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
0
votes
0
answers
32
views
Spectral measures of a family of parameter-dependent self-adjoint contractions on an $L^2$-space
I have a self-adjoint linear contraction $A_g$ on an $L^2$-space of the form $$A_gf=\int\gamma(f,g),$$ where $\gamma$ is Lipschitz continuous and $g$ is an a priori fixed function. Assuming $1-A_g$ is ...
2
votes
2
answers
746
views
Spectral theory in non-separable Hilbert Spaces
I am wondering about what can be said about the spectral theorem for unbounded, self-adjoint operators in a non-separable Hilbert space. There is a comment in this sense to the question "Does spectral ...
0
votes
0
answers
79
views
$ 0 $ locates in the continuous spectrum of Schrodinger operators?
This is question is motivated by Non-closed range space of Laplace operators?. We aim to determine what kind of potential will make corresponding schrodinger operators possess non-closed range.
For ...
2
votes
2
answers
517
views
Graph with complex eigenvalues
The question I am wondering about is:
Can the discrete Laplacian have complex eigenvalues on a graph?
Clearly, there are two cases where it is obvious that this is impossible.
1.) The graph is ...
1
vote
1
answer
189
views
A question involving a summation of eigenvalues of the Laplacian operator on $\mathbb{S}^2$
Infinite series involving eigenvalues of the Beltrami-Laplace operator on Riemannian manifolds as well as $L^p$-estimates of eigenfunctions arise in the study of the nonlinear Schrödinger equation (...
14
votes
1
answer
990
views
Computing spectra without solving eigenvalue problems
There is a rather remarkable conjecture formulated in this paper, "Computing spectra without solving eigenvalue problems," https://arxiv.org/pdf/1711.04888.pdf and in this talk by Svitlana Mayboroda ...
4
votes
0
answers
71
views
Tight bound on spectral gap of compact homogeneous manifold?
This paper by Peter Li proves a bound on the spectral gap of the Laplacian on a compact homogeneous manifold of diameter $d$:
$$ \lambda_1 \ge c/d^2, $$
where $c=\pi^2/4$. Can this bound be ...
1
vote
0
answers
52
views
Spectrum of a $1$-parameter family of symmetric linear operators
I am working with certain submanifolds of symmetric spaces and, using a construction in Terng-Thorbergson, we ended up in the following Hilbert space problem:
Let $H$ be a (real) Hilbert Space and $...
4
votes
2
answers
761
views
Decay of eigenfunctions for Laplacian
Consider the discrete second derivative with Dirichlet boundary conditions on $\mathbb C^n$.
Its eigendecomposition is fully known:
see wikipedia
It seems like the largest eigenvalue $\lambda_1$ is ...
2
votes
1
answer
524
views
Understanding a proof about limit of a sequence of open sets
We are reading a proof about the following limit
\begin{equation}\tag{1}
\lim_{n \to \infty} \sigma_1(T_n)= \sigma_1(T),
\end{equation}
where $T:D(T) \subseteq H \to H$ and $T_n:D(T_n) \subseteq H \to ...
7
votes
1
answer
354
views
Spectral gaps for spin manifold Laplace spectrum
For a (compact) spin manifold, we know that the eigenvalues $\lambda_n$ of the Dirac operator are countable, with finite multiplicity, and satisfy
$$
|\lambda_n| \to \infty, ~~~ \text{ as } n \to \...
1
vote
1
answer
113
views
Spectral bound for maximum clique $k(G)$ in a permutation graph
Let $\pi \in S_n$ be an arbitrary permutation. By permutation graph, we refer to a simple graph with nodes $[n]$ and edges that connect pairs of nodes that appear sorted in $\pi$. Formally, $G=(V=[n],...
3
votes
1
answer
1k
views
Different definitions of a relatively compact operator
(Cross-post from Math Stackexchange, where some work has been done in the comments)
Let $T,K$ be unbounded operators on a Hilbert space $H$.
I've seen the following definition of a relatively compact ...
4
votes
2
answers
1k
views
What's the full assumption for Laplacian matrix $L=BB^T=\Delta-A$?
Graph with no-selfloop, no-multi-edges, unweighted.
directed
For directed graph Adjacency matrix is a non-symmetric matrix $A_{in}$ considering indegree or $A_{out}$ considering outdegree. Degree ...
1
vote
1
answer
860
views
Regarding essential spectrum of the unilateral shift operator
This is with context to Example 4.10 in Section 11 of the book : A course in functional Analysis by J.B Conway. Let $\sigma_{le}(S)$ and $\sigma_{re}(S)$ denote the left and right essential spectrum ...
2
votes
0
answers
64
views
Inclusion of the spectrum of two differential operators defined on $L^2[-a,a]$ and $L^2[0, \infty)$
Let $T$ be the formal operator defined by $$Tu:= \sum_{j=0}^{2n} a_j\frac{d^ju}{dx^j}$$ where $a_j \in \mathbb{C}$. Consider the differential operators $T_a: D(T_a)\subseteq L^2[-a,a] \to L^2[-a,a]$ ...
