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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2
votes
1answer
49 views

Limit-circle and limit-point at endpoints

I was wondering if the following holds: If you have an ODE $$-y''(x) + q(x) y(x) = \lambda y(x)$$ on a finite interval $(a,b)$ and you know that this equation is limit-circle or limit-point at the ...
0
votes
0answers
42 views

Estimation of growth rate of spectral radius

I have following problem: Let the spectral radius of $S=(a_{ij})_{n\times n}$ be $\lambda>1$, where each $a_{i,j}$ is a positive integer, then we have that $$\lim_{k\to ...
0
votes
0answers
52 views

Hill's discriminant and spectral properties of Schrödinger operator

I am currently reading this paper on Schrödinger operators see here. On page 6 and 7 they talk about Hill's discriminant and how this is connected with the spectral properties. They also show some ...
3
votes
1answer
104 views

Has uniform ellipticity implications on the spectrum?

Let $X$ be a complete Riemann surface with a smooth metric, and $L$ a line bundle on it also equipped with a smooth metric; associated to this data there is a Laplace-Beltrami operator $D_L$ acting on ...
6
votes
0answers
80 views

Weyl law for Maass forms with nontrivial character

The classical Weyl law for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$ counts the number of Maass cusp forms on $\Gamma \backslash \mathbb{H}$ with Laplace eigenvalue less than $T$. This is originally due to ...
0
votes
0answers
86 views

Log of heat kernel for positive time

A well-known theorem by Varadhan relates the logarithm of the heat kernel on a manifold and the geodesic distance function. In particular, if $d(x,y)$ is geodesic distance from $x$ to $y$ and ...
2
votes
0answers
152 views

Eigenvalues of this matrix

I have a linear map that is defined by $$T:\text{lin}(1,...,x^m) \rightarrow \text{lin}(1,...,x^m) \text{ with}$$ $$x^k \mapsto 2w(k-m)x^{k+1}+(k^2-k-w^2)x^k-2kwx^{k-1}+(k-k^2)x^{k-2}$$ Let me give a ...
1
vote
1answer
85 views

Can we count isospectral graphs?

On n-vertices, how many isospectral graphs exist? [..I saw this previous "historic" discussion between two of the stalwarts in this field - Operation on Isospectral graphs ] Given a graph are ...
2
votes
0answers
96 views

Examples for Markov generators with pure point spectrum

I'm looking at symmetric diffusion Markov generators $L$ with pure point spectrum, i.e. infinitesimal generators of symmetric diffusion Markov semigroups, which are defined on $L^2(\mu)$ where $\mu$ ...
5
votes
1answer
138 views

Laplacian eigenfunction $L^p$ norms

Suppose I have a compact surface (possibly with boundary), and consider the eigenfunctions of the Laplacian, normalized so that their $L^2$ norms are $1.$ Is there some general result or conjecture on ...
7
votes
1answer
244 views

Error in Maurins proof for the nuclear spectral theorem?

I am currently studying the nuclear spectral theorem as presented in K. Maurins Monograph "General Eigenfunction Expansions and Unitary Representations of Topological Groups", second chapter or ...
7
votes
1answer
99 views

Can the graph Laplacian be well approximated by a Laplace-Beltrami operator?

It seems rather well known that given a Laplace-Beltrami operator $\mathcal{L}_{M}$ on a manifold $M$ we can approximate its spectrum by that of a graph Laplacian $L_{G}$ for some $G$ (where $G$ is ...
2
votes
1answer
117 views

Lp estimate for resolvent of Laplace operator

Consider for $1<p<\infty$ operator $A_p:L_p(0,1)\to L_p(0,1), \ D(A_p)=\{u\in W^2_p(0,1): u'(0)=u'(1)=0\}, \ A_pu=u''$, i.e. $L_p$-realisation of the Laplace operator with Neumann bcd on the ...
6
votes
1answer
217 views

Spectrum of this ODE

I noticed something interesting studying this Sturm-Liouville Problem: $$ \frac{d}{dx}\left(\sqrt{(1-x^{2})}\frac{df}{dx} \right)+\frac{\left(n \alpha x+\alpha^2 x^{2} + ...
0
votes
0answers
93 views

Equivalence of Positive Matrix in Infinite Dimensional Vector Space

What is the corresponding linear operator on an infinite dimensional vector space, say a Banach space or Hilbert space, to the nonnegative matrix on a finite dimensional vector space? What is the ...
5
votes
1answer
118 views

Origins of the Jacobi matrix

I have several questions concerning history of Jacobi matrices. Does anybody know why the Jacobi matrix (=symmetric tridiagonal matrix) is named by Carl Gustav Jacob Jacobi? What was his contribution ...
2
votes
0answers
121 views

