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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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 (...
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100 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 ...
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278 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 ...
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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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Analytic formula for the eigenvalues of kernel integral operator induced by Laplace kernel $K(x,x') = e^{-c\|x-x'\|}$ on unit-sphere in $\mathbb R^d$

Let $d \ge 2$ be an integer and let $X=\mathcal S_{d-1}$ the unit-sphere in $\mathbb R^d$. Let $\tau_d$ be the uniform distribution on $X$. Define a function $K:X \times X \to \mathbb R$ by $K(x,y) := ...
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257 views

Exponential convergence of Ricci flow

I've been trying to understand the asymptotic behavior of Ricci flow, and there are two facts which I am unable to square away. I'm interested in higher dimensional manifolds, but my question is ...
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144 views

Convergence of discrete Laplacian to continuous one

I make the following observation: Let $\Delta^{(n)}$ be the discrete Laplacian on $\mathbb{C}^n$ (ie the $n\times n $ matrix with diagonal $-2$ and upper/lower diagonal $1$.) This one has eigenvalues ...
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Sobolev spaces and spectral theorem

Consider a generalised harmonic oscillator of the form $$ A(x,D_x) = \langle x \rangle (1-\Delta_x)\langle x \rangle, \quad x \in \mathbb{R}^n, $$ where $\langle x \rangle = (1+|x|^2)^{1/2}.$ The ...
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535 views

Analytical origins of the Stone duality

I've asked this question in the HSM community, but by the nature of my question, some user told me to ask this question here. This is the original post https://hsm.stackexchange.com/q/13087/14296 ...
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Spectral gaps and convergence of solutions

I'm reading the following paper by C. Villani on the Fokker-Planck equation $$u_t = Lu$$ with $L = \Delta - \nabla V\cdot \nabla$ in which he states (pg. 3) that the existence of a spectral gap for ...
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Multiplication of a Riesz basis

Let ${(\phi_n(.),\psi_n(.))}_{n\geq 1}$ be a Riesz basis in $H^1_0(0,1) \times L^2(0,1)$. My question is the following: If we multiply the basis by the matrix $e^{Mx}$, $x \in (0,1)$ where $M$ is a ...
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1answer
140 views

Perron-Frobenius and Markov chains on countable state space

The following question naturally arises in the theory of Markov chains with countable state space to which I would be curious to know the answer: Let $A:\ell^1 \rightarrow \ell^1$ be a contraction, i....
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References to the theory of scattering resonances in dimension 1

Does anyone know a good reference to study the theory of scattering resonances in dimension 1 ? I would also like to add that I don't know anything about this theory, so I'm starting from zero. Thanks ...
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How to prove that there are infinitely many eigenvalues below $1$ of the following differential operator?

The question from Remark (2) below Lemma 7 in Enno Lenzmann: uniqueness of ground states for the pseudorelativistic Hartree equations. Let $l \ge 1$, and consider the following operator \begin{...
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52 views

Non-regular cospectral graphs with same degree sequences

I am looking for a large family (infinite pairs) of cospectral graphs with these condtions: The graphs are non-regular, Minimum degree is greater than $1$, The degree sequences of these cospectral ...
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Eigenvalues of splitting scheme

In numerical analysis it is common to approximate a solution to a PDE $$u'(t) = (A+B) u(t), \quad u(0)=u_0$$ which is just given by $e^{t(A+B)}u_0$ by the splitting $e^{tB/2} e^{tA} e^{tB/2}u_0.$ Here,...
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Heat equation damps backward heat equation?

In a previous question on mathoverflow, I was wondering about the following: Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions. ...
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determine the spectrum of the operator [closed]

determine the spectrum of the operator $$-\Delta-5\phi^4,\phi=3^{1/4}(1+|x|^2)^{-\frac{1}{2}},x\in R^3$$
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122 views

Mapping properties of backward and forward heat equation

In a previous question on mathoverflow, I asked about the following: Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions. The ...
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1answer
67 views

Spectrum of an elliptic operator in divergence form with a reflecting boundary condition

Let $\Omega$ be a bounded open domain and $v:\Omega\to\mathbb{R}^n$. Consider the following elliptic operator in divergence form, defined on smooth functions $u: \Omega \to \mathbb{R}$ \begin{align} L ...
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Strongly continuous semigroups on weighted $\ell^1$ space

Let $x=(x_i)$ be a sequence in $\ell^1$ such that all $x_i>0.$ Let $T(t):\ell^1 \rightarrow \ell^1$ be a strongly continuous semigroup of, i.e. $t \mapsto T(t)y$ is continuous for every $y \in \ell^...
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Show that the Laplacian on these domains is isospectral

Let $\Omega_i\subseteq\mathbb R^d$ be bounded and open, $A_i$ denote the weak Laplacian with domain $\mathcal D(A):=\{u\in H_0^1(\Omega_i):\Delta u\in L^2(\Omega_i)\}$ on $L^2(\Omega_i)$ and $$T_i(t)f:...
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Are Weyl sequences polynomially bounded?

