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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1
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2answers
108 views

Concentration of matrix norms under random projection.

Let X be a given matrix of dimension $p \times q$. Let $G$ be a $s \times p$ dimensional matrix of standard normal/Gaussian random variables. Are there cases where one can been able to quantify $...
0
votes
1answer
81 views

Exact formula for computing n-step transition probability of random walks with self-transitions

Consider a semi-infinite random walks $X_n$, $n=0,1,2,\ldots$, whose state space is a set of consecutive integers and whose one-step transition probabilities are $P_{ij}=\mathrm{Pr}\{X_{n+1}=j|X_n=i\}$...
30
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7answers
4k views

Good references for Rigged Hilbert spaces?

Every now and then I attempt to understand better quantum mechanics and quantum field theory, but for a variety of possible reasons, I find it very difficult to read any kind of physicist account, ...
0
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0answers
50 views

estimate of smallest eigenvalue of Schrodinger operator

I am looking for references on estimate of first nonzero Dirichlet eigenvalue for Schrodinger operator $-\Delta + V$, if sharp bounds exist, that would be better, here for simplicity, we can assume ...
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0answers
42 views

Min-Max Principle - $\inf_{u \in H_{n-1}} \rho (u)= \sup_{u \in \text{span}(u_1, \dots, u_n)} \rho (u)$ [closed]

In the document Weyl's law of Matt Stevenson, it is indicated at the Minimax Principle, Version $1$, that $$\inf_{u \in H_{n-1}} \rho (u)= \sup_{u \in \text{span}(u_1, \dots, u_n)} \rho (u).$$ ...
0
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0answers
47 views

Min-max theorem - Subspaces over $H_0^1$ [closed]

I am trying to understand what is going on (in using the Minimax Principle; Weyl's law) in the Dirichlet eigenvalue problem $$\left\{ \begin{aligned} -u''(x)&=\lambda u(x)\quad 0<x<\...
0
votes
1answer
200 views

The spherical harmonics are the EIGENVECTORS of Beltrami operator [closed]

In the well-known book "THE PRINCETON COMPANION TO MATHEMATICS" page 296, it is indicated that the spherical harmonics are the EIGENVECTORS of the Beltrami operator. In the document Spectral Geometry ...
1
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1answer
45 views

On the limit set of eigenvalues of banded Toeplitz Hessenberg matrices

Let $T_{n}(b)$ be the $n\times n$ Toeplitz matrix determined by the symbol $$ b(z)=\frac{1}{z}+\sum_{j=0}^{k}a_{j}z^{j} $$ where $k\in\mathbb{N}$ and $a_{0},\dots,a_{k}\in\mathbb{R}$, $a_{k}\neq0$. ...
1
vote
1answer
165 views

analytic continuation argument

In "Pseudo-spectra, the harmonic oscillator and complex resonances" (login required), the author says Sections $2$ and $3$ of this paper concern the operator $Hf(x)=(-\frac{d^{2}}{dx^{2}}+cx^{2})...
7
votes
2answers
227 views

Why $M_1 \subset M_2 \not \Rightarrow N_{M_1} (\lambda) \leq N_{M_2} (\lambda)$ for eigenvalue problem? (EDIT)

We know that for a direct problem with Dirichlet Boundary Condition (with Laplacian operator) that if two domains $M_1$ and $M_2$ are such that $M_1 \subset M_2$, then $\lambda(M_2) \leq \lambda(M_1)$,...
3
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0answers
97 views

quasi-nilpotent part of a dual operator

Definitions and notation. Let $X$ be a complex Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator on $X$. We define the quasi-nilpotent part of $T$ as \begin{equation*}H_0(T):=\left\{...
5
votes
2answers
151 views

Lower bound for smallest eigenvalue of symmetric doubly-stochastic Metropolis-Hasting transition matrix

For my master's thesis research, I stumbled upon a question concerning the Metropolis-Hasting transition matrix $W$. Context $\quad$ Let me start with some context. I consider connected undirected ...
2
votes
1answer
294 views

Is there any way to compare between diagonals of a resolvent and a Cauchy transform?

Say $A$ is a symmetric matrix of $n$ dimensions. Then let the ``resolvent" of $A$ be the matrix valued function $R_A(z) = \frac{1}{z-A}$ and its Cauchy transform be the real valued function $C_A(z) = ...
1
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0answers
42 views

Dirichlet conditions - Proof of theorem $4$ [migrated]

I have a few difficulties understanding the first part (Dirichlet conditions) of the proof of theorem $4$ in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$...
4
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0answers
67 views

Geometrically-explicit upper bound for on-diagonal heat kernel

Let $M$ be a compact Riemannian manifold, and $K(t;z,w)$ the heat kernel associated to the usual Laplace-Beltrami operator on functions. There are results of the form $$K(t;z,z) \leq \frac{C_M}{f_z(t)...
0
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0answers
34 views

