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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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Multiplicity of Laplace eigenvalues and symmetry

Let $S$ be a smooth closed connected hyperbolic surface. On $S$ we have the Laplace operator $\Delta$, whose eigenvalues form a discrete sequence \begin{equation} 0=\lambda_0<\lambda_1\leq \...
Claudius's user avatar
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121 views

On the spectrum of Fokker–Planck with linear drift

The paper by Liberzon and Brockett, "Spectral analysis of Fokker–Planck and related operators arising from linear stochastic differential equations." SIAM Journal on Control and ...
mathamphetamine's user avatar
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Computing the eigenvalues of $A+E$ where $A$ is an upper triangular matrix whose diagonal entries are all zero and $E$ is a rank one matrix

Let us consider the backward-shift matrix $B=(b_{ij})\in M_n(\mathbb{R})$ whose entries are given by $b_{k,k+1}=1$ and the other entries are all 0. We also consider $X=(x_{ij})\in M_n(\mathbb{R})$ ...
ABB's user avatar
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3 votes
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186 views

The definition of simple eigenvalue

This question was posted a long time ago on the mathexchange, but I didn't get any answers there, and despite having discussed it with some colleagues, I don't think I have a definitive answer. I am ...
Mrcrg's user avatar
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Given $\sigma(AB-BA) = \{0\}$, what can be said about $\sigma(A)$ and $\sigma(B)$?

Let $\mathcal H$ be a separable Hilbert space, and $\mathfrak B(\mathcal H)$ denote the algebra of bounded linear operators on $\mathcal H$. Furthermore, let $A,B \in \mathfrak B(\mathcal H)$ be two ...
stoic-santiago's user avatar
3 votes
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63 views

How do I show countable additivity of (proposed) spectral measure in the proof of the spectral theorem?

I'm currently writing a Bachelors thesis based on the following paper: Douglas, R., & Pearcy, C. (1970). On the spectral theorem for normal operators. Mathematical Proceedings of the Cambridge ...
Abhijeet Vats's user avatar
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12 views

Spectral universality of sample covariance from unit sphere

If $x_1,\dots,x_n\sim \mu$ are mean-zero iid samples in $R^d$ that are drawn from unit sphere, with covariance $E x x^\top = I_d,$ and $C_n:=\frac1n \sum_i^n x_i x_i^\top$ is the sample covariance, ...
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Positive definitness of $f(|x|^\gamma)$, $0<\gamma<1$

Let $f(x)$ be a positive definite function on $x \in R^d$. Assume $f(x)$ is radial , so $f(x)$ is a function of $|x|$, let's say $g(|x|):=f(x)$. How can I show that $g(|x|^\gamma)$ is positive ...
Alexey S's user avatar
1 vote
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106 views

Quiver representations and the standard matrix decompositions

Many matrix decompositions - like the Jordan Normal Form, the SVD, the spectral theorem, the Takagi decomposition - have the property that they express a matrix $M$ as the form: $$M = A D B$$ where $D$...
wlad's user avatar
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On a compact operator in the plane

Let $\Omega \subset \mathbb R^2$ be a bounded domain with a smooth boundary. Let $$\bar{\partial}= \frac{1}{2} ( \partial_{x^1} + i \,\partial_{x^2}),$$ and let $G: L^2(\Omega)\to H^2(\Omega)$ be the ...
Ali's user avatar
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1 vote
0 answers
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On spectral representation of solutions to wave equations with impulse initial data

Let $\Omega \subset \mathbb R^n$ be a bounded domain with a smooth boundary that contains the origin. Let us consider the following classical linear wave equation $$ \begin{cases} \partial^2_t u -\...
Ali's user avatar
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0 answers
51 views

The limit spectral distribution of the random matrix $(\hat{\Sigma}_1+\hat{\Sigma}_2)^{-1}\hat{\Sigma}_1$

Let $S_1$ and $S_2$ be the collection of i.i.d. copies of $X\sim\mathcal{N}(0,I_p)$, where $|S_1|=n_1,|S_2|=n_2$. Let $\hat{\Sigma}_1$ and $\hat{\Sigma}_2$ be the covariance matrix using samples in $...
aurora_borealis's user avatar
2 votes
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67 views

On Dirichlet eigenfunctions of a domain

Given any bounded domain $\Omega\subset \mathbb R^n$, $n\geq 2$, with a Lipschitz boundary, let $\{(\lambda_k,\phi_k)\}_{k\in \mathbb N}$ be the Dirichlet eigenvalues and eigenfunctions of $-\Delta$ ...
Ali's user avatar
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4 votes
1 answer
112 views

Existence of a domain with simple Dirichlet eigenvalues

Let $g$ be a smooth Riemannian metric on $\mathbb R^3$ that coincides with the Euclidean metric outside a compact set $K$. Does there exist some domain $\Omega$ with smooth boundary such that $K \...
Ali's user avatar
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3 votes
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Is there a space of smooth functions dense in the domain of Coulomb-like potentials in dimension two?

