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Classes for which the Spectrum determines a Convex Shape

Given a planar domain $\Omega \subset \Bbb{R}^2$ bounded and open we can associate to it the spectrum of the Laplace operator with Dirichlet boundary condition. It is known that there are planar ...
Beni Bogosel's user avatar
  • 2,222
9 votes
0 answers
802 views

Positive definiteness of matrix

This question is about the positive definiteness of a (non-random) matrix that is defined using random variables as follows: We fix the vector $v=(1,1)$ (yet, it seems the final result does not ...
Kung Yao's user avatar
  • 192
8 votes
0 answers
197 views

Subleading terms in Weyl's Law

The two term Weyl's conjecture states that $$N(\lambda)\sim\frac{\operatorname{area}(\Omega)}{4\pi}\lambda-\frac{\operatorname{perimeter}(\partial\Omega)}{4\pi}\sqrt\lambda$$ where $\Omega$ is a ...
antrep1234's user avatar
7 votes
0 answers
195 views

Hölder continuity of spectrum of matrices

Endow $\mathbb{C}^{d \times d}$ with the norm induced by the Euclidean norm on $\mathbb{C}^d$. It is well-known (to those who know it well, I guess) that the spectrum $\sigma(A)$ of a matrix $A \in \...
Jochen Glueck's user avatar
7 votes
0 answers
354 views

Locally symmetric spaces: spectrum of the Laplacian

Let $M = \Gamma\backslash X$ denote a locally symmetric space of non-compact type and $\Delta$ the Laplacian on $L^2(M)$. It is known that the spectrum of $\Delta$ decomposes into finitely many ...
espressionist's user avatar
6 votes
0 answers
201 views

Dependence of Neumann eigenvalues on the domain

I have the following problem in hands, in the context of a broader investigation: Let $V\in L^{n/2}$ compactly supported, where $n\geq 3$ is the dimension. I want to prove the following: For any $\...
Manuel Cañizares's user avatar
6 votes
0 answers
151 views

Gap between consecutive Dirichlet eigenvalues

Suppose $\Omega \subset \mathbb R^2$ is a domain with a Lipschitz boundary and let $\{\lambda_k\}_{k=0}^n$ be the eigenvalues for the Laplacian operator on $\Omega$, that is to say $$ -\Delta \phi_k = ...
Ali's user avatar
  • 4,135
6 votes
0 answers
107 views

Eigenvalues of splitting scheme

In numerical analysis it is common to approximate a solution to a PDE $$u'(t) = (A+B) u(t), \quad u(0)=u_0$$ which is just given by $e^{t(A+B)}u_0$ by the splitting $e^{tB/2} e^{tA} e^{tB/2}u_0.$ Here,...
Sascha's user avatar
  • 536
6 votes
0 answers
137 views

Spectrum of perturbed differential operators

I am looking for a reference that could help me with the following two questions: Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with Lipschitz boundary. Consider a sequence of differential ...
Dominick's user avatar
4 votes
0 answers
95 views

Minimal eigenvalue of infinite dimensional matrix

Consider the following symmetric, positive-definite matrix $$ H_{nm}=-\frac{(4 m+4 n+1)}{(4 m-4 n-1) (4 m-4 n+1)}\sqrt{\frac{(4 m-1)!! (4 n-1)!!}{(4 m)!! (4 n)!!}} $$ where $n,m=0,1,2\dots$ (Here $!!$ ...
AccidentalFourierTransform's user avatar
4 votes
0 answers
151 views

Eigenvalues of Laplacian and eigenvalues of curvature operator

Let $(M^n,g)$ be a compact Riemannian manifold (without boundary). The symmetries of the curvature $R$ of (the Levi-Civita connection associated to) $g$ allow one to realise $R$ as a self-adjoint (...
GradStudent's user avatar
4 votes
0 answers
98 views

Spectrum of Laplace-Beltrami with piecewise constant coefficients

By the Laplace-Beltrami with piecewise constant coefficients I means the operator $-div (f\, \nabla .)$ in the 2-sphere. Where $f$ is a piecewise constant function that takes two values $1$ and $a>...
rihani's user avatar
  • 61
4 votes
0 answers
155 views

Schrodinger operator with magnetic field: eigenvalues

Consider the self-adjoint operator on $L^{2}(\mathbb{R}^{N})$, $$H=-\frac{1}{2}(\nabla-iA)^{2}+V,$$ where $A\in C^{\infty}(\mathbb{R}^{N}, \mathbb{R}^{N} )$, $V\in C^{\infty}(\mathbb{R}^{N})$, $V\...
user152385's user avatar
3 votes
0 answers
282 views

