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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6
votes
1answer
569 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
1answer
226 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
1answer
253 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 - ...
7
votes
2answers
469 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 ...
7
votes
3answers
537 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
1answer
354 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
1answer
198 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
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!} ...
1
vote
0answers
172 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
0answers
113 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
3answers
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
2answers
235 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
4answers
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 $f:\left[a,b\right]\rightarrowā„¯$ such that $\int^{b}_{a}\left|f\left(x\right)\right|^{2}dx<\infty$ (Lebesgue integral). Then we can ...
1
vote
2answers
292 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
2answers
448 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
3answers
912 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 ...
14
votes
2answers
437 views

Is there a spectral theory approach to non-explicit Plancherel-type theorems?

Teaching graduate analysis has inspired me to think about the completeness theorem for Fourier series and the more difficult Plancherel theorem for the Fourier transform on $\mathbb{R}$. There are ...
6
votes
1answer
153 views

numerically track spectrum curves of a parameter dependent linear operator

Hi, I am interested in how to numerically track spectrum curves of a parameter dependent linear operator. Given a linear operator in square matrix form $M(t)$, where the matrix is smooth dependent of ...
4
votes
2answers
499 views

How do you solve linear systems whose solutions decay exponentially?

Consider the heat equation $$\dot{u} = \Delta u$$ with initial conditions $$u_0 = \delta(x)$$ for some point $x$ in the domain $\Omega$ of the problem. If $\Omega$ is $\mathbb{R}^n$, then this ...
5
votes
1answer
657 views

Generalising Gelfand's spectral theory

This is primarily a request for references and advices. Question (edited on 10/29/2011). What's known about comprehensive generalisations of Gelfand's spectral theory for unital [associative] ...
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 ...
2
votes
2answers
229 views

Smooth dependence of the spectrum on the operator

I would like to know if there are theorems that state under which circumstances spectra of operator families depend smoothly on the parameter. To clarify, suppose I have a 1-parameter family $T_h$ of ...
3
votes
3answers
330 views

Estimates for the diameter of a (nice) surface?

The Question Let $M$ be a compact, connected, orientable surface without boundary of bounded genus smoothly embedded in $\mathbb{R}^3$; define the diameter $d_M$ of $M$ as the maximum minimal ...
0
votes
1answer
383 views

Simple system of ODEs with periodic coefficients

I am stuck with a little problem that I cannot solve mith the standard methods I learn at university. I have a system of coupled ODEs: $f'(t) = P \cos(k t + \Phi_1) g(t)$ $g'(t) = Q \cos(k t + ...
1
vote
0answers
304 views

Definition of spectral gradient

Consider this differential operator $$ \mathcal{H}(\phi(\mathbf{x})) = -\triangle + V(\mathbf{x})H_\epsilon (\phi(\mathbf{x})) $$ where $\mathbf{x} \in \mathbb{R}^2$, $\phi : \mathbb{R}^2 \rightarrow ...
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 ...
0
votes
2answers
911 views

Diagonalization of a matrix of differential operators

Dear community, i have a question regarding differential operators acting on vector valued functions and how to "diagonalize" them. To explain my question i will use an example: Let $V^k$ be the ...
5
votes
1answer
758 views

How to construct a scalar differential operator having the same spectrum as a non-scalar differential operator exploiting symmetries?

I am interested in eigenvalue problems for differential operators acting on one forms on closed two-dimensional manifolds and how they relate to eigenvalue problems of associated operators acting on ...
5
votes
2answers
616 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 ...
0
votes
1answer
393 views

Spectral theory of real symmetric matrices with random diagonal elements

Can you point me in the direction of any research done on the spectral theory (i.e. eigenvalues and eigenvectors) of real symmetric matrices with random (Gaussian or Levy) diagonal elements and fixed ...
1
vote
0answers
178 views

Joint Convexity of Spectral functions of several matrices

$\{A_1 \ldots A_K \}$ is a set of matrices in $\mathbb{R}^{m \times n}$. Let $f (A_1,\ldots,A_K)$ be a function of the singular values of all matrices. For e.g., $f$ is just summation of singular ...
2
votes
0answers
446 views

Comparision of cubic hermite finite element and cubic B-spline finite element (in condition nunmbers of stiffness matrix, or sth else)

Background Consider the one dimensional second order elliptic PDE, $$ \left\{\!\! \begin{aligned} & -(a(x)u'(x))'+b(x)u(x)=f(x)\qquad x\in[0,1]\\ & u(0)=u(1)=0 \end{aligned} ...
2
votes
2answers
780 views

Perturbative solution to an Eigenvalue Problem with a continuous spectrum

I am trying to find an approximate solution to an eigenvalue equation using techniques from perturbation theory. Roughly speaking, the problem is as follows $L^s \phi_q^s = \lambda_q^s \phi_q^s$ ...
4
votes
1answer
615 views

What does $L^\infty_\varepsilon$ mean?

