# Questions tagged [sp.spectral-theory]

Schrodinger operators, operators on manifolds, general differential operators, numerical studies, integral operators, discrete models, resonances, non-self-adjoint operators, random operators/matrices

714
questions

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3 views

### 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 ...

**3**

votes

**1**answer

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 $$...

**2**

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**0**answers

54 views

### 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 $\rho(A) < 1$, ...

**1**

vote

**1**answer

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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59 views

### 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 ...

**2**

votes

**1**answer

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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**1**answer

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} \...

**8**

votes

**1**answer

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
$$\...

**10**

votes

**6**answers

1k views

### 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 ...

**5**

votes

**1**answer

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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**0**answers

22 views

### 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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**0**answers

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^...

**2**

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**0**answers

32 views

### 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\,)\...

**0**

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**0**answers

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{...

**4**

votes

**1**answer

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 ...

**14**

votes

**1**answer

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 ...

**4**

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**1**answer

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
\...

**3**

votes

**1**answer

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 ...

**10**

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**1**answer

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 ...

**8**

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**0**answers

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, ...

**7**

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**1**answer

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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**0**answers

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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49 views

### 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
$$
\...

**2**

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**0**answers

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) ...

**6**

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**0**answers

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-...

**3**

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**1**answer

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 ...

**1**

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**1**answer

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 ...

**2**

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**2**answers

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(\...

**1**

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**0**answers

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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**0**answers

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 = ...

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**2**answers

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 ...

**3**

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**0**answers

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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90 views

### 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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**0**answers

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\...

**2**

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**0**answers

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 ...

**9**

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**0**answers

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 ...

**1**

vote

**1**answer

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 ...

**3**

votes

**2**answers

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**

votes

**1**answer

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
$$A_{ij}= i \delta_{ij} - \frac{\varepsilon}{\sqrt{i}\sqrt{j}}\,.$$
...

**1**

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**1**answer

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 ...

**9**

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**0**answers

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 ...

**1**

vote

**1**answer

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**

votes

**1**answer

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, $\...

**0**

votes

**2**answers

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 ...

**4**

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**0**answers

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 ...