Joint distribution of eigenvalue and matrix entry

Let $X=(X_{n,m})_{n,m=1}^N$ be a $N\times N$ GUE random matrix, and let $\lambda_1,\dots,\lambda_N$ denote its unordered eigenvalues. What can be said about the distribution of, say, $$\lvert\lambda_1-...
1
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0answers
38 views

The ground state energy of an atom as a function of an external electric field

I think this question belongs to mathematical physics. The Hamiltonian of an N-electron atom in a homogeneous electric field is $$ H =\left( \sum_{i=1}^N \frac{p_i^2}{2m } - \frac{Z e^2}{r_i} - E_z ...
7
votes
0answers
120 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 ...
0
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1answer
52 views

Request for references about computing or estimating Rademacher complexity

Is Rademacher complexity defined for any space of functions? Or are there restrictions on the function space over which this can be defined? For example is the Rademacher complexity defined or has ...
4
votes
1answer
128 views

Functional Calculus of closed operators

I learned that there is a holomorphic functional calculus for closed operators: If $T$ is a closed operator on a Hilbert space, and $f$ is a function that is holomorphic on some open subset $\Omega$ ...
4
votes
2answers
109 views

The square root of Laplacian with nonconstant coefficent

I am still a newbie to $\Psi$DO-Operators. As far as i understood, one can easily compute the square root of the Laplace operator $\Delta$ by $$(-\Delta)^{1/2} \ u=\mathcal{F}^{-1}(\|\xi\| \widehat{u}...
3
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1answer
134 views

Bounded solutions for Schrödinger equation at the edge of the essential spectrum

Let $V:R^d\to R_+$ be with a compact support. The Schrödinger operator $H_a=-\Delta - a V$ acting in $L^2(R^d)$ has then (at most) finitely many negative eigenvalues. Denote the number of negative ...
1
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0answers
93 views

Weyl's law for minimal surfaces

I wanted to know if there was some equivalent of Weyl law for the spectrum of the Jacobi operator of a minimal surface in the non-compact case. If the minimal surface is not closed, for example in $\...
5
votes
2answers
142 views

significance of the Fučík spectrum

The Fučík spectrum seems to gain momentum among people working on spectral theory, with almost 300 articles published on this topic over the last 5 years, according to Google scholar. There exist ...
5
votes
1answer
228 views

Convergence of Riemannian metrics spectra

Consider a one-parameter real analytic family of metrics $g_t$ on a compact manifold $M$ converging to a metric $g$ in $C^k$-norm, for some $k$. It is known that the Laplace spectrum of $g_t$ will ...
0
votes
1answer
86 views

Eigenvectors of symmetric positive semidefinite matrices as measurable functions

I'm currently interested in how discontinuous can get the eigenprojections of a continuous function taking values in a particular subspace of symmetric matrices. I've been searching everywhere for an ...
1
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0answers
91 views

Eigenfunctions of Laplacian, what's wrong in the following argument?

Can anyone tell me what's wrong in the following argument?? Consider a closed riemannian manifold $M$. Let $f$ be an eigenfunction which corresponds to the first nonzero eigenvalue of Laplacian on $M$...
1
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0answers
123 views

On isolated points of the approximate point spectrum of a bounded operator

Let $X$ be a complex Banach space and $T$ be a bounded operator acting on $X$. Let $\sigma(T)$ and $\sigma_{ap}(T)$ denote the spectrum and approximate point spectrum of $T$, respectively. Let $\...
1
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1answer
67 views

Spectral radius's relation with row sum

Let $A$ be a non-negative $N \times N$ square matrix with $a_{i,i}=0, 1 \leq i \leq N$. Also, let $r_i$ be the $i$-th row sum of $A$. I know that $\rho(A)$, the spectral radius of $A$, is bounded as ...
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1answer
56 views

Does $\|(I-A)^{-1}\|_{2} \leq 1/(1-\|A\|_{2})$ holds for matrices with spectral radius smaller then 1?

Given a symmetric positive semidefinite matrix matrix $A$, if its spectral radius $0<\rho(A)<1$, does the inequality $\|(I-A)^{-1}\|_{2} \leq 1/(1-\|A\|_{2})$ hold true? $\|A\|_{2}$ denotes ...
1
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1answer
87 views

Largest element in inverse of a positive definite symmetric matrix [closed]

If I have an $n \times n$ positive definite symmetric matrix $A$, with eigenvalues $\lambda_{1}>\lambda_{2}\cdots>\lambda_{n}$, can I claim that the highest value which matrix $A^{-1}$ can have ...
0
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0answers
57 views

What equals $\ker[(A-\lambda I)^+]$ for a negative unbounded operator $A$?

We have the following result: $\{ E_{\lambda}; \, -\infty <\lambda < + \infty\}$ is a spectral family, where $E_{\lambda}$ is the projection of $H$ onto the null space $\mathscr N \left(A_\...
2
votes
0answers
47 views

When does the ground state energy continuously depend on a parameter?

