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Konstantin Pankrashkin

Publications and source records attributed to Konstantin Pankrashkin.

At least 37 records · Page 2Linked to original sources
arXiv2020

An eigenvalue estimate for a Robin $p$-Laplacian in $C^1$ domains

Let $Ω\subset \mathbb{R}^n$ be a bounded $C^1$ domain and $p>1$. For $α>0$, define the quantity \[ Λ(α)=\inf_{u\in W^{1,p}(Ω),\, u\not\equiv 0} \Big(\int_Ω|\nabla u|^p\,\mathrm{d}x - α\int_{\partialΩ} |u|^p \,\mathrm{d} s\Big)\Big/ \int_Ω|u|^p\,\mathrm{d} x \] with $\mathrm{d} s$ being the hypersurface measure, which is the lowest eigenvalue of the $p$-laplacian in $Ω$ with a non-linear $α$-dependent Robin boundary condition. We show the asymptotics $Λ(α) =(1-p)α^{p/(p-1)}+o(α^{p/(p-1)})$ as $α$ tends to $+\infty$. The result was only known for the linear case $p=2$ or under stronger smoothness assumptions. Our proof is much shorter and is based on completely different and elementary arguments, and it allows for an improved remainder estimate for $C^{1,λ}$ domains.

math.AP↗
arXiv2020

Effective operators for Robin eigenvalues in domains with corners

We study the eigenvalues of the Laplacian with a strong attractive Robin boundary condition in curvilinear polygons. It was known from previous works that the asymptotics of several first eigenvalues is essentially determined by the corner openings, while only rough estimates were available for the next eigenvalues. Under some geometric assumptions, we go beyond the critical eigenvalue number and give a precise asymptotics of any individual eigenvalue by establishing a link with an effective Schrödinger-type operator on the boundary of the domain with boundary conditions at the corners.

math.SP↗
arXiv2019

Strong coupling asymptotics for $δ$-interactions supported by curves with cusps

Let $Γ\subset \mathbb{R}^2$ be a simple closed curve which is smooth except at the origin, at which it has a power cusp and coincides with the curve $|x_2|=x_1^p$ for some $p>1$. We study the eigenvalues of the Schrödinger operator $H_α$ with the attractive $δ$-potential of strength $α>0$ supported by $Γ$, which is defined by its quadratic form \[ H^1(\mathbb{R}^2)\ni u\mapsto \iint_{\mathbb{R}^2} |\nabla u|^2\,\mathrm{d}x-α\int_Γu^2\, \mathrm{d}s, \] where $\mathrm{d}s$ stands for the one-dimensional Hausdorff measure on $Γ$. It is shown that if $n\in\mathbb{N}$ is fixed and $α$ is large, then the well-defined $n$th eigenvalue $E_n(H_α)$ of $H_α$ behaves as \[ E_n(H_α)=-α^2 + 2^{\frac{2}{p+2}} \mathcal{E}_n \,α^{\frac{6}{p+2}} + \mathcal{O}(α^{\frac{6}{p+2}-η}), \] where the constants $\mathcal{E}_n>0$ are the eigenvalues of an explicitly given one-dimensional Schrödinger operator determined by the cusp, and $η>0$. Both main and secondary terms in this asymptotic expansion are different from what was observed previously for the cases when~$Γ$ is smooth or piecewise smooth with non-zero angles.

math.SP↗
arXiv2019

Two-dimensional Dirac operators with singular interactions supported on closed curves

We study the two-dimensional Dirac operator with an arbitrary combination of electrostatic and Lorentz scalar $δ$-interactions of constant strengths supported on a smooth closed curve. For any combination of the coupling constants a rigorous description of the self-adjoint realizations of the operators is given and the qualitative spectral properties are described. The analysis covers also all so-called critical combinations of coupling constants, for which there is a loss of regularity in the operator domain. In this case, if the mass is non-zero, the resulting operator has an additional point in the essential spectrum, and the position of this point inside the central gap can be made arbitrary by a suitable choice of the coupling constants. The analysis is based on a combination of the extension theory of symmetric operators with a detailed study of boundary integral operators viewed as periodic pseudodifferential operators.

math.AP↗
arXiv2019

Attractive conical surfaces create infinitely many bound states

In this paper we study spectral properties of a three-dimensional Schrödinger operator $-Δ+V$ with a potential $V$ given, modulo rapidly decaying terms, by a function of the distance of $x \in \mathbb{R}^3$ to an infinite conical hypersurface with a smooth cross-section. As a main result we show that there are infinitely many discrete eigenvalues accumulating at the bottom of the essential spectrum which itself is identified as the ground-state energy of a certain one-dimensional operator. Most importantly, based on a result of Kirsch and Simon we are able to establish the asymptotic behavior of the eigenvalue counting function using an explicit spectral-geometric quantity associated with the cross-section. This shows a universal character of some previous results on conical layers and $δ$-potentials created by conical surfaces.

math-ph↗
arXiv2018

Bound states of a pair of particles on the half-line with a general interaction potential

