**1**

vote

**2**answers

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

**1**answer

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**

votes

**7**answers

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**

votes

**0**answers

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 ...

**-1**

votes

**0**answers

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**

votes

**0**answers

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

**1**answer

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**

vote

**1**answer

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

**1**answer

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

**2**answers

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**

votes

**0**answers

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

**2**answers

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

**1**answer

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**

vote

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

vote

**0**answers

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

**0**answers

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**

votes

**1**answer

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

**1**answer

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

**2**answers

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**

votes

**1**answer

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**

vote

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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**

vote

**0**answers

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**

vote

**0**answers

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**

vote

**1**answer

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 ...

**-6**

votes

**1**answer

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**

vote

**1**answer

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**

votes

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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**

vote

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**3**answers

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**

vote

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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**

vote

**0**answers

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 $g=(-+++)$. ...