Luc Hillairet, Chris Judge
We study the Laplace operator with Dirichlet or Neumann boundary condition on polygons in the Euclidean plane. We prove that almost every simply connected polygon with at least four vertices has simple spectrum. We also address the more general case of geodesic polygons in a constant curvature space form.
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We prove that the spectral radius of even order nonnegative irreducible tensors is real geometrically simple. In the case when the order of the tensor is odd, or in the complex field, some conditions are given to guarantee the geometric simplicity of the spectral radius.
This paper studies the inverse Steklov spectral problem for curvilinear polygons. For generic curvilinear polygons with angles less than $π$, we prove that the asymptotics of Steklov eigenvalues obtained in arXiv:1908.06455 determines, in a constructive manner, the number of vertices and the properly ordered sequence of side lengths, as well as the angles up to a certain equivalence relation. We also present counterexamples to this statement if the generic assumptions fail. In particular, we show that there exist non-isometric triangles with asymptotically close Steklov spectra. Among other te…
Let $Ω$ be a curvilinear polygon and $Q^γ_Ω$ be the Laplacian in $L^2(Ω)$, $Q^γ_Ωψ=-Δψ$, with the Robin boundary condition $\partial_νψ=γψ$, where $\partial_ν$ is the outer normal derivative and $γ>0$. We are interested in the behavior of the eigenvalues of $Q^γ_Ω$ as $γ$ becomes large. We prove that the asymptotics of the first eigenvalues of $Q^γ_Ω$ is determined at the leading order by those of model operators associated with the vertices: the Robin Laplacians acting on the tangent sectors associated with $\partial Ω$. In the particular case of a polygon with straight edges the first eigenp…
Let us say that an $n$-sided polygon is semi-regular if it is circumscriptible and its angles are all equal but possibly one, which is then larger than the rest. Regular polygons, in particular, are semi-regular. We prove that semi-regular polygons are spectrally determined in the class of convex piecewise smooth domains. Specifically, we show that if $Ω$ is a convex piecewise smooth planar domain, possibly with straight corners, whose Dirichlet or Neumann spectrum coincides with that of an $n$-sided semi-regular polygon $P_n$, then $Ω$ is congruent to $P_n$.
The Steklov spectrum of convex polygonal domains I: spectral finiteness
本文研究凸多边形的 Steklov 反问题，证明几乎所有凸多边形至多有有限多个等谱但不全等的区域，并给出数量上界及特征值等周界。
Spectral invariants of integrable polygons
本文研究了内角为π分数的可积多边形在Dirichlet和Neumann边界条件下的Laplace谱，获得了谱zeta函数、zeta正则化行列式及热迹短时渐近展开中的新谱不变量，并探讨了普通多边形与光滑边界域热迹不变量的关系。
The Steklov spectrum of convex polygonal domains II: investigating spectral determination
研究凸多边形的Steklov谱如何确定其几何特性，证明几乎所有三角形由其Steklov谱唯一确定，并探讨四边形和正n边形的谱确定性，揭示Steklov谱对凸多边形与光滑边界域的区别。
We consider the spectrum of a second-order elliptic operator in divergence form with periodic coefficients, which is known to be completely described by Bloch eigenvalues. We show that under small perturbations of the coefficients, a multiple Bloch eigenvalue can be made simple. The Bloch wave method of homogenization relies on the regularity of spectral edge. The spectral tools that we develop, allow us to obtain simplicity of an internal spectral edge through perturbation of the coefficients. As a consequence, we are able to establish Bloch wave homogenization at an internal edge in the pres…
Spectral Equality for Tropical Totally Nonnegative Matrices: Dominant Cancellation and Newton Polygons
本文研究热带全非负矩阵的经典与热带特征值赋值相等的条件，指出主导消去是唯一障碍，并给出基于牛顿多边形断点一致性的充要判据及例外提升的显式方程。
We describe a method for comparing the real analytic eigenbranches of two families of quadratic forms that degenerate as t tends to zero. One of the families is assumed to be amenable to `separation of variables' and the other one not. With certain additional assumptions, we show that if the families are asymptotic at first order as t tends to 0, then the generic spectral simplicity of the separable family implies that the eigenbranches of the second family are also generically one-dimensional. As an application, we prove that for the generic triangle (simplex) in Euclidean space (constant cur…
Spectral Simplicity and Joint Eigenvalue Densities for a Non-Gaussian Brownian Time Change
该研究针对单位正方形上带独立非高斯系数狄利克雷特征函数展开构造的速度测度的布朗时间变换，通过鞅论证、酉变换等方法证明其谱简单性并得到联合特征值密度。
We consider Sturm-Liouville equation $y^{\prime\prime}+(λ-q)y=0$ where $q\in L^{1}[0,a]$. We obtain various conditions on the Fourier coefficients of q such that the periodic eigenvalues having the form given by Titchmarsh are asymptotically simple. Under these conditions, we give some asymptotic estimates for the spectral gaps.