Let $V : \mathbb{R}^2 \to \mathbb{R}$ be compactly supported, bounded away from the origin, and obey $$ |V(x)| \lesssim r^{-\delta_0}, \qquad 0 < |x| \le 1, \qquad r : =|x|,$$ for some $0 < \...
JZS's user avatar
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0 answers
104 views

Eigenvalues of Schrödinger operator with Robin condition on the boundary

Let $(M^2,g)$ be a compact Riemannian surface with boundary and let $L = \Delta_g + q$ be a Schrödinger operator, where $\Delta_g = -\operatorname{div} \nabla$ is the Laplacian with respect to the ...
Eduardo Longa's user avatar
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48 views

Commutativity of spectral projection with linear opeartor

For $A \in L(X)$, with A a being a closed operator and $X$ is a Banach space (with bounded $\sigma(A)$), define $\mathbb{P} = \frac{1}{2 \pi i} \int_\gamma R(\lambda,A) \, d\lambda$ to be the spectral ...
LeiLi7865's user avatar
3 votes
1 answer
131 views

Must solutions to the time-independent Schrodinger equation that have discrete or negative eigenvalues be square-integrable?

This is a very straightforward question (although the answer may not be!) about a ubiquitous textbook physics situation. I assume that the answer is probably well-known, but I asked it on both Math ...
tparker's user avatar
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1 answer
60 views

Orthogonality to a one parameter family of eigenfunctions

Let $\rho>0$ be a smooth realvalued function such that $\rho=1$ outside the unit interval $(-1,1)$. For each $t>0$, let us denote by $\{\lambda_n(t)\}_{n=1}^{\infty}$ and $\{\phi_n(t;x)\}_{n=1}^{...
Ali's user avatar
  • 3,863
2 votes
0 answers
32 views

A mapping property for fractional Laplace--Beltrami operator

Suppose that $g$ is a smooth Riemannian metric on $\mathbb R^3$ that is equal to the Euclidean metric outside the unit ball. Let us define the fractional Laplace--Beltarmi operator on $\mathbb R^3$ of ...
Ali's user avatar
  • 3,863
0 votes
0 answers
108 views

About the proof of Lebesgue decomposition theorem for Hilbert spaces

Let $\mu$ be a Borel measure on $\mathbb{R}$. By the Lebesgue decomposition theorem, there exists measures $\mu_\text{pp}$, $\mu_\text{ac}$ and $\mu_\text{sing}$ such that $\mu = \mu_\text{pp}+\mu_\...
MathMath's user avatar
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2 votes
0 answers
47 views

Examples of elementary group of isometries of the ideal boundary of hyperbolic plane

A Riemannian manifold $(X,g)$ is called Hadamard manifold if it is complete and simply connected and has everywhere non-positive sectional curvature. An example of such a manifold would be the ...
Quanta's user avatar
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0 answers
70 views

Uniqueness of Borel functional calculus for unbounded self-adjoint operators

I was reading these short notes on the Borel functional calculus where the author discusses the uniqueness property of this calculus for both bounded and unbounded self-adjoint operators. When it ...
MathMath's user avatar
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2 votes
1 answer
100 views

Existence of eigen basis for elliptic operator on compact manifold

Let $M$ be a compact Riemannian manifold. Let $E$ be a vector bundle over $M$ equipped with a Hermitian (or Euclidean) metric on its fibers. Let $D$ be a linear elliptic differential operator acting ...
asv's user avatar
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3 votes
1 answer
203 views

Spectral Radius and Spectral Norm for Markov Operators

My question concerns differences between the spectral radius $\rho$ and norm $\| \cdot \|$ of Markov operators in infinite-dimensional Banach spaces. This is far from my area of expertise, that is ...
Sam OT's user avatar
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0 answers
29 views

