Schrodinger operators, operators on manifolds, general differential operators, numerical studies, integral operators, discrete models, resonances, non-self-adjoint operators, random operators/matrices

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5
votes
3answers
979 views

Spectral theorem for self-adjoint differential operator on Hilbert space

I need a reference concerning a theorem that shows the following result, stated very roughly: Given a self-adjoint differential operator densely defined on a Hilbert space, then the given Hilbert ...
3
votes
1answer
203 views

What is the spectrum of the Rado graph?

Isn't this question self-explanatory? There is a lot of literature about the Rado graph $R$ in various places. This graph is also known as the "Random Graph" because a countable random graph is ...
2
votes
0answers
239 views

eigenvalues of the sum of a stochastic matrix and a diagonal matrix

Let $D$ be a real diagonal matrix $D=diag(a_1,a_2,\ldots,a_n)$ with $a_1\le a_2\le\ldots\le a_n$. Assume that at least one of the $a_i$ is positive. Let $P$ be an irreducible, real, row-stochastic ...
13
votes
6answers
586 views

Invertibility of a certain matrix indexed by the Hamming cube

For reasons which the margin of this page is too small to hold, I have been reading parts of a recent paper by O. Selim On submeasures on Boolean algebras, arXiv 1212.6822v3 and in Section 7 the ...
0
votes
2answers
493 views

Eigenvalues of principle minors Vs. eigenvalues of the matrix

Say I have a positive semi-definite matrix with least positive eigenvalue x. Are there always principal minors of this matrix with eigenvalue less than x? (Here "semidefinite" can not be taken to ...
2
votes
1answer
690 views

Eigenvalues of directed Laplacian matrix $L$ and $DL$, where $D$ is a diagonal matrix with positive entries

I have a weighted Laplacian matrix $L$ of a strongly connected directed graph and a diagonal matrix $D$ with positive entries. Since the graph is directed, $L$ is non-symmetric real. Further, since ...
1
vote
0answers
202 views

bivariate polynomial

Hello, Let $p(x,y) = \sum_{m=1}^M\sum_{n=1}^N a_{m,n}x^{m-1}y^{n-1}$ be a bivariate polynomial where $\{a_{m,n}\}$ are complex. If $(x_k, y_k), k=1,2,\cdots, MN-1$ are roots of $p(x,y)=0$ where ...
3
votes
1answer
254 views

First eigenvalue of $\Delta$ on Kaehler manifold with $Ricci\ge k$.

Let $M$ be a Kaehler manifold of complex dimension $n$. Let $\Delta$ be the real Laplacian of the underline Riemannian manifold. Let's assume the Ricci curvature of $M$ satisfies $\text {Ric}\ge ...
2
votes
1answer
148 views

Moments of random matrices - when are they finite

I need to evaluate the moment $$\mathbb{E} (AX)^n,$$ where A is an NxN Hermitian square matrix, and X is $$X=ZZ^{\ast},$$ where $Z=\mu+Y$, where $\mu$ is mean of $Z$ and $Y$ is a zero-mean complex ...
7
votes
1answer
246 views

Eigenfunctions restricted on closed geodesics

Consider the flat torus $T^2=\frac{\mathbb{R}^2}{l_1\mathbb{Z}\oplus l_2\mathbb{Z}}$. It is easy to see that the eigenvalues of the Laplacian on torus, $-\frac{\partial^2}{\partial ...
25
votes
10answers
4k views

real symmetric matrix has real eigenvalues - elementary proof

Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers?
2
votes
1answer
224 views

Asymptotic Behavior of Non-Analytic Function of the Eigenvalues

Hello, Let $A_n = (a_{k-j};\;k,j = 0,1,\ldots,n-1)$ be a sequence of $n\times n$ Toeplitz matrices, with eigenvalues $(\lambda_{n,i};\;i = 0,1,\ldots,n-1)$. If $A_n$ were a sequence of Hermitian ...
3
votes
0answers
150 views

spectrum of a polygon and zeta function

Let $\Delta(x,y) = 1,0$ according to whether $(x,y)$ is in some polygon (symmetric with respect to the diagonal axis). E.g. The convex hull of three points (taken from a paper on dominoes) $$ ...
2
votes
0answers
188 views

Spectrum of the Normal Operator associated to compact supported spectral measures

Let $\mathcal{H}$ be a Hilbert space and $E:\Sigma\to\mathcal{L}(\mathcal{H})$ be a compactly supported spectral on the Borel $\sigma$-algebra $\Sigma$ of $\mathbb{C}$. Then we can form the bounded, ...
2
votes
1answer
670 views

Condition numbers of Vandermonde matrices

Denote by $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bT}{\mathbb{T}}$ $\bT^N$ the real torus $$\mathbb{T}^N :=\bigl\lbrace\vec{z}\in\bC^N;\;\;|z_1|=\cdots =|z_N|=1\bigr\rbrace$$ To each ...
10
votes
1answer
344 views

relationship between eigenvalues of (A-B) and eigenvalues of (A^2-B^2)

Let us suppose that $A_{n}$ and $B_n$ are sequences of positive definite matrices satisfying $c\leq \lambda_{\min}(A_n)\leq \lambda_{\max}(A_n)\leq C$ and $c\leq \lambda_{\min}(B_n)\leq ...
3
votes
3answers
1k views

Integral kernel for the resolvent of the laplace operator

Consider the Laplace operator defined in the biggest possible subset of $L^2(\mathbb{R}^2)$ and let $z \in \mathbb{C}\backslash\mathbb{R}$. Therefore $z \notin \sigma (\Delta)$ the spectrum of ...
7
votes
2answers
511 views

What are first eigenfunctions of Laplacian for $CP^n$ with Fubini-Study metric?

