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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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8 votes
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Reference request: Software for producing sounds of drums of specified shapes

Is there software that, when the input is the shape of a drum, will produce the corresponding audible sound?
Michael Hardy's user avatar
21 votes
0 answers
737 views

Can two drums almost sound the same?

Let $D\subset \mathbb R^2$ be a region and let $\Lambda=\{\lambda_1,\lambda_2,\dots\}$ be the set of eigenvalues of the Laplacian $-\Delta$ (with boundary condition $\psi=0$ on $\partial D$). Mark Kac,...
Kenta Suzuki's user avatar
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1 vote
0 answers
94 views

$(\lambda I-A)^{-1}-(\lambda I-B)^{-1}$ compact implies $\sigma_\text{ess}(A)=\sigma_\text{ess}(B)$

Suppose $H$ is a Hilbert space and $A$, $B$ are two adjoint operators on it (not necessarily bounded), satisfying $D(A)=D(B)$. Question: If $\exists \lambda\in \rho(A)\cap\rho(B)$ such that $(\lambda ...
Richard's user avatar
  • 613
0 votes
1 answer
71 views

Spatially localised solution to the Schrödinger equation with potential is a combination of eigenfunctions

In this Terry Tao's blog post, he claims that if one has a solution to the Schrödinger equation $$i\,\partial_t u +\Delta u=Vu $$ with a "reasonably smooth and localised $V$", $u$ has ...
Earl Jones's user avatar
0 votes
1 answer
27 views

Expectation of spectral norm of a diagonal stochastic matrix

There is a diagonal matrix $G(t)\in R^{M\times M}$ where the diagonal elements are independent Bernoulli stochastic variables, satisfying $\mathbb{E}(g_i(t) = 1) = b$, and $ \mathbb{E}(g_i(t) = 0) = 1 ...
Keven Sun's user avatar
1 vote
0 answers
24 views

Class of semi-algebraic potentials satisfying growth condition for the sublevel sets

Recently I have studied the paper: Rozenbljum, G. V. (1974). Asymptotics of the eigenvalues of the Schrödinger operator. Mathematics of the USSR-Sbornik, 22(3), 349. In there Rozenbljum gives ...
Severin Schraven's user avatar
4 votes
1 answer
94 views

Accumulation points of point spectrum of Schrödinger operator in one dimension

Consider a Schrödinger operator $H=-\partial_x^2+V(x)$, with $x\in\mathbb R$, $V(x)$ tending monotonically to $V_\pm$ as $x\to\pm\infty$, and $\min V(x)<V\pm$. Intuitively, the only accumulation ...
user3476591's user avatar
1 vote
1 answer
80 views

Where does $V$ from the spectral decomposition $A = VDV^*$ lie, if $A$ has only imaginary entries?

The spectral theorem says that for every Hermitian matrix $A \in \mathbb{C}^{n \times n}$ there is a unitary matrix $V \in U(n)$ and a diagonal matrix $D \in \mathbb{R}^{n \times n}$ such that $A = ...
Tardis's user avatar
  • 1,253
2 votes
0 answers
125 views

Error in an argument using spectral theory

Let $Z=\sqrt{2}\mathbb Z$, and consider the sequence on $\mathbb{Z}$ $$\xi(k)= 1_{Z\cap [k,k+1]\neq\emptyset}.$$(remark that the intersection is either empty or with one point.) Thanks to the comments,...
kaleidoscop's user avatar
  • 1,332
0 votes
1 answer
100 views

Spectral theory: a key to unlocking efficient insights in network datasets

In the context of directed or undirected graphs, matrices such as adjacency and Laplacian matrices are commonly used. The eigenbasis of these matrices addresses some practical implications, such as ...
ABB's user avatar
  • 4,030
2 votes
0 answers
59 views

Regularity and decay of Fourier-like series on a manifold

Let $D$ be a first-order self-adjoint elliptic differential operator acting on sections of a vector bundle $S$ over a closed manifold $M$. Then it is well-known that the various eigenspaces $E_\lambda$...
geometricK's user avatar
  • 1,901
1 vote
0 answers
63 views

Equidistribution on subvarieties of $\mathrm{SL}_n(\mathbb{Z})$

$\DeclareMathOperator\SL{SL}$Let $\Gamma_{\infty}$ be a maximal parabolic subgroup of $\SL_n(\mathbb{Z})$. I believe there are various equidistribution theorems, which say that Iwasawa $x$ coordinates ...
kindasorta's user avatar
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3 votes
1 answer
292 views

