**1**

vote

**1**answer

148 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

**1**answer

256 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

**1**answer

114 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

**1**answer

135 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

**0**answers

158 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

**3**answers

672 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

**1**answer

289 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

**4**answers

629 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

**1**answer

684 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

**2**answers

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

**2**answers

722 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

**3**answers

483 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

**0**answers

244 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

**1**answer

342 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

**3**answers

181 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

**0**answers

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 ...

**5**

votes

**1**answer

366 views

### About the quantum spectrum of a certain potential.

Intuitively one understands that if one is solving the Schroedinger's equation for energies $E$ such that $\{ x \vert U(x)\leq E \}$ is compact (..is there a weaker criteria?..) then the spectrum ...

**13**

votes

**2**answers

1k 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 ...

**13**

votes

**1**answer

1k 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

**1**answer

177 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

**2**answers

403 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 ...

**9**

votes

**2**answers

1k 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 ...

**6**

votes

**3**answers

3k 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

**0**answers

233 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

**3**answers

220 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

**0**answers

82 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

**0**answers

85 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.)
...

**3**

votes

**2**answers

576 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

**0**answers

253 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

**1**answer

196 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

**0**answers

150 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

**2**answers

625 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

**1**answer

516 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

**0**answers

484 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 + ...

**7**

votes

**1**answer

657 views

### 0 eigenvalue for a symmetric tridiagonal matrix

Let $T\in \mathbb{R}^{n\times n}$ be a symmetric tridiagonal matrix having the off--diagonal entries equal to -1. The diagonal entries are all positive, $a_i>0$, $i=\overline{1,n}$, and there ...

**3**

votes

**1**answer

237 views

### Estimating spectral radius with a Gaussian vector

Suppose I'm trying to estimate the spectral radius of a square $n \times n$ matrix $A$,
and let $N$ be a distribution over Gaussian i.i.d. vectors of length $n$.
Is the following lemma true:
If the ...

**4**

votes

**1**answer

263 views

### Spectral Properties of $A(I-A)^{-1}$

I am working with a class of matrices $A$ which are non-negative-definite, not symmetric, and have maximum eigenvalue less than 1. I am interested in the spectral properties of the matrix $H = A(I - ...

**8**

votes

**2**answers

488 views

### Efficiently computing a few localized eigenvectors

Let $H = \triangle + V(x) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$. I am interested in domain decomposition for an eigenproblem involving $H$.
The lowest 1000 eigenfunctions of $H$, $ \psi_i $, can ...

**8**

votes

**3**answers

636 views

### Functions of Pseudodifferential Operators

Suppose I have a self-adjoint pseudo-differential operator $A$ on $\mathbb{R}^n$ and a continuous function $f$ (possibly bounded, or Schwartz, or compactly supported) on its spectrum. Then I can ...

**7**

votes

**1**answer

393 views

### Is the Cheeger constant of an induced subgraph of a cube at most 1?

It is known that the
Cheeger constant
of a
hypercube graph $Q_n$
is exactly $1$, regardless of its dimension $n$. Is $1$ also an upper bound
on the Cheeger constant of nontrivial induced connected ...

**0**

votes

**1**answer

209 views

### Spectrum of the operator PAP, with A self-adjoint and P strictly positive

Let $A$ be an unbounded self-adjoint operator with spectrum $\sigma(A)=\mathbb R$ in a Hilbert space $\mathcal H$. Let $P$ be a bounded operator in $\mathcal H$ satisfying $P\ge1$ and
$$
{\rm ...

**5**

votes

**3**answers

440 views

### Traceless GUE : Four Centered Fermions

The proof of the Wigner Semicircle Law comes from studying the GUE Kernel
\[ K_N(\mu, \nu)=e^{-\frac{1}{2}(\mu^2+\nu^2)} \cdot \frac{1}{\sqrt{\pi}} \sum_{j=0}^{N-1}\frac{H_j(\lambda)H_j(\mu)}{2^j j!} ...

**1**

vote

**0**answers

177 views

### Norm related to diophantine approximation?

I'm trying to read this paper:
http://www.springerlink.com/content/g0046660260825x3/
or http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.20.1813&rep=rep1&type=pdf
But I don't ...

**0**

votes

**0**answers

116 views

### singular values

Hello everyone,
Are there any known conditions to ensure that the singular value of a matrix A is smaller than 1 ?
More specifically, in my case A is the product of an M-Matrix and an inverse ...

**9**

votes

**3**answers

1k views

### The first eigenvalue of the laplacian for complex projective space

What is the exact value of the first eigenvalue of the laplacian for complex projective space viewed as $SU(n+1)/S(U(1)\times U(n))$?

**3**

votes

**2**answers

238 views

### Analogue of PSD matrices for permanents?

Let's begin with a few observations. Suppose we consider the set of $N\times N$ matrices and consider the matrices with positive determinant. There are several connected components in this set; let ...

**21**

votes

**4**answers

1k views

### Can $L^{2}$ be represented as a space of functions (not equivalence classes)?

Let $X$ be the vector space of all Lebesgue-measurable functions $f:\left[a,b\right]\rightarrowℝ$ such that $\int^{b}_{a}\left|f\left(x\right)\right|^{2}dx<\infty$ (Lebesgue integral). Then we can ...

**1**

vote

**2**answers

310 views

### Showing a solution of elliptic PDe is non-degenerate

Dear Mathoverflowers:
I am interested in radial positive solutions of
$-\Delta u(r) = r^\alpha u(r)^p$ in the unit ball in $ R^N$ with $ u=0$ on the boundary.
Here $p>1$ and $ \alpha >0$. ...

**3**

votes

**2**answers

459 views

### Number of perturbations of the Jordan form

I am looking for information about the number of Jordan forms that can be obtained from a given Jordan form of a small perturbation.
For example, if a Jordan form consists of a single cell 2x2
...

**1**

vote

**3**answers

982 views

### Generalization of eigenvalues/vectors to modules?

What is the generalization of eigenvalues/vectors to modules?
To be specific, given a "vector" v in a module over some ring, and a linear "operator" O from the module to itself (please feel free to ...