**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

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

**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

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

**2**answers

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

**3**answers

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

**2**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**3**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**3**answers

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

**4**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**2**answers

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

**3**answers

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

**5**answers

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

**4**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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 ...