**1**

vote

**1**answer

126 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

315 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

192 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 ...

**1**

vote

**0**answers

109 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, ...

**1**

vote

**0**answers

76 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

85 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)\| ...

**15**

votes

**1**answer

260 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

209 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

69 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

69 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

129 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

138 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

172 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

274 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

83 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 ...

**7**

votes

**3**answers

211 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

96 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

68 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

191 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

187 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

198 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

156 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

70 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

221 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

307 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

152 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 ...

**4**

votes

**1**answer

193 views

### Lower bounds on matrix eigenvalues

Let $A$ be a real $n\times n$ matrix and let $\mu_1, \dots, \mu_n$ the (generalized, complex) eigenvalues of $A$. Assume that
$$ 0 < \alpha < \mathrm{Re}(\mu_1) < \dots < ...

**9**

votes

**3**answers

390 views

### Why is this operator compact?

Let $D$ be the Dirac-Operator on $\mathbb{R}^n$ or more generally the Dirac spinor bundle $\mathcal{S}\to M$ of a (semi-)Riemannian spin manifold $M$. Then we consider $D$ as an unbouded Operator on ...

**2**

votes

**1**answer

171 views

### Eigenvalues for elements of (infinite) Coxeter groups

My current research requires some knowledge on the eigenvectors of elements (of infinite order) of Coxeter groups view as reflections in their geometric representation.
After some reading, my ...

**2**

votes

**2**answers

92 views

### Eigenfunctions of Schrödinger Operators on the boundary

Hello,
let's consider a compact and connected Riemannian manifold with the SchrÃ¶dinger Operator $L=-\Delta +V:dom(H)\subset L^2(M)\rightarrow L^2(M)$ whereas $dom(L):=\lbrace f\in ...

**1**

vote

**1**answer

374 views

### square root of a certain matrix [closed]

Hello,
I'd like to know the square root of the following $n$ by $n$ matrix, for $n > 2$ and $r>0$:
$R_{ii}=r+1$ for $i < n$
$R_{ij}=r$ otherwise
The $2$ by $2$ case is given by
...

**1**

vote

**0**answers

65 views

### Second eigenvalue of a weighted tree

Hello,
I am interested in upper bounding the second largest eigenvalue of the adjacency matrix of a graph $T$ with the following property:
1. $T$ contains self loops.
2. $T$ contains multiple edges ...

**1**

vote

**0**answers

120 views

### Recovering a partition from spectral properties of the graph Laplacian

Let $G$ be a weighted graph with vertices $V$. Let $W$ be its real-valued, non-negative, $|V|\times|V|$ adjacency/affinity matrix. Let $L = \mathrm{diag}(W\mathbf1)-W$ be the (unnormalized) graph ...

**5**

votes

**2**answers

324 views

### Generalized basis

In quantum mechanics, people introduce the notion of "continuous basis" (I actually don't know the mathematical denomination of it). It is not a Schauder basis. I would like to know what could be a ...

**5**

votes

**2**answers

238 views

### Bound on the order of a finite group generated by elements $a$ and $b$ of order 2 and $n \geq 3$ such that the sum of the images of $a$, $b$ and $b^{-1}$ under any ordinary representation has only rational eigenvalues

Assume that $G = \langle a, b \rangle$ is a finite non-abelian group which is generated by an
involution $a$ and an element $b$ of order $n$ ($n\geq 3$) such that for every (complex) representation
...

**1**

vote

**1**answer

325 views

### Eigenvalues of Sum of non-singular matrix and diagonal matrix

Suppose $D={\rm diag}(d_i)$ is a diagonal matrix with all diagonal entries $d_i=\pm 1$. This implies $D^2=I$.
Suppose $A$ is a non-singular Hermitian matrix. If we know that $A+A^{-1}+D$ has rational ...

**2**

votes

**0**answers

111 views

### Optimization over Spectral Laplacian in cycles and trees

Is there any idea on how one can deal with an optimization problem of sum of k largest eigenvalues(min) of Laplacian matrix of a simple cycle or tree?
I would like to use semidefinite programming for ...

