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
988
questions
0
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23
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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. ...
0
votes
0
answers
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
0
answers
233
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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
0
answers
158
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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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0
answers
16
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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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0
answers
39
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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
0
answers
96
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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 $...
0
votes
0
answers
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
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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
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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
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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
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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
0
answers
136
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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
0
answers
82
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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)\...
0
votes
0
answers
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
$$L(\varphi) = \varphi_{rr} - \frac{1}{r} \varphi_r - [V + \frac{m}{r^2}] \varphi$$
$$\varphi(0) = \...
1
vote
0
answers
37
views
Gradient estimate of the eigenfunction of Laplacian on hyperbolic space
I am trying to understand the asymptotic behaviors of the gradient of the eigenfunction function of the Laplace-Beltrami operator on the hyperbolic plane $\mathbb{H}^2$. Specifically, my focus lies on ...
1
vote
0
answers
341
views
Calculating frequency of sound of ringing metal coin
I would like to reproduce the results of Manas - The music of gold: Can gold counterfeited coins be detected by ear?, but it skips a lot of steps, and the mathematics behind it is a bit advanced for ...
1
vote
0
answers
174
views
Connection of eigenspace of finite Hilbert matrix and its continuous operator counterpart
I am trying to understand the connection between the eigenspace of the continuous operator
$$
H(x,y) = \frac{1}{x+y}
$$
which is nothing but the square of the Laplace operator, and its discrete ...
1
vote
0
answers
95
views
If we have $\mu_{xy}$, why can we only construct the spectral measure if $\| \mu_{xy} \| \le \| x \| \|y \|$?
Definitions
Representation
Let $X \subset \mathbb{C}^N$ and $\mathcal{A}$ be an algebra in $\mathcal{C}(X)$. Also, we denote $L(H)$ as the set of all linear operators on HIlbert space $H$.
We call $\...
3
votes
2
answers
172
views
How to diagonalize this tridiagonal difference operator with unbounded coefficients?
Problem: I have a self-adjoint operator in $\ell^2(\mathbb{Z})$ which acts as
$$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 ...
0
votes
0
answers
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 ...
0
votes
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 ...
0
votes
0
answers
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
0
answers
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'...
0
votes
0
answers
47
views
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$...
0
votes
0
answers
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)\,...