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
967
questions
1
vote
0
answers
76
views
Graph energy and spectral radius
Suppose $G$ is a simple graph of order $n$ with eigenvalues $\lambda_1\geq \cdots\geq \lambda_n$. I've encountered the quantity $L=\big\vert |\lambda_1|-|\lambda_2|-\cdots-|\lambda_n|\big\vert$. Note ...
- 171
4
votes
0
answers
147
views
Roots of smoothing operators
Suppose that $(M,g)$ is a smooth, compact Riemann manifold and $K:M\times M\to\mathbb{R}$ is a smooth, symmetric nonnegative function. We regard is as the Schwartz kernel of a smoothing operator. In ...
- 33.5k
0
votes
0
answers
36
views
Regarding significance of spectral variation under algebraic operations
I have been reading the paper Determining elements in $C^∗$-algebras through spectral properties.
The paper discusses about what would be the relation be between two elements $a$ and $b$ of a Banach ...
- 27
2
votes
0
answers
434
views
Eigenvalues of the sum of matrices, where matrices are tensor products of Pauli matrices
recently I've been studying the toric code (a squared lattice in the context of quantum computation). I want to calculate the energy of the ground state and of all the excitations, with the respective ...
- 21
-1
votes
1
answer
74
views
Applications and motivations of resolvent for elliptic operator
Let $ A(x)=(a_{ij}(x)):\mathbb{R}^d\to\mathbb{R}^{d\times d} $ be a matrix satisfying ellipticity condition, that is
\begin{align}
\mu^{-1}|\xi|^2\geq \sum_{i,j=1}^da_{ij}(x)\xi_i\xi_j\geq\mu|\xi|^2
\...
- 1,091
-1
votes
1
answer
101
views
Closure of the point spectrum of an unbounded diagonalizable operator
Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to $H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...
- 939
1
vote
1
answer
85
views
Are these $L_2$-spectral radii approximations strictly increasing?
Suppose that $V$ is a finite dimensional complex Hilbert space. Let $L(V)$ denote the collection of all linear mappings from $V$ to $V$. Let $A_1,\dots,A_r:V\rightarrow V$ be linear operators. Then ...
- 27.4k
5
votes
1
answer
162
views
Spectral theory of infinite volume hyperbolic manifolds
I have a question about the discrete spectrum of the Laplace operator on hyperbolic manifolds with infinite volume. I understand the case of infinite area surfaces: see Chapter 7 (Sections 1 and 2) of ...
- 1,293
2
votes
1
answer
394
views
Positiveness of Banach limit [closed]
I‘m currently reading Arveson’s “A Short Course on Spectral Theory”, and I’m stuck at Exercise 3.1 (1). The question is:
Let $l^{\infty}(\mathbb{N})$ be the set of all bounded sequences of complex ...
- 23
7
votes
1
answer
331
views
Deriving Sommerfeld radiation condition from limiting absorption principle
For the Helmholtz equation
$$
-(\Delta + k ^2) u = f, \label{1}\tag{1}
$$
imposing the Sommerfeld radiation condition
$$
\lim_{r\to\infty} r ^{\frac{m-1}2} \left( u_r - i k u\right) = 0
$$
on $u$ ...
- 71
2
votes
0
answers
87
views
Eigenvalues of two positive-definite Toeplitz matrices
Consider two positive-definite Toeplitz matrices $M_1$ and $M_2$ both with dimension $2^j \times 2^j$. Their matrix elements are:
$$M_1[x,y] = \frac{\text{sin}(\pi(x-y)/2^j)}{\pi(x-y)} \qquad M_2[x,y] ...
- 21
5
votes
1
answer
377
views
Fokker-Planck: equivalence between linear spectral gap and nonlinear displacement convexity?
In a smooth, bounded and convex domain $\Omega\subset \mathbb R^d$ consider the usual linear Fokker-Planck equation with Neumann (some would say Robin) boundary conditions
\begin{equation}
\label{FP}
\...
- 4,846
1
vote
1
answer
116
views
Spectrum invariant under (generalised) transpose as operator on trace class operators
For matrices $A$ it is well known that the spectrum is invariant under transpose $\sigma(A^T) = \sigma(A)$. Furthermore, the spectrum of the adjoint matrix $\sigma(A^*) = \overline{ \sigma(A)}$ the ...
2
votes
0
answers
72
views
Second order differential operator with a Lipschitz coefficient
Let $a(x) \in W^{1, \infty}(\mathbb{R})$ be real-valued such that $a(x) \ge a_0 > 0$. Let $A^2$ denote the second order differential operator $A^2 : = -\partial_x (a(x) \partial_x) + 1 : L^2(\...
- 459
13
votes
1
answer
375
views
Why are we interested in spectral gaps for Laplacian operators
Let $M$ be a Riemannian manifold and let $\Delta$ be its Laplacian operator. There is a large literature on a spectral gap for such a $\Delta$, that is, finding an interval $(0,c)$ which does not ...
- 175
15
votes
3
answers
2k
views
Why is the set of Hermitian matrices with repeated eigenvalue of measure zero?
The Hermitian matrices form a real vector space where we have a Lebesgue measure. In the set of Hermitian matrices with Lebesgue measure, how does it follow that the set of Hermitian matrices with ...
- 53
4
votes
0
answers
156
views
Hodge theory in higher eigen-spaces?
