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

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Density of zero modes

Let $(M,g)$ be a compact smooth Riemannian manifold with a smooth boundary. Let $\{(\lambda_k,\phi_k)\}_{k\in\mathbb N}$ be the spectral data on $(M,g)$, namely an orthonormal basis for $L^2(M)$ ...
2 votes
1 answer
78 views

On compactly supported functions with prescribed sparse coordinates

Let $\{\phi_n\}_{n=1}^{\infty}$ be an orthonormal basis for $L^2((0,1))$ consisting of Dirichlet eigenfunctions for the operator $-\partial^2_x + q(x)$ where $q \in C^{\infty}_c((0,1))$ is fixed. ...
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65 views

Convergence of metric implies convergence of eigenvalues?

Consider the metric $g_\varepsilon=d\theta^2 + \varepsilon^2 \sin(\theta)^2 \, d\varphi^2$ on the hemisphere $S^2_+.$ I had two naive questions: Does $g_\varepsilon$ converge to the flat metric on ...
2 votes
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233 views

Convergence of metric and eigenvalues on a tubular neighbourhood

Background: Consider the sphere $S^2$ with the round metric $g$ and let $\gamma$ be one half of a great circle of length $\pi.$ Let $T_\epsilon$ denote a geodesically convex tube around $\gamma$ of ...
3 votes
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Bourgain-Gamburd-like theorems in the non-algebraic case

For $\mu$ a Borel probability measure on the compact group $G=\operatorname{SU}(d)$, Bourgain-Gamburd prove that the spectral radius of the associated operator on $L^2(G)$ is strictly less than one, ...
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Bound the $\infty$-norm of the eigenvector of the second minimum eigenvalue of normalized Laplacian from below

I meet the above problem while reading a paper. The problem can be stated as below. Consider an undirected graph $G$. Let $\mathbf{v}$ be a vector such that $\mathbf{D}^{1/2}\mathbf{v}$ is the ...
3 votes
2 answers
376 views

Monotonicity of matrix conjugation

Let $A$ and $B$ be positive-definite matrices such that $A \le B.$ By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$ I am now curious under ...
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39 views

Computation of Brown measure of the shift operator on $\ell^2(\mathbb N)$?

This looks an extremely simple question - I am just trying to give an example of Brown measure, https://en.wikipedia.org/wiki/Brown_measure, so I try to compute it for the left/right-shift operator on ...
1 vote
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Discrete spectrum of $A \otimes 1+ 1 \otimes B$ [migrated]

Let $A, B$ be unbounded self-adjoint operators on Hilbert spaces $\mathcal{H_1}, \mathcal{H_2}$, with both non-empty discrete spectra. Let us say, for instance, $\inf \, \sigma(A) = \lambda_1^A$ and $...
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190 views

Eigenvalue multiplicity of tensor product of positive operator with itself

Let $H$ be a separable complex Hilbert space and let $A\in B(H)$ be positive with $||A||=1$ and have eigenvalue 1 with multiplicity 1. Suppose $A=T^*T$ for some $T\in B(H)$. Denote the spectrum of $A$ ...
2 votes
1 answer
130 views

Difference in essential spectrum between Schrodinger operators

I am considering two Schrodinger operators on $\mathbb{Z}^2$ and compare their essential spectrum. The operators are both of the form $H=A+V$ where $A$ is the adjacency operator on the $\mathbb{Z}^2$-...
1 vote
1 answer
711 views

A bounded operator $T$ is compact if and only if $\sigma_{\mathrm{ess}}(T)=\{0\}$ [closed]

Theorem: Let $T$ be a bounded self-adjoint operator on a complex infinite-dimensional Hilbert space $H$. Then $T$ is compact if and only if $\sigma_{\mathrm{ess}}(T)=\{0\}$. Proof: If $T$ is compact ...
3 votes
1 answer
101 views

