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
940
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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 \...
- 196
3
votes
0
answers
121
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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 ...
0
votes
0
answers
42
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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 $X=(x_{ij})\in M_n(\mathbb{R})$ ...
- 3,774
3
votes
1
answer
186
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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 ...
- 223
0
votes
0
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83
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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 ...
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3
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0
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63
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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 ...
- 131
0
votes
0
answers
12
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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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3
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0
answers
79
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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 ...
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1
vote
0
answers
106
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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 $D$...
- 4,511
3
votes
1
answer
147
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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 ...
- 3,863
1
vote
0
answers
23
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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
$$
\begin{cases}
\partial^2_t u -\...
- 3,863
0
votes
0
answers
51
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The limit spectral distribution of the random matrix $(\hat{\Sigma}_1+\hat{\Sigma}_2)^{-1}\hat{\Sigma}_1$
Let $S_1$ and $S_2$ be the collection of i.i.d. copies of $X\sim\mathcal{N}(0,I_p)$, where $|S_1|=n_1,|S_2|=n_2$. Let $\hat{\Sigma}_1$ and $\hat{\Sigma}_2$ be the covariance matrix using samples in $...
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2
votes
0
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67
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On Dirichlet eigenfunctions of a domain
Given any bounded domain $\Omega\subset \mathbb R^n$, $n\geq 2$, with a Lipschitz boundary, let $\{(\lambda_k,\phi_k)\}_{k\in \mathbb N}$ be the Dirichlet eigenvalues and eigenfunctions of $-\Delta$ ...
- 3,863
4
votes
1
answer
112
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Existence of a domain with simple Dirichlet eigenvalues
Let $g$ be a smooth Riemannian metric on $\mathbb R^3$ that coincides with the Euclidean metric outside a compact set $K$. Does there exist some domain $\Omega$ with smooth boundary such that $K \...
- 3,863
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0
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118
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Is there a space of smooth functions dense in the domain of Coulomb-like potentials in dimension two?
Let $V : \mathbb{R}^2 \to \mathbb{R}$ be compactly supported, bounded away from the origin, and obey
$$ |V(x)| \lesssim r^{-\delta_0}, \qquad 0 < |x| \le 1, \qquad r : =|x|,$$
for some $0 < \...
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4
votes
0
answers
104
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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 ...
- 2,001
0
votes
0
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48
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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 ...
3
votes
1
answer
131
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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 ...
- 1,213
0
votes
1
answer
60
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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}^{...
- 3,863
2
votes
0
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32
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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 ...
- 3,863
0
votes
0
answers
108
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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_\...
- 1,057
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 ...
- 31
1
vote
0
answers
70
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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 ...
- 1,057
2
votes
1
answer
100
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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 ...
- 20.3k
3
votes
1
answer
203
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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 ...
- 540
0
votes
0
answers
29
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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 &...
- 447
7
votes
0
answers
157
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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 ...
- 71
3
votes
1
answer
120
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$\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$ ...
- 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)$...
3
votes
1
answer
133
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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 ...
- 888
1
vote
0
answers
44
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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 ...
- 11.4k
8
votes
2
answers
227
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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 $...
- 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 ...
- 11
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_{...
- 109
4
votes
1
answer
141
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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| \...
- 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,
$$S = -\frac{1}{2\pi i} ...
- 447
0
votes
0
answers
114
views
Question on possibility of uniquely defining the FRFT via certain properties
I was working around with the fractional Fourier transform (FRFT) when the mathematics undergrad found out, by brute-force computations, that the derivative of the FRFT with respect to the parameter ...
- 151
5
votes
1
answer
149
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Domains with discrete Laplace spectrum
Let $\Omega \subset \mathbb{R}^n$ be a domain. Assume that the Laplacian $-\Delta=-(\partial^2/\partial x_1^2+\cdots+\partial^2/\partial x_n^2)$ has a discrete spectrum on $L^2(\Omega)$ (i.e., we are ...
- 413
3
votes
0
answers
134
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Spectrum of large Hilbert matrices
Let $x_k>0$ be a increasing sequence of real numbers, such that
$$\sum_0^\infty\frac1{x_k}<+\infty.$$
Let us form the (infinite) Hilbert matrix $A\in{\bf Sym}({\mathbb N};{\mathbb R})$ with
$$a_{...
- 49.5k
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 \...
- 229
2
votes
0
answers
78
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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 ...
- 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 ...
- 53
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 ...
- 419
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 $A$ ...
- 211
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 $f_0 \in W^{1,2}(\mathbb R^d,\gamma_d)$.
...
- 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 \...
- 10.4k
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 ...
- 452
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 ...
- 8,446
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 ...
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 ...
- 4,511