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
990
questions
2
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68
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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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0
answers
458
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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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2
votes
0
answers
80
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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0
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180
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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
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0
answers
68
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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 ...
- 4,089
3
votes
0
answers
199
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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?
$$ V(f) (x) =\...
7
votes
1
answer
908
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On eigenfunctions of the Laplace Beltrami operator [closed]
How can we generate the eigenspace for the Laplace Beltrami operator on SU(2)?
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2
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0
answers
129
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How to prove that a finite rank perturbation on an infinite matrix does not change its continuous spectrum?
I have the discrete Laplace operator on an infinite Hilbert space with an orthonormal basis $\psi_x$ ($\forall x \in \mathbb Z$), given by $\Delta \psi_x=\psi_{x-1}+\psi_{x+1}$. If I introduce a ...
- 539
1
vote
1
answer
117
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Adjoint operator of OU generator
The generator an OU process is given by
$$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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1
vote
0
answers
149
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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
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0
answers
63
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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
112
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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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4
votes
1
answer
147
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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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2
votes
1
answer
209
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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 ...
2
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0
answers
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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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0
answers
63
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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
146
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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 ...
0
votes
0
answers
108
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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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0
answers
100
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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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votes
1
answer
356
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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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4
votes
1
answer
190
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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
$$(Lf)(x) = \...
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9
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0
answers
504
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Why is spectral theory developed for $\mathbb C$
Spectral theory is a fundamental part of operator theory and the spectrum of many operators is investigated throughout the existing literature. And that is for a good reason: If $A$ is some closed ...
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1
vote
1
answer
72
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A property for generic pairs of functions and metrics
Let $M$ be a compact smooth manifold with a smooth boundary. Given a smooth Riemannian metric $g$ on $M$, we denote by $\{\phi_k\}_{k=1}^{\infty}$ an $L^2(M)$--orthonormal basis consisting of ...
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2
votes
1
answer
241
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Compactness for initial-to-final map for heat equation
Let $M$ be a compact smooth manifold without boundary. Let $T>0$ and let $g$ be a smooth Riemannian metric on $M$. Given any $f \in L^2(M)$ let $u$ be the unique solution to the equation
$$\...
- 4,089
4
votes
1
answer
287
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First eigenvalue of the Laplacian on the traceless-transverse 2-forms
Let $(S^3/\Gamma, g)$ be a spherical space form with constant sectional curvature $1$, where $\Gamma$ is a finite subgroup of $SO(4)$ acting freely on $S^3$.
Consider the first nonzero eigenvalue ...
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votes
0
answers
99
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The non empty set of accumulation points of a bounded linear operator is the spectrum of another operator
Let $X$ be an infinite dimensional Banach space, and let $T \in L(X)$ such that the set of accumulation points of $T$ is non empty, i-e $\mbox{acc}\,\sigma(T)\neq 0.$\
Is there a Banach space $Y$ ...
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2
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1
answer
224
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On the Schrödinger equation and the eigenvalue problem
Li-Yau 1983_Article
The second part of above paper used the discrete eigenvalues of $\frac{-\Delta}{q}$ where $q>0$ to proof the the number of non-positive eigenvalues of
Schrödinger operator $-\...
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votes
0
answers
53
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A question about the choice of a special harmonc spinor
Let $X$ be a complete Riemannian manifold and $H$ be the kernel of generalized Dirac operator $D$ on $L(S)$, where $S$ is the Dirac bundle. Let $K$ be a compact subset of $X$ and $K\subset \Omega$ be ...
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2
votes
1
answer
99
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Orthogonal decomposition of $L^2(SM)$
I have been stuck on the following problem for a long time but I could not get the answer. Would you please help me? I was reading one paper [Paternian Salo Uhlmann: Tensor tomography on the surface] ...
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5
votes
1
answer
326
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Lower bound on the first eigenvalue of the Lichnerowicz Laplacian on positive Einstein manifolds
Suppose $(M^n,g)$ is an $n$-dimensional Einstein manifold with $Ric=(n-1)g$. Let $\lambda$ be the minimal eigenvalue of the Lichnerowicz Laplacian $\Delta_L$ defined on all transverse-traceless ...
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0
answers
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Convergence in the resolvent sense and spectral properties
Let $\{T_k\}_k, T$ be unbounded selfadjoint operators on a Hilbert space $H$. If $T_k\to T$ in the norm-resolvent sense, then for any $(a,b)\subset \mathbb R$ with $\{a,b\}\cap \sigma(T)=\emptyset$, ...
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2
votes
1
answer
130
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Expressing the singular values of a 2-by-2 real-valued matrix by the norm of the two columns and the angle between them
I'm looking for an elegant way to show the following claim.
Claim: Let $m_1, m_2 \in \mathbb{R}^2$ be the two columns of matrix $M \in \mathbb{R}^{(2 \times 2)}$. The singular values of the matrix are ...
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1
vote
1
answer
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Eigenvalues of operator
In the question here
the author asks for the eigenvalues of an operator
$$A = \begin{pmatrix} x & -\partial_x \\ \partial_x & -x \end{pmatrix}.$$
Here I would like to ask if one can extend ...
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3
votes
1
answer
156
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Explicit eigenvalues of matrix?
Consider the matrix-valued operator
$$A = \begin{pmatrix} x & -\partial_x \\ \partial_x & -x \end{pmatrix}.$$
I am wondering if one can explicitly compute the eigenfunctions of that object on ...