Uniqueness of solution to abstract wave equation with unsigned energy

Let $H$ be a self-adjoint operator on a Hilbert space $(\mathcal{H}, \langle \cdot, \cdot \rangle$). Suppose the spectrum of $H$ in $(-\infty, 0)$ consists of only finitely many eigenvalues $\mu^2_k &...
JZS's user avatar
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7 votes
0 answers
157 views

Spectral decomposition of $\Gamma\backslash X$

Let $X$ be a reasonable manifold of non-positive curvature (could be $\mathbb{H}^n$, symmetric or locally symmetric space, homogeneous Hadamard manifold etc.), and let $\Gamma$ be a reasonable group ...
SKNEE's user avatar
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3 votes
1 answer
120 views

$\tau$-measurable operator

Problem: Let $M$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $\tau$. Let $m$ be a positive element in $M$ and let $e_{(0,\infty)}(m)$ be the spectral projection of $m$ ...
John's user avatar
  • 45
1 vote
0 answers
35 views

$L^2$ norm of a kernel with a variable width

Suppose you have a kernel operator on a torus, with a kernel of a spatially varying width $\epsilon(x)$, which might be zero at certain points. That is to say, for some approximate identity $\psi_h(x)$...
Caroline Wormell's user avatar
3 votes
1 answer
133 views

Characters of algebra of Schwartz functions

Consider the (non-unital) $\mathbb{C}$-algebra (point-wise multiplication) of $\mathcal{S}$ of Schwartz functions on $\mathbb{R}$. Question: Does there exist some character (non-zero multiplicative ...
Hua Wang's user avatar
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1 vote
0 answers
44 views

What do you call this class of matrices with a unique positive eigenvalue associated to a graph?

I am looking for the name of a class of symmetric matrices $M\in\Bbb R^{n\times n}$ that I can associate to a (finite simple) graph $G=(V,E)$ with $V=\{1,...,n\}$ and that have the following ...
M. Winter's user avatar
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8 votes
2 answers
227 views

Is the $n/2$-th heat kernel coefficient topological?

I have asked the same question on math.SE, without much success so I'm trying my luck here too. Let $M$ be an $n$-dimensional manifold, with $n$ even and consider the heat kernel of the Laplacian on $...
brick's user avatar
  • 91
1 vote
0 answers
42 views

The spectral measure of the integral of an operator-valued function with respect to a projection-valued measure

Let $\mathcal{H}$ be a Hilbert space, $T$ a bounded self-adjoint operator, and $F:\left[a,b\right]\to\mathcal{B}\left(\mathcal{H}\right)$ such that for any $t\in\left[a,b\right]$, $F\left(t\right)$ is ...
user122321's user avatar
0 votes
0 answers
18 views

How to calculate the power transformation of a spectral density function

There is a problem I have been trying to solve for a while. Let $X_t$ be a stationary (univariate) time series. The spectral density of the moving average process $$X_t=\sum^{\infty}_{j=-\infty}a_je_{...
Saïd Maanan's user avatar
4 votes
1 answer
141 views

Spectrum near zero of $-\partial^2_x + V : L^2(\mathbb{R}) \to L^2(\mathbb{R})$, where $V = O(|x|^{-2 - \delta})$

Let $H = -\partial^2_x + V(x) : L^2(\mathbb{R}) \to L^2(\mathbb{R})$ be a one dimensional Schrödinger operator, where the potential $V$ is real-valued, belongs to $L^\infty(\mathbb{R})$, and, as $|x| \...
JZS's user avatar
  • 447
1 vote
1 answer
56 views

Show $A_0 + S$ is invertible, where $S$ is the Riesz projection of bounded $A_0$

Let $A_0$ be a bounded linear operator on a Hilbert space $H$. Suppose $0$ is an isolated point of the spectrum of $A_0$. Let $S$ be the corresponding Riesz projection, namely, $$S = -\frac{1}{2\pi i} ...
JZS's user avatar
  • 447
0 votes
0 answers
114 views

Question on possibility of uniquely defining the FRFT via certain properties

I was working around with the fractional Fourier transform (FRFT) when the mathematics undergrad found out, by brute-force computations, that the derivative of the FRFT with respect to the parameter ...
Kanghun Kim's user avatar
5 votes
1 answer
149 views

