We conduct extensive numerical experiments aimed at finding the admissible range of the ratios of the first three eigenvalues of a planar Dirichlet Laplacian. The results improve the previously known theoretical estimates of M Ashbaugh and R Benguria. We also prove some properties of a maximizer of the ratio $λ_3/λ_1$.
We derive a two-terms asymptotics for eigenvalues of the Dirichlet Laplacian in a narrow strip of variable width. The asymptotics is taken with respect to a small paprameter that characterizes the width of the strip.
In the present article we will give a new proof of the ground state asymptotics of the Dirichlet Laplacian with a Neumann window acting on functions which are defined on a two-dimensional infinite strip or a three-dimensional infinite layer. The proof is based on the analysis of the corresponding Dirichlet-to-Neumann operator as a first order classical pseudo-differential operator. Using the explicit representation of its symbol we prove an asymptotic expansion as the window length decreases.
We consider the Dirichlet Laplacian in a family of narrow unbounded domains. As the width of these domains goes to 0, we study the asymptotic behavior of the eigenvalues that lie below the essential spectrum and the asymptotic behavior of the corresponding eigenfunctions.
This paper is about a shape optimization problem related to the Dirichlet Laplacian eingevalues in the Euclidean plane. More precisely we study the shape of the minimizer in the class of open sets of constant width. We prove that the disk is not a local minimizer except for a limited number of eigenvalues.
The aim of the paper is to derive spectral estimates on the eigenvalue moments of the magnetic Dirichlet Laplacian defined on the two-dimensional disk with a radially symmetric magnetic field.
We prove lower bounds for the Dirichlet Laplacian on possibly unbounded domains in terms of natural geometric conditions. This is used to derive uncertainty principles for low energy functions of general elliptic second order divergence form operators with not necessarily continuous main part.
We prove a two-term Weyl-type asymptotic formula for sums of eigenvalues of the Dirichlet Laplacian in a bounded open set with Lipschitz boundary. Moreover, in the case of a convex domain we obtain a universal bound which correctly reproduces the first two terms in the asymptotics.
This paper is dedicated to the classical Hadamard formula for asymptotics of eigenvalues of the Dirichlet-Laplacian under perturbations of the boundary. We prove that the Hadamard formula still holds for $C^1$-domains with $C^1$-perturbations. We also derive an optimal estimate for the remainder term in the $C^{1,α}$-case. Furthermore, if the boundary is merely Lipschitz, we show that the Hadamard formula is not valid.
We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet-Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact 1-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.
We study the optimality of the remainder term in the two-term Weyl law for the Dirichlet Laplacian within the class of Lipschitz regular subsets of $\mathbb{R}^d$. In particular, for the short-time asymptotics of the trace of the heat kernel we prove that the error term cannot be made quantitatively better than little-$o$ of the second term.
We investigate Dirichlet Laplacian in a straight twisted tube of a non-circular cross section, in particular, its discrete spectrum coming from a local slowdown of the twist. We prove a Lieb-Thirring-type estimate for the spectral moments and present two examples illustrating how the bound depends on the tube cross section.
Corrigendum to "Ramification of multiple eigenvalues for the Dirichlet-Laplacian in perforated domains" published in Journal of Functional Analysis 283 (2022) 109718
Let $Ω\subset \mathbb R^3$ be a waveguide which is obtained by translating a cross-section in a constant direction along an unbounded spatial curve. Consider $-Δ_Ω^D$ the Dirichlet Laplacian operator in $Ω$. Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of $-Δ_Ω^D$. Then, we state sufficient conditions that give rise to a non-empty discrete spectrum for $-Δ_Ω^D$; in particular, we show that the number of discrete eigenvalues can be arbitrarily large since the waveguide is thin enough.
We study the spectrum of the Dirichlet Laplacian on an unbounded twisted tube with twisting velocity exploding to infinity. If the tube cross section does not intersect the axis of rotation, then its spectrum is purely discrete under some additional conditions on the twisting velocity (D.Krejcirik, 2015). In the current work we prove a Berezin type upper bound for the eigenvalue moments.
In this paper, we study eigenvalues and eigenfunctions of $p$-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of $p$-Laplacian, as $p\to 1,$ we identify the Cheeger constant of a symmetric graph with that of the quotient graph. By this approach, we calculate various Cheeger constants of spherically symmetric graphs.