In this paper we study an interacting two-particle system on the positive half-line. We focus on spectral properties of the Hamiltonian for a large class of two-particle potentials. We characterize the essential spectrum and prove, as a main result, the existence of eigenvalues below the bottom of it. We also prove that the discrete spectrum contains only finitely many eigenvalues.

math-ph↗
arXiv2018

Dirac operators on hypersurfaces as large mass limits

We show that the eigenvalues of the intrinsic Dirac operator on the boundary of a Euclidean domain can be obtained as the limits of eigenvalues of Euclidean Dirac operators, either in the domain with a MIT-bag type boundary condition or in the whole space, with a suitably chosen zero order mass term.

math-ph↗
arXiv2018

Robin eigenvalues on domains with peaks

Let $Ω\subset\mathbb{R}^N$, $N\ge 2,$ be a bounded domain with an outward power-like peak which is assumed not too sharp in a suitable sense. We consider the Laplacian $u\mapsto -Δu$ in $Ω$ with the Robin boundary condition $\partial_n u=αu$ on $\partialΩ$ with $\partial_n$ being the outward normal derivative and $α>0$ being a parameter. We show that for large $α$ the associated eigenvalues $E_j(α)$ behave as $E_j(α)\sim -ε_j α^ν$, where $ν>2$ and $ε_j>0$ depend on the dimension and the peak geometry. This is in contrast with the well-known estimate $E_j(α)=O(α^2)$ for the Lipschitz domains.

math.AP↗
arXiv2017

Dirac operators with Lorentz scalar shell interactions

This paper deals with the massive three-dimensional Dirac operator coupled with a Lorentz scalar shell interaction supported on a compact smooth surface. The rigorous definition of the operator involves suitable transmission conditions along the surface. After showing the self-adjointness of the resulting operator we switch to the investigation of its spectral properties, in particular, to the existence and non-existence of eigenvalues. In the case of an attractive coupling, we study the eigenvalue asymptotics as the mass becomes large and show that the behavior of the individual eigenvalues and their total number are governed by an effective Schrödinger operator on the boundary with an external Yang-Mills potential and a curvature-induced potential.

math.SP↗
arXiv2017

Spectral asymptotics for $δ$-interactions on sharp cones

We investigate the spectrum of three-dimensional Schrödinger operators with $δ$-interactions of constant strength supported on circular cones. As shown in earlier works, such operators have infinitely many eigenvalues below the threshold of the essential spectrum. We focus on spectral properties for sharp cones, that is when the cone aperture goes to zero, and we describe the asymptotic behavior of the eigenvalues and of the eigenvalue counting function. A part of the results are given in terms of numerical constants appearing as solutions of transcendental equations involving modified Bessel functions.

math.SP↗
arXiv2017

Eigenvalues of Robin Laplacians in infinite sectors

For $α\in(0,π)$, let $U_α$ denote the infinite planar sector of opening $2α$, \[ U_α=\big\{ (x_1,x_2)\in\mathbb R^2: \big|\arg(x_1+ix_2) \big|<α\big\}, \] and $T^γ_α$ be the Laplacian in $L^2(U_α)$, $T^γ_αu= -Δu$, with the Robin boundary condition $\partial_νu=γu$, where $\partial_ν$ stands for the outer normal derivative and $γ>0$. The essential spectrum of $T^γ_α$ does not depend on the angle $α$ and equals $[-γ^2,+\infty)$, and the discrete spectrum is non-empty iff $α<\fracπ2$. In this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle $α$. In particular, there is just one discrete eigenvalue for $α\ge \fracπ{6}$. As $α$ approaches $0$, the number of discrete eigenvalues becomes arbitrary large and is minorated by $κ/α$ with a suitable $κ>0$, and the $n$th eigenvalue $E_n(T^γ_α)$ of $T^γ_α$ behaves as \[ E_n(T^γ_α)=-\dfrac{γ^2}{(2n-1)^2 α^2}+O(1) \] and admits a full asymptotic expansion in powers of $α^2$. The eigenfunctions are exponentially localized near the origin. The results are also applied to $δ$-interactions on star graphs.

math.SP↗
arXiv2016

Self-adjoint indefinite Laplacians

Let $Ω_-$ and $Ω_+$ be two bounded smooth domains in $\mathbb{R}^n$, $n\ge 2$, separated by a hypersurface $Σ$. For $μ>0$, consider the function $h_μ=1_{Ω_-}-μ1_{Ω_+}$. We discuss self-adjoint realizations of the operator $L_μ=-\nabla\cdot h_μ\nabla$ in $L^2(Ω_-\cupΩ_+)$ with the Dirichlet condition at the exterior boundary. We show that $L_μ$ is always essentially self-adjoint on the natural domain (corresponding to transmission-type boundary conditions at the interface $Σ$) and study some properties of its unique self-adjoint extension $\mathcal{L}_μ:=\overline{L_μ}$. If $μ\ne 1$, then $\mathcal{L}_μ$ simply coincides with $L_μ$ and has compact resolvent. If $n=2$, then $\mathcal{L}_1$ has a non-empty essential spectrum, $σ_\mathrm{ess}(\mathcal{L}_{1})=\{0\}$. If $n\ge 3$, the spectral properties of $\mathcal{L}_1$ depend on the geometry of $Σ$. In particular, it has compact resolvent if $Σ$ is the union of disjoint strictly convex hypersurfaces, but can have a non-empty essential spectrum if a part of $Σ$ is flat. Our construction features the method of boundary triplets, and the problem is reduced to finding the self-adjoint extensions of a pseudodifferential operator on $Σ$. We discuss some links between the resulting self-adjoint operator $\mathcal{L}_μ$ and some effects observed in negative-index materials.

