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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Applications and motivations of resolvent for elliptic operator

Let $ A(x)=(a_{ij}(x)):\mathbb{R}^d\to\mathbb{R}^{d\times d} $ be a matrix satisfying ellipticity condition, that is \begin{align} \mu^{-1}|\xi|^2\geq \sum_{i,j=1}^da_{ij}(x)\xi_i\xi_j\geq\mu|\xi|^2 \...
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Closure of the point spectrum of an unbounded diagonalizable operator

Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to $H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...
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1 vote
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Are these $L_2$-spectral radii approximations strictly increasing?

Suppose that $V$ is a finite dimensional complex Hilbert space. Let $L(V)$ denote the collection of all linear mappings from $V$ to $V$. Let $A_1,\dots,A_r:V\rightarrow V$ be linear operators. Then ...
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4 votes
1 answer
114 views

Spectral theory of infinite volume hyperbolic manifolds

I have a question about the discrete spectrum of the Laplace operator on hyperbolic manifolds with infinite volume. I understand the case of infinite area surfaces: see Chapter 7 (Sections 1 and 2) of ...
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2 votes
1 answer
338 views

Positiveness of Banach limit [closed]

I‘m currently reading Arveson’s “A Short Course on Spectral Theory”, and I’m stuck at Exercise 3.1 (1). The question is: Let $l^{\infty}(\mathbb{N})$ be the set of all bounded sequences of complex ...
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6 votes
2 answers
95 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 $u$ ...
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Eigenvalues of two positive-definite Toeplitz matrices

Consider two positive-definite Toeplitz matrices $M_1$ and $M_2$ both with dimension $2^j \times 2^j$. Their matrix elements are: $$M_1[x,y] = \frac{\text{sin}(\pi(x-y)/2^j)}{\pi(x-y)} \qquad M_2[x,y] ...
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1 vote
1 answer
109 views

Fokker-Planck: equivalence between linear spectral gap and nonlinear displacement convexity?

In a smooth, bounded and convex domain $\Omega\subset \mathbb R^d$ consider the usual linear Fokker-Planck equation with Neumann (some would say Robin) boundary conditions \begin{equation} \label{FP} \...
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Spectrum invariant under (generalised) transpose as operator on trace class operators

For matrices $A$ it is well known that the spectrum is invariant under transpose $\sigma(A^T) = \sigma(A)$. Furthermore, the spectrum of the adjoint matrix $\sigma(A^*) = \overline{ \sigma(A)}$ the ...
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2 votes
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Second order differential operator with a Lipschitz coefficient

Let $a(x) \in W^{1, \infty}(\mathbb{R})$ be real-valued such that $a(x) \ge a_0 > 0$. Let $A^2$ denote the second order differential operator $A^2 : = -\partial_x (a(x) \partial_x) + 1 : L^2(\...
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Why are we interested in spectral gaps for Laplacian operators

Let $M$ be a Riemannian manifold and let $\Delta$ be its Laplacian operator. There is a large literature on a spectral gap for such a $\Delta$, that is, finding an interval $(0,c)$ which does not ...
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14 votes
3 answers
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Why is the set of Hermitian matrices with repeated eigenvalue of measure zero?

The Hermitian matrices form a real vector space where we have a Lebesgue measure. In the set of Hermitian matrices with Lebesgue measure, how does it follow that the set of Hermitian matrices with ...
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Hodge theory in higher eigen-spaces?

Hodge theory for elliptic complexes $E$ identifies the space of harmonic sections with cohomology $$\mathcal{H}(E) \simeq H(E).$$ A classical example with differential forms ($E = (\Omega,d)$) ...
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Question about Dirac operator

Let $D$ be a generalized Dirac operator on a complete Riemannian manifold. I'm a little confused to prove that there exists a constant $c>0$ such that $$\|D\sigma\|^2\geq c^2\|\sigma\|^2$$ for $\...
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On the convergence of the spectral decomposition of a harmonic function

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$ with a smooth boundary. Denote by $0<\lambda_1\leq \lambda_2\leq\ldots$ the Dirichlet eigenvalues of $-\Delta_g$ on $(M,g)$...
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3 votes
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Minimal eigenvalue of infinite dimensional matrix

Consider the following symmetric, positive-definite matrix $$ H_{nm}=-\frac{(4 m+4 n+1)}{(4 m-4 n-1) (4 m-4 n+1)}\sqrt{\frac{(4 m-1)!! (4 n-1)!!}{(4 m)!! (4 n)!!}} $$ where $n,m=0,1,2\dots$ (Here $!!$ ...
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1 vote
1 answer
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Regarding variation of spectra

I have been reading the article The variation of spectra by J.D Newburgh. in this article and all related reference/ articles, the term 'variation of spectra' keeps coming in, but I nowhere find a ...
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1 vote
1 answer
170 views

Monotonicity of eigenvalues II

In a previous question here, I asked the question below for block matrices and received an answer showing the question is true if $\mathcal B$ is hermitian and false, in general if $\mathcal B$ is non-...
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6 votes
1 answer
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Monotonicity of eigenvalues