Dimension of spectral projection subspaces under local convergence

I'm interested in estimates on dimension of spectral projection subspaces of some limit operator. I recently asked a related question in the thread Dimension of spectral projection subspaces under ...
3 votes
2 answers
148 views

Dimension of spectral projection subspaces under strong convergence of operators

I have a possibly simple question regarding estimating bounds on spectral projection subspace. Let $H_n$ be a sequence of bounded self-adjoint operators on $\ell^2(\mathbb{Z}^2)$ converging in the ...
2 votes
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136 views

Spectrum of an almost Hamiltonian matrix

I have a complex-valued block matrix $N=\begin{bmatrix} A & B \\ C & -A^* \end{bmatrix}$, where $A$ is diagonal, $B=B^*$, and $C$ is rank-1 but not Hermitian. If $C$ were Hermitian, $N$ would ...
1 vote
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82 views

Spectral gap of a Markov operator on $L^2$ with a symmetric $L^\infty$ kernel

Let $I$ be a compact interval, say $I:=(0,1)$, and $k\in L^\infty(I\times I)$ a symmetric Markov kernel, i.e. $k(x,y)=k(y,x)$ and $$\int_I k(x,y) d y = 1\qquad\mbox{for almost all } x\in I.$$ Let $K:L^...
7 votes
1 answer
812 views

Why is the length spectrum called a spectrum?

Given a hyperbolic surface $X$, one considers the multiset of lengths of closed primitive geodesics. This multiset is called the length spectrum $\mathcal{L}(X)$. Question: is $\mathcal{L}(X)$ a ...
3 votes
2 answers
217 views

Question about the Bessel operator

For $\nu>-1$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the succesive positive zeros of the Bessel function of the first kind $J_{\nu}$. The Bessel operator is given by \begin{equation*} L_\...
7 votes
1 answer
382 views

Deriving Sommerfeld radiation condition from limiting absorption principle

For the Helmholtz equation $$ -(\Delta + k ^2) u = f, \label{1}\tag{1} $$ imposing the Sommerfeld radiation condition $$ \lim_{r\to\infty} r ^{\frac{m-1}2} \left( u_r - i k u\right) = 0 $$ on $u$ ...
2 votes
1 answer
153 views

Estimate for the operator $A A_D^{-1}$

Let $O\subset\mathbb{R}^d$ be a bounded domain of the class $C^{1,1}$ (or $C^2$ for simplicity). Let the operator $A_D$ be formally given by the differential expression $A=-\operatorname{div}g(x)\...
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47 views

Induced higher Gershgorin estimate

I have a problem which I suspect appears in literature under a name I haven't found yet. Let $H:\ell^2(\mathbb{Z}^2)\to \ell^2(\mathbb{Z}^2)$ given by $H=\Delta + D$, where $\Delta$ is the graph ...
5 votes
1 answer
181 views

In what sense does the Laplacian on compact intervals converge to one on all of $\mathbb{R}$?

I guess this topic may have been addressed somewhere but I cannot really find a reference myself, so I ask here. For each $N \in \mathbb{N}$, consider the Laplacian $\Delta$ on the interval $[-N,N]$ ...
1 vote
1 answer
230 views

Eigenvalues of a Schrödinger operator

I'm interested in the existence of eigenfunctions and finding eigenvalues of the following operator $$L(\varphi) = \varphi_{rr} - \frac{1}{r} \varphi_r - [V + \frac{m}{r^2}] \varphi$$ $$\varphi(0) = \...
1 vote
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37 views

Gradient estimate of the eigenfunction of Laplacian on hyperbolic space

I am trying to understand the asymptotic behaviors of the gradient of the eigenfunction function of the Laplace-Beltrami operator on the hyperbolic plane $\mathbb{H}^2$. Specifically, my focus lies on ...
1 vote
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341 views

Calculating frequency of sound of ringing metal coin

I would like to reproduce the results of Manas - The music of gold: Can gold counterfeited coins be detected by ear?, but it skips a lot of steps, and the mathematics behind it is a bit advanced for ...
1 vote
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174 views

Connection of eigenspace of finite Hilbert matrix and its continuous operator counterpart

I am trying to understand the connection between the eigenspace of the continuous operator $$ H(x,y) = \frac{1}{x+y} $$ which is nothing but the square of the Laplace operator, and its discrete ...
1 vote
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95 views

If we have $\mu_{xy}$, why can we only construct the spectral measure if $\| \mu_{xy} \| \le \| x \| \|y \|$?