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1
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0
answers
157
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Relation between the spectrum of 𝐷𝐴 and 𝐴 where 𝐴 is a bounded linear self-adjoint positive operator and 𝐷 is a constant diagonal positive matrix
Let $D$ be a $3\times 3 $ constant real positive-definite diagonal matrix and $\Omega\subset\mathbb{R}^3$ be a bounded lipschitz domain.
Denote by $(L^2(\Omega))^3$ the set of square integrable ...
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0
votes
1
answer
100
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Essential spectrum of constant invertible diagonal matrix acting on a product of Hilbert spaces [closed]
Let $M$ be a $3\times 3$ real invertible diagonal matrix and $H$ a Hilbert space of infinite dimension (for example, we can take $H$ as the space of square integrable functions over a bounded ...
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3
votes
0
answers
243
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Extending Ky Fan's eigenvalues inequality to kernel operators
--Migrating from MSE since it might fit better here--
Base result
The following result in Terry Tao's book, 'p. 47, Ky Fan inequality' reads as:
$$\sum_i\lambda_i(A+B) \leq \sum_i \lambda_i(A) + \...
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2
votes
0
answers
93
views
Spectrum of a Lax Pair and conservation laws of a PDE
I would like to ask a question that I had asked a few days ago on the site math.stackexchange
and I still have not received an answer.
If we have a Lax operator, we know that the spectrum of this ...
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14
votes
1
answer
820
views
Spectrum of matrix involving quantum harmonic oscillator
The quantum harmonic oscillator relies on two classical objects, the so-called creation and annihilation operator
$$a ^* = x- \partial_x \text{ and }a = x+\partial_x.$$
Fix two numbers $\alpha,\beta \...
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4
votes
0
answers
147
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Products of eigenfunctions on compact Riemann surfaces
Let $M$ be a compact Riemann surface with genus $g\geq 2$, endowed with the Riemannian metric with constant sectional curvature $-1$. Let $f_1, f_2$ be two (global) eigenfunctions for the Laplace-...
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votes
1
answer
231
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Essential spectrum under perturbation
Given a Banach space $X$ and a bounded linear operator $T$ on $X$.
It's well known that the essential spectrum of $T$ is invariant under additive compact perturbation.
My question is about minimal ...
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1
vote
1
answer
332
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Spectral theorem and diagonal expansion for self adjoint operators
Asked by a physicist:
In physics we use the fact that, given a self adjoint operator $A$, every element in Hilbert space can be written as "generalized linear combination" of the eigenstates ...
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1
vote
1
answer
100
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uniform convergence of $H^r$ projectors on compact sets?
Let $\Omega\subset \mathbb R^d$ be a smooth, bounded domain. Let $(e_n)_{n\geq 0}\subset L^2(\Omega)$ be the Hilbert basis generated by the Dirichlet-Laplacian eigenfunctions, i-e $-\Delta e_n=\...
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1
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0
answers
95
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Counting number of distinct eigenvalues
Let $\Omega$ be a Lipschitz domain in $\mathbb{R}^n$, and let $N(\lambda)$ be the number of Dirichlet Laplacian eigenvalues less than or equal to $\lambda$. The famous Weyl's law says that as $\...
- 1,320
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votes
0
answers
50
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Relation between the eigenvalues of the weighted laplacian and fractional laplacian?
Consider the eigenvalue problem $-\Delta u = \lambda u\rho$ for $u\in \dot{H}^{1}(\mathbb{R}^n)$ with $n\geq 3$ and weight $\rho\in L^{n/2}(\mathbb{R}^n).$ Let $(\lambda_k, \psi_k)$ be the increasing ...
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2
votes
1
answer
102
views
Finiteness of Schatten $p$-norm of truncated free resolvent
Consider the resolvent operator $ R(z) := (-\Delta - z)^{-1}$ of the Laplace operator on $L^2(\mathbb R^d)$, where $z\in \rho(-\Delta) = \mathbb C \setminus \mathopen [0, \infty)$.
For $p \geq 1$, let ...
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3
votes
0
answers
136
views
Is the term "Borel transform" appropriate here?
Does anyone knows a reference for the name "Borel transform" given to the map $\mu\to F_\mu$ where $\mu$ is a probability measure on the real line and $F_\mu$ is the holomorphic function ...
0
votes
0
answers
99
views
Are there any known results for the spectrum of $(-\Delta)^s/V^{p-1}$?
I am interested in generalizing some results known for the $\frac{-\Delta}{U^{p-1}}$ where $U$ is a Talenti bubble to the non-local operator $\frac{(-\Delta)^s}{V^{p-1}}$ where $U$ and $V$ are bubbles ...
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2
votes
2
answers
384
views
Does this operator have a continuous, localized eigenfunction with negative eigenvalue?
I am looking at a class of operators
$$
L[f](x)=af_{xxxx}-bf_{xx}+\frac{d}{dx}(\delta(x)f_x)
$$ , a<0,b<0,
on the real line, where $\delta$ is Dirac-delta.
I am interested in ruling out the ...
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3
votes
1
answer
420
views
Hilbert-Schmidt integral operator with missing eigenfunctions
I'm having some issues with the spectral decomposition of the integral operator
\begin{equation}
(Af)(x)=\int_0^1|x-y|f(y)dy,\text{ with $f\in L^2[0,1]$}.
\end{equation}
Since
\begin{equation}
...
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