22
votes
5
answers
1k
views
Rigorous justification for this formal solution to $f(x+1)+f(x)=g(x)$
Let $g\in C(\Bbb R)$ be given, we want to find a solution $f\in C(\Bbb R)$ of the equation
$$
f(x+1) + f(x) = g(x).
$$
We may rewrite the equation using the right-shift operator $(Tf)(x) = f(x+1)$...
2
votes
3
answers
953
views
Sum of Square of the Eigenvalues of Wishart Matrix
Let $A\in\mathbb{R}^{m\times d}$ matrix with iid standard normal entries, and $m\geqslant d$, and define $S=A^T A$.
I want to have a tight upper bound for $\sum_{k=1}^d \lambda_k^2$, where $\...
1
vote
0
answers
134
views
Shift operator on a Banach space
I have been the paper titled Dual Piecewise Analytic Bundle Shift Models of Linear Operators by Dmitry Yakubovich.
In the second paragraph of the introduction it says "Let $T$ be a bounded Linear ...
4
votes
1
answer
403
views
Graph Fourier transform definition
I have a question about the definition of the graph Fourier transform. Let me start with definition.
Let $A$ be the adjacency matrix of a graph $G$ with vertex set $V = \{1, 2, \dots, n\}$. The ...
2
votes
0
answers
57
views
Zero in the spectrum of an elliptic second order operator
This might be considered as a continuation of my previous question Spectrum of a linear elliptic operator
but is independent. I have another question on V. Gribov's paper "Quantization of non-Abelian ...
2
votes
0
answers
160
views
Spectrum of a linear elliptic operator
In the paper in quantum fields theory by
Gribov,V.; (1978) "Quantization of non-Abelian gauge theories". Nuclear Physics B. 139: 1–19;
in Section 3 the author makes the following claim from PDE and ...
2
votes
0
answers
102
views
Spectrum of a Hamiltonian which is a perturbation of Laplacian
Let $\Delta =\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\frac{\partial^2}{\partial x_3^2}$ be the Laplacian on $\mathbb{R}^3$.
Consider a self adjoint operator $H$ on complex ...
0
votes
0
answers
206
views
Show convergence of a sequence of resolvent operators
Let
$E$ be a locally compact separable metric space
$(\mathcal D(A),A)$ be the generator of a strongly continuous contraction semigroup on $C_0(E)$
$E_n$ be a metric space for $n\in\mathbb N$
$(\...
1
vote
0
answers
64
views
Strong Differentiability of Spectral Projections
Let $H$ be a Hilbert space and $W$ be a dense subspace, equipped with a different norm that turns it into a Hilbert space. Let $(A(t))_{t\in[0,T]}$ be a family of Operators in $B(W,H)$ (bounded ...
18
votes
2
answers
2k
views
Eigenvalues of the Laplace-Beltrami operator on a compact Riemannnian manifold
Let $(M,g)$ be a compact Riemannian manifold, and let $\Delta_g$ be its Laplace-Beltrami operator. A "well-known fact" is that the eigenvalues of $\Delta_g$ have finite multiplicity and tend to ...
3
votes
1
answer
159
views
How does $E$ closed follow from the upper semicontinuity of the spectrum?
Let $f$ be an analytic function for a domain $D$ of $\mathbb{C}$ into a Banach algebra $A$. Suppose that, for all $\lambda \in D$, $\text{Sp}f(\lambda)$ is finite or a sequence converging to $0$.
...
1
vote
0
answers
110
views
Bounds on spectral radius using chromatic number
I am struggling with this question:
If I have a connected graph $G$ on $n$ vertices and $m$ edges with chromatic number $d$ then how can I give a bound(lower and upper) on its spectral radius in ...
-1
votes
2
answers
572
views
Invariance of spectrum under conjugation
Let $T$ be a self-adjoint invertible operator on $\mathcal{H}$ with a continuous spectrum, means the spectral measure is nonatomic. For which class of invertible operators $V$( with continuous ...
5
votes
1
answer
129
views
Stable region of minimal hypersurfaces with finite Morse index
In this Inventiones Mathematicae paper, Fischer-Colbrie proved the following result (Proposition 1):
Proposition: Let $ M$ be a complete two-sided minimal surface in a three manifold $N$. Then if $M$...
1
vote
0
answers
58
views
Sturm-Liouville-like Eigenproblem
Consider the piecewise-deterministic Markov process on $\mathbf{R}$ which
moves according to the vector field $\phi (x) = 1$,
experiences events at rate $\lambda(x) = 1$, and
at events, jumps ...
10
votes
2
answers
799
views
Weyl law for (non-semiclassical) Schrodinger operator
The Weyl law for a semiclassical Schrodinger operator
$$ A_h\ := \ -h^2\Delta+V(x) $$
on an $d$-dimensional complete Riemannian manifold $M$
says that the number $N(A_h,1)$ of eigenvalues of $A_h$ ...
3
votes
1
answer
345
views
Oscillatory integrals
Consider the integrals
$$I_n(\zeta,\epsilon)=\int_{-\zeta}^\zeta \left|(t-i\epsilon)^{-n}-(t+i\epsilon)^{-n}\right|\,dt$$
I would like to know the asymptotic behavior of $I_n(\zeta,\epsilon)$ for ...