**0**

votes

**1**answer

161 views

### Spectral decomposition function [closed]

Once I met a notation of "spectral decomposition function" (for a self-adjoint operator). No definition was given.
Could someone give me a clue what can that be, cause I can't find this exact phrase ...

**3**

votes

**1**answer

243 views

### The first eigenvalue of the Schrödinger operator is simple.

Hello,
let $(M,g)$ be a compact and connected Riemannian manifold (possibly with $\partial M\neq \emptyset$). We consider the Friedrichs extension of $L=-\Delta +V: C^{\infty}(M,\mathbb{R})\subset ...

**1**

vote

**1**answer

110 views

### regularity of eigenfunctions of Schrödinger Operator

Hello,
I consider a compact and connected (smooth) Riemannian manofold $(M,g)$. I'm interested in the eigenfunctions of the SchrÃ¶dinger Operator $L=-\Delta+ V$ acting on (smooth) functions. Do you ...

**1**

vote

**0**answers

48 views

### Possible diagonal values of a product of matrices with some specific characteristics

Hello all,
This is a question that might or might not be related to my previous one.
Imagine you have two matrices:
Matrix $\mathbf{\Phi}=[\Phi_1,\ldots,\Phi_M]\in\mathbb{R}^{L\times M}$ where ...

**0**

votes

**0**answers

177 views

### A tricky optimization problem over matrices

Hi
I have the following problem whose solution has lured me for some months now....
All matrices are complex $N\times N$.
Let $A$ be a positive definite matrix with all eigenvalues strictly smaller ...

**3**

votes

**1**answer

193 views

### A spectral radius inequality

Define $\rho(A)$ to be the spectral radius of a square matrix $A$. Let $S$ and $T$ be two non-negative square matrices and $h$ a real number such that $\rho(S+T) < h$. Show that $\rho((hI-S)^{-1}T) ...

**4**

votes

**2**answers

313 views

### spectral radius monotonicity

I encountered an inequality when reading a paper. Can someone help to show how to prove it?
Let be the spectral radius of matrix $A$ or $\rho(A)=\max\{|\lambda|, \lambda \text{ are eigenvalues of ...

**1**

vote

**0**answers

88 views

### null controllability of linear wave equation

Consider the linear wave equation :
$$z_{tt}=\Delta z + k(x) z + h(t) , \; in \; \Omega\times (0,T)$$
Are there sufficient conditions on the functions $k(x)$ and $h(t)$ for which $(z,z_t)$ vanish ...

**2**

votes

**2**answers

250 views

### Weyl law for arithmetic Fuchsian groups known?

For congruence subgroups of $PSL(2,\mathbb{Z})$, the Weyl law for the eigenvalues of Maass cusp forms had been proven by Selberg. How is the status of such a Weyl law for eigenvalues of Maass cusp ...

**2**

votes

**2**answers

253 views

### Gap between first two nonzero Laplacian eigenvalues on closed compact surface?

Much has been said about bounds on Laplacian eigenvalues, and the literature can be tough to sort through! I am specifically interested in the case where the domain is a closed compact surface, and am ...

**1**

vote

**1**answer

98 views

### Spectrum of composition of graphs( lexicographic product)

I was wondering how to relate the spectra of the composition of two graphs in term of the factors...someone can help me?

**1**

vote

**0**answers

54 views

### strong stability for the wave equation

Consider the $n-$dimensional wave equation
$$z_{tt}=\Delta z + k(x) z - \epsilon {1}_\omega z_t, \; in \; \Omega\times (0,T)$$
where $\omega\subset \Omega.$ Can I have $z(t) \to 0,$ as $t\to+\infty$ ...

**0**

votes

**0**answers

88 views

### Global solution for spectral clustering

I used spectral clustering for directed graphs suggested by Dengyong Zhou paper to partition the graph.I selected the eigen vectors corresponding to k largest eigen values and then I use kmeans or FCM ...