Domains with discrete Laplace spectrum

Let $\Omega \subset \mathbb{R}^n$ be a domain. Assume that the Laplacian $-\Delta=-(\partial^2/\partial x_1^2+\cdots+\partial^2/\partial x_n^2)$ has a discrete spectrum on $L^2(\Omega)$ (i.e., we are ...
mmen's user avatar
  • 413
3 votes
0 answers
134 views

Spectrum of large Hilbert matrices

Let $x_k>0$ be a increasing sequence of real numbers, such that $$\sum_0^\infty\frac1{x_k}<+\infty.$$ Let us form the (infinite) Hilbert matrix $A\in{\bf Sym}({\mathbb N};{\mathbb R})$ with $$a_{...
Denis Serre's user avatar
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2 votes
1 answer
182 views

Spectrum of $(Jx)_n =i((2n+1)x_{n+1}-(2n-1)x_{n-1})$ on $\ell^2(\mathbb{Z})$

I've been working on the spectrum of the closure of the operator $J: \mathcal{D}(J)= \mbox{span}\{ e_n: n \in \mathbb{Z}\} \subseteq \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ defined for $x=(x_n)_{n \...
Mainkit's user avatar
  • 229
2 votes
0 answers
78 views

On Fredholm alternative for Neumann conditions

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and $f \in L^2(\Omega)$. It is well known that if $\lambda$ is a Dirichlet Laplacian eigenvalue, then the equation $$\begin{cases} -\Delta ...
student's user avatar
  • 1,320
4 votes
1 answer
475 views

Left and right eigenvectors are not orthogonal

Consider a compact operator $T$ on a Hilbert space with algebraically simple eigenvalue $\lambda$. Is it then true that left (the eigenvector of the adjoint with complex-conjugate eigenvalue) and ...
Guido Li's user avatar
4 votes
1 answer
99 views

Uniform decay of operator norm for smooth family of operators

Let $\mathscr{H}$ be a Hilbert space and let $\mathbb{R} \to B(\mathscr{H}), r \mapsto S_r$ be a continuous (or smooth) family of operators, where $B(\mathscr{H})$ is the space of bounded operators on ...
Constantin K's user avatar
0 votes
0 answers
111 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 $A$ ...
Dasherman's user avatar
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0 votes
0 answers
43 views

A question about approximation in Gaussian Sobolev space

Let $\gamma_d$ be the standard Gaussian measure in $\mathbb R^d$, and let $W^{1,2}(\mathbb R^d,\gamma_d)$ be the Gaussian Sobolev space on $\mathbb R^d$. Fix $f_0 \in W^{1,2}(\mathbb R^d,\gamma_d)$. ...
dohmatob's user avatar
  • 6,338
7 votes
0 answers
128 views

Hölder continuity of spectrum of matrices

Endow $\mathbb{C}^{d \times d}$ with the norm induced by the Euclidean norm on $\mathbb{C}^d$. It is well-known (to those who know it well, I guess) that the spectrum $\sigma(A)$ of a matrix $A \in \...
Jochen Glueck's user avatar
0 votes
0 answers
21 views

Minimizing $\|Ax\|_2^2$ over the simplex for a very fat matrix

Suppose I have a really fat $mxN$ matrix (with $N>>m$). I want to find an $x$ in the simplex (i.e $x’1=1$ and $x_i\geq 0$) that minimizes $\|Ax\|_2^2$. The only operations I can practically do ...
AspiringMat's user avatar
9 votes
1 answer
660 views

Counterexamples to weak dispersion for the Schrödinger group

Let $A$ be a selfadjoint operator on some Hilbert space $H$, let $U(t)=e^{itA}$ be the corresponding continuous group, and let $f\in H$ be orthogonal to all eigenvectors of $A$. Are there examples ...
Piero D'Ancona's user avatar
0 votes
0 answers
22 views

Generalization of covariance of random (eigen)vectors to $k-$variance

Recently I got interested in "random Gaussian eigenvectors": Fix a large matrix $A\in \text{GL}_d(\mathbb{C})$ and denote the orthogonal projector to a fixed eigenspace by $P$. One can ...
Denis Marcinkov's user avatar
3 votes
0 answers
168 views

Where could a paper on a unification of matrix decompositions be published?

I've got a paper which shows that when the spectral theorem (as a statement that every self-adjoint matrix can be unitarily diagonalised) is naively generalised to $*$-algebras other than the complex ...
wlad's user avatar
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