We consider block matrices $$\mathcal A = \begin{pmatrix} 0 & A\\A^* & 0 \end{pmatrix}$$ and $$\mathcal B = \begin{pmatrix} 0 & B\\C & 0 \end{pmatrix}.$$ Then we define the new matrix $...
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3 votes
4 answers
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Prove that $(v^Tx)^2−(u^Tx)^2\leq \sqrt{1−(u^Tv)^2}$ for any unit vectors $u, v, x$

I believe I found a complicated proof by bounding the spectral norm $||uu^T-vv^T||^2_2:=\max_{||x||=1}|(u^Tx)^2-(v^Tx)^2|$. Using the fact that $dist(x,y):=\sin|x-y|$ is a distance function over unit ...
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2 votes
0 answers
41 views

Weyl asymptotic in towers

Let $M$ be a compact Riemannian manifold. The Weyl law gives the asymptotic of the counting function $N(T)$ of Laplace eigenvalues $|\lambda|\le T$ as $T\to\infty$. Now suppose you are given a tower ...
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Perturbation of Neumann Laplacian

Consider the $N \times N$ matrix $$A_{\alpha}=\begin{pmatrix} \lambda_1 & -1 & -\alpha & 0 & \cdots & 0\\ -1 & \lambda_2 & -1 & -\alpha & \cdots & 0\\ -\alpha &...
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Class of spectral zeta functions whose analytic extension takes a particular form

In quantum field theory the one-loop effective action is expressed in terms of the functional determinant of the (elliptic and self-adjoint) operator of small disturbances. Since the real eigenvalues ...
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2 votes
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Mathematical reason for scatter states being special?

In infinite spectral theory, we have the discrete and continuous spectrum, which are called "bound" and "scatter states" in physics. My understanding is, if $O \in B(H)$ is a self-...
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The definition of essential spectrum for general closed operators

I've asked this problem in MSE several days ago, see here. But there is no reply up until now. Maybe I wrote things too complicated there and so I'll write a very clean problem here. For background ...
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Can we describe the transition affected by the measurement of a quantum mechanical observable in the language of probability theory?

Consider a quantum mechanical system $S$ with the state space being given by a $\mathbb C$-Hilbert space $H$. It is assumed each physically measurable quantity $\mathcal A$ is described by an ...
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2 votes
0 answers
71 views

Proving an eigenvalue bound without resorting to Weyl's law

Suppose $(M,g)$ is a smooth compact Riemannian manifold of dimension $n\geq 2$ with smooth boundary and denote by $\{\phi_k,\lambda_k\}_{k\in \mathbb N}$ its Dirihclet spectral decomposition for the ...
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Reed-Simon Vol. IV: Question regarding convergence of eigenvalues

I am reading through Chapter XIII.16 of Reed and Simon's Methods of Modern Mathematical Physics IV: Analysis of Operators about Schrödinger operators with periodic potentials. Since the topic is kind ...
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2 votes
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Weyl's law and eigenfunction bounds for weighted Laplace-Beltrami operator

I would appreciate any answers or even references for the following problem. Let $(M,g)$ be a complete smooth Riemannian manifold with an asymptotically Euclidean metric (let's even say that the ...
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3 votes
0 answers
104 views

Extended adjoint of Volterra operator

Let $V$ be a Volterra operator on $L^2 [0,1]$. Does there exist a nonzero operator $X $ satisfying the following system $VX=XV^∗$, where $V^∗$ is the adjoint of the Volterra operator? $$ V(f) (x) =\...
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4 votes
1 answer
297 views

On eigenfunctions of the Laplace Beltrami operator [closed]

How can we generate the eigenspace for the Laplace Beltrami operator on SU(2)?
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0 answers
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Finding the kernel of the discrete Laplace operator?

I have the discrete Laplace operator on an infinite Hilbert space H, and wanted to find if its a Fredholm operator or not. So I want to check if its kernel or cokernel are finite dimensional. First, ...
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0 answers
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Regularity conditions involving fractional spectral Laplacian

I want to check the regularity conditions $$ (1) \quad\| (-\Delta)^{-1/2}F'(u)(-\Delta)^{1/2} w \| \leq L\cdot \| w \|,\quad u \in L^2(\Omega),\ w\in D((-\Delta)^{1/2})$$ and $$ (2)\quad \| \Delta^{-...
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2 votes
0 answers
77 views

How to prove that a finite rank perturbation on an infinite matrix does not change its continuous spectrum?