Definitions Representation Let $X \subset \mathbb{C}^N$ and $\mathcal{A}$ be an algebra in $\mathcal{C}(X)$. Also, we denote $L(H)$ as the set of all linear operators on HIlbert space $H$. We call $\...
3 votes
2 answers
172 views

How to diagonalize this tridiagonal difference operator with unbounded coefficients?

Problem: I have a self-adjoint operator in $\ell^2(\mathbb{Z})$ which acts as $$T g(x)=q^{-2 x -3/2} g(x+1)+(1+q) q^{-2 x-1} g(x)+q^{-2 x +1/2} g(x-1),$$ and I am looking to diagonalize it. The ...
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127 views

Self-adjoint operator with pure point spectrum

Suppose that A is a self-adjoint (possible unbounded) operator from a separable Hilbert space H to itself. I would like to know if the following statement is true: A has pure point spectrum (i.e., the ...
3 votes
1 answer
114 views

Spectra of products variously permutated

Let $x,y$ be elements of a Banach algebra $A$ and $\lambda\in\mathbb C\setminus\{0\}$. If $\lambda-xy $ is invertible, then $\frac1{\lambda}\big[1+y(\lambda-xy)^{-1}x \big]$ is clearly an inverse of $...
5 votes
1 answer
589 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 = ...
4 votes
2 answers
715 views

Decay of eigenfunctions for Laplacian

Consider the discrete second derivative with Dirichlet boundary conditions on $\mathbb C^n$. Its eigendecomposition is fully known: see wikipedia It seems like the largest eigenvalue $\lambda_1$ is ...
3 votes
1 answer
138 views

Operator Semigroup: Resolvent estimates and stabilization, a detail in the paper of Nicoulas Burq and Patrick Gerard

In Appendix A of the paper Stabilization of wave equations on the torus with rough dampings https://msp.org/paa/2020/2-3/p04.xhtml or https://arxiv.org/abs/1801.00983 by Nicoulas Burq and Patrick ...
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1 answer
138 views

Relating classic spectral decomposition with Euclidean Jordan algebras

I'm currently getting into studying optimization problems over symmetric cones (NSCP) and I'm having some trouble to understand something. Let me first give some context, sorry if it is repetitive to ...
2 votes
1 answer
172 views

On spectral calculus and commutation of operators

Let $\mathcal{H}$ be a Hilbert space, $B\in\mathcal{B}(\mathcal{H})$ be bounded and self-adjoint and $A:\mathcal{D}(A)\to\mathcal{H}$ closed (but not necessarily self-adjoint or bounded). The ...
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89 views

Operator identity

Let $T:\mathcal{D}(A)\to\mathcal{H}$ be a unbounded, self-adjoint, operator with positive spectrum $\sigma(T)\subset [\varepsilon,\infty)$ for $\varepsilon>0$. Hence $T$ is bijective with bounded ...
2 votes
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89 views

Dimension of Laplacian eigenspaces along a smooth 1-parameter family of metrics

Let $(M^n,g)$ be a closed Riemannian manifold, $n \geq 2$. For a smooth 1-parameter family $g_t$, $t \in (-\varepsilon, \varepsilon)$, of Riemannian metrics on $M$ with $g_0 = g$, let $\lambda_k(t)$, $...
4 votes
1 answer
224 views

