**2**

votes

**1**answer

258 views

### Is the structure constant additive on connected components?

This is the reanimation of a question which already got an answer, that I did not fully understand. Coming back to it, after let it sit in a corner for some time, I keep not getting the point. I would ...

**15**

votes

**4**answers

936 views

### Who first used the multiplication operator version of spectral theory

This is another history question.
Hilbert phrased the spectral theorem in terms of resolutions of the identity.
While this remained the form of Stone and von Neumann, they did also have the ...

**2**

votes

**1**answer

98 views

### Spectral norm tail bound of a correlated random matrix

I am looking for the tail bound of spectral norm for certain type of random matrix.
Let's say we have a $n\times n$ symmetric random matrix $R$, and for each entry $R_{ij}$, we have that
$$
...

**-2**

votes

**1**answer

70 views

### On a characterization of some subsets [closed]

Let $X$ be a Banach space. $A$ is a linear closed and densely defined operator and $S$ is a bounded invertible operator. What' s the relation between $\sigma_{e,S}(\lambda S - A)$ and ...

**0**

votes

**1**answer

54 views

### Using Marchenko - Pastur type Theorems on Regression Analysis

Sometimes when doing regression analysis, we estimate our function $g(x) = E(Y |X =x )$ using an orthonormal series, and in particular we use an approximate series $g_{p_n}(x) = \sum_{k=1}^{p_n} ...

**3**

votes

**1**answer

158 views

### Spectral gap of unitary representation

Does anyone know any book or article proving that the unitary representation $\pi$ of $SL(2,\mathbb{R})$ into $L^2(SL(2,\mathbb{R}))$ has spectral gap? And what happens if we replace ...

**0**

votes

**1**answer

161 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 ...

**4**

votes

**1**answer

151 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

242 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

83 views

### interpretation of generalized eigenvalue/vectors in spectral graph theory [closed]

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

156 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 ...

**0**

votes

**2**answers

567 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 ...

**8**

votes

**0**answers

202 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

110 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. ...

**4**

votes

**0**answers

169 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

**0**answers

183 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 ...

**2**

votes

**0**answers

43 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

148 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

107 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

**1**answer

129 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

202 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.)
...

**17**

votes

**2**answers

458 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 ...

**2**

votes

**2**answers

387 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 ...

**-1**

votes

**1**answer

187 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 ...

**2**

votes

**2**answers

349 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 ...

**7**

votes

**0**answers

102 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

**1**answer

138 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 ...

**9**

votes

**1**answer

509 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 ...

**0**

votes

**1**answer

317 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 ...

**0**

votes

**0**answers

90 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

116 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)\| ...

**17**

votes

**1**answer

427 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 ...

**5**

votes

**1**answer

235 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
...

**0**

votes

**0**answers

91 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

83 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

165 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 ...

**4**

votes

**1**answer

163 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

207 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 ...

**1**

vote

**1**answer

305 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

88 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 ...

**9**

votes

**3**answers

389 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 ...

**2**

votes

**0**answers

81 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 ...

**3**

votes

**2**answers

233 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 ...

**0**

votes

**1**answer

236 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, ...

**2**

votes

**0**answers

248 views

### Versions of the spectral theorem

Since any $C^*$-algebra can be represented as an algebra of bounded operators $\mathcal{B(H)}$ on a Hilbert space $\mathcal{H}$, the spectral theorem applies to all $C^*$-algebras:
($*$) ...

**4**

votes

**1**answer

176 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, ...

**1**

vote

**0**answers

83 views

### What is a good reference for periodic Jacobi matrices?

I have been using Barry Simon's text "Szego's Theorem and its Descendants" and Gerald Teschl's text on Jacobi matrices to learn about the theory of periodic Jacobi matrices, but I would like to have ...

**1**

vote

**1**answer

285 views

### Reference request: Spectral analysis of advection diffusion PDE

As the title says, I am looking for a authoritative reference/monograph on this topic. My interest is in spectral properties of this PDE, and
NOT on existence/uniqueness etc. which is usually the ...

**3**

votes

**1**answer

519 views

### numerical range of a column-zero-sum matrix

I am trying to produce an example of a (necessarily non-normal) matrix that has only eigenvalues with positive real part, but whose numerical range contains elements with strictly negative real part. ...

**6**

votes

**0**answers

180 views

### Spectral theory for Dirac Laplacian on a funnel

I would like to study the spectral theory of the Dirac Laplacian on a non-compact quotient of the hyperbolic plane by a discrete group (I am particularly interested in the simple case where the ...