1
vote
0answers
48 views

Decay of Eigenfunctions for the 1D Discrete Random Schrodinger Operators

Consider the operator on $\ell^2(\mathbb{Z})$ $$ H = \Delta + v. $$ Here $\Delta$ is the nearest neighbour Laplacian on $\mathbb{Z}$, $\Delta_{k, \ell} =1 $ if $|k - \ell| =1 $ and zero otherwise, ...
4
votes
0answers
110 views

Solving a Fredholm equation with a piecewise kernel : Karhunen-Loeve of a stopped Brownian motion

Is there a way to solve analytically the Fredholm integral equation of the second kind $$ \int_0^{100} K(s, t) f(s) ds = \lambda f(t) $$ where the kernel has the piecewise 'linear' form \begin{align} ...
1
vote
0answers
109 views

expected inverse of circulant plus random diagonal

I have a deterministic circulant matrix $R$ and a random diagonal matrix $X$ where all elements are IID and positive. I need to determine the expected inverse of $R+X$, that is: Evaluate, in closed, ...
1
vote
0answers
83 views

Distribute Monte Carlo samples among dimensions

Simplified problem: Given a $d$-times nested convolution of an input function $g(x):\mathbb{R}\mapsto \mathbb{R}$ with the same band-limited smooth function $f(x):\mathbb{R}\mapsto \mathbb{R}$. I am ...
2
votes
1answer
144 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 ...
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
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 ...
3
votes
1answer
227 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 ...
5
votes
3answers
420 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!} ...
6
votes
1answer
182 views

Brownian particle with jump boundary condition

I would like to find a function $f(s)$, which solves the following equation: $ \int_0^t \int_0^L f(s,x) p(t-s,x,y) dy ds = 1 $ The function $p(\tau,x,y)$ is $p(\tau,x,y) = \sum_n e^{-\lambda_n ...
0
votes
1answer
293 views

Robust entropy-like measure for analyzing uncertainity

I'm looking for a measure to analysis the uncertainty observed in a set of variables (with multivariate Gaussian distribution). So, I've tried conventional Shanon entropy (differential entropy) which ...
5
votes
2answers
617 views

Literature on behaviour of eigenfunctions under multiplication?

Dear community, I would be happy about any literature or comments on the behaviour of the pointwise product of eigenfunctions of a self-adjoint operator with discrete spectrum, acting on a separable ...
4
votes
2answers
294 views

Is independence meaningful for commutative $C^*$-algebras?

I don't know very much about spectral theory so probably the answer to my question has a basic reference which I would appreciate. Let's say I have two self-adjoint operators on a Hilbert space and ...
27
votes
3answers
2k views

Perron-Frobenius “inverse eigenvalue problem”

The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...
5
votes
1answer
516 views

Spectrum of a generic integral matrix.

My collaborators and I are studying certain rigidity properties of hyperbolic toral automorphisms. These are given by integral matrices A with determinant 1 and without eigenvalues on the unit ...