Spectral density of symmetrized Haar matrix

Let $O$ be a random orthogonal matrix (according to Haar measure) of size $n$. I found by simulations that the spectral density of $O+O^\top$ is the arcsin law rescaled to the interval $[-2,2]$. I can'...
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Rotational invariance of Laplace-Beltrami eigenvalue problem on smooth manifolds

I am currently looking at the eigenvalue problems of the Laplace-Beltrami operator. Let $(M,g$) be a smooth and oriented Riemann manifold. I am investigating the eigenvalue problem of the Laplace-...
1 vote
0 answers
145 views

Generalization of Borel functional calculus

[Repost from https://math.stackexchange.com/questions/4802593/generalization-of-borel-functional-calculus] Let $A$ be a normal operator on a Hilbert space $V$. The continuous functional calculus gives ...
15 votes
2 answers
1k views

Error in Maurins proof for the nuclear spectral theorem?

I am currently studying the nuclear spectral theorem as presented in K. Maurins Monograph [2], second chapter or alternatively his paper [1] which contains basically the same proof. Let $\Phi\subset H\...
2 votes
2 answers
1k views

Lebesgue integral with respect to vector measures?

Good evening, I'm reading some papers of Jim Agler and Nicholas Young, in which they prove a formula of integral representation with respect to a vector measure, but the integration is in the sense ...
1 vote
1 answer
72 views

Spectrum below zero for $-\beta(x) \partial^2_x : L^2(\mathbb{R}) \to L^2(\mathbb{R})$

Let $\beta \in L^\infty(\mathbb{R} ; (0, \infty))$ be bounded from above and below by positive constants. Consider the self-adjoint operator $ -\beta^{-1} \partial^2_x : L^2(\mathbb{R}; \beta dx) \to ...
3 votes
2 answers
198 views

Domain of spectral fractional Laplacian

Let $(M,g)$ be a complete Riemannian manifold with Laplacian $\Delta:C^{\infty}_{c}(M)\to C^{\infty}_{c}(M)$ (think of $\mathbb{R}^{d}$ if you wish). This operator is essentially self-adjoint in $L^{2}...
6 votes
0 answers
166 views

Dependence of Neumann eigenvalues on the domain

I have the following problem in hands, in the context of a broader investigation: Let $V\in L^{n/2}$ compactly supported, where $n\geq 3$ is the dimension. I want to prove the following: For any $\...
5 votes
1 answer
128 views

Stable region of minimal hypersurfaces with finite Morse index

In this Inventiones Mathematicae paper, Fischer-Colbrie proved the following result (Proposition 1): Proposition: Let $ M$ be a complete two-sided minimal surface in a three manifold $N$. Then if $M$...
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102 views

Reducing subspaces of unitary operators

Let $U$ be a unitary operator on a Hilbert space $\mathcal{H}$. We can assume $\mathcal{H}$ is an $L^2$ space and $U$ acts as multiplication by a function $u$ with $|u(x)| = 1$ a.e (by the spectral ...
1 vote
1 answer
113 views

Sobolev-type estimate for irrational winding on a torus

Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. Let $\mathscr{C}^{\infty}_0(\mathbb{T})$ be the subset of $\mathscr{C}^{\infty}(\mathbb{T})$ of ...
2 votes
1 answer
2k views

Eigenvalues of directed Laplacian matrix $L$ and $DL$, where $D$ is a diagonal matrix with positive entries

I have a weighted Laplacian matrix $L$ of a strongly connected directed graph and a diagonal matrix $D$ with positive entries. Since the graph is directed, $L$ is non-symmetric real. Further, since ...
3 votes
1 answer
226 views

Quadrature domains for arc length

Is ellipse a quadrature domain for arc-length measure? More precisely does there exist points $z_1,\cdots,z_n$ inside an ellipse $E$ and non zero constants $c_1,\cdots,c_n$ such that $$\int _Ep(z)\,...

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