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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322 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 ...
7
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0answers
148 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
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0answers
90 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
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0answers
121 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} ...
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0answers
177 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
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0answers
38 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
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1answer
133 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
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1answer
90 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 ...
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0answers
66 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
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1answer
95 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
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1answer
150 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.) ...
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39 views

Gordon Lemma for Jacobi operators

The Gordon lemma is a useful tool in the theory of 1-D discrete time-independent Schrödinger operators that exploits local repetition in the potential to prove absence of point spectrum. Has anyone ...
17
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2answers
369 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 ...
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0answers
33 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
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2answers
180 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 ...
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1answer
152 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
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2answers
220 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 ...
3
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0answers
78 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 ...
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1answer
127 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
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1answer
334 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
1answer
211 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 ...
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0answers
111 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, ...
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0answers
78 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
1answer
88 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
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1answer
270 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
1answer
210 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
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0answers
75 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
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0answers
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 ...
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0answers
132 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
1answer
142 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
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1answer
178 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 ...
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1answer
283 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 ...
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0answers
85 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 ...
8
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3answers
218 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
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0answers
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
2answers
197 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
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1answer
190 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
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0answers
202 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
1answer
157 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
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0answers
71 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
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1answer
233 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
1answer
335 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
0answers
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
1answer
197 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
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3answers
396 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
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1answer
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
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2answers
95 views

Eigenfunctions of Schrödinger Operators on the boundary

Hello, let's consider a compact and connected Riemannian manifold with the Schrödinger Operator $L=-\Delta +V:dom(H)\subset L^2(M)\rightarrow L^2(M)$ whereas $dom(L):=\lbrace f\in ...
1
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1answer
402 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 ...
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0answers
68 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
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0answers
124 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 ...