All Questions
Tagged with sp.spectral-theory reference-request
108 questions
0
votes
0
answers
55
views
reference request: conditions for pointwise and operator-norm convergence of kernel projections
At a very high level, I’m interested in the following question. Suppose $X$ is a (separable) Hilbert space, and $T_n : X \rightarrow X$ is a sequence of finite rank self-adjoint maps that converges (...
- 101
0
votes
1
answer
73
views
Computing spectrum of very simple Schrödinger operator
I asked this question recently on a thread in math stack exchange, but with no real answers suggested. I think this is a relatively simple variation on the classical free Laplacian spectrum, so I ...
- 633
8
votes
1
answer
584
views
Reference request: Software for producing sounds of drums of specified shapes
Is there software that, when the input is the shape of a drum, will produce the corresponding audible sound?
0
votes
1
answer
100
views
Spatially localised solution to the Schrödinger equation with potential is a combination of eigenfunctions
In this Terry Tao's blog post, he claims that if one has a solution to the Schrödinger equation
$$i\,\partial_t u +\Delta u=Vu $$
with a "reasonably smooth and localised $V$", $u$ has ...
- 94
3
votes
0
answers
186
views
Bourgain-Gamburd-like theorems in the non-algebraic case
For $\mu$ a Borel probability measure on the compact group $G=\operatorname{SU}(d)$, Bourgain-Gamburd prove that the spectral radius of the associated operator on $L^2(G)$ is strictly less than one, ...
- 31
7
votes
1
answer
930
views
Why is the length spectrum called a spectrum?
Given a hyperbolic surface $X$, one considers the multiset of lengths of closed primitive geodesics. This multiset is called the length spectrum $\mathcal{L}(X)$.
Question: is $\mathcal{L}(X)$ a ...
- 1,095
4
votes
1
answer
237
views
Spectral density of symmetrized Haar matrix
Let $O$ be a random orthogonal matrix (according to Haar measure) of size $n$. I found by simulations that the spectral density of $O+O^\top$ is the arcsin law rescaled to the interval $[-2,2]$. I can'...
- 1,608
3
votes
0
answers
135
views
the second largest eigenvalue of transfer operators
A Gauss map $T$ is mixing and satisfies Lasota-York inequalities. By Henon's theorem, we know that the transfer operator $\hat{T}$ associated with $T$ has a spectral gap. This means there exists a ...
user509763
1
vote
0
answers
163
views
Reference on spectral theory of self-adjoint operators
I am reading this paper on comparing different moments of independent random variables. A initial step in their approach is designing an operator $L$ over smooth functions (and extended to an self ...
- 13
3
votes
0
answers
151
views
Reference request: trace norm estimate
In a paper I am currently reading, the author uses that if $T$ is an operator given by the kernel $$T(x,y) = \int_{\mathbb R} p(x,z) q(z,y) dz,$$
then $$\lvert \operatorname{tr} T \rvert \leq \lVert T ...
- 101
8
votes
0
answers
197
views
Subleading terms in Weyl's Law
The two term Weyl's conjecture states that
$$N(\lambda)\sim\frac{\operatorname{area}(\Omega)}{4\pi}\lambda-\frac{\operatorname{perimeter}(\partial\Omega)}{4\pi}\sqrt\lambda$$
where $\Omega$ is a ...
- 81
5
votes
2
answers
321
views
Multiplicity of Laplace eigenvalues and symmetry
Let $S$ be a smooth closed connected hyperbolic surface. On $S$ we have the Laplace operator $\Delta$, whose eigenvalues form a discrete sequence
\begin{equation}
0=\lambda_0<\lambda_1\leq \...
- 218
8
votes
1
answer
245
views
Spectral decomposition of $\Gamma\backslash X$
Let $X$ be a reasonable manifold of non-positive curvature (could be $\mathbb{H}^n$, symmetric or locally symmetric space, homogeneous Hadamard manifold etc.), and let $\Gamma$ be a reasonable group ...