Extending Ky Fan's eigenvalues inequality to kernel operators

--Migrating from MSE since it might fit better here-- Base result The following result in Terry Tao's book, 'p. 47, Ky Fan inequality' reads as: $$\sum_i\lambda_i(A+B) \leq \sum_i \lambda_i(A) + \...
user43389's user avatar
  • 255
3 votes
0 answers
190 views

Error term in the Euclidean Weyl law

Let $\Omega\subset \mathbb R^n$ be an open bounded set with smooth boundary. The Laplacian on $\Omega$ with Dirichlet boundary conditions has discrete spectrum $\lambda_1\le \lambda_2\le \ldots$ that ...
Dario's user avatar
  • 381
2 votes
0 answers
94 views

Dimension of Laplacian eigenspaces along a smooth 1-parameter family of metrics

Let $(M^n,g)$ be a closed Riemannian manifold, $n \geq 2$. For a smooth 1-parameter family $g_t$, $t \in (-\varepsilon, \varepsilon)$, of Riemannian metrics on $M$ with $g_0 = g$, let $\lambda_k(t)$, $...
Eduardo Longa's user avatar
2 votes
0 answers
538 views

Eigenvalues of the sum of matrices, where matrices are tensor products of Pauli matrices

recently I've been studying the toric code (a squared lattice in the context of quantum computation). I want to calculate the energy of the ground state and of all the excitations, with the respective ...
MarcPN's user avatar
  • 21
2 votes
0 answers
121 views

Eigenvalues of two positive-definite Toeplitz matrices

Consider two positive-definite Toeplitz matrices $M_1$ and $M_2$ both with dimension $2^j \times 2^j$. Their matrix elements are: $$M_1[x,y] = \frac{\text{sin}(\pi(x-y)/2^j)}{\pi(x-y)} \qquad M_2[x,y] ...
Chriscrosser's user avatar
2 votes
0 answers
102 views

Spectrum of a Lax Pair and conservation laws of a PDE

I would like to ask a question that I had asked a few days ago on the site math.stackexchange and I still have not received an answer. If we have a Lax operator, we know that the spectrum of this ...
Niser's user avatar
  • 93
2 votes
0 answers
75 views

Unimodality of a function of a non-negative matrix

I am taking an interest in the following problem: Consider a real matrix with non-negative entries $\boldsymbol{A} \in \mathbb{R}_+^{d \times d}$, with $d \in \mathbb{N}$. For $k \in \mathbb{N}$, ...
geo.wolfer's user avatar
2 votes
0 answers
150 views

Limit circle/point of an ODE with finite eigenvalues

Consider the following Sturm–Liouville (SL) eigenvalue problem defined in $(-\infty,0]$ or $[0,\infty)$ or $(-\infty,+\infty)$ $$(py')'-qy=-\lambda^2wy,$$ in which $p(x)=x^2$, $w(x)=1$, and $q(x)=(x/2+...
xiaohuamao's user avatar
2 votes
0 answers
111 views

Li-Yau inequality $\frac{4\pi}{|D|} k<\lambda_{k}<\frac{4\pi}{|D|} k+c\sqrt{k}$ for large enough k

For proving another interesting question: Tauberian theorem $\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} $ I need the following inequality for Dirichlet ...
Thomas Kojar's user avatar
  • 5,474
2 votes
0 answers
452 views

Largest eigenvalues distribution of tridiagonal symmetric random matrix

I would like to find the largest eigenvalue distribution of the following tridiagonal symmetric random matrix in an analytic way. All the ${\lambda}_i$ are distributed the same way with chi-square (...
user1065320's user avatar
2 votes
0 answers
279 views

Eigenvalues of this matrix

I have a linear map that is defined by $$T:\text{lin}(1,...,x^m) \rightarrow \text{lin}(1,...,x^m) \text{ with}$$ $$x^k \mapsto 2w(k-m)x^{k+1}+(k^2-k-w^2)x^k-2kwx^{k-1}+(k-k^2)x^{k-2}$$ Let me give a ...
BaoLing's user avatar
  • 329
1 vote
0 answers
183 views

Connection of eigenspace of finite Hilbert matrix and its continuous operator counterpart

I am trying to understand the connection between the eigenspace of the continuous operator $$ H(x,y) = \frac{1}{x+y} $$ which is nothing but the square of the Laplace operator, and its discrete ...
knuth's user avatar
  • 33
1 vote
0 answers
62 views

What do you call this class of matrices with a unique positive eigenvalue associated to a graph?

