Diana Barseghyan, Francoise Truc
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.
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We investigate spectral properties of Kirchhoff Laplacians on radially symmetric antitrees. This class of metric graphs enjoys a rich group of symmetries, which enables us to obtain a decomposition of the corresponding Laplacian into the orthogonal sum of Sturm--Liouville operators. In contrast to the case of radially symmetric trees, the deficiency indices of the Laplacian defined on the minimal domain are at most one and they are equal to one exactly when the corresponding metric antitree has finite total volume. In this case, we provide an explicit description of all self-adjoint extensions…
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.
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…
Inequalities between Neumann and Dirichlet Laplacian eigenvalues on planar domains
本文推广了平面区域上 Neumann 与 Dirichlet Laplacian 特征值间的经典不等式，通过新的变分原理证明该不等式对所有单连通平面 Lipschitz 区域均成立，证实了相关猜想。
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.
Inequalities between Dirichlet and Neumann eigenvalues of the magnetic Laplacian
该研究针对二维和三维均匀磁场下的磁拉普拉斯算子，证明了凸区域上磁Neumann特征值与磁Dirichlet特征值的两类不等式关系，方法基于对Levine和Weinberger策略的改进，丰富了磁算子谱理论的相关结果。
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…
We derive spectral estimates of the Lieb-Thirring type for eigenvalues of Dirichlet Laplacians on strictly shrinking spiral-shaped domains.
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.
Trace formulas for the magnetic Laplacian and Dirichlet to Neumann operator -- Explicit expansions --
本文受 Liu 和 Tan 启发，通过显式示例概述磁热迹公式，并计算 Steklov 热迹渐近中的非局部项和对数项，获得新公式。
Inequalities for the eigenvalues of the (negative) Laplacian subject to mixed boundary conditions on polyhedral and more general bounded domains are established. The eigenvalues subject to a Dirichlet boundary condition on a part of the boundary and a Neumann boundary condition on the remainder of the boundary are estimated in terms of either Dirichlet or Neumann eigenvalues. The results complement several classical inequalities between Dirichlet and Neumann eigenvalues due to Pólya, Payne, Levine and Weinberger, Friedlander, and others.
Weighted heat traces of the Dirichlet Laplacian on Lipschitz domains
本文证明 Lipschitz 域上 Dirichlet Laplacian 加权热迹的渐近展开，并应用于加权 Riesz 均值和特征函数局部化测度的收敛。
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 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.
Some universal inequalities for Dirichlet eigenvalues of the Laplacian on a Euclidean convex domain
本文建立了欧几里得凸域上拉普拉斯算子狄利克雷本征值的两个通用不等式，探讨了相关数学性质。
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.
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…
Curl curl versus Dirichlet Laplacian eigenvalues
该研究给出有界三维区域上curl curl算子特征值的Dirichlet拉普拉斯特征值上界估计，补充了其与Neumann拉普拉斯特征值的不等式，对应常参数Maxwell特征值问题。
Numerical Optimization of Eigenvalues of the magnetic Dirichlet Laplacian with constant magnetic field
本文提出结合梯度下降法和基本解方法，对常数磁场下磁 Dirichlet Laplacian 前七个特征值进行数值优化，发现当磁通量超过目标特征值指标时极小化子始终为圆盘。
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.
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.
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.
The Weyl law for the Dirichlet Laplacian
本文回顾 Dirichlet Laplacian 特征值的渐近分布，引入相关谱量，并基于 Fourier Tauberian 定理给出 Weyl 定律的证明。
Riesz means asymptotics for Dirichlet and Neumann Laplacians on Lipschitz domains
本文证明 Lipschitz 有界域上 Dirichlet 和 Neumann Laplacian 任意正阶 Riesz 均值的两项渐近，并在凸域上给出仅依赖简单几何特征的非渐近通用界，利用非渐近 Tauberian 定理。
Semiclassical inequalities for Dirichlet and Neumann Laplacians on convex domains
本文研究凸域上Dirichlet和Neumann拉普拉斯算子特征值Riesz均值的半经典不等式，将Berezin-Li-Yau和Kröger不等式推广到$\gamma<1$，并分析Pólya猜想失效的影响及谱形状优化问题的渐近行为。
Corrigendum to "Ramification of multiple eigenvalues for the Dirichlet-Laplacian in perforated domains" published in Journal of Functional Analysis 283 (2022) 109718
修正了原定理1.17的证明，该定理在域中移除小孔时未能正确给出Dirichlet-Laplacian特征值渐近行为的系数。
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 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.
In this paper, we prove two new Weyl-type upper estimates for the eigenvalues of the Dirichlet Laplacian. As a consequence, we obtain the following {\em lower} bounds for its counting function. For $\la\ge \la_1$, one has N(\la) > \dfrac{2}{n+2} \dfrac{1}{H_n} (\la-\la_1)^{n/2} \la_1^{-n/2}, and N(\la) > (\dfrac{n+2}{n+4})^{n/2} \dfrac{1}{H_n} (\la-(1+4/n) \la_1)^{n/2} \la_1^{-n/2}, where H_n=\dfrac{2 n}{j_{n/2-1,1}^2 J_{n/2}^2(j_{n/2-1,1})} is a constant which depends on $n$, the dimension of the underlying space, and Bessel functions and their zeros.
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.
