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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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About concentration of eigenvalues values of a random symmetric matrix in a specific interval

Given a random symmetric matrix $M$ and two numbers $\lambda_\min$ and $\lambda_\max$ how do we calculate the expected or high probability value of the fraction of its eigenvalues which lie in the ...
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1answer
73 views

Approximation of vectors using self-adjoint operators

Let $T$ be an unbounded self-adjoint operator. Does there exist, for any $\varphi$ normalized in the Hilbert space, a constant $k(\varphi)>0$ and a sequence of normalized $(\varphi_n)$ such that $$...
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Convergence to equilibrium of a nonlinear dynamical system

Consider the following dynamical system in $\mathbb{R}^n$ $$ \dot{x} = -x + A\tanh(x)=:f(x) $$ where $x = (x_1,...,x_n) \in \mathbb{R}^n$, $A$ is a real matrix with spectral radius $\rho(A) < 1$, ...
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1answer
41 views

Uniform boundedness of resolvents on the imaginary axis

Let $A \colon \mathcal{D}(A) \subset \mathbb{H} \to \mathbb{H}$ be a closed linear operator in a Hilbert space $\mathbb{H}$, which generates a $C_{0}$-semigroup. Suppose that in a $\varepsilon$-...
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Motivation for persistent homology with respect to eigenfunctions of distance kernel operator in a recent preprint

I have a question about a recent preprint https://arxiv.org/pdf/1912.02225.pdf by Maria, Oudot, and Solomon. As far as I understand, in Section 8 they prove that persistent homology (persistence ...
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1answer
74 views

Spectral representation of closed operators with finite spectral bound

Assume $A$ is a closed linear operator on a Banach space $X$ and is densely defined. Assume the spectral bound $s(A) = \sup\{Re\lambda: \lambda\in \sigma(A)\}$ is finite. For example, if $A$ is the ...
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135 views

Fourier transform of Green function and its derivative

Consider a real Sturm-Liouville operator $L$ on $[0,+\infty)$ and use the following notations : https://www.encyclopediaofmath.org/index.php/Titchmarsh-Weyl_m-function Assume $a = 0$, $\alpha \in [0,\...
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1answer
46 views

Determine the singular values of a compact operator in terms of the eigenvalues of an alternating tensor product of operators

Let $H$ be a $\mathbb R$-Hilbert space, $A\in\mathfrak L(H)$ be compact and $$|A|:=\sqrt{A^\ast A}$$ denote the square-root of $A$. By definition, the $k$th largest singular value $\sigma_k(A)$ of $A$ ...
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52 views

The eigenvalue of Schrodinger opeartor

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$, consider $L=\Delta+V$ where $\Delta=div\nabla$. According to section 8.12 in Gilbarg and Trudinger's book, if $V\in L^\infty(\Omega)$, then ...
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63 views

Lyapunov indices of a product of operators

The deterministic part of the proof of the multiplicative ergodic theorem can be proven using Proposition 1.3 in the paper Lyapunov indices of a product of random matrices.$^1$ They consider a ...
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91 views

Spectrum of Laplacian-like operator

Let $\kappa_1, \kappa_2>0$ be fixed. Consider the unbounded operator $A: D(A) \rightarrow L^2(-1,1)\times\mathbb{R}$ defined by $$ A\begin{bmatrix} y \\ h \end{bmatrix} = \begin{bmatrix} \...
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262 views

A question about comparison of positive self-adjoint operators

I have the following question but have no idea on its proof (one direction is trivial): Let $A$ and $B$ be (bounded) positive self-adjoint operators on a complex Hilbert space $H$. Prove that $$\...
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6answers
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Differentiability of eigenvalues of positive-definite symmetric matrices

Let $A\in M(n,\mathbb{R})$ be an invertible matrix. Consider the (real) eigenvalues $\lambda_1,\cdots,\lambda_n$, in increasing order, of the positive-definite symmetric matrix $A^t A$. We shall ...
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1answer
158 views

Nonnegativity implies $\langle Lf,f\rangle\geq \int f^2-(\int fg)^2$ for $g\geq 0$

I have been a lot of time trying to understand a key step on a paper about spectral analysis but I have no clue how to prove it (and the authors only said "by standard analysis"). Let me state the ...
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stability of NLS solutions with $N$ nodes

I have a question regarding the stability of standing wave solutions to nonlinear Schrodinger equations (NLS) with $N$ nodes on the real line. In case $N=1$, the stability result is known due result ...
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74 views

Laplacian spectrum

I have found (in lecture notes) a method to calculate the spectrum of the operator $$ A:D(A)\subset L^2([0,\pi])\longrightarrow L^2([0,\pi])\text{, such that} $$ $$ Au=\dfrac{\partial^2u}{\partial^2x^...
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Cwikel–Lieb–Rosenbljum inequality including zero resonances

The Cwikel–Lieb–Rosenbljum inequality asserts that, for any potential $V:\mathbb{R}^n\to\mathbb{R}$, we have $$(\mbox{number of eigenvalue} \leq 0\mbox{ , counted with multiplicity, of }-\Delta+V\,)\...
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34 views