We develop the complex scaling method for the Dirichlet Laplacian in a domain with asymptotically cylindrical end. We define resonances as discrete eigenvalues of non-selfadjoint operators, obtained as deformations of the selfadjoint Dirichlet Laplacian $Δ$ by means of the complex scaling. The resonances are identified with the poles of the resolvent matrix elements $((Δ-μ)^{-1}F, G)$ meromorphic continuation in $μ$ across the essential spectrum of $Δ$, where $F$ and $G$ are elements of an explicitly given set of partial analytic vectors. It turns out that the Dirichlet Laplacian has no singul…
We explicitly express the spectral determinant of Friederichs Dirichlet Laplacians on the 2-dimensional hyperbolic (Gaussian curvature -1) cones in terms of the cone angle and the geodesic radius of the boundary. The related results in the recent paper "Riemann-Roch isometries in the non-compact orbifold setting," J. Eur. Math. Soc. 22 (2020) by G. Freixas i Montplet and A. von Pippich turn out to be incorrect.
We consider a spectral stability estimate by Burenkov and Lamberti concerning the variation of the eigenvalues of second order uniformly elliptic operators on variable open sets in the N-dimensional euclidean space, and we prove that it is sharp for any dimension N. This is done by studying the eigenvalue problem for the Dirichlet Laplacian on special open sets inscribed in suitable spherical cones.
We consider the Laplace operator with Dirichlet boundary conditions on a planar domain and study the effect that performing a scaling in one direction has on the spectrum. We derive the asymptotic expansion for the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This method allows us, for instance, to obtain an approximation for the first Dirichlet eigenvalue for a large class of planar domains, under very mild assumptions.
We consider Dirichlet Laplacian in a thin curved three-dimensional rod. The rod is finite. Its cross-section is constant and small, and rotates along the reference curve in an arbitrary way. We find a two-parametric set of the eigenvalues of such operator and construct their complete asymptotic expansions. We show that this two-parametric set contains any prescribed number of the first eigenvalues of the considered operator. We obtain the complete asymptotic expansions for the eigenfunctions associated with these first eigenvalues.
We study spectral properties of Dirichlet Laplacian on the conical layer of the opening angle $π-2θ$ and thickness equal to $π$. We demonstrate that below the continuum threshold which is equal to one there is an infinite sequence of isolated eigenvalues and analyze properties of these geometrically induced bound states. By numerical computation we find examples of the eigenfunctions.
Given an Euclidean domain with very mild regularity properties, we prove that there exist perturbations of the Dirichlet Laplacian of the form $-(I+S_ε)Δ$ with $\|S_ε\|_{L^2\to L^2}\leq ε$ whose high energy eigenfunctions are quantum uniquely ergodic (QUE). Moreover, if we impose stronger regularity on the domain, the same result holds with $\|S_ε\|_{L^2\to H^γ}\leq ε$ for $γ>0$ depending on the domain. We also give a proof of a local Weyl law for domains with rough boundaries.
We consider the Laplace operator with Dirichlet boundary conditions on a domain in R^d and study the effect that performing a scaling in one direction has on the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This generalizes our previous results in two dimensions and, as in that case, allows us to obtain an approximation for Dirichlet eigenvalues for a large class of domains, under very mild assumptions. As an application, we derive a three--term asymptotic expansion for the first eigenvalue of d-dimensional ellipsoids.
We consider the Dirichlet Laplacian in the half-plane with constant magnetic field. Due to the translational invariance this operator admits a fiber decomposition and a family of dis- persion curves, that are real analytic functions. Each of them is simple and monotically decreasing from positive infinity to a finite value, which is the corresponding Landau level. These finite limits are thresholds in the purely absolutely continuous spectrum of the magnetic Laplacian. We prove a limiting absorption principle for this operator both outside and at the thresholds. Finally, we establish analytic …
How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot of techniques involved can be traced back to a newly rediscovered manuscript of Hörmander from the 1950s. We present Hörmander's approach and its applications, with an emphasis on eigenvalue estimates and spectral asymptotics. In particular, we obtain results for the DtN maps on non-smooth boundaries in the Riemannian setting, the DtN operators for the Helmholtz equation and the DtN operators on di…
We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the hypersurfaces tends to zero. The asymptotics are uniform and local in the sense that the coefficients depend only on the extremal points where the ratio of the area of the Neumann boundary to the Dirichlet one is locally the biggest.
We establish sharp pointwise estimates for the ground states of some singular fractional Schrödinger operators on relatively compact Euclidean subsets. The considered operators are of the type $(-Δ)^{α/2}|_\Om-c|x|^{-α}$, where $(-Δ)^{α/2}|_\Om$ is the fraction-Laplacien on an open subset $\Om$ in $\R$ with zero exterior condition and $0<c\leq(\frac{d-α}{2})^2$. The intrinsic ultracontractivity property for such operators is discussed as well and a sharp large time asymptotic for their heat kernels is derived.