Book on Hilbert spaces, including non-separable

I am looking for a book that develops the theory of Hilbert spaces, including the spectral theorems and unitary representations, but includes non-separable Hilbert spaces in the main exposition. Any ...
1 vote
0 answers
54 views

On an estimate in the paper by Donnelly and Fefferman

I was reading the following paper by Donnelly and Fefferman https://link.springer.com/content/pdf/10.1007/BF01393691.pdf which essentially deals with the Hausdorff dimension bound of the nodal sets ...
Emmie's user avatar
  • 61
2 votes
1 answer
161 views

Feynman–Kac formula for other operators

I recently came across the Feynman-Kac formula which states that given an open domain $\Omega\in\mathbb{R}^n$ and $f \in L^2(\Omega)$ where $x \in \Omega$ and $t > 0$, then $e^{t\Delta_D}f(x) = ...
Emmie's user avatar
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5 votes
2 answers
435 views

Question about Neumann eigenvalues on manifolds

Question: Let $\Omega\subset \mathbb{S}^2_+$ be any geodesically convex subset of the hemisphere $\mathbb{S}^2_+$. Then is it true that $\mu_1(\Omega)\geq \mu_1(\mathbb{S}^2_+)$ where $\mu_1$ is the ...
Student's user avatar
  • 655
2 votes
0 answers
60 views

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 $L^2(M)$ ...
Ali's user avatar
  • 4,113
2 votes
1 answer
105 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. ...
Ali's user avatar
  • 4,113
0 votes
0 answers
77 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 ...
Student's user avatar
  • 655
0 votes
0 answers
17 views

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 ...
Lasting Howling's user avatar
3 votes
2 answers
386 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 ...
António Borges Santos's user avatar
3 votes
0 answers
175 views

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, ...
John Rached's user avatar
2 votes
0 answers
241 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 ...
Student's user avatar
  • 655
0 votes
0 answers
42 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 ...
Ma Joad's user avatar
  • 1,683
2 votes
1 answer
154 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 $\mathbb{Z}^2$-...
Keen-ameteur's user avatar
3 votes
1 answer
106 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 ...
Keen-ameteur's user avatar
3 votes
2 answers
167 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 ...
Keen-ameteur's user avatar
2 votes
0 answers
146 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 ...
mathamphetamine's user avatar
1 vote
0 answers
93 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^...
Johny B's user avatar
  • 11
7 votes
1 answer
866 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 ...
Andrey Ryabichev's user avatar
3 votes
2 answers
240 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_\...
Tony419's user avatar
  • 401
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 ...
Keen-ameteur's user avatar
5 votes
1 answer
197 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 $[-N,N]$ ...
Isaac's user avatar
  • 3,113
1 vote
1 answer
261 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) = \...
JMK's user avatar
  • 299
1 vote
0 answers
45 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 ...
Atlas Tasilli's user avatar
1 vote
0 answers
98 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 $\...
S-F's user avatar
  • 155
2 votes
1 answer
169 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)\...
Yulia Meshkova's user avatar
1 vote
0 answers
177 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 ...
knuth's user avatar
  • 33
0 votes
0 answers
172 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 ...
user3476591's user avatar
4 votes
2 answers
183 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 ...
Leonid Petrov's user avatar
3 votes
1 answer
117 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 $...
Pietro Majer's user avatar
  • 58.1k
3 votes
1 answer
150 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 ...
monotone operator's user avatar
2 votes
1 answer
209 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 ...
B.Hueber's user avatar
  • 1,077
0 votes
0 answers
93 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 ...
B.Hueber's user avatar
  • 1,077
2 votes
0 answers
92 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 $\lambda_k(t)$, $...
Eduardo Longa's user avatar
4 votes
1 answer
229 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'...
Pluviophile's user avatar
  • 1,576
0 votes
0 answers
53 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-...
SebastianP's user avatar
1 vote
0 answers
160 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 ...
oggius's user avatar
  • 85
1 vote
1 answer
75 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 ...
JZS's user avatar
  • 469
3 votes
2 answers
234 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}...
B.Hueber's user avatar
  • 1,077

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