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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1answer
108 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) = \...
8
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0answers
395 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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1answer
52 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 ...
2
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1answer
200 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 $$\...
4
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1answer
202 views

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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50 views

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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69 views

Existence of solution of a difference equation

Consider the operator $T$ acting from $L^2(0,1+r)$ to it self defined by $$Tu(x)=a1_{(0,1)}(x)u(x+r)+b1_{(1,1+r)}(x)u(x-1)$$ where $r\in (0,1)$ and $a,b$ are real, and $1_A(x)$ is the characteristic ...
2
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1answer
165 views

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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33 views

Analytic extensions of heat equation with time-dependent coefficients

Let us consider a closed manifold $M$ (compact smooth manifold with no boundary) and let $\{g_t\}_{t\in [0,T]}$ be a family of smooth Riemannian metrics on $M$ that depend smoothly on $t$. We consider ...
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0answers
50 views

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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1answer
71 views

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] ...
4
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1answer
91 views

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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43 views

Does the eigenvectors of a self-adjoint operator whose spectrum consists of simple eigenvalues construct a Hilbert basis?

I already asked this question on math.stackexchange a few days ago, but I still haven't received an answer, so I'm asking it here. Some PDE's have what is called a Lax pair i.e. there exists two ...
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42 views

Lack of conformal invariance for fractional Laplace operators on a closed Riemannian surface

Let $(\Sigma,g)$ be a compact two-dimensional Riemannian manifold with no boundary. Let us denote by $\{(\lambda_k,\phi_k)\}_{k=0}^{\infty}$ the spectral data for the Laplace--Beltrami operator on $(\...
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41 views

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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1answer
73 views

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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1answer
161 views

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 ...
3
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1answer
140 views

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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0answers
141 views

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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1answer
69 views

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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0answers
90 views

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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0answers
53 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 ...
14
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1answer
711 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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97 views

Fredholm Operator

Let $$\Psi : h\in L^2(\mathbb T)\longmapsto \left(\langle f_{n}(h)|e^{inx}\rangle\right)_{n\in \mathbb N}\in \ell^2(\mathbb N, \mathbb C),$$ and suppose that $f_n(h)=C(n,x) h\in L^2(\mathbb T)$ ...
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0answers
99 views

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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1answer
117 views

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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1answer
197 views

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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1answer
76 views

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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0answers
75 views

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 $\...
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34 views

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 ...
2
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1answer
65 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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0answers
74 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 ...
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0answers
70 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 ...
2
votes
2answers
356 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 ...
3
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1answer
174 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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0answers
48 views

The advantages of having a discrete spectrum of the Lax operator

Suppose that we have an equation which admits a pair of Lax $(L_u, B_u)$ and that the Lax operator $L_u$ admits a spectrum that is formed of eigenvalues all simple. What is the advantage to have a ...
7
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0answers
255 views

"Non-critical" zeros of $\zeta$ and the $\zeta$-cycles of Connes and Consani

In the recent preprint of Connes and Consani https://arxiv.org/abs/2106.01715 a new spectral realization of the critical zeros of $\zeta$ (edit: defined as being those on the critical line only, see ...
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1answer
117 views

Are $\log(\sigma(A(z))$ subharmonic functions?

Let $A$ be a matrix-valued entire function. It is then well-known that $\log \Vert A(z)\Vert$ is subharmonic. In particular, the operator norm is just the largest singular value of $A$. Is it ...
0
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2answers
97 views

Computation to differentiate a determinant [closed]

Consider a positive Hermitian $N \times N$ matrix $A$ with complex valued coefficients. We list its eigenvalues in increasing order and with their multiplicities, $\mu_{1} \leq \mu_{2} \leq \cdots \...
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1answer
75 views

Relation between the solutions $v_t=Lv$ and $v_t=Av$ if $A$ is a relatively compact perturbation of the linear operator $L$

In a nutshell, here is my question. I read and know about the relation between the spectra of $L$ and $A$ if $A$ is a relatively compact perturbation of $L$. However, for my purpose, I am interested ...
11
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1answer
738 views

Imaginary eigenvalues

Consider the matrix $$A(\mu) = \begin{pmatrix} 0 & 1& 0 & 0 \\ -1 & -i\mu & 0 & i \\ 0 & 0 & 0 & 1 \\ 0 &i & -1 & i\mu \end{pmatrix}.$$ This matrix is ...
13
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3answers
1k views

Eigenvalue pattern

We consider a matrix $$M_{\mu} = \begin{pmatrix} 1 & \mu & 1 & 0 \\ -\mu & 1 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 &-1 & 0 & 0 \end{pmatrix}$$ One easily ...
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0answers
181 views

Inverse of block matrix II

This is a follow-up question on a previous question of mine that had a negative answer. I tried some examples and believe the following has a chance to be true. Let $V$ be a finite-dimensional vector ...
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0answers
52 views

Methods to find the spectrum of an operator

Suppose we have a bounded, self-adjoint operator $T$ on a set of functions $\mathcal{F}$. What kinds of methods are there to find the spectrum of $T$? Here is the setting I'm wondering about: consider ...
6
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0answers
118 views

Gap between consecutive Dirichlet eigenvalues

Suppose $\Omega \subset \mathbb R^2$ is a domain with a Lipschitz boundary and let $\{\lambda_k\}_{k=0}^n$ be the eigenvalues for the Laplacian operator on $\Omega$, that is to say $$ -\Delta \phi_k = ...
4
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0answers
89 views

Fourier transform without characters (Eigenfunctions of an operator)

Let's consider a very simple problem in quantum mechanics: We have, in $\mathbb R,$ a potential barrier of the form $$ V(x) = V_0 \mathbf 1_{[-a,a]}(x), $$ where $\mathbf 1_{[-a,a]}$ denotes the ...
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0answers
15 views

Singular values of a matrix that its rows have (1) a tightly bounded angle, and (2) at least some norm

We're looking for a connection between a matrix's singular values and some information we hold about its rows. We wish to find a tight bound on the singular values. We know on the rows that they have (...
3
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2answers
122 views

Massive dirac operator symmetric spectrum

Consider the Dirac operator $$ H = \begin{pmatrix} m & -i\partial_z \\ -i\partial_{\bar z} & -m \end{pmatrix},$$ where $\partial_{\bar z}$ is the Cauchy-Riemann operator and $m \ge 0.$ It is ...
5
votes
1answer
300 views

Existence of periodic solution to ODE

We shall consider the matrix-valued differential operator $$(L u)(x) :=u'(x) - \begin{pmatrix} 0 & \sin(2\pi x-\frac{\pi}{6})\\ - 2\sin(2\pi x+\frac{\pi}{6}) & 0 \end{pmatrix} u(x).$$ This is ...
3
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0answers
82 views

Eigenvalues of Laplacian and eigenvalues of curvature operator

Let $(M^n,g)$ be a compact Riemannian manifold (without boundary). The symmetries of the curvature $R$ of (the Levi-Civita connection associated to) $g$ allow one to realise $R$ as a self-adjoint (...

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