**0**

votes

**2**answers

227 views

### Spectral decomposition of compact operators

Let T be a compact operator on an infinite dimensional hilbert space H. I am proving the theorem which says that $Tx=\sum_{n=1}^{\infty}{\lambda}_{n}\langle x,x_{n}\rangle y_{n}$ where ($x_{n}$) is an ...

**3**

votes

**1**answer

132 views

### A.C. spectrum of the non additive perturbation BAB of a self-adjoint operator A where B is strictly positive

If have the following problem:
Let $A : \mathcal{H} \to \mathcal{H}$ be a bounded, self-adjoint operator on some Hilbert space $\mathcal{H}$. Let $B: \mathcal{H} \to \mathcal{H}$ be a bounded, ...

**0**

votes

**1**answer

66 views

### Spectrum of an angular-momentum related operator

Could someone please give me a reference for the eigenvalues and eigenstates of operators related to the angular momentum of a spinless, non-relativistic 2-D quantum particle?
In particular, I'm ...

**1**

vote

**0**answers

57 views

### Recovering Spherical Harmonics from Discrete Samples

Consider a collection of $N$ points on the 2-sphere chosen uniformly at random. Let's say that there's an edge between two such vertices if their geodesic distance is less than $r_N$. The resulting ...

**1**

vote

**1**answer

73 views

### Neighborhood overlap matrix for a bipartite graph

Let $G$ be an undirected, simple, bipartite graph with parts $V$ (having $n$ vertices) and $W$ (having $m$ vertices). Define the following $n$-by-$n$ matrix: for any $i,j \in V$, $$a_{ij} = |N_i \cap ...

**1**

vote

**0**answers

49 views

### interpretation of generalized eigenvalue/vectors in spectral graph theory

Let us say I have a symmetric graph adjacency matrix A, a degree matrix D, a laplacian L (D-A). I have a generalized eigenvalue equation $Av=\lambda Lv$. Does the eigenvalue/vectors produced in this ...

**2**

votes

**0**answers

68 views

### How can I find the spectrum of this operator?

I've posted this now in on /r/math on math.se to no avail. Maybe this problem is harder than I thought and you folks can help me out.
I'm working on a variational problem in elasticity which ...

**7**

votes

**0**answers

123 views

### Phillips-Sarnak conjecture in higher dimension

The Phillips-Sarnak conjecture states that for a generic Fuchsian lattice the space of Maass cusp forms is finite-dimensional. Generic here means in particular non-uniform, non-arithmetic, no special ...

**2**

votes

**0**answers

77 views

### Laplacian on manifolds with corners

So far I've studied smooth riemannian manifolds and the Laplace-Beltrami operator associated to it. Therefore I know basic theorems like Stoke's theorem, divergence theorem, green's identities etc. ...

**1**

vote

**0**answers

163 views

### An optimization problem on the sphere

Let $S$ be a sphere centered at origin in $\Bbb R^{2n}$ of radius $\sqrt{2n}$. Let $D$ be a diagonal matrix. Let $U$ be an orthogonal matrix. Let $r\in\Bbb Z_+$ be a fixed integer.
Let vector ...

**4**

votes

**0**answers

95 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}
...

**3**

votes

**0**answers

142 views

### spectrum of a polygon and zeta function

Let $\Delta(x,y) = 1,0$ according to whether $(x,y)$ is in some polygon (symmetric with respect to the diagonal axis).
E.g. The convex hull of three points (taken from a paper on dominoes)
$$ ...

**2**

votes

**0**answers

35 views

### Anderson localization for non-i.i.d. potentials

Consider a random potential $V$ on edges of $\mathbb{Z}$ defined in the following way:
a) each edge starts with an i.i.d. $0$ or $1$
b) each edge chooses its new potential value according to some ...

**0**

votes

**1**answer

120 views

### Showing there is a unique spectral measure

All the books I have seen have proved that, for a normal bounded operator $T$, there is a unique spectral measure $E$ such that $\int_{\sigma(T)}^{}\lambda\,dE=T$ by first proving in it for a general ...

**1**

vote

**1**answer

78 views

### wavefront is a coisotropic

I'm thinking about the following example: Let $u_k:=e^{i(kx_1+k^2x_2)}$, where $k\in\mathbb{N}$ and $x_1,x_2\in\mathbb{R}$. Then for the sequence $h_k:=1/(k\sqrt{1+k^2})$ ($h_k\rightarrow0$ as ...

**0**

votes

**0**answers

52 views

### Fractional power and norm's operators

Let $T$ be a linear densely defined operator on a Hilbert space $H$ and $L$ be a selfadjoint operator with discrete spectrum such that $L^{-1}$ is bounded
and $$\|Tf\| \leq \Phi_{\eta}(f) + \eta ...