In Volume 4 of Reed and Simon on page 83 the authors refer to the space $(L^\infty(\mathbb{R}^3))_\varepsilon$, and later on page 119 they use $L^\\infty_\varepsilon$. Are these two spaces the same? ...
4
votes
1answer
533 views

Multiplicity of eigenvalues of the Laplacian on quaternionic projective space

Using the classic spherical harmonics theory, one obtains the $k$-th eigenvalue of the $n$-dimensional round sphere $S^n$ to be $k(k+n-1)$, and its multiplicity is $\binom{n+k}{k}-\binom{n+k-1}{k-1}$, ...
6
votes
2answers
931 views

What is the relationship amongst all the different kinds of spectra?

The word "spectrum" gets tossed around a lot in mathematics, and there seem to be a number of different concepts to which it applies. There is of course a physical connotation to the word which is ...
0
votes
1answer
838 views

laplacian for metrics on $S^n$

It is true that the restiction of the Laplace operator on $\mathbb R^n$ to functions on the sphere is the Laplacian for the round metric on the sphere. Is this true for any Riemannian metric $g$ on ...
4
votes
1answer
672 views

Growth of Laplacian eigenvalues on a compact domain?

Let $\mathcal{M}$ be a compact Riemannian manifold and let $\Delta$ be the (scalar) Laplace-Beltrami operator on $\\mathcal{M}$. Then $\Delta$ has a discrete spectrum and if we order its distinct ...
7
votes
1answer
778 views

First eigenvalue of the Laplacian on Berger spheres

Consider the Hopf fibrations $S^1\to S^{2n+1}\to CP^n$ and $S^3\to S^{4n+3}\to HP^n$. These are Riemannian submersions with totally geodesic fibers. Consider now their canonical variations (the ...
4
votes
3answers
646 views

Homogeneous linear differential equation system with simple periodical coefficient matrix

Hello, I encountered the following system of linear first-order differential equations: $y'(z)=A(z) y(z)$ where $y(z): R \rightarrow R^2$ and $A(z)=\begin{pmatrix} 0 & B Cos(\alpha z + \Phi_b) ...
12
votes
4answers
1k views

High multiplicity eigenvalue implies symmetry?

It is well known that on any compact Riemannian symmetric space $X$, the eigenvalues of the Laplacian have very high multiplicity (comparable with the Weyl bound), and the resulting actions ...
7
votes
2answers
376 views

Symmetric spaces, Horocycle spaces and intertwining operators

Let $G=KAN$ be an Iwasawa decomposition of a connected semisimple Lie group with finite center. Let us assume for simplicity that the associated symmetric space $G/K$ has rank 1. Harish-Chandras ...
2
votes
1answer
524 views

spectra of sums in (Banach) algebras

A similar question was already asked in question titled "Spectra of sums and products in (Banach) algebras [was: Spectrum in Banach Algebra]". Answer there led me to the following question. If for ...
1
vote
1answer
282 views

Can be this operator extended to an unbounded self-adjoint operator ?

Consider an enumeration $\{q_1,q_2,\ldots\}$ of $\mathbb{Q}\cap [1,\infty)$ and a orthogonal Schauder basis $\{e_1,e_2,\ldots\}$ of $\ell^2(\mathbb{N})$. Define $Ae_{2k-1}=e_{2k-1}$ and ...
5
votes
1answer
1k views

The Guinand-Weil explicit formula without entire function theory

I'll admit from the outset that this question is slightly vague. The actual question appears at the end of the post. The explicit formula of Guinand and Weil can be written in the following way: For ...
3
votes
1answer
409 views

eigenvalue problem on the geodesic ball of sphere

I have a question about eigenvalue problem on the geodesic ball in $n$-dimensional sphere $\mathbb{S}^n\subset\mathbb{R}^{n+1}$. Consider the eigenvalue problem in the geodesic ball ...
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 ...
4
votes
1answer
620 views

Why is the cuspidal spectrum discrete?

Hi, I have a short question concerning the spectral theory of automorphic forms. What is the main property of the unipotent group $N$, which consist of matrices in the form ...
3
votes
2answers
439 views

Localization of Laplacian eigenfunction on the unit square?

Let A be the unit square, $\{u_k\}$ is the set of all L2-normalized Laplacian eigenfunctions with Dirichlet boundary condition. Is it true that for any open subset V, $C_V = \inf\limits_k ...
3
votes
3answers
515 views

Boundness of Laplacian eigenfunctions

Let $A$ be a bounded domain in $\mathbb R^d$, $d>1$, and $\{u_k\}$ is the set of all $L^2$-normalized Laplacian eigenfunctions on $A$ with Dirichlet boundary condition (i.e., $\|u_k\|_2 = 1$). Is ...