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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2
votes
1answer
494 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
335 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
915 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
434 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
2k 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
170 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
140 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
249 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
232 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
206 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 ...
4
votes
1answer
277 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
236 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
156 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
190 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 ...
6
votes
4answers
2k 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
158 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
77 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
130 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
131 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
140 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
220 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
108 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
130 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
152 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
556 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
268 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
3answers
468 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
532 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
469 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{| ...
3
votes
2answers
595 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
411 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 $B=AA^T$. $\lambda_1$ ...
4
votes
1answer
290 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 $X$. ...
1
vote
0answers
115 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 ...
13
votes
2answers
902 views

Minimum off-diagonal elements of a matrix with fixed eigenvalues

Hello, I am en engineer working in radar research. I came accross a problem I cannot seem to find math literature on it. I can ask it in two different ways. Perhaps depending on the reader, the ...
11
votes
1answer
921 views

Relationship between Green's function and geodesic distance?

I am interested in showing that a certain Green's function can be used to approximate the distance function on a Riemannian manifold in the following sense. Let $(M,g)$ be a Riemannian manifold and ...
2
votes
1answer
144 views

Generalizing the spectral radius of a unistochastic matrix

Consider a square matrix $A$, and from it construct $B$ whose entries are the squared magnitudes of those in $A$. What can we say about the spectral radius of $B$? I know that for a unitary matrix ...
1
vote
2answers
345 views

Lebesgue integral with respect to vector measures?

Good evening, I'm reading some papers of Jim Agler and Nicholas Young, in which they prove a formula of integral representation with respect to a vector measure, but the integration is in the sense ...
8
votes
2answers
929 views

What is a good reference that compact resolvent implies Fredholm operator?

Suppose $A \in \mathcal{L}(E_1, E_0)$ is a bounded linear operator between Banach spaces $E_1$ and $E_0$, and we also have that $E_1$ is densely, continuously embedded in $E_0$ (i.e. $A$ can be ...
3
votes
2answers
2k views

Interesting relationships between Cholesky decomposition and diagonalization

Let $\Sigma$ be a hermitian positive definite matrix and $L$ be it's Cholesky decomposition so that $LL^\ast=\Sigma$. Furthermore, let's diagonalize $\Sigma$ as $\Sigma = P\Lambda P^\ast$. $\Lambda$ ...
1
vote
0answers
216 views

Matrix conditions under which spectral radius is smaller than 1?

Hello everyone, I would like to find out which conditions are necessary so that the spectral radius $\rho(M)<1$ where $M$ represents the following matrix: $M = \left( \begin{array}{ccc} W & 0 ...
4
votes
3answers
217 views

Conditions ensuring an order betweenthe smallest eigenvalues of two positive definite Jacobi matrices

Let $J, L$ be two symmetric positive definite tridiagonal matrices of positive diagonal entries, $\mbox{diag}(J)=(a_1, a_2, \ldots, a_n)$, $\mbox{diag}(L)=(\alpha_1, \alpha_2, \ldots, \alpha_n)$, ...
1
vote
0answers
78 views

Given the Fourier coefficient moduli, how to choose the phases to have integer components?

Take $n\geq 1$, and let $c_1,...,c_n$ be $n$ non-negative numbers. For every $\phi_1,...,\phi_n$, the formulae $$v_k=\sum_{j=1}^n c_k \omega^{jk+\phi_k}$$ define a vector $v\in \mathbb{R}^n$, where ...
2
votes
0answers
78 views

Searching for inequalities relating a convolution-type integral of functions of modulus less than but close to one.

Suppose $f(x,y)$ and $g(x,y)$ are both measurable functions from $[0,1]\times[0,1]\to \mathbb{C}$ with $|f|,|g|<1$, and let $h(x,y)=\int_{0}^1 f(x,z)g(z,y) \ dz$. (So $|h(x,y)|<1$ also.) ...
2
votes
2answers
543 views

Exotic spectrum of Laplace operator

Given a closed Riemannian manifold and a generalized Laplace $\Delta$ operator, it is well known that $\Delta$ has discrete spectrum $(\lambda_n)_n$ (arranged in a increasing way, not counting ...
1
vote
0answers
221 views

weak mixing and spectral theorem

I'm trying to prove an equivalent statement about weak mixing transformations that relies on the spectral theorem, but I can't find a reference to fill in the last details. A hint for solving it or ...
7
votes
1answer
190 views

Singular values of $X+iY$ where $X$ and $Y$ are Hermitian

Lets have two Hermitian $n\times n$ matrices $X$ and $Y$. Are there any known properties of the singular values of $$Z = X + i Y.$$ I am the most interested in bounding from above a few first ...
1
vote
0answers
146 views

Inequalities between self-adjoint operators

Let $T_s$ ($s\ge0$) be a smooth family of non-negative self-adjoint operators in a separable Hilbert space $H$. Suppose that, for some $C'>C>0$, we have $T_0+Cs^2\le T_s\le T_0+C's^2$ for all ...
6
votes
2answers
617 views

analytic approximation of a non-negative matrix by a sequence of positive matrices

Let $L \in \{0,1\}^{n \times n}$ be a non-negative matrix whose row sum is 1. ($L$ stands for "limit"). It is known that there exists a unique $r \times r$ principal minor of $L$ that is a permutation ...
2
votes
1answer
441 views

Global Lichnerowicz Formula Proof (in the Kahler case)?

For a Kahler manifold $M$, let us denote its Dirac operator $\overline{\partial} + \overline{\partial}^\ast$, with respect to a metric $g$, by $D$. Moreover, let us dentoe the Levi-Civita connection ...
8
votes
0answers
447 views

Bounding sum of first singular values squared for Kronecker sum of traceless matrices

Let $A$ and $B$ be $4\times4$ traceless matrices with Hilbert-Schmidt norms summing up to $1/4$, i.e. $$\text{Tr}\left[ A\right]=\text{Tr}\left[ B\right] = 0,\qquad\text{Tr}\left[ A^\dagger A + ...