**0**

votes

**1**answer

88 views

### Does spectral theory assume separability

On an infinite dimensional space, the spectral theorem for compact normal operators says that the eigenvectors form an orthonormal basis which, from wikipedia, is equivalent to the space being ...

**5**

votes

**1**answer

116 views

### Is irreducibility sufficient for uniqueness of invariant distribution for a Feller Semigroup?

Let $(T_t)$ be a strongly continuous semigroup of positive operators on $C(K)$, where $K$ is a compact space. Assume also that $T_t1 =1 $ for every $t\geq 0$.
(This is also called a Feller Semigroup.)
...

**0**

votes

**0**answers

32 views

### Gordon Lemma for Jacobi operators

The Gordon lemma is a useful tool in the theory of 1-D discrete time-independent SchrÃ¶dinger operators that exploits local repetition in the potential to prove absence of point spectrum.
Has anyone ...

**15**

votes

**2**answers

302 views

### Has Witten's perturbation on de Rham complex been studied on other elliptic complexes?

In his famous work, Supersymmetry and Morse theory, Witten perturbs de Rham complex by perturbing the exterior derivative
$$d_h=e^{-ht}de^{ht}.$$
And he proves Morse inequality using some spectral ...

**7**

votes

**3**answers

191 views

### Spectrum of Dirichlet Laplacian on a Parallelogram

Let $ M \subset \mathbb{R}^2 $ be parallelogram constructed by putting together two equilateral triangles (so that all sides of the parallelogram have length 1, and the internal angles are 60 and ...

**0**

votes

**0**answers

21 views

### Vanishing of non commutative ( Wodzicki) residue on pseudo differential projections

Its a known fact that the non-commutative (Wodzicki) residue of a pseudo-differential projection is always zero.
My question is:
Is it possible to get this result by looking at structure of the ...

**2**

votes

**2**answers

113 views

### What is the right initial domain for the Dirichlet-Laplacian on a bounded domain?

Given some bounded domain $\Omega\subset \mathbb{R}^n$ with sufficiently regular boundary (e.g. smooth boundary). Then I saw two slightly different definitions for the Dirichlet-Laplacian.
Some books ...

**2**

votes

**1**answer

452 views

### Condition numbers of Vandermonde matrices

Denote by $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bT}{\mathbb{T}}$ $\bT^N$ the real torus
$$\mathbb{T}^N :=\bigl\lbrace\vec{z}\in\bC^N;\;\;|z_1|=\cdots =|z_N|=1\bigr\rbrace$$
To each ...

**-1**

votes

**1**answer

143 views

### Relation of Hodge Theorem to Eigenfunction Basis of Laplacian

The classical Hodge theorem I know of relates the de Rham cohomology groups isomorphically to the space of harmonic forms and shows that $Id=\pi+\Delta G$, where $\pi$ is the harmonic projection of ...

**3**

votes

**0**answers

74 views

### Classes for which the Spectrum determines a Convex Shape

Given a planar domain $\Omega \subset \Bbb{R}^2$ bounded and open we can associate to it the spectrum of the Laplace operator with Dirichlet boundary condition. It is known that there are planar ...

**2**

votes

**2**answers

174 views

### Schrodinger's equation via Spectral Theorem

How do you prove basic facts on the Schrodinger equation using the spectral theorem? More precisely, here is what I have in mind.
The version of the Spectral Theorem I am familiar with is the ...

**1**

vote

**1**answer

114 views

### Relation of spectrum

Let $X$ denotes a complex $C^*$-algebra and $\{Z(t)\}_{t\geq 0}$ is a $C_0$-Semigroup of operators on $X$. If for $x\in X$, I have $x=x^*$ (x is selfadjoint), and its spectrum $\sigma(x)\subset ...

**0**

votes

**1**answer

173 views

### Spectral measure and Stone's theorem

Let $T$ be an unbounded self-adjoint operator on a Hilbert space and let $E(\lambda )$ be the associated spectral measure and $R(\lambda ) = (T-\lambda )^{-1}$ the resolvent. By Stone's theorem we ...

**9**

votes

**1**answer

283 views

### Spectrum of Laplacian in non-compact manifolds

What can be said about the spectrum of the Laplace-Beltrami operator on a non-compact, complete Riemannian manifold of finite volume? For example, is the point spectrum non-empty?
What would be a ...

**1**

vote

**1**answer

245 views

### Asymptotic of the heat kernel

This is the same question I asked in stackexchange:
http://math.stackexchange.com/questions/519152/asymptotic-of-the-heat-kernel
I read Proposition 3.23(page 101) in Rosenberg's book "The Laplacian ...

**1**

vote

**0**answers

101 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, ...