Look at the Hilbert space $l^2( \mathbb{Z}) $ and let $A$ be a translation invariant band operator. I.e. if $\{ e_n \}_{n \in \mathbb Z} $ is the standard basis for $l^2( \mathbb{Z}) $ then it holds ...
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77 views

Spectral decomposition of idele class group $L^2(\mathbb{I}_F/ F^{\times})$

Let $F$ be a number field. Let $\mathbb{A}_F$ be the ring of adeles. The group of units of $\mathbb{A}_F$ is called the group of ideles $\mathbb{I}_F=\mathbb{A}_F^{\times}= GL_1(\mathbb{A}_F)$. The ...
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1answer
63 views

Sum of positive self-adjoint operator and an imaginary “potential”: literature request

To keep things simple, let us consider the following: $L$ is a positive, unbounded S.A. operator on $L_2(\mathbb{R},f(x))$, where $f(x)$ is a Gaussian. Assume that we know the spectrum and ...
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1answer
83 views

Characterization of absolutely-continuous spectrum

The essential spectrum of a bounded linear operator $A$ on a separable Hilbert space $\mathcal{H}$ is defined as $$ \sigma_{\mathrm{ess}}(A) \equiv\left\{z\in\mathbb{C}\left.\right|A-z\mathbb{1}\text{ ...
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71 views

spectral decomposition for harmonic oscillator

Consider the Hermitian operator (Hamilton operator for the harmonic oscillator) $H=-\Delta+|x|^2$. It is known that the Hermite functions are eigenfunctions for $H$: $$H(\Phi_\alpha)= (2|\alpha| +d) \...
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On the boundary integral of Neumann eigenfunctions

Let $v$ be an eigenfunction corresponding to the first nonzero Neumann Laplacian eigenvalue on a domain $\Omega \subset \mathbb{R}^2$. By definition, we know that $\int_{\Omega} v \, dx=0$. If $\Omega$...
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284 views

Essential spectrum of multiplication operator

Let $a\in \mathcal{L}(L^2([0, 1], \mathbb{R}))$ be a multiplication operator. I wonder whether there is any work on calculating its essential spectrum. Is there any way to explicitly compute its ...
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128 views

Proof of the analytic Fredholm theorem in Borthwick

I've stumbled across a proof of the analytic Fredholm theorem given in Theorem 6.1 in Spectral Theory of Infinite-Area Hyperbolic Surfaces by David Borthwick (see below). Given the notion of being &...
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73 views

Spectrum of generalized Ornstein-Uhlenbeck generator?

At some point, I stumbled over the work by Giorgio Metafune who basically solved, with collaborators, the problem of understanding the spectrum of the Ornstein-Uhlenbeck operator on $L^p$ with respect ...
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1answer
68 views

Spectrum decomposition of the scaling operator on weighted spaces

Consider the bounded linear operator $M_a$ defined by $M_au(x)=\frac{1}{\sqrt{a}}u\left(\frac{x}{a}\right)$, for $a>1$. On $L^2(\mathbb{R})$, it is easy to see that this is a unitary operator and ...
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Can the existence of geodesics be deduced from properties of the Laplacian?

As I understand it, the semiclassical trace formula in particular relates lengths of geodesics to eigenvalues of the Laplacian. Is it possible to prove that every compact Riemannian manifold has a ...
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1answer
194 views

Diagonalise self-adjoint operator explicitly?

Consider the linear constant coefficient differential operator $P$ on the Hilbert space $L^2([0,1]^2;\mathbb C^2)$ $$P= \begin{pmatrix} D_{z}+c & a \\ b & D_{z}+c \end{pmatrix}$$ where $D_z=-...
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1answer
107 views

Laplace eigenfunction on a polygonal domain symmetric about an axis

Consider a polygon $\Omega \subseteq \mathbb{R}^2$, and let us consider the usual Laplacian operator $\Delta = \partial_x^2 + \partial_y^2$, with Dirichlet boundary conditions. My question comes from ...
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58 views

Finiteness of a constrained min/max

Take a bounded subset $\Omega \subset \mathbb{R}^d$ with smooth boundary (let's say a ball), take a ball of radius $r$, $B_r \subset \Omega$, take $\alpha \in ]0,1[$. How to prove that \begin{align*} ...
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Is something known about the third term in Weyl law?