I have the discrete Laplace operator on an infinite Hilbert space with an orthonormal basis $\psi_x$ ($\forall x \in \mathbb Z$), given by $\Delta \psi_x=\psi_{x-1}+\psi_{x+1}$. If I introduce a ...
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1 vote
1 answer
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Adjoint operator of OU generator

The generator an OU process is given by $$A = \operatorname{tr}(QD^2)+\langle Bx,D\rangle.$$ This one possesses an invariant measure given by $$d\mu(x) = b(x) \ dx \text{ with } b(x) = \frac{1}{(4\pi)^...
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  • 369
1 vote
0 answers
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Eigenvalues and eigenvectors of non-symmetric elliptic operators

We know that the operator $A=\Delta$ with domain $D(A)=\{u\in W^{2, 2}(\Omega): u=0 \ \ \text{on } \partial\Omega\}$ (say $\Omega$ is a bounded nice domain) has eigenvalues $\lambda_1>\lambda_2\ge \...
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3 votes
0 answers
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Eigenvalues of an elliptic operator on shrinking domains

This was probably done somewhere 100 times, but I can't find a reference. Assume that we have a bounded star-shaped domain $\Omega\subset \mathbb{R}^n$ with piece-wise smooth boundary and a general ...
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4 votes
1 answer
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Domain of Friedrichs extension of $-\partial^2_r + mr^{-2} : L^2(0,\infty) \to L^2(0,\infty)$

Consider the second order differential operator $$ A = -\partial^2_r + mr^{-2} : L^2(0, \infty) \to L^2(0,\infty), \qquad m \ge -\frac{1}{4}, $$ equipped with domain $C^\infty_0(0, \infty)$. Since $\|...
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  • 273
4 votes
1 answer
119 views

Resource on spectral theory for differential operators with symmetry groups

In Methods of Mathematical Physics IV by Reed and Simon, the authors cover Floquet theory in detail in Section XIII.16. On page 280, they note that "A part of the analysis of [the periodic ...
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3 votes
1 answer
184 views

Reference request for spectral theory of elliptic operators [closed]

I want to learn the spectral theory of linear elliptic operators in bounded and unbounded domains in $R^n$, in particular for Laplacian and Schrodinger operators. Please suggest me some reference. I ...
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2 votes
0 answers
34 views

Number of small eigenvalues for flat unitary bundles

It is known that the number of small eigenvalues (eigenvalues less than $1/4$) of the Laplacian on hyperbolic surfaces may be topologically bounded from above. If $X$ is a finite area hyperbolic ...
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1 vote
0 answers
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Show that an integral operator with Bessel function kernel is bounded on $L^2(0,\infty)$

Let $J_0$ denote the Bessel function of the first kind of order $\nu = 0$ (see DLMF 10.2), $$ J_0(z) = \sum_{k = 0}^\infty (-1)^k \frac{(\tfrac{1}{4} z^2)^k }{k! \Gamma(k + 1)}. $$ Put $u_0(r) = r^{1/...
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3 votes
0 answers
113 views

Non-emptiness of spectrum $\sigma(a)$ in non-Archimedean Banach algebras

I'm trying to determine whether or not the standard proof that the spectrum of a point in a unital Banach algebra is non-empty can be adapted to prove the same thing over certain non-Archimedean ...
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0 votes
0 answers
66 views

Hessian estimates of eigenfunctions without Bochner

let $\Omega$ be a bounded domain in a Riemannian manifold $(M,g)$. Consider the Dirichlet eigenvalues and eigenfunctions of Laplacian on $\Omega$, that are, the $\lambda_i>0$ and $\phi_i\in H^{1}_0(...
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2 votes
0 answers
95 views

Existence and uniqueness for $\Delta f + \lambda f = g$ on $S^2$ for $\lambda>0$ [closed]

Consider the PDE $$\Delta f + \lambda f = g$$ on $S^2$, where $\Delta$ is with respect to the round metric, $g \in L^2(S^2)$ and $\lambda>0$. I wish to study the existence and uniqueness of this ...
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6 votes
1 answer
186 views

Criteria for operators to have infinitely many eigenvalues

Normal compact linear operators on Hilbert spaces have infinitely many (counting multiplicities) eigenvalues by the spectral theorem. For non-normal operators this no longer has to be true. There ...
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  • 417
4 votes
1 answer
128 views

Spectrum Cauchy-Euler operator

A Cauchy-Euler operator is an operator that leaves homogeneous polynomial of a certain degree invariant, named after the Cauchy-Euler differential equations We consider the operator $$(Lf)(x) = \...
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  • 417
8 votes
0 answers
423 views

Why is spectral theory developed for $\mathbb C$

Spectral theory is a fundamental part of operator theory and the spectrum of many operators is investigated throughout the existing literature. And that is for a good reason: If $A$ is some closed ...
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  • 351
1 vote
1 answer
58 views

A property for generic pairs of functions and metrics

Let $M$ be a compact smooth manifold with a smooth boundary. Given a smooth Riemannian metric $g$ on $M$, we denote by $\{\phi_k\}_{k=1}^{\infty}$ an $L^2(M)$--orthonormal basis consisting of ...
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  • 2,858
2 votes
1 answer
213 views

Compactness for initial-to-final map for heat equation

Let $M$ be a compact smooth manifold without boundary. Let $T>0$ and let $g$ be a smooth Riemannian metric on $M$. Given any $f \in L^2(M)$ let $u$ be the unique solution to the equation $$\...
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