Given a family of Schrödinger operators $H_\gamma=-\Delta+V_\gamma$, under which condition is the map $\gamma\mapsto\inf\sigma(H_\gamma)$ continuous? This is surely the case for many textbook ...
4
votes
1answer
156 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 ...
5
votes
1answer
817 views

Why is the cuspidal spectrum discrete?

I have a short question concerning the spectral theory of automorphic forms. What is the main property of the unipotent group $N$, which consist of matrices in the form \begin{pmatrix} 1 & t \\ 0 &...
2
votes
0answers
35 views

Nonnegative Inverse Eigenvalue Problem (NIEP),

Does the NIEP, currently open for $n\ge 5$, have any good, practical applications? For the easy case, $n=2$, I am able to prove some of the results that agree with the current literature. In some of ...
1
vote
1answer
130 views

Can we claim that all the terms in a matrix are less than equal to 1 if spectral radius is less than 1?

I have a a full column rank matrix A, and using this I want to construct a matrix with spectral radius less than 1. I do that using, H = $I-\alpha A^{T} A$ ($I$ is identity matrix), where the term $\...
1
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0answers
52 views

inverse of asymptotic Toeplitz matrix with band limited associated function

I am reviewing a controversial paper, and the main result, a revolution within my field, comes down to whether or not the following is true. I strongly believe it is not, but would need confirmation. ...
3
votes
1answer
95 views

Restrictions on spectral measure

Given any Borel measure $\mu$ on $\mathbb{R}$, define a map that sends any $f\in C_c(\mathbb{R})$ to $$T_\mu(f)(y)=\int \langle\exp(-i x \lambda),f(x)\rangle\exp(iy\lambda)d\mu(\lambda).$$ Here $\...
52
votes
2answers
862 views

Can you hear the shape of a drum by choosing where to drum it?

I find the problem of hearing the shape of a drum fascinating. Specifically, given two connected subsets of $\mathbb R^2$ with piecewise-smooth boundaries (or a suitable generalization to a riemannian ...
0
votes
1answer
98 views

Spectrum of compact operator between different Banach spaces

Let $X, Y$ be two different Banach spaces, and let $T: X \to Y$ be a compact linear operator. Suppose the identity $I : X \to Y$ is well-defined. (For example, we could have $X = L^2([0,1])$ and $Y = ...
6
votes
3answers
246 views

Non-self adjoint Sturm-Liouville problem

I'm new to this site, but I felt the need to post when I recently came in to an ordinary differential equation/boundary value problem with this form: $(1)- \frac{d^2 y}{d x^2} + \frac{m(m+1)} {x^2(...
1
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0answers
61 views

inverse problem to resolution of the identity

Suppose $f_\lambda(x),\lambda \in \mathbb{R}$ is a continuous/smooth family of bounded function on $\mathbb{R}$. Given a measurable set $\mathfrak{m}$ on $\mathbb{R}$, define $$E_\mathfrak{m}:L^2\...
20
votes
2answers
1k views

In a Banach algebra, do ab and ba have almost the same exponential spectrum?

Let $A$ be a complex Banach algebra with identity 1. Define the exponential spectrum $e(x)$ of an element $x\in A$ by $$e(x)= \{\lambda\in\mathbb{C}: x-\lambda1 \notin G_1(A)\},$$ where $G_1(A)$ is ...
0
votes
1answer
132 views

Continuity of the largest eigenvalue with respect to length

Let $k:\mathbb{R}^+\to\mathbb R^+$ be a continuous function. For $a>0$, define $T_a$ acting on $L^2[0,a]$, by $$T_af(x) = \int_0^a k(|x-y|)f(y)\,dy.$$ Clearly for each $a>0$, the operator $T_a$...
1
vote
1answer
127 views

Zero set of eigenfunction along a sub manifold

Let $M$ be a 2-dimensional closed Riemannian manifold and let $$\phi:M\rightarrow M$$ be an isometry with $\phi^2=Id_M$. Consider the fixed point set $$F:=\lbrace x\in M: \phi(x)=x \rbrace\subset M,$$ ...
10
votes
2answers
218 views

Spectral properties of the Laplace operator and topological properties

Suppose that $M$ is a closed Riemannian manifold: one can construct the so called Laplace-Beltrami operator on $M$. Its spectrum contains some information of the underlying manifold: for example its ...
4
votes
1answer
180 views

Progress on isospectral plane domains

Has there been any progress on the smooth isospectral plane domains for Laplacian problem with Dirichlet data? In particular, are there known examples of domains which are isospectral to the unit ...
2
votes
1answer
126 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 ...
1
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0answers
97 views

Eigenvalues of the D'Alembertian operator

My question about the spectral theory of the D'Alembertian operator on a Lorentzian manifolds (say the spacetime $M^{3+1}$) given by $$\square = -\partial_{t}^2 + \Delta$$ for the metric $g=(-+++)$. ...