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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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
237 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 $f:\left[a,b\right]\rightarrowℝ$ such that $\int^{b}_{a}\left|f\left(x\right)\right|^{2}dx<\infty$ (Lebesgue integral). Then we can ...
1
vote
2answers
300 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
454 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
940 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
449 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
159 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
503 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
676 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
183 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
233 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
336 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
405 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
311 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
297 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 ...
1
vote
2answers
1k 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
794 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
629 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
397 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
180 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
483 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
785 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
626 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
554 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
944 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
852 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
695 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
826 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
780 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
388 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
527 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
291 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
431 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
308 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
655 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
453 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
557 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 ...
10
votes
5answers
853 views

Can always a family of symmetric real matrices depending smoothly on a real parameter be diagonalized by smooth similarity transformations?

This question is related to another question, but it is definitely not the same. Is it always possible to diagonalize (at least locally around each point) a family of symmetric real matrices $A(t)$ ...
3
votes
4answers
973 views

Boundaries of the eigenvalues of a symmetric matrix (or of its Lapacian)

Given the adjacency matrix $A_{ij}$ of a graph with $N$ vertices and $M$ links (or any binary symmetric matrix of size $N \times N$), is it possible to establish lower and upper boundaries of its ...
2
votes
1answer
385 views

orthonormal basis of eigenvectors for laplacian on a concave polygon

I am interested in the Laplace operator $\Delta$ on a concave polygon. When the polygon is convex, it is known that $\Delta: H^2(\Omega) \rightarrow L^2(\Omega)$ is boundedly invertible. In addition, ...
0
votes
2answers
641 views

Convergence of eigenvectors

Let $T$ be a compact operator on $l^2$. Let $T_n$ be finite rank operators and $T_n \to T$ in the operator norm. Is it true that the eigenvalues and eigenvectors of $T_n$ converge to eigenvalues and ...
0
votes
0answers
658 views

Spectral decomposition of the shift operator on $\ell_p(N)$

In this article http://en.wikipedia.org/wiki/Decomposition_of_spectrum_(functional_analysis) spectrum decomposition of the shift operator on $\ell_p(N)$ has been discussed. Question: Is it possible ...
3
votes
1answer
357 views

Estimating laplace-beltrami spectra for a graph surface in $R^3$

Consider a surface $\Gamma$ in $R^3$. The surface $\Gamma$ is a graph, i.e. $\Gamma = (x,y, h(x,y))$, for $x \in R^2$ and some smooth function $h$, where $h$ and all its derivatives are periodic on ...
3
votes
1answer
608 views

Laplace-deRham operator for 1-forms on the sphere

What do the eigenforms of the 1-form Laplace-de Rham operator look like on the 2-sphere, seen as vector fields via the inner product? For the standard Laplace-de Rham operator on 0-forms (functions) ...
4
votes
1answer
571 views

Tridiagonal Matrix

What is the most efficient way to calucate the dominant eigenvector of a real symmetric tridiagonal matrix, and what's the corresponding time complexity bound? Could someone give me a reference for ...
4
votes
2answers
422 views

An analogue of Hilbert-Schmidt theorem for multilinear forms

Let $H$ be a (the) real separable Hilbert space. The Hilbert--Schmidt theorem says that a compact self-adjoint operator $A$ has an eigenfunction expansion. Instead of operator, we can think of a ...
2
votes
1answer
225 views

Singular values of differences of square matrices

Suppose $A, B \in \mathbb{R}^{n \times n}$. Let $\sigma_1(A),\ldots,\sigma_n(A)$ be the singular values of $A$, and let $\sigma_1(B),\ldots,\sigma_n(B)$ be the singular values of $B$. If I know these ...