First eigenfunction of the p-Laplacian in an interval $(a,b) \subset \mathbb R$

What is the explicit expression of the first eigenfunction $u$ of the $p$-Laplacian ($p>1$) in a bounded interval $(a,b) \subset \mathbb R$ (up to multiplicative constant)? \begin{equation} \begin{...
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1answer
126 views

Elliptic estimates for self-adjoint operators

Let $A$ be a symmetric matrix in $\mathbb R^n$ such that $A$ is positive definite and hence satisfies $0< \lambda \le A \le \Lambda < \infty.$ Let $T$ be a densely defined and closed operator ...
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1answer
396 views

Eigenfunctions of the laplacian on $\mathbb{CP}^n$

I want to find explicit formulas for the eigenfunctions of the Laplacian on $\mathbb{CP}^n$ endowed with the Fubini Study metric. For the first eigenvalue $\lambda_1 = 4(n+1)$, the eigenfunctions ...
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1answer
88 views

Singular Sturm-Liouville problems: criterion for discrete spectrum for zero potential ($q=0$) and Hermite Polynomials

There are some known criteria for the Sturm-Liouville Problem \begin{equation} \tag{1} \frac {\mathrm {d} }{\mathrm {d} x}\left[p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=-\lambda w(x)y \...
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1answer
93 views

When is rank-1 perturbation to a positive operator still positive?

Let $A : \mathcal{H} \to \mathcal{H}$ and $B : \mathcal{H} \to \mathcal{H}$ be trace-class (hence compact) Hermitian operators on a separable Hilbert space. Assume that $A$ is strictly positive and ...
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1answer
284 views

Is the spectrum of a “self adjoint” operator real on $\ell^p$?

There might be an obvious answer to the question, but it doesn't come to mind. Suppose we have an infinite matrix $A=(a_{ij})$, which defines a bounded linear operator on $\ell^p$, i.e. for all ...
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244 views

Connections between spectral geometry and critical point/Morse theory

I am researching electrostatic knot theory, which is essentially the theory of harmonic functions on knot complements. I want to understand the number of critical points of the electric potential, ...
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1answer
316 views

Spectrum of “classical” operators

Lately, I've been reading a couple of papers from different one-dimensional PDE contexts on which operators like $\mathcal{L}:=-\partial_x^2+c_*+\Phi$ repeatedly appear. Usually, on these contexts $\...
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52 views

Polar decomposition of the Volterra integral operator

Repost of this Math.SE question due to a lack of answers (No one was able to help me find the closed form of $U_T$ and $|T|$ after two bounties). I also searched extensively online but couldn't find ...
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20 views

Spectral resolutions of a second-order orthogonal tensor

Page 37 of Continuum mechanics by C. S. Jog lists the following formulae as the "spectral resolutions" of an orthogonal tensor $\bf R$ as having the eigenvectors ${\bf e} , \, {\bf n} , \, {\bf \hat{n}...
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On various versions of the harmonic oscillator

The standard $n$-dimensional harmonic oscillator is the operator $ \mathcal H=\frac{1}{2}\sum_{1\le j\le n}(D_j^2+x_j^2), \text{ $D_j=-i\partial_{x_j}$}, $ and its spectral decomposition is $$ \...
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67 views

First Dirichlet eigenvalue below second Neumann eigenvalue?

Let $\Omega$ be a bounded domain in $\mathbb R^n $ with smooth boundary. I was wondering if there exist any known conditions on $\Omega$ such that the 1st Dirichlet eigenvalue of the (positive) ...
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254 views

Recent work on Pseudo-Laplacian and Pseudo-cuspform in the spirit of Riemann Hypothesis after the work of Bombieri and Garrett

( This is my first MO question . I'm totally inexperienced on MO so, forgive me for my mistakes .) Paul Garrett and Enrico Bombieri were (are?) Secretly Working on Pseudo-Laplacians and Pseudo-...
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1answer
279 views

Field extensions in Grothendieck rings

Let $k$ be a field, and consider the Grothendieck ring of $k$-varieties, $K_0(V_k)$. Let $K/k$ and $K'/k$ be field extensions of $k$. We view $\mathrm{Spec}(K)$ and $\mathrm{Spec}(K')$ as $k$-schemes ...
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1answer
109 views

Spectral properties of operators mapped to zero by some polynomial

Let $T$ be a bounded operator on a Banach space $X$ and suppose that there is a non-constant polynomial $p$ such that $p(T) = 0$. It seems to be well known that the spectrum of such an operator ...
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2answers
101 views

Eigenvalue problem of Schrodinger equation with polynomial growth potential on the real line

Consider the operator $L=-\frac{d^2}{dx^2}+q(x)$, where $q(x)$ is the potential with polynomial-type growth, say $|x|^s,s>1$. The eigenvalue problem $$L\phi=\lambda\phi$$ where $\phi$ is in $L^2(\...
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41 views