For any bounded regular domain $Ω$ of a real analytic Riemannian manifold $M$, we denote by $λ_{k}(Ω)$ the $k$-th eigenvalue of the Dirichlet Laplacian of $Ω$. In this paper, we consider $λ_k$ and as a functional upon the set of domains of fixed volume in $M$. We introduce and investigate a natural notion of critical domain for this functional. In particular, we obtain necessary and sufficient conditions for a domain to be critical, locally minimizing or locally maximizing for $λ_k$. These results rely on Hadamard type variational formulae that we establish in this general setting.
We study the eigenvalue problem for the Dirichlet Laplacian in bounded simply connected plane domains $Ω\subset\mathbb{C}$ using conformal transformations of the original problem to the weighted eigenvalue problem for the Dirichlet Laplacian in the unit disc $\mathbb{D}$. This allows us to estimate the variation of the eigenvalues of the Dirichlet Laplacian upon domain perturbation via energy type integrals for a large class of "conformal regular" domains which includes all quasidiscs, i.e. images of the unit disc under quasiconformal homeomorphisms of the plane onto itself. Boundaries of such…
The goal of this paper is to derive estimates of eigenvalue moments for Dirichlet Laplacians and Schrödinger operators in regions having infinite cusps which are geometrically nontrivial being either curved or twisted; we are going to show how those geometric properties enter the eigenvalue bounds. The obtained inequalities reflect the essentially one-dimensional character of the cusps and we give an example showing that in an intermediate energy region they can be much stronger than the usual semiclassical bounds.
In this paper, we prove the upper and lower bounds for normal derivatives of spectral clusters $u=χ_λ^s f$ of Dirichlet Laplacian $Δ_M$, $$c_s λ\|u\|_{L^2(M)} \leq \| \partial_νu \|_{L^2(\partial M)} \leq C_s λ\|u\|_{L^2(M)} $$ where the upper bound is true for any Riemannian manifold, and the lower bound is true for some small $0<s<s_M$, where $s_M$ depends on the manifold only, provided that $M$ has no trapped geodesics (see Theorem \ref{Thm3} for a precise statement), which generalizes the early results for single eigenfunctions by Hassell and Tao.
For $d \in \N$ and $Ω\ne \emptyset$ an open set in $\R^d$, we consider the eigenfunctions $Φ$ of the Dirichlet Laplacian $-Δ_Ω$ of $Ω$. If $Φ$ is associated with an eigenvalue below the essential spectrum of $-Δ_Ω$ we provide estimates for the $L_1$-norm of $Φ$ in terms of its $L_2$-norm and spectral data. These $L_1$-estimates are then used in the comparison of the heat content of $Ω$ at time $t>0$ and the heat trace at times $t' > 0$, where a two-sided estimate is established. We furthermore show that all eigenfunctions of $-Δ_Ω$ which are associated with a discrete eigenvalue of $H_Ω$, belo…
We consider the Laplace operator in a planar waveguide, i.e., an infinite two-dimensional straight strip of constant width, with particular types of Robin boundary conditions. We study the essential spectrum of the corresponding Laplacian when the boundary coupling function has a limit at infinity. Furthermore, we derive sufficient conditions for the existence of discrete spectrum.
We perform quantitative spectral analysis of the self-adjoint Dirichlet Laplacian $\mathsf{H}$ on an unbounded, radially symmetric (generalized) parabolic layer $\mathcal{P}\subset\mathbb{R}^3$. It was known before that $\mathsf{H}$ has an infinite number of eigenvalues below the threshold of its essential spectrum. In the present paper, we find the discrete spectrum asymptotics for $\mathsf{H}$ by means of a consecutive reduction to the analogous asymptotic problem for an effective one-dimensional Schrödinger operator on the half-line with the potential the behaviour of which far away from th…
Let -Δdenote the Dirichlet Laplace operator on a bounded open set in \mathbb{R}^d. We study the sum of the negative eigenvalues of the operator -h^2 Δ- 1 in the semiclassical limit h \to 0+. We give a new proof that yields not only the first term of the asymptotic formula but also the second term involving the surface area of the boundary of the set. The proof is valid under weak smoothness assumptions on the boundary.
We study the survival probability of a particle diffusing in a two-dimensional domain, bounded by a smooth absorbing boundary. The short-time expansion of this quantity depends on the geometric characteristics of the boundary, whilst its long-time asymptotics is governed by the lowest eigenvalue of the Dirichlet Laplacian defined on the domain. We present a simple algorithm for calculation of the short-time expansion for an arbitrary "star-shaped" domain. The coefficients are expressed in terms of powers of boundary curvature, integrated around the circumference of the domain. Based on this ex…
We prove that for any domain in the Heisenberg group the (k+1)'th Neumann eigenvalue of the sub-Laplacian is strictly less than the k'th Dirichlet eigenvalue. As a byproduct we obtain similar inequalities for the Euclidean Laplacian with a homogeneous magnetic field.
