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
792
questions
0
votes
0answers
12 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
votes
2answers
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 ...
5
votes
1answer
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 ...
3
votes
0answers
66 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 (...
3
votes
0answers
89 views
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) := ...
6
votes
2answers
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 ...
4
votes
1answer
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 ...
3
votes
0answers
138 views
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 ...
11
votes
2answers
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
...
2
votes
0answers
43 views
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 ...
0
votes
0answers
52 views
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 ...
6
votes
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....
0
votes
0answers
34 views
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 ...
3
votes
1answer
126 views
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{...
1
vote
1answer
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 ...
6
votes
0answers
88 views
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,...
3
votes
0answers
284 views
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. ...
1
vote
0answers
54 views
determine the spectrum of the operator [closed]
determine the spectrum of the operator
$$-\Delta-5\phi^4,\phi=3^{1/4}(1+|x|^2)^{-\frac{1}{2}},x\in R^3$$
4
votes
1answer
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 ...
5
votes
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 ...
2
votes
0answers
87 views
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^...
1
vote
0answers
57 views
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 $$T_i(t)f:...
2
votes
0answers
128 views
Are Weyl sequences polynomially bounded?
Look at the Hilbert space $l^2( \mathbb{Z}) $ and let $A$ be a translation invariant band operator. I.e. if $\{ e_n \}_{n \in \mathbb Z} $ is the standard basis for $l^2( \mathbb{Z}) $ then it holds ...
2
votes
0answers
77 views
Spectral decomposition of idele class group $L^2(\mathbb{I}_F/ F^{\times})$
Let $F$ be a number field. Let $\mathbb{A}_F$ be the ring of adeles. The group of units of $\mathbb{A}_F$ is called the group of ideles $\mathbb{I}_F=\mathbb{A}_F^{\times}= GL_1(\mathbb{A}_F)$. The ...
1
vote
1answer
63 views
Sum of positive self-adjoint operator and an imaginary “potential”: literature request
To keep things simple, let us consider the following: $L$ is a positive, unbounded S.A. operator on $L_2(\mathbb{R},f(x))$, where $f(x)$ is a Gaussian. Assume that we know the spectrum and ...
2
votes
1answer
83 views
Characterization of absolutely-continuous spectrum
The essential spectrum of a bounded linear operator $A$ on a separable Hilbert space $\mathcal{H}$ is defined as $$ \sigma_{\mathrm{ess}}(A) \equiv\left\{z\in\mathbb{C}\left.\right|A-z\mathbb{1}\text{ ...
0
votes
0answers
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 $H$:
$$H(\Phi_\alpha)= (2|\alpha| +d) \...
1
vote
0answers
33 views
On the boundary integral of Neumann eigenfunctions
Let $v$ be an eigenfunction corresponding to the first nonzero Neumann Laplacian eigenvalue on a domain $\Omega \subset \mathbb{R}^2$. By definition, we know that $\int_{\Omega} v \, dx=0$. If $\Omega$...
6
votes
3answers
284 views
Essential spectrum of multiplication operator
Let $a\in \mathcal{L}(L^2([0, 1], \mathbb{R}))$ be a multiplication operator. I wonder whether there is any work on calculating its essential spectrum. Is there any way to explicitly compute its ...
0
votes
1answer
128 views
Proof of the analytic Fredholm theorem in Borthwick
I've stumbled across a proof of the analytic Fredholm theorem given in Theorem 6.1 in Spectral Theory of Infinite-Area Hyperbolic Surfaces by David Borthwick (see below).
Given the notion of being &...
1
vote
0answers
73 views
Spectrum of generalized Ornstein-Uhlenbeck generator?
At some point, I stumbled over the work by Giorgio Metafune who basically solved, with collaborators, the problem of understanding the spectrum of the Ornstein-Uhlenbeck operator on $L^p$ with respect ...
1
vote
1answer
68 views
Spectrum decomposition of the scaling operator on weighted spaces
Consider the bounded linear operator $M_a$ defined by $M_au(x)=\frac{1}{\sqrt{a}}u\left(\frac{x}{a}\right)$, for $a>1$. On $L^2(\mathbb{R})$, it is easy to see that this is a unitary operator and ...
5
votes
0answers
91 views
Can the existence of geodesics be deduced from properties of the Laplacian?
As I understand it, the semiclassical trace formula in particular relates lengths of geodesics to eigenvalues of the Laplacian.
Is it possible to prove that every compact Riemannian manifold has a ...
2
votes
1answer
194 views
Diagonalise self-adjoint operator explicitly?
Consider the linear constant coefficient differential operator
$P$ on the Hilbert space $L^2([0,1]^2;\mathbb C^2)$
$$P= \begin{pmatrix} D_{z}+c & a \\ b & D_{z}+c \end{pmatrix}$$
where $D_z=-...
3
votes
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 ...
1
vote
0answers
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*}
...
10
votes
0answers
163 views
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 ...
0
votes
0answers
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 ...
1
vote
0answers
43 views
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 $\...
5
votes
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 $-\...
1
vote
0answers
91 views
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 ...
1
vote
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 ...
0
votes
0answers
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 $\...
0
votes
0answers
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 ...
0
votes
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 ...
0
votes
0answers
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 $\...
2
votes
0answers
23 views
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} - ...
0
votes
0answers
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 ...
0
votes
0answers
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 $f(A)=Q^{-1} M_{f\circ \alpha} Q$. (...
4
votes
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 ...