Spectral theorems for generalized Hermitian matrices

Let $k$ be a field, and let $\sigma$ be a nontrivial involutory automorphism of $k$. Let $A$ be a square matrix with entries in $k$, such that $(A^{\sigma})^T = A$; here $A^\sigma$ means the matrix $(...
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146 views

Eigenvalue and eigenfunction convergence

Consider a bounded Euclidean domain $\Omega \subset \mathbb{R}^n$ (for simplicity, let's say, $\Omega$ has smooth boundary and is simply connected). Let $p \in \Omega$ be a point, and call $\Omega_n = ...
2
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2answers
147 views

Significance of the length of the Perron eigenvector

Let $A$ be a positive square matrix. Perron-Frobenius theory says that there exist $\lambda,v$ with $Av=\lambda v$ and $\lambda$ equals the spectral radius of $A$, $\lambda$ is simple, and $v$ is ...
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52 views

What's the essential definition of resonance of Schrodinger operator?

Rencently, I am reading some articles about time decay estimates or Strichartz estimates for Schrodinger equations with potential. When considering Strichartz estimates for potential $V$ with decay $|...
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Common eigenvector of the Wigner D matrices for the eigenvalue $\lambda = 1$?

Consider the family of spherical harmonics $Y_{n}^m$ on the sphere $\mathbb{S}^2$, with $n \geq 0$ and $-n \leq m \leq n$. The Wigner matrices are matrices $\mathrm{D}_{\mathrm{R},n} \in \mathbb{R}^{(...
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137 views

Schrodinger operator with magnetic field: eigenvalues

Consider the self-adjoint operator on $L^{2}(\mathbb{R}^{N})$, $$H=-\frac{1}{2}(\nabla-iA)^{2}+V,$$ where $A\in C^{\infty}(\mathbb{R}^{N}, \mathbb{R}^{N} )$, $V\in C^{\infty}(\mathbb{R}^{N})$, $V\...
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0answers
77 views

Lippmann-Schwinger equation for the Coulomb potential

Let $H=H_0+V$ be a Hamiltonian on $\mathbb{R}^3$ where $H_0=-\frac{\Delta}{2m}$ is the free Hamiltonian and $V$ is a potential. Let us assume first that $V$ decays sufficiently fast at infinity and ...
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0answers
705 views

Positive definiteness of matrix

This question is about the positive definiteness of a (non-random) matrix that is defined using random variables as follows: We fix the vector $v=(1,1)$ (yet, it seems the final result does not ...
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1answer
65 views

Anderson localization for Bernoulli potentials on half-line

Anderson localisation for (discrete) Schrödinger operators with Bernoulli potentials on $l^2(\mathbb{Z})$ was proven in https://link.springer.com/article/10.1007/BF01210702 I am wondering if there ...
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2answers
236 views

Random matrix is positive

This is a follow up question on my previous question here that was on solved in the deterministic setting by Denis Serre, when the perturbation can be separated. Therefore, I decided to split the ...
6
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1answer
243 views

Phase transition in matrix

Playing around with Matlab I noticed something very peculiar: Take the symmetric matrix $A \in \mathbb R^{n \times n}$ defined by $$A_{ij}= i \delta_{ij} - \frac{\varepsilon}{\sqrt{i}\sqrt{j}}\,.$$ ...
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1answer
160 views

Positive matrix and diagonally dominant

There is a well-known criterion to check whether a matrix is positive definite which asks to check that a matrix $A$ is a) hermitian b) has only positive diagonal entries and c) is diagonally ...
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165 views

Non real eigenvalues for elliptic equations

I am looking for an example of a pure second order uniformly elliptic operator $L=\sum_{i,j=1}^da_{ij}(x)D_{ij}$ in a bounded domain $\Omega$ (with Dirichlet boundary conditions, for example) having a ...
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1answer
127 views

Some properties of fractional Dirichlet heat kernel

Let $\Omega\subset \mathbb{R}^n (n\geq 2)$ be a bounded open domain with smooth boundary $\partial\Omega$. Consider the fractional heat equation with Dirichlet boundary condition： \begin{equation} \...
2
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1answer
106 views

Courant nodal domain theorem for fractional Laplacian

Let $\lambda_k$ and $\varphi_k$ be the $k$-th eigenvalue and a corresponding eigenfunction of the fractional Laplacian in a bounded domain $\Omega \subset \mathbb{R}^N$, $N \geq 2$. That is, $\...
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2answers
121 views

Spectrum of a Markov kernel acting on $L^2$

Let $P$ be a Markov kernel on a measurable space $(E,\mathcal E)$ admitting an invariant probability measure $\pi$. $P$ acts on $L^2(\pi)$ via $$Pf:=\int\kappa(\;\cdot\;{\rm d}y)f(y).$$ The invariance ...
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0answers
322 views

Eigenvalues and spectrum of the adjoint

In a finite-dimensional Hilbert space, the eigenvalues of the adjoint $A^*$ of an operator $A$ are the complex conjugates of the eigenvalues of $A$. But in infinite dimensions this need no longer be ...

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