I know the round $n$-sphere has $f_i=\cos(dist(e_i, x))$ as the set of first eigenfunctions for $e_i=(0, \cdots, 1, \cdots, 0)\in \mathbb R^{n+1}$. i.e. $\Delta f_i=\lambda_1 f$, where $\lambda_1$ is ...
1
vote
2answers
6k views

What does multiplying a matrix by its transpose have to do with spectral theorem? [closed]

What does multiplying a matrix by its transpose have to do with spectral theorem? I basically am trying to understand what this would mean with regards to spectra of waves. I think it give you a ...
4
votes
1answer
198 views

Simplicity of eigenvalues of an elliptic operator under a symmetry assumption

A striking difference in the spectral analysis of 2nd order elliptic boundary-value problems between one and several space dimensions is the following. In one space dimension, the eigenvalues are ...
1
vote
1answer
146 views

Maximal spectrum of a complex, unital and commutative Banach-algebra

Let $A$ be a complex, unital and commutative Banach-algebra. Question: Is the maximal spectrum $Max(A)$ of $A$ endowed with the topology induced by the prime spectrum $Spec(A)$ of $A$, Hausdorff? ...
1
vote
2answers
291 views

Exponential stability in nonlinear differential equations

I have this nonlinear differential equation $d\textbf{x}/dt=f(\textbf{x})$, where $\textbf{x}\in \mathbb{R}^n$. There are results which guarantee the convergence of the dynamical system to ...
6
votes
0answers
268 views

Paving conjecture for Toeplitz matrices

Let me first recall what is the so-called paving conjecture: for any $\epsilon >0$, there exists $r\in \mathbb N$ such that for any bounded operator $A$ on $\ell^2(\mathbb Z)$, there exists a ...
2
votes
1answer
225 views

Partial order on self-adjoint extensions?

Is there a natural partial order and/or lattice structure on the set of closed symmetric or self-adjoint extensions of a densely defined, unbounded, symmetric operator on a Hilbert space? Any ...
6
votes
1answer
265 views

A doubt about the parts of the spectrum of tensor products

Let $\mathcal{H}$ be any complex Hilbert space of infinite dimensional. By an operator $T$ I mean a linear bounded transformation from $\mathcal{H}$ into $\mathcal{H}$, i.e, ...
4
votes
1answer
347 views

Finding the spectrum of the composition of a projection with a multiplication operator

In reading a paper on numerical quadrature I've come across a result that is proved in a manner that is very clever: Let $X \subset \mathbb{C}$ be a compact, convex set. If $U$ is a ...
1
vote
1answer
308 views

A is a nonnegative matrix; the only principal submatrix having spectral radius above 1 is A itself

Let $\rho(M)$ denote the spectral radius (modulus of the largest eigenvalue) of a square matrix $M$. I am looking for a characterization or anything else interesting about the set of matrices $A$ ...
3
votes
1answer
163 views

Effects of unitarian multiplication into the spectrum of a finite matrix.

I am interested in the following problem: Let $P$ be a $n\times n$ complex finite matrix such as $PP^\dagger =W$. Given $W$, what can I say about the spectrum of $P$? This matrix "square-root" has ...
4
votes
1answer
195 views

Well defined Tensoring of spectral triples

Hi, I have a misunderstanding that I am hoping is really quite trivial. I will give my question directly and context below for those that need/want it. Question: In connes standard model he takes ...
5
votes
4answers
3k views

Eigenvalues of infinite matrices [closed]

I am trying to find some literature on infinite matrices because I want to know how to get the eigenvalues of infinite matrices. Seriously, it seems there are very few references available. Can ...
3
votes
1answer
173 views

Avalanche Principle for higher dimensional unimodular matrices ?

Hello everyone, I have a quick question for people working on quasi-periodic Schrodinger operators, Lyapunov exponents for Schrodinger cocycles or in other fields that might make them aware of this ...
1
vote
0answers
83 views

Decay rate of Discrete Prolate Spheroidal Sequences in frequency

What is the decay rate of DPSS sequences in frequency? Consider an interval $T\subset\mathbb{Z}$ of length N in time. Consider another interval $[-W,W]$ in frequency with $W<1/2$. Let $\phi_0$ ...
1
vote
0answers
145 views

Distance between probability amplitude functions

Suppose we have two probability measures $P_1$ and $P_2$ on some Riemannian manifold $(\Sigma,g)$. There are many potential distance measures between $P_1$ and $P_2$: The Wasserstein distance For ...
1
vote
0answers
142 views

Weyl quantization and convexity

Let $C$ be a convex subset of $\mathbb R^{2n}$ and $\mathbf 1_C$ be the characteristic function of $C$. Is it true that $$\forall u\in\mathscr S(\mathbb R^n),\quad \langle\mathbf ...
1
vote
1answer
147 views

Weyl asymptotics vs. form perturbations

Consider Hilbert spaces $V$,$H$; a closed quadratic form $a$ with domain $V$; and its associated operator $A$ on $H$. (If necessary, the form can be assumed to be coercive.) For the sake of ...
1
vote
1answer
246 views

Fourier inversion formula for complex-valued random variables?