- 81
4
votes
1
answer
167
views
Spectrum near zero of $-\partial^2_x + V : L^2(\mathbb{R}) \to L^2(\mathbb{R})$, where $V = O(|x|^{-2 - \delta})$
Let $H = -\partial^2_x + V(x) : L^2(\mathbb{R}) \to L^2(\mathbb{R})$ be a one dimensional Schrödinger operator, where the potential $V$ is real-valued, belongs to $L^\infty(\mathbb{R})$, and, as $|x| \...
- 481
7
votes
0
answers
195
views
Hölder continuity of spectrum of matrices
Endow $\mathbb{C}^{d \times d}$ with the norm induced by the Euclidean norm on $\mathbb{C}^d$. It is well-known (to those who know it well, I guess) that the spectrum $\sigma(A)$ of a matrix $A \in \...
- 12.6k
3
votes
2
answers
264
views
Regularity of eigenfunctions of a self-adjoint differential operator in Gilbarg-Trudinger
Let $\Omega$ be a bounded smooth domain,
$Lu = D_i \left( a^{ij} (x) D_ju \right)$, and two constants
$\lambda, \Lambda > 0$. Suppose the coefficient $a$ is measurable,
symmetric, and satisfies
$$
...
- 33
1
vote
1
answer
294
views
Spectral perturbation theory of discrete spectra in presence of continuous spectrum
This is a 2 part question:
1). I am looking for a (hopefully accessible to beginning grad student who knows matrix perturbation theory) reference for doing concrete calculations of perturbed discrete ...
- 2,281
9
votes
0
answers
210
views
Why and how is a representation "continuously decomposable"?
What I am asking may apply to a much more general setting and I am interested in the underlying level of generality of these statements, mostly with canonical references. However I state the question ...
- 5,424
1
vote
0
answers
85
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
5
votes
1
answer
224
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 ...
- 1,407
0
votes
0
answers
540
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
4
votes
1
answer
155
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 ...
- 854
2
votes
1
answer
215
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 ...
1
vote
1
answer
116
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=\...
- 5,380
1
vote
0
answers
99
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 $\...
- 1,350
2
votes
0
answers
145
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 ...
- 1,054
2
votes
0
answers
110
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 ...
- 421
1
vote
1
answer
111
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 ...
- 13
1
vote
0
answers
42
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 $\Omega$...
- 1,350
7
votes
3
answers
2k
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 ...
- 119
3
votes
1
answer
190
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 ...
- 33
2
votes
0
answers
43
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} - ...
- 593
0
votes
0
answers
122
views
Isolated points of the spectra of self-adjoint operators on Hilbert spaces
Let $T$ be a (everywhere defined) self-adjoint operator on a complex Hilbert space $\mathcal{H}$.
I am interested in results that give (non-trivial, possibly mild) sufficient conditions on $T$ to ...
6
votes
0
answers
85
views
Finding approximate eigenvectors: quantitative results
Let $X$ be a complex Banach space and $T \colon X \to X$ be a bounded operator. For every $x \in X \setminus \{0\}$, denote by $Y_x$ the smallest closed $T$-invariant subspace of $X$ containing $x$. ...
- 1,005
3
votes
1
answer
145
views
Reference for "the algebra of multiplication by all measurable bounded functions acts in Hilbert space in a unique manner"
I read a paper of Alain Connes on "Duality between shapes and spectra" and in page 4, he says
Due
to a theorem of von Neumann the algebra of
multiplication by all measurable bounded ...
- 6,853
4
votes
0
answers
98
views
Spectrum of Laplace-Beltrami with piecewise constant coefficients
By the Laplace-Beltrami with piecewise constant coefficients I means the operator $-div (f\, \nabla .)$ in the 2-sphere. Where $f$ is a piecewise constant function that takes two values $1$ and $a>...
- 61
2
votes
0
answers
145
views
When does a one-dimensional Schrödinger operator have a threshold resonance?