I am looking for the name of a class of symmetric matrices $M\in\Bbb R^{n\times n}$ that I can associate to a (finite simple) graph $G=(V,E)$ with $V=\{1,...,n\}$ and that have the following ...
M. Winter's user avatar
  • 13.6k
1 vote
0 answers
67 views

Spectral theorems for generalized Hermitian matrices

Let $k$ be a field, and let $\sigma$ be a nontrivial involutory automorphism of $k$. Let $A$ be a square matrix with entries in $k$, such that $(A^{\sigma})^T = A$; here $A^\sigma$ means the matrix $(...
THC's user avatar
  • 4,547
1 vote
0 answers
52 views

Laplacian eigenvalue problem for systems coupled along the boundary

I am looking for references on eigenvalue problems for systems of the following type: Let $\Omega$ be the region enclosed by a right triangle with legs $\Gamma_1$, $\Gamma_2$, and hypotenuse $\...
Justin Erik Katz's user avatar
1 vote
0 answers
259 views

An estimate for the solution of an elliptic PDE depending on a parameter

Let $\Omega\subset\mathbb R^n$ be a bounded domain with a sufficiently smooth boundary $\partial\Omega$. We assume $\lambda_1\in\mathbb R$ is the principle eigenvalue of the operator $$ -\Delta:\ H^...
CooLee's user avatar
  • 375
1 vote
0 answers
439 views

estimate of smallest eigenvalue of Schrodinger operator

I am looking for references on estimate of first nonzero Dirichlet eigenvalue for Schrodinger operator $-\Delta + V$, if sharp bounds exist, that would be better, here for simplicity, we can assume ...
Yimin's user avatar
  • 151
1 vote
0 answers
119 views

inverse of asymptotic Toeplitz matrix with band limited associated function

I am reviewing a controversial paper, and the main result, a revolution within my field, comes down to whether or not the following is true. I strongly believe it is not, but would need confirmation. ...
Max Hamper's user avatar
1 vote
0 answers
158 views

Interpreting (Fiedler) spectral bisectioning

I would appreciate help on how to interpret the results of spectral bisectioning of a graph. Given a $G=(V,E)$ with size $N$ represented by $Q$ its Laplacian matrix where the eigenvalues are ordered ...
Val K's user avatar
  • 355
1 vote
0 answers
631 views

Bounding the largest Singular value

D is a $n \times n$ diagonal matrix whose diagonal entries lies in $(0,1]$. B is any $n \times n$ n.n.d. matrix. What will be the sharpest upper bound on the largest eigenvalue of: $(D+B)^{-1}D^2(...
Gourab Mukherjee's user avatar
0 votes
0 answers
50 views

Induced higher Gershgorin estimate

I have a problem which I suspect appears in literature under a name I haven't found yet. Let $H:\ell^2(\mathbb{Z}^2)\to \ell^2(\mathbb{Z}^2)$ given by $H=\Delta + D$, where $\Delta$ is the graph ...
Keen-ameteur's user avatar
0 votes
0 answers
74 views

Computing the eigenvalues of $A+E$ where $A$ is an upper triangular matrix whose diagonal entries are all zero and $E$ is a rank one matrix

Let us consider the backward-shift matrix $B=(b_{ij})\in M_n(\mathbb{R})$ whose entries are given by $b_{k,k+1}=1$ and the other entries are all 0. We also consider $X=(x_{ij})\in M_n(\mathbb{R})$ ...
ABB's user avatar
  • 4,058
0 votes
0 answers
220 views

Eigenvalue multiplicity of tensor product of positive operator with itself

Let $H$ be a separable complex Hilbert space and let $A\in B(H)$ be positive with $||A||=1$ and have eigenvalue 1 with multiplicity 1. Suppose $A=T^*T$ for some $T\in B(H)$. Denote the spectrum of $A$ ...
Dasherman's user avatar
  • 203
0 votes
0 answers
109 views

The non empty set of accumulation points of a bounded linear operator is the spectrum of another operator

Let $X$ be an infinite dimensional Banach space, and let $T \in L(X)$ such that the set of accumulation points of $T$ is non empty, i-e $\mbox{acc}\,\sigma(T)\neq 0.$\ Is there a Banach space $Y$ ...
Luffy's user avatar
  • 1
0 votes
0 answers
122 views

Methods to find the spectrum of an operator

Suppose we have a bounded, self-adjoint operator $T$ on a set of functions $\mathcal{F}$. What kinds of methods are there to find the spectrum of $T$? Here is the setting I'm wondering about: consider ...
900edges's user avatar
  • 153
0 votes
0 answers
107 views

Numerical error on the spectrum of a matrix

Let $Q=(q_{j,k})_{1\le j,k\le N}$ be a (Hermitian) $N\times N$ matrix with complex-valued entries. The matrix $Q$ is given numerically and the absolute error on each entry is bounded above by a (small)...
Bazin's user avatar
  • 16.2k