Hodge theory for elliptic complexes $E$ identifies the space of harmonic sections with cohomology
$$\mathcal{H}(E) \simeq H(E).$$
A classical example with differential forms ($E = (\Omega,d)$) ...
- 4,760
1
vote
0
answers
96
views
Question about Dirac operator
Let $D$ be a generalized Dirac operator on a complete Riemannian manifold. I'm a little confused to prove that there exists a constant $c>0$ such that
$$\|D\sigma\|^2\geq c^2\|\sigma\|^2$$
for $\...
- 483
4
votes
0
answers
74
views
On the convergence of the spectral decomposition of a harmonic function
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$ with a smooth boundary. Denote by $0<\lambda_1\leq \lambda_2\leq\ldots$ the Dirichlet eigenvalues of $-\Delta_g$ on $(M,g)$...
- 3,987
4
votes
0
answers
85
views
Minimal eigenvalue of infinite dimensional matrix
Consider the following symmetric, positive-definite matrix
$$
H_{nm}=-\frac{(4 m+4 n+1)}{(4 m-4 n-1) (4 m-4 n+1)}\sqrt{\frac{(4 m-1)!! (4 n-1)!!}{(4 m)!! (4 n)!!}}
$$
where $n,m=0,1,2\dots$ (Here $!!$ ...
1
vote
1
answer
112
views
Regarding variation of spectra
I have been reading the article The variation of spectra by J.D Newburgh. in this article and all related reference/ articles, the term 'variation of spectra' keeps coming in, but I nowhere find a ...
- 27
1
vote
1
answer
216
views
Monotonicity of eigenvalues II
In a previous question here, I asked the question below for block matrices and received an answer showing the question is true if $\mathcal B$ is hermitian and false, in general if $\mathcal B$ is non-...
- 496
6
votes
1
answer
524
views
Monotonicity of eigenvalues
We consider block matrices
$$\mathcal A = \begin{pmatrix} 0 & A\\A^* & 0 \end{pmatrix}$$ and
$$\mathcal B = \begin{pmatrix} 0 & B\\C & 0 \end{pmatrix}.$$
Then we define the new matrix
$...
- 496
3
votes
4
answers
353
views
Prove that $(v^Tx)^2−(u^Tx)^2\leq \sqrt{1−(u^Tv)^2}$ for any unit vectors $u, v, x$
I believe I found a complicated proof by bounding the spectral norm $||uu^T-vv^T||^2_2:=\max_{||x||=1}|(u^Tx)^2-(v^Tx)^2|$.
Using the fact that $dist(x,y):=\sin|x-y|$ is a distance function over unit ...
- 143
2
votes
0
answers
54
views
Weyl asymptotic in towers
Let $M$ be a compact Riemannian manifold. The Weyl law gives the asymptotic of the counting function $N(T)$ of Laplace eigenvalues $|\lambda|\le T$ as $T\to\infty$.
Now suppose you are given a tower ...
- 1,900
5
votes
0
answers
204
views
Perturbation of Neumann Laplacian
Consider the $N \times N$ matrix
$$A_{\alpha}=\begin{pmatrix} \lambda_1 & -1 & -\alpha & 0 & \cdots & 0\\
-1 & \lambda_2 & -1 & -\alpha & \cdots & 0\\
-\alpha &...
- 53
1
vote
0
answers
55
views
Class of spectral zeta functions whose analytic extension takes a particular form
In quantum field theory the one-loop effective action is expressed in terms of the functional determinant of the (elliptic and self-adjoint) operator of small disturbances. Since the real eigenvalues ...
- 87
2
votes
0
answers
68
views
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-...
- 477
0
votes
0
answers
392
views
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 ...
- 1
2
votes
0
answers
78
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 ...
- 3,987
0
votes
0
answers
159
views
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 ...
- 71
2
votes
0
answers
63
views
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 ...
- 3,987
3
votes
0
answers
174
views
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?
$$ V(f) (x) =\...
7
votes
1
answer
763
views
On eigenfunctions of the Laplace Beltrami operator [closed]
How can we generate the eigenspace for the Laplace Beltrami operator on SU(2)?
- 85
2
votes
0
answers
122
views
How to prove that a finite rank perturbation on an infinite matrix does not change its continuous spectrum?
I have the discrete Laplace operator on an infinite Hilbert space with an orthonormal basis $\psi_x$ ($\forall x \in \mathbb Z$), given by $\Delta \psi_x=\psi_{x-1}+\psi_{x+1}$. If I introduce a ...
- 477
1
vote
1
answer
107
views
Adjoint operator of OU generator
The generator an OU process is given by
$$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)^...
- 192
1
vote
0
answers
136
views
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 \...
- 11
3
votes
0
answers
60
views
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 ...
- 445
4
votes
1
answer
111
views
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 $\|...
- 459
4
votes
1
answer
144
views
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 ...
- 824
3
votes
1
answer
208
views
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
votes
0
answers
40
views
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 ...
- 163
1
vote
0
answers
58
views
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/...
- 459
3
votes
0
answers
139
views
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
101
views
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(...
- 183
2
votes
0
answers
99
views
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 ...
- 761
6
votes
1
answer
320
views
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 ...
- 496
4
votes
1
answer
184
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) = \...
- 496
8
votes
0
answers
491
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 ...
- 361
1
vote
1
answer
71
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 ...
- 3,987