The well-known Weyl law states that if $\Omega$ is a open subset of $\mathbb R^d$ that is "nice enough" (I don't want to enter into details on this point) and if $N(\lambda)$ is the number ...
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31 views

Relating spectral guarantee to L2 guarantees

Let $A \in \mathbb{R}^{d\times d}$ be a real, symmetric matrix. Let $U_k \in \mathbb{R}^{d\times k}$ denote the matrix containing the first top $k$-components (eigenvectors) $u_i$ of $A$ as its ...
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Lower-bounding the eigenvalues of a certain positive-semidefinite kernel matrix, as a function of the norm of the input matrix

Let $\phi:[-1,1] \to \mathbb R$ be a function such that $\phi$ is $\mathcal C^\infty$ on $(-1,1)$. $\phi$ is continuous at $\pm 1$. For concreteness, and if it helps, In my specific problem I have $\...
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1answer
214 views

A better version of Weyl's Law or uniform estimates of Laplacian higher eigenvalues

Let $(M^n,g)$ be a closed $n$ dimensional Riemannian manifold with $\mathrm{Ric}_g\ge -K$, $(K\ge 0)$. Weyl's law(along with Karamata Tauberian Theorem) asserts that the eigenvalue $\lambda_i$ of $-\...
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Calculating frequency of sound of ringing metal coin

I would like to reproduce the results of Manas - The music of gold: Can gold counterfeited coins be detected by ear?, but it skips a lot of steps, and the mathematics behind it is a bit advanced for ...
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1answer
42 views

Relating classic spectral decomposition with Euclidean Jordan algebras

I'm currently getting into studying optimization problems over symmetric cones (NSCP) and I'm having some trouble to understand something. Let me first give some context, sorry if it is repetitive to ...
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82 views

How to use resolvent identity to check the following continuity argument?

Let $H(\alpha)=\left( \begin{matrix} 0 & i L_- \\ -i L_+ & 0 \end{matrix} \right) $, where $L_+=-\Delta +\alpha^2-3\phi^2(\cdot,\alpha)$, $L_ -=-\Delta +\alpha^2-\phi^2(\cdot,\alpha)$, and $\...
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46 views

Isolated eigenvalues of “bipartite” operators

Please note: This is a reformulation of a previous question of mine. The old question has been already answered to, so I prefer asking a new one. However, it looked like the old formulation did not ...
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1answer
112 views

Detecting isolated eigenvalues from local spectral measures

Please note: This question has been edited after it became clear from Christian Remling's answer that the original formulation was far from what I really meant to ask. Let $T\ne 0$ be a self-adjoint ...
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80 views

Comparison of the spectrum decomposition

In the spectral theorem, we learned that the spectrum of a linear operator $A$ is a disjoint union of: point spectrum (eigenvalues), continuous spectrum (the kernel of $zI-A$ is zero, the closure of $\...
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References on discrete Sturm-Liouville eigenvectors convergence

Let $ L : u_n \mapsto a_n u_{n + 1} + b_n u_n + a_{n - 1} u_{n -1} = \nabla ( a_n \Delta u_n ) + (b_n + a_n + a_{n - 1}) u_n $ be a discrete Sturm-Liouville operator, with $ \nabla u_n := u_{n + 1} - ...
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110 views

A gap in the proof of uniqueness of functional calculus based on a spectral theorem

This question considers the proof of a fundamental theorem of functional calculus, given in the book Spectral Theory - Basic Concepts and Applications by David Borthwick (Theorem 5.9). Firstly we have ...
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53 views

“Trade-off” between bound on the function and on the spectrum for functional calculus in spectral theory

Let $A$ be a self-adjoint (unbounded) operator on a separable Hilbert space $H$. From the following form of spectral theorem, we may define a functional calculus by $f(A)=Q^{-1} M_{f\circ \alpha} Q$. (...
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1answer
108 views

Canonical multiplication representation of self-adjoint operator in quantum chemistry and coding theory research

In my applied math research group, we are studying and going over functional analysis results from papers and theses from our institution to generalize their results and apply them in our discrete ...

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