math.SP↗
arXiv2016

Discrete spectrum of interactions concentrated near conical surfaces

We study the spectrum of two kinds of operators involving a conical geometry: the Dirichlet Laplacian in conical layers and Schrödinger operators with attractive $δ$-interactions supported by infinite cones. Under the assumption that the cones have smooth cross-sections, we prove that such operators have infinitely many eigenvalues accumulating below the threshold of the essential spectrum and we express the accumulation rate in terms of the eigenvalues of an auxiliary one-dimensional operator with a curvature-induced potential.

math.SP↗
arXiv2016

Eigenvalue inequalities and absence of threshold resonances for waveguide junctions

Let $Λ\subset \mathbb{R}^d$ be a domain consisting of several cylinders attached to a bounded center. One says that $Λ$ admits a threshold resonance if there exists a non-trivial bounded function $u$ solving $-Δu=νu$ in $Λ$ and vanishing at the boundary, where $ν$ is the bottom of the essential spectrum of the Dirichlet Laplacian in $Λ$. We derive a sufficient condition for the absence of threshold resonances in terms of the Laplacian eigenvalues on the center. The proof is elementary and is based on the min-max principle. Some two- and three-dimensional examples and applications to the study of Laplacians on thin networks are discussed.

math.SP↗
arXiv2016

On the $p$-Laplacian with Robin boundary conditions and boundary trace theorems

Let $Ω\subset\mathbb{R}^ν$, $ν\ge 2$, be a $C^{1,1}$ domain whose boundary $\partialΩ$ is either compact or behaves suitably at infinity. For $p\in(1,\infty)$ and $α>0$, define \[ Λ(Ω,p,α):=\inf_{\substack{u\in W^{1,p}(Ω)\\ u\not\equiv 0}}\dfrac{\displaystyle \int_Ω|\nabla u|^p \mathrm{d} x - α\displaystyle\int_{\partialΩ} |u|^p\mathrm{d}σ}{\displaystyle\int_Ω|u|^p\mathrm{d} x}, \] where $\mathrm{d}σ$ is the surface measure on $\partialΩ$. We show the asymptotics \[ Λ(Ω,p,α)=-(p-1)α^{\frac{p}{p-1}} - (ν-1)H_\mathrm{max}\, α+ o(α), \quad α\to+\infty, \] where $H_\mathrm{max}$ is the maximum mean curvature of $\partialΩ$. The asymptotic behavior of the associated minimizers is discussed as well. The estimate is then applied to the study of the best constant in a boundary trace theorem for expanding domains, to the norm estimate for extension operators and to related isoperimetric inequalities.

math.SP↗
arXiv2016

Eigenvalue counting function for Robin Laplacians on conical domains

We study the discrete spectrum of the Robin Laplacian $Q^Ω_α$ in $L^2(Ω)$, \[ u\mapsto -Δu, \quad \dfrac{\partial u}{\partial n}=αu \text{ on }\partialΩ, \] where $Ω\subset \mathbb{R}^{3}$ is a conical domain with a regular cross-section $Θ\subset \mathbb{S}^2$, $n$ is the outer unit normal, and $α>0$ is a fixed constant. It is known from previous papers that the bottom of the essential spectrum of $Q^Ω_α$ is $-α^2$ and that the finiteness of the discrete spectrum depends on the geometry of the cross-section. We show that the accumulation of the discrete spectrum of $Q^Ω_α$ is determined by the discrete spectrum of an effective Hamiltonian defined on the boundary and far from the origin. By studying this model operator, we prove that the number of eigenvalues of $Q^Ω_α$ in $(-\infty,-α^2-λ)$, with $λ>0$, behaves for $λ\to0$ as \[ \dfrac{α^2}{8πλ} \int_{\partialΘ} κ_+(s)^2d s +o\left(\frac{1}λ\right), \] where $κ_+$ is the positive part of the geodesic curvature of the cross-section boundary.

math.SP↗
arXiv2015

Variational proof of the existence of eigenvalues for star graphs

We provide a purely variational proof of the existence of eigenvalues below the bottom of the essential spectrum for the Schrödinger operator with an attractive $δ$-potential supported by a star graph, i.e. by a finite union of rays emanating from the same point. In contrast to the previous works, the construction is valid without any additional assumption on the number or the relative position of the rays. The approach is used to obtain an upper bound for the lowest eigenvalue.

math-ph↗