Error in trace of operator

Imagine you have a Schroedinger operator $H:=-\frac{d^2}{dx^2}+V$ with $V \in C[a,b]$ on some compact interval $[a,b]$. The boundary conditions are supposed to be taken in such a way that this ...
6
votes
3answers
595 views

Some explanation about Dynin's formalism

I have seen this claim on the Wikipedia page for the Yang-Mills Millenium problem by Alexander Dynin. He is a mathematician working at the Department of Mathematics of Ohio State University and so, I ...
1
vote
0answers
59 views

Decay of Eigenfunctions for the 1D Discrete Random Schrodinger Operators

Consider the operator on $\ell^2(\mathbb{Z})$ $$ H = \Delta + v. $$ Here $\Delta$ is the nearest neighbour Laplacian on $\mathbb{Z}$, $\Delta_{k, \ell} =1 $ if $|k - \ell| =1 $ and zero otherwise, ...
4
votes
1answer
165 views

Asymptotic behavior of Schrödinger operators

I am currently dealing with $1$ or at most $2$-dimensional Schrödinger operators on compact domains. A classical result of spectral theory is the Weyl approximations for this operator $H = -\Delta ...
1
vote
1answer
120 views

Cauchy-Schwarz type formula for positive integral operator

This question arises when I am reading Klainerman&Machedon's paper "On the Uniqueness of Solutions to the Gross-Pitaevskii Hierarchy". The author made a comment on page 3, which in effect is as ...
1
vote
1answer
48 views

Integral representation of joint projection valued measures

Given two positive $\sigma$-finite measures $\mu_{1/2}$ on the spaces $X_{1/2}$ one can define the product measure $\mu_1\otimes\mu_2$ on the product space $X_1\times X_2$. It can be proved that the ...
1
vote
0answers
45 views

Reference Request: Generalization of spectral theory to symmetric KL divergence type metrics?

Spectral theory(Courant Fischer Theorem) provides a definition of the spectrum in term of the minima/maxima of the rayleigh coefficient of a matrix. So I can say that kth eigenvector and associated ...
0
votes
1answer
135 views

Legendre differential equation with additional term

In an application I encountered the ODE $$ \left( {x}^{2}-1 \right) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}f \left( x \right) +x \left( {\frac {\rm d}{{\rm d}x}}f \left( x \right) \right) \left( ...
4
votes
0answers
154 views

Adjoint of sum of two operators. Kato-Rellich

Let $A$ be self-adjoint and $B$ be symmetric with $A$-bound less than $1$. By Kato-Rellich, I know that $(A+B)^*=A+B$. Could I also get something like $(A+iB)^*=A-iB$ or is there a counterexample to ...
5
votes
2answers
218 views

When is the group algebra $L^1(G)$ semisimple?

Let $G$ be locally compact group. Define group algebra as $$L^1(G)=\{f\colon G\to\Bbb{C}\mid\int\lvert f(x)\rvert\, dx<\infty\}$$ with convolution product. When is the group algebra $L^1(G)$ ...
0
votes
0answers
75 views

Is the exponential Mathieu operator trace-class?

Let $H \psi(x) = -\frac{d^2}{dx^2} \psi(x) - \alpha \cos(x) \psi(x)$ on $[0,2\pi]$ be the Mathieu operator ( according to Mathieu's ODE). My question is: Do we know whether $U(t):=e^{-tH}$ for some ...
1
vote
1answer
205 views

Existence of a real eigenvalue

I have a matrix $M \in \mathbb{R}^{(n+1) \times (n+1)}$ that is tridiagonal. In numerical computations I found out that I always find a real eigenvalue. My question is: Is there a theorem that ...
1
vote
2answers
355 views

Spectrum of Mathieu equation

I have the differential equation $-f''(x)-q \cos(x) f(x) = \lambda f(x)$ and I want to find all the eigenvalues of this equation analytically on $[0,2\pi]$ that satisfy the boundary condition $f(0) = ...
1
vote
1answer
128 views

Is the structure constant additive on connected components?

Let $M$ be a Riemann surface and $\mu$ a metric on it, which could be non-compact. Moreover let $\Delta_{\mu,\,M}$ be the Laplacian on $M$ induced by $\mu$ and $\mathrm{det}^*(\Delta_{\mu,\,M})$ its ...
16
votes
4answers
705 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
1answer
52 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 $$ ...
0
votes
1answer
60 views

On a characterization of some subsets

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
1answer
32 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
1answer
89 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
1answer
83 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
0answers
64 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
1answer
103 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
0answers
64 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
0answers
79 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
2answers
294 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
votes
0answers
145 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
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
votes
0answers
115 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
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
votes
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
votes
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
vote
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 ...
0
votes
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
votes
1answer
94 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 ...