Schrodinger operators, operators on manifolds, general differential operators, numerical studies, integral operators, discrete models, resonances, non-self-adjoint operators, random operators/matrices
7
votes
2answers
538 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-...
4
votes
1answer
144 views
Intuitive proof of Golden-Thompson inequality
Sutter et al. [1] in their paper "Multivariate Trace Inequalities" give an intuitive proof of the following Golden-Thompson inequality:
For any hermitian matrices $A,B$:
$$
\text{tr}(\exp{(A+B)}) \...
-2
votes
1answer
124 views
sum of positive definite matrix
sum of positive definite matrix $A+B $is positive definite. I want to look at the spectrum of $C=A+B$
can we say the ith largest eigenvalue of $C$ is no less than the ith largest eigenvalue of $A$ i....
5
votes
0answers
46 views
Eigenvalue lower bounds for manifold with positive Ricci curvature
For closed $n$-manifold with Ricci curvature $\ge (n-1)$, it is known that the first eigenvalue $\lambda_1\ge n$ with equality holds if and only if $M$ is isometric to the Euclidean sphere $S^n$. My ...
4
votes
0answers
187 views
Spectral Gap of Elliptical Operator
Under what conditions on $a(x)$ and domain $D$, the spectral gap of elliptical operator $ \nabla \cdot(a(x)\nabla)$ defined on $D$, can be controlled?
The boundary condition is that the solution at ...
2
votes
2answers
186 views
Non-linear Basis for PDE's
Asked this on stack exchange and got no response, so I'll try here.
An idea popped into my head awhile ago while doing a project on non-linear effects of systems of coupled oscillators, but I'm not ...
3
votes
1answer
79 views
Non-point spectrum for diagonalisable self-adjoint unbounded 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 ...
0
votes
0answers
154 views
Why is $E$ closed? [migrated]
I am reading through the following statement and proof in Aupetit’s A Primer on Spectral Theory
He provides the following proof:
Towards the end of the proof, Aupetit says that the set $E$ as ...
5
votes
1answer
124 views
Hilbert representation of a bilinear form
Let $\sigma:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}$ be a bilinear symmetric form which is non-degenerate in the sense that for every $0\neq u\in \mathbb{R}^n$ there is $v\in \mathbb{R}^n$ with $...
4
votes
0answers
98 views
Spectral gap for the Brownian motion with drift on a compact manifold
Let $M$ be a compact Riemannian manifold without boundary, $X$ a smooth vector field on $M$. Consider the Brownian motion $t\mapsto B_t$ on $M$ with drift $X$, so that its generator is $L=\Delta +X$. ...
4
votes
2answers
108 views
Radial symmetry of the first eigenfunction
Let $M$ be a simply connected space form (i.e. $\mathbb R^n$, sphere, or hyperbolic space) and $B$ be a ball in $M$. Let $\phi$ be the first Laplacian eigenfunction on $B$, with respect to the ...
1
vote
0answers
43 views
Nodal domains on a surface
What is it known about the topology of nodal domains of eigenfunctions of self-adjoint operators?
In particular I'm interested in self-adjoint operators on a complete, non-compact, surface $\Sigma \...
1
vote
0answers
34 views
First eigenvalue for domains in hyperbolic space
I am interested in examples of bounded open subsets of the hyperbolic space, for which the first eigenvalue of the Dirichlet Laplace operator (acting on functions) is known. In Euclidean space several ...
4
votes
1answer
173 views
Can this self-adjoint operator have an infinite-dimensional compression with compact inverse?
The following might be quite straightforward, but I very rarely work in detail with unbounded operators, so I thought it would be worth seeing quickly if I have overlooked an example that is obvious ...
2
votes
0answers
181 views
Absence of fixed points
Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \frac{xy}{(x^2+y^2+1)} \ dx$$
where $x_0$ is an arbitrary but fixed ...
2
votes
1answer
81 views
Local-Global Principle in Graph Spectrum
The question is a bit vague, but any ideas/directions will be appreciated.
Let us fix an $n$-vertex $d$-regular graph $G=(V,E)$. As I understand it, the eigenvalues of the adjacency matrix $A$ of $G$ ...
8
votes
2answers
292 views
Does a spectral gap lift to covering spaces?
Let $M$ be a complete Riemannian manifold. Denote $\Delta_M\ge0$ the unique self-adjoint extension of the Laplace-Beltrami operator in $L^2(M)$ and $\sigma(\Delta_M)\subset [0,\infty)$ its spectrum. ...
3
votes
0answers
134 views
Does the zeta regularized Laplacian determinant measure the volume of some parameter space? How many “spanning trees” on a manifold?
Let $(M,g)$ be a Riemannian manifold, with Laplacian $\Delta$. If $\lambda_i$ are the nonzero eigenvalues of $\Delta$, we can define the zeta function $\zeta(s) = \Sigma \lambda_i^{-s}$. By analytic ...
4
votes
1answer
239 views
What is $e^{- \zeta_{\Delta} '(0)}$ for a $\Delta$ the Laplacian of a manifold?
For a connected, finite graph $G$, let $\lambda_1, \ldots, \lambda_n$ denote the nonzero eigenvalues of the graph Laplacian. We define $\zeta_G = \Sigma_{i = 1}^n \lambda_i^s$.
Then Kirkoffs Matrix-...
2
votes
2answers
99 views
Spectrum of finite-band random matrices?