We prove Szegő-type trace asymptotics for translation-invariant operators on polygons. More precisely, consider a Fourier multiplier $A=\mathcal{F}^\ast σ\mathcal{F}$ on $\mathsf{L}^2(\mathbb{R}^2)$ with a sufficiently decaying, smooth symbol $σ:\mathbb{C}\to\mathbb{C}$. Let $P\subset \mathbb{R}^2$ be the interior of a polygon and, for $L\geq 1$, define its scaled version $P_L:=L\cdot P$. Then we study the spectral asymptotics for the operator $A_{P_L}=χ_{P_L}Aχ_{P_L}$, the spatial restriction of $A$ onto $P_L$: for entire functions $h$ with $h(0)=0$ we provide a complete asymptotic expansion …
Response Calculus for Spectral Simplicity and Joint Eigenvalue Densities
该研究针对狄利克雷刘维尔布朗运动建立响应演算，证明其生成元在0<γ<2内几乎必然有简单谱，得到预解式可微性及沃德恒等式，解决了相关公开问题。
On spectral simplicity of the Hodge Laplacian and Curl Operator along paths of metrics
本文证明在闭定向3-流形上，旋度算子的特征空间在一般度量下是一维的，即使沿C^k度量族也成立，并分析了Hodge Laplacian非简单谱的两种来源。
We describe the spectrum of a non-self-adjoint elliptic system on a finite interval. Under certain conditions we find that the eigenvalues form a discrete set and converge asymptotically at infinity to one of several straight lines. The eigenfunctions need not generate a basis of the relevant Hilbert space, and the larger eigenvalues are extremely sensitive to small perturbations of the operator. We show that the leading term in the spectral asymptotics is closely related to a certain convex polygon, and that the spectrum does not determine the operator up to similarity. Two elliptic systems w…
The heat trace for domains with curved corners
研究了具有弯曲角的平面多边形的热迹，发现其在阶$ t^{1/2} $处首次出现角和曲率的相互作用，并证明了Dirichlet角因子的符号法则，表明其符号仅由角是否为凸或凹决定。
We announce a new result which shows that under either Dirichlet, Neumann, or Robin boundary conditions, the corners in a planar domain are a spectral invariant of the Laplacian. For the case of polygonal domains, we show how a locality principle, in the spirit of Kac's "principle of not feeling the boundary" can be used together with calculations of explicit model heat kernels to prove the result. In the process, we prove this locality principle for all three boundary conditions. Albeit previously known for Dirichlet boundary conditions, this appears to be new for Robin and Neumann boundary c…
Optimization Inverse Spectral Problem for the Schrödinger Operator on a Compact Graph
该研究针对紧图上薛定谔算子前m个特征值的优化逆谱问题，证明了解的存在性，给出无环、有环图下最优修正/势的表示，还构造反例并推导势的解析性条件。
Discussed here are criteria for the existence of continuous components in the spectra of operators with random potential. First, the essential condition for the Simon-Wolff criterion is shown to be measurable at infinity. By implication, for the iid case and more generally potentials with the K-property the criterion is boosted by a zero-one law. The boosted criterion, combined with tunneling estimates, is then applied for sufficiency conditions for the presence of continuous spectrum for random Schrödinger operators. The general proof strategy which this yields is modeled on the resonant delo…
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