**27**

votes

**3**answers

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

**1**answer

202 views

### Separating the spectrum of a Hermitian matrix

Given Hermitian matrix $A$, I would like to perturbate it so that its eigenvalues become well-separated.
Specifically, let $A$ be some Hermitian matrix, and let $G$ be a Gaussian matrix, with each
...

**1**

vote

**0**answers

71 views

### “Almost orthogonalizing” matrices using a signature matrix

Suppose $A$ and $B$ are two real symmetric $n \times n$ matrices (If simpler, consider $A$ and $B$ to be 0/1 matrices, say, adjacency matrices of d-regular graphs).
Then $||AB||_{op} \leq ...

**4**

votes

**1**answer

73 views

### Examples of optimal ultracontractivity estimates for a Markovian semigroup $T_t$ that do not depend polynomialy on $t$

Let $(X,\mu)$ be a measure space and $T_t : L_2(\mu) \to L_2(\mu)$ for $t \geq 0$ a symmetric Markovian semigroup. Local ultracontractivity estimates of the form:
$$
\| T_t : L_p(\mu) \to L_q(\mu)\| ...

**9**

votes

**0**answers

137 views

### Avoiding integers in the spectrum of the Laplacian of a Riemann surface

Let $\Sigma^2$ be an orientable compact surface of genus $gen(\Sigma)\geq2$, and denote by $\mathcal M(\Sigma)$ the moduli space of hyperbolic metrics on $\Sigma$, i.e., Riemannian metrics of constant ...

**14**

votes

**2**answers

569 views

### Eigenvalues of an “oblique diagonal” matrix

I am looking for guidance about the behavior of powers of a particular matrix (call it $A_n$ for $n\ge2$), which has come up in a counting problem about quantum knot mosaics (a good reference for ...

**0**

votes

**0**answers

65 views

### Spectrum of an operator from Transpose sum

I was wondering if there is anything we can know about the spectrum of an operator $A$ if we know that $M = A + A^{T}$ is a positive operator?

**3**

votes

**0**answers

62 views

### Origin of spectral theory on infinite-area hyperbolic surfaces

The study of spectral theory of finite-area hyperbolic surfaces is intimately related to number theory, in particular by the importance of Maass cusp forms. The counting of resonances is of ...

**1**

vote

**0**answers

114 views

### Constructing an $\epsilon$-net for a Lipschitz subspace of $L^2$

Let $X$ be a subset of $L^2([0,1])$ which contains only Lipschitz function.
Also, the member of $X$ are uniformly bounded
$$
|x(t)| < M, \text{ for all $x \in X$ and $t \in [0,1]$}.
$$
Let $F: X ...

**5**

votes

**1**answer

123 views

### Eigenvalues vs resonances

I understand that for infinite-area hyperbolic surfaces, there are no $L^2$-eigenfunctions of the Laplace-Beltrami operator but there are a lot resonances.
But I am confused about the notion of ...

**2**

votes

**1**answer

155 views

### positive semidefinite matrix condition

There is a great work of Alizadeh that in section 4 speaks about Minimizing sum of the first few(k-largest) eigenvalues of a symmetric matrix. Instead of a symmetric model we use the weighted ...

**2**

votes

**0**answers

78 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 ...

**0**

votes

**1**answer

174 views

### Trace, eigenvalues and functional calculus

Let $T$ be a (possibly unbounded) self-adjoint operator on a Hilbert space. Assume that we for some reason know that the point spectrum of $T$ consists of a finite number of eigenvalues $\lambda _1, ...

**0**

votes

**0**answers

92 views

### Perturbation of spectrum and eigenspaces

Let $A \in \mathbb{C}^{n \times n}$ be an $n \times n$ matrix. Consider the rank-$1$ perturbation $A'$ of $A$ given by replacing a column $v$ of $A$ by $\alpha \cdot v$, where $\alpha \in [0, 1)$. Can ...

**2**

votes

**0**answers

61 views

### What are the boundary asymptotics of harmonic symmetric transverse traceless rank-s tensors on $\mathbb{H}^n$ in the Poincare upper-half-space model? [closed]

This question is motivated by the results in this paper, http://calvino.polito.it/~camporesi/JMP94.pdf
In this paper some of its most important results about the asymptotics of symmetric traceless ...

**19**

votes

**9**answers

2k views

### real symmetric matrix has real eigenvalues - elementary proof

Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers?

**3**

votes

**2**answers

179 views

### A version of the spectral theorem for group actions

Suppose $G$ is a sufficiently nice (maybe locally compact and abelian) group which acts on the separable Hilbert space $\mathcal{H}$ by unitary transformations. Is there a generalization of the ...

**5**

votes

**1**answer

642 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] ...