Let
$X_n=(X_{ij})_{1 \leq i,j \leq n}$ such that :
$$ \begin{cases}
&X_{ij} = 0 \quad \text{if}\quad \vert i - j \vert > k\\
& X_{ij} \sim P_X \quad \text{otherwise}
\end{cases}$$
And ...
0
votes
0answers
54 views
Numerical error on the spectrum of a matrix
Let $Q=(q_{j,k})_{1\le j,k\le N}$ be a (Hermitian) $N\times N$ matrix with complex-valued entries. The matrix $Q$ is given numerically and the absolute error on each entry is bounded above by a (small)...
15
votes
1answer
271 views
Approximate eigenvectors for a set of non-commuting self-adjoint operators
This problem is motivated by finding the right mathematical setting for expressing the compatibility of classical physics with quantum mechanics.
Let $\mathcal H$ be a Hilbert space and $S$ a ...
2
votes
2answers
147 views
iid random operator and its spectrum
consider an insteresting question:
given Banach Space $ \mathcal{B}$, independent identical distribution random operator on $ \mathcal{B}$: $ (T_i)_{i \ge 1} $, where operator space is endowed with ...
3
votes
1answer
56 views
Simultaneous diagonalization on spaces with constant curvature
I have an operator $C$ that I wish to diagonalize on a Riemmanian manifold $M$ with constant curvature $\Lambda$
$$C = A + B$$
Now I know that these operators $A$ and $B$ commute in flat space, but on ...
0
votes
0answers
61 views
What is spectral multiplicity for multiplication operators in general von Neumann algebra set up?
When two multiplication operators $M_{f}$ and $M_{g}$ acting on $L^2(X,\mu) $and $L^2(Y,\nu)$ are unitary equivalent? How multiplicity function look like here? What is the spectral multiplicity in ...
20
votes
2answers
2k views
Can one hear the (topological) shape of a drum?
Let $(M,g)$ be a (say closed) Riemannian manifold. One can try to understand the geometry/topology of $(M,g)$ by studying the eigenvalues of the Laplacian (this I guess has two versions: when ...
2
votes
1answer
114 views
A nodal theorem in 1D
Consider a 1D zero-energy Schrödinger equation on the half-line,
$(-\partial_x^2 + V(x))\psi(x)=0, \quad x \in (0, \infty)$
with a zero boundary condition $\psi(0) = 0$.
Is it true that if the zero-...
5
votes
1answer
134 views
Spectrum of the product of operators
Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space $F$.
Let $A,B\in \mathcal{B}(F)^+:=\left\{T\in \mathcal{B}(F);\,\langle Tx, x\...
0
votes
0answers
67 views
Discrete spectrum of Schrodinger operator
Assume $\Omega$ is a non-compact region or manifold with dimension $\geq4$. Let $H=-\Delta+V$ be Schrodinger operator. Here $V$ is a (smooth)function.
I know that if $V\geq c>0$ or $V\to c>0$,...
2
votes
0answers
53 views
Density in the graph norm
Let $T\in B(H)$, where $H$ is a (separable) complex Hilbert space. Let $A$ be an unbounded self-adjoint operator acting in $H$. We denote by $D(A)$ its domain. We endow it with the graph norm, e.g., $\...
1
vote
1answer
68 views
Moment generating function of spectral norm of iid N(0,1) data matrix
Let $W^{p\times p}$ be a normal data matrix with $W_{ij}$ i.i.d. $N(0,1)$. Are there any results on the evaluation, or upper bound for the Moment Generating Function of the spectral norm of W, that is,...
0
votes
0answers
21 views
sensitive perturbation approximation
I was reading paper which associated with perturbation approximation. paper1 paper2.
In paper1:
$\bar{R}=R+\epsilon C$, first order: when $\Lambda_1\gg\Lambda_2$, $\Delta\Lambda_{max}=\frac{\vec{v}^...
1
vote
0answers
105 views
Perturbation of Elliptic operator
Let $\Omega$ be an open region or a non-compact complete manifold, $L$ be an elliptic operator with possibly non vanishing zero-order term, e.g. $-\Delta+q$. Suppose $W$ is an operator such that $W(...
1
vote
0answers
35 views
Bound of analytic torsion for a line bundle
Let $E\rightarrow X$ be a line bundle over a compact Riemann surface. Let $\Delta_{E}$ be the Dolbeault Laplacian $\Delta_{\overline{\partial}}$ extended to sections of $E$. Let $g_1, g_2$ be two ...
4
votes
1answer
94 views
Reference for Weyl's law for higher order operators on closed Riemannian manifolds
I am looking at page 32 (beginning of Chapter 5) here. We are given a formally self-adjoint, metrically defined differential operator $A$ on $(M^n,g)$ of order $2l$ with positive definite leading ...
0
votes
0answers
57 views
The connection between the (inverse) Rayleigh quotient and discreteness of the spectrum
Let $T:D(T)\rightarrow H$ be a (densely-defined) self adjoint operator. If $0\not \in \sigma(T)$, then $T$ is invertible, and we know that if $T^{-1}$ is compact, then $T$ has discrete spectrum. I ...
0
votes
0answers
49 views
Finite rank perturbations of matrices - eigenvalues and eigenvectors
Assume we know the spectral decomposition (eigenvalues and eigenvectors) of a $n\times n$ matrix $