Consider the operator
$$ L = -\partial_x^2 + V(x),$$
for some bounded, decaying potential, i.e. $V(x)\to 0$ as $x\to \pm \infty$. I'm interested in the $L^2(\mathbb R)$ spectrum of $L$. We know that $...
- 21
4
votes
0
answers
134
views
What is known about the density of states for the Anderson Model?
The Anderson Model is given by the random Hamiltonian (as an operator on $l^2(\mathbb{Z}^d)$)
$$
H_\omega = - \triangle + V(\omega)
$$
where $V(\omega) \mid x \rangle = \omega(x) \mid x \rangle$ ...
- 1,054
1
vote
0
answers
128
views
Angular excitations and Schrodinger operators with radial potential in N-dimensions
Can someone please explain the following in mathematical language?
"First of all, angular excitations only push the energy up, never down, so it is enough to analyze spherically symmetric s-waves....
- 141
2
votes
0
answers
212
views
Fourier mode decomposition and eigenvalues of Schroedinger operators with radial potential in N-dimensions
In the study the stability of minimal hipersurfaces $\Sigma \subset \mathbb{R}^{N+1}$ one is lead to study the Morse index of a Schroedinger operator $J := - \Delta_g + |A|^2$ (usually called Jacobi ...
- 141
9
votes
0
answers
331
views
Connections between spectral geometry and critical point/Morse theory
I am researching electrostatic knot theory, which is essentially the theory of harmonic functions on knot complements. I want to understand the number of critical points of the electric potential, ...
- 217
1
vote
1
answer
153
views
Spectral properties of operators mapped to zero by some polynomial
Let $T$ be a bounded operator on a Banach space $X$ and suppose that there is a non-constant polynomial $p$ such that $p(T) = 0$. It seems to be well known that the spectrum of such an operator ...
- 381
5
votes
1
answer
719
views
Eigenvalue and eigenfunction convergence
Consider a bounded Euclidean domain $\Omega \subset \mathbb{R}^n$ (for simplicity, let's say, $\Omega$ has smooth boundary and is simply connected). Let $p \in \Omega$ be a point, and call $\Omega_n = ...
- 51
1
vote
0
answers
59
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 ...
- 801
10
votes
2
answers
938
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 $A_h$ ...
- 658
2
votes
1
answer
114
views
Spectrum of the Magnetic Stark Hamiltonians $H(\mu,\epsilon)$
I am looking for a document where I can find a proof the spectrum of the of the Magnetic Stark
Hamiltonians $H(\mu,\epsilon)=\big(D_x-\mu y)^2+D^2_y+\epsilon x+V(x,y)$ cited on the article below for $\...
- 81
0
votes
0
answers
215
views
Resolvent estimate of compact perturbation of self-adjoint operator
Let $T$ be a selfadjoint operator on Hilbert space $H$. Then we know that there is a resolvent estimation $$\left\lVert (\lambda-T)^{-1}\right\rVert = \frac{1}{dist(\lambda,\sigma(T))}, \ \lambda \in \...
- 31
5
votes
0
answers
132
views
Laplace Beltrami eigenvalues on surface of polytopes
The recently posted arxiv paper Spectrum of the Laplacian on Regular Polyhedra
by Evan Greif, Daniel Kaplan, Robert S. Strichartz, and Samuel C. Wiese, collects numerical evidence for conjectured ...
- 3,209
7
votes
2
answers
2k
views
Eigenvalues of Laplace-Beltrami on half sphere
Let $ \Delta_\theta$ denote the Laplace-Beltrami operator on $S^{N-1}$. The eigenvalues of this are well known. I assume the same is the case of this operator on the upperhalf sphere; say $ S^{N-...
- 1,385
2
votes
2
answers
242
views
iid random operator and its spectrum
consider an insteresting question:
given Banach Space $ \mathcal{B}$, independent identical distribution random operator on $ \mathcal{B}$: $ (T_i)_{i \ge 1} $, where operator space is endowed with ...
- 553