2
votes
0
answers
95
views
Smallest singular value distribution
Let $G_\mathbb{R}\in\mathbb{R}^{n\times n}$ and $G_\mathbb{C}\in\mathbb{C}^{n\times n}$ denote the real and complex Ginibre random matrices, i.e. random matrices with independent real/complex Gaussian ...
11
votes
2
answers
533
views
Smoothness of finite-dimensional functional calculus
Assume that $f:\mathbb R\to\mathbb R$ is continuous.
Given a real symmetric matrix $A\in\text{Sym}(n)$, we can define $f(A)$ by applying $f$ to its spectrum. More explicitly,
$$ f(A):=\sum f(\lambda)...
3
votes
2
answers
746
views
Is the ring of $p$-adic integers extremally disconnected?
We call a topological space $X$ extremally disconnected if the closures of its open sets remain open. Obviously, Hausdorff extremally disconnected spaces are totally disconnected in the sense that ...
2
votes
0
answers
77
views
Existence of a fixed point for this operator
I'm looking for some mathematical results that I might be able to apply to see if an operator I'm considering has a fixed point.
In particular consider,
$$ Ag(x) = \Big\{ \xi(x) + K(g(x))^\frac{1}{\...
1
vote
0
answers
46
views
The Morse Index of a $T$- periodic geodesics is a integer number?
It is well known that compact Riemannian manifolds $(M, g)$ with
periodic geodesic flows have ( Besse Book) exceptional spectral properties: the spectrum
of $ \sqrt{ - \Delta}$, the square root of ...
3
votes
1
answer
182
views
Quasinilpotent vectors of Newton potential vanish
Suppose $\Omega$ is a smooth bounded domain in $\mathbb{R}^3$. Consider the Newton potential
\begin{equation}
T [\phi](x) = \int_{\Omega} \frac{1}{|x-y|} \phi(y)dy.
\end{equation}
It is well know ...
4
votes
1
answer
363
views
The exceptional eigenvalues and Weyl's law in level aspect
The Weyl law for Maass cusp forms for $SL_2(\mathbb Z)$ was obtained by Selberg firstly and has been generalized to various caces (Duistermaat-Guillemin, Lapid-Müller and others). For example, one of ...
3
votes
0
answers
156
views
Perturbation theory compact operator
Let $K$ be a compact self-adjoint operator on a Hilbert space $H$ such that for some normalized $x \in H$ and $\lambda \in \mathbb C:$
$\Vert Kx-\lambda x \Vert \le \varepsilon.$
It is well-known ...
2
votes
0
answers
144
views
Off-diagonal estimates for Poisson kernels on manifolds
Let $(M,g)$ be a complete Riemannian manifold, $\Delta$ its Laplace-Beltrami operator and $T_t = (e^{t \Delta})_{t \geq 0}$ the associated heat semigroup. We can define the subordinated Poisson ...
1
vote
0
answers
183
views
One question about Schrodinger Semigroups-(B. Simon)
This question comes from the paper: B. Simon, Schrodinger Semigroups, Bull. A.M.S., (1982) Vol. 7 (3).
On the Theorem C.3.4(subsolution estimate) of the paper, it says that: Let $Hu=Eu$ and $u\in L^...
2
votes
1
answer
422
views
Compact operators on Banach spaces and their spectra
I have a question about compact operators on Banach spaces.
Let $B$ be a real Banach space and $L$ a closed linear operator on $B$.
We assume that $L$ generates a contraction semigroup $\{T_t\}_{t>...
4
votes
1
answer
209
views
Non-isolated ground state of a Schrödinger operator
Question. Does there exist a dimension $d \in \mathbb{N}$ and a measurable function $V: \mathbb{R}^d \to [0,\infty)$ such that the smallest spectral value $\lambda$ of the Schrödinger operator $-\...
3
votes
1
answer
204
views
Eigenvalue estimates for operator perturbations
I edited the question to a general mathematical question, since I found the answer in Carlo Beenakker's reference and think that my initial question was mathematically misleading.
What was behind ...
0
votes
1
answer
243
views
Perturbing a normal matrix
Let $N$ be a normal matrix.
Now I consider a perturbation of the matrix by another matrix $A.$
The perturbed matrix shall be called $M=N+A.$
Now assume there is a normalized vector $u$ such that $\...
5
votes
1
answer
243
views
Stable matrices and their spectra
I am a graduate student in engineering and we work a lot with so-called Hurwitz (or stable) matrices.
A matrix in our terminology is called stable if the real part of the eigenvalues is strictly ...
3
votes
1
answer
770
views
Real part of eigenvalues and Laplacian
I am working on imaging and I am a bit puzzled by the behaviour of this matrix:
$$A:=\left(
\begin{array}{cccccc}
1 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 &...
3
votes
1
answer
422
views
Spectrum of this block matrix
Consider the following block matrix
$$A = \left(\begin{matrix} B & T\\ T & 0 \end{matrix} \right)$$
where all submatrices are square and
matrix $B = \mbox{diag}\left(b_1 ,0,0,\dots,0,b_n \...