Leonid Friedlander, Michael Solomyak
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.
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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.
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。
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。
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.
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。
Fractional Laplacian in bended strip
本文研究了在弯曲波导中分数阶拉普拉斯算子的谱性质，证明了存在低于连续谱阈值的本征值，并建立了确保受困模式存在的曲率条件。
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 derive spectral estimates of the Lieb-Thirring type for eigenvalues of Dirichlet Laplacians on strictly shrinking spiral-shaped domains.
Eigenvalues of the magnetic Dirichlet Laplacian with constant magnetic field on disks in the strong field limit
本文研究强磁场下磁 Dirichlet Laplacian 的特征值渐近行为，证明圆盘上特征值分支的线性渐近及指数小余项，并揭示强场会以与区域无关的尖锐因子违背 Pólya 谱界。
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.
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…
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…
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。
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.
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。
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。
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。
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。
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.
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。
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 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.
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 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 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 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.
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。
Comparisons of Dirichlet, Neumann and Laplacian eigenvalues on graphs and their applications
本文研究图上Dirichlet、Neumann和Laplacian特征值之间的比较关系，讨论刚性情形，并给出Lichnerowicz型、Fiedler型和Friedman型估计等应用，推广了Hua-Huang-Wang及作者先前的工作。
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 …
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 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.
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.
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。
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 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 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.
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…
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。
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 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.
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 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.
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…
Taking advantage from the so-called "Lemma on small eigenvalues" by Colin de Verdière, we study ramification for multiple eigenvalues of the Dirichlet Laplacian in bounded perforated domains. The asymptotic behavior of multiple eigenvalues turns out to depend on the asymptotic expansion of suitable associated eigenfunctions. We treat the case of planar domains in details, thanks to the asymptotic expansion of a generalization of the so-called u-capacity which we compute in dimension 2. In this case multiple eigenvalues are proved to split essentially by different rates of convergence of the pe…
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.
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。
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。
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.
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.
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。
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。
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。
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…
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。
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