We consider the Laplacian in a domain squeezed between two parallel curves in the plane, subject to Dirichlet boundary conditions on one of the curves and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the curves 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 curvature radii of the Neumann boundary to the Dirichlet one is the biggest. We also show that the asymptotics can be obtained from a form of norm-resolvent con…
Optimizing the ground state energy of the three-dimensional magnetic Dirichlet Laplacian with constant magnetic field
针对固定体积三维区域中恒定磁场下磁性狄利克雷拉普拉斯算子基态能量极小化问题，通过柱形区域分析揭示几何约束（磁场增强时区域沿磁场方向特定速率伸长），结合数值结果验证预测并提出猜想。
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.
We study the Laplacian in deformed thin (bounded or unbounded) tubes in ?$\R^3$, i.e., tubular regions along a curve $r(s)$ whose cross sections are multiplied by an appropriate deformation function $h(s)> 0$. One the main requirements on $h(s)$ is that it has a single point of global maximum. We find the asymptotic behaviors of the eigenvalues and weakly effective operators as the diameters of the tubes tend to zero. It is shown that such behaviors are not influenced by some geometric features of the tube, such as curvature, torsion and twisting, and so a huge amount of different deformed tub…
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.
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.
Discrete approximations to Dirichlet and Neumann Laplacians on a half-space and norm resolvent convergence
该研究将$\boldsymbol{R}^d$中拉普拉斯算子离散逼近的范数预解收敛结果扩展到半空间上的Dirichlet和Neumann拉普拉斯算子，证明了网格尺寸二次速率的收敛性，并推广到含特定势的算子及拉普拉斯算子函数。
We consider the Laplacian in a curved two-dimensional strip of constant width squeezed between two curves, subject to Dirichlet boundary conditions on one of the curves and variable Robin boundary conditions on the other. We prove that, for certain types of Robin boundary conditions, the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Laplacian in a Dirichlet-Robin annulus determined by the geometry of the strip. Moreover, we show that an appropriate combination of the geometric setting and boundary conditions leads to a Hardy-type inequality in infi…
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.
Dirichlet Fractional Laplacian in multi-tubes
本文研究了多管区域中受限Dirichlet分数阶拉普拉斯算子的谱结构，发现了与局部情形相比的新效应，并实质性改进了定理4。
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.
Lieb-Thirring type estimates for Dirichlet Laplacians on spiral-shaped domains
本文推导了渐近 Archimedean 螺旋形区域上 Dirichlet Laplacian 本质谱阈值以下特征值的 Lieb-Thirring 估计。
Basisness of Fucik eigenfunctions for the Dirichlet Laplacian
本文改进了Fucik特征值序列的充分条件，证明其对应的特征函数在$ L^2(0,π) $中构成Riesz基。
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.
Monotonicity of discrete spectra of Dirichlet Laplacian in 3-dimensional layers
研究Dirichlet拉普拉斯算子在固定宽度多面体层中特征值的单调性性质，证明了低于本质谱阈值的特征值随几何参数单调变化，并展示了不对称几何扰动导致的非单调谱行为，给出了展开多面体层时出现离散特征值的例子。
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 …
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.
Spectral invariants of the Dirichlet-to-Neumann map associated to the Witten-Laplacian with potential
本研究针对带光滑边界的紧连通黎曼流形，提出计算带位势的Witten-Laplace算子相关Dirichlet-to-Neumann映射谱渐近公式全系数的有效方法，显式给出前四个谱不变量，可提供流形体积、曲率等信息。
Magnetic Dirichlet Laplacian on a perturbed twisted tube
本文研究了带有磁场的磁狄拉克拉普拉斯算子在小扰动下的稳定性，证明其谱在局部变形下保持稳定。
Magnetic Dirichlet Laplacian in curved waveguides
本文研究弯曲波导中磁Dirichlet拉普拉斯算子的谱稳定性，证明在小而非局部形变下谱保持稳定，与无磁场情形形成对比。
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.
Magnetic Dirichlet Laplacian on deformed waveguides
研究磁性狄利克雷拉普拉斯算子在非局部波导边界扰动下的谱稳定性，证明小扰动下谱稳定性。
Limiting absorption principle and perfectly matched layer method for Dirichlet Laplacians in quasi-cylindrical domains
针对拟柱状区域中的Dirichlet Laplacians，通过微分同胚变换和有限长完美匹配层，建立了极限吸收原理并证明了近似解指数收敛于出射/入射解。
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.
Spectra of Dirichlet Laplacian in 3-dimensional polyhedral layers
本文研究三维多面体层中Dirichlet拉普拉斯算子的谱，发现本质谱由最小二面角决定，离散谱有限，并构造了离散谱为空集的例子，同时证明正则多面体层中谱非空。
We give a description of the lower part of the spectrum of the Dirichlet Laplacian in an unbounded 3D periodic lattice made of thin bars (of width $\varepsilon\ll1$) which have a square cross section. This spectrum coincides with the union of segments which all go to $+\infty$ as $\varepsilon$ tends to zero due to the Dirichlet boundary condition. We show that the first spectral segment is extremely tight, of length $O(e^{-δ/\varepsilon})$, $δ>0$, while the length of the next spectral segments is $O(\varepsilon)$. To establish these results, we need to study in detail the properties of the Dir…
Range of the first three eigenvalues of the planar Dirichlet Laplacian
通过数值实验确定平面Dirichlet拉普拉斯算子前三个特征值之比的允许范围，改进理论估计，并证明比率最大化者的性质。
Convergence of graph Dirichlet energies and graph Laplacians on intersecting manifolds of varying dimensions
针对不同内在维数相交流形并集，论文证明归一化图 Dirichlet 能量可同时适应各维数并发生 Γ-收敛与谱收敛，而未归一化能量仅反映最高维流形。
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