a bounded Lipschitz domain}, \end{align*} for some $δ>0$ and $1<p<q<2$. If a set satisfies the first condition one says that it has a non-isotropic peak at $0$. Now consider the operator $Q_Ω^α$ acting as the Laplacian $u\mapsto-Δu$ on $Ω$ with the Robin boundary condition $\partial_νu=αu$ on $\partialΩ$, where $\partial_ν$ is the outward normal derivative. We are interested in the strong coupling asymptotics of $Q_Ω^α$. We prove that for large $α$ the $j$th eigenvalue $E_j(Q_Ω^α)$ behaves as $E_j(Q_Ω^α)\approx \mathcal{A}_jα^{\frac{2}{2-q}}$, where the constants $\mathcal{A}_j<0$ are eigenvalues of a one dimensional Schrödinger operator which depends on $p$ and $q$.

math.SP↗

arXiv2022

On Schrödinger operators with $δ'$-potentials supported on star graphs

The spectral properties of two-dimensional Schrödinger operators with $δ'$-potentials supported on star graphs are discussed. We describe the essential spectrum and give a complete description of situations in which the discrete spectrum is non-trivial but finite. A more detailed study is presented for the case of a star graph with two branches, in particular, the small angle asymptotics for the eigenvalues is obtained.

math.SP↗