The characteristic function of a complex-valued random variable $X$ with pdf $\mu$ is given by $$ \phi(t) = \int \exp[i \Re(\bar{t} X)] \; d\mu $$ (or, so says Wikipedia). How does one recover the ...
1
vote
1answer
112 views

Morse index and permutation of diagonal entries of a symmetric matrix

Do there exist results concerning preservation or not of the Morse index of a symmetric matrix $A$, after permuting its diagonal entries, and keeping fixed the off--diagonal ones? Thanks!
1
vote
1answer
134 views

LSI for Gaussian measure in $({\mathbb{R}^d})^{\mathbb{Z}^d}$

I am looking for a reference: Does Gaussian measure satisfy Logarithmic Sobolev Inequality (LSI) in $\({\mathbb{R}^d}\)^{\mathbb{Z}^d}$. Thanks.
1
vote
0answers
156 views

Banach Algebras and the peripheral spectrum

Here is a little theorem that I'm trying to prove. I haven't seen it in literature before, but I think the applications will be quite useful, particularly in the context of Banach algebras. Denote ...
5
votes
3answers
647 views

Why do we distinguish the continuous spectrum and the residual spectrum?

As we know, continuous spectrum and residual spectrum are two cases in the spectrum of an operator, which only appear in infinite dimension. If $T$ is a operator from Banach space $X$ to $X$, $aI-T$ ...
3
votes
1answer
286 views

Weyl law for SL(2,C)

Are there any estimates for the eigenvalues of the Laplace operator for $\Gamma \backslash SL(2, \mathbb{C})/SU(2)$ known beyond the main term? Here, $\Gamma$ should be congruence subgroup in ...
6
votes
4answers
609 views

Multiplicity one conjecture

I recently became interested in Maass cusp forms and heared people mentioning a "multiplicity one conjecture". As far as I understood it, it says that the dimension of the space of Maass cusp form for ...
1
vote
1answer
657 views

Some Functional Analysis Questions (Laplace Operator And Fourier Transform)

Given a set of the k first eigenvalues $ (\lambda_i)_ 1 ^k $ of some operator , and a set of the first k orthomormal eigenfunctions for these eigenvalues : $ ( \phi_i ) $ . Define: $ \Phi(x,y) = ...
2
votes
2answers
472 views

prove that flat shape maximizes a functional

The following functional arises in an information theoretic problem that I work on currently. $I(G(\omega)) = \int_{-\kappa\pi}^{\kappa\pi} \frac{A}{G(\omega)+A}d\omega-\frac{| ...
4
votes
2answers
692 views

Sum of two essentially self-adjoint operators

Hi, I hope this question will make more sense than the one I posted yesterday. I have two operators $p$ and $q$ which are essentially self-adjoint on a common domain $D$. Now I define $A = c_1 p + ...
5
votes
3answers
466 views

Has the largest-to-rest eigenvalue ratio of real symmetric matrices been researched before?

I'm investigating the eigenvalue ratios $$ \frac{\lambda_1}{\sum_{j=2}^N\lambda_j} \quad\mbox{and}\quad \frac{\sum_{j=1}^N\lambda_j}{\sum_{j=2}^N\lambda_j} $$ of the NxN matrix $B=AA^T$. $\lambda_1$ ...
2
votes
0answers
241 views

Eigen-decomposition perturbation

Let $A$, $B$ and $A_k + B$ be symmetric matrices with eigenvalues $\sigma_1 \geq \sigma_2 \ldots \geq \sigma_n$, $\rho_1 \geq \rho_2 \ldots \geq \rho_n$ and $\lambda_1 \geq \lambda_2 \ldots \geq ...
4
votes
1answer
332 views

Spectrum of $L^\infty(X,\mu)$

Suppose that $(X,\Sigma,\mu)$ is a measured set with respect to $\sigma$-algebra $\Sigma$. Suppose that $L^\infty(X,\mu)$ is the set of all $\mu$-equal bounded $\Sigma$-measurable functions on $X$. ...
6
votes
3answers
177 views

Stability of the spectrum for perturbations of the boundary

Consider the Laplace operator on a smooth bounded open set with Dirichlet boundary conditions. I need some result of the following type: if one perturbs the boundary in a suitable sense to be ...
1
vote
0answers
116 views

showing convergence of a function recursion relation

I have obtained (formally) a perturbative solution $$ H(y) = \sum_{n=0}^\infty \delta^n H_n(y) $$ to the following integro-differential equation ($\delta$ is a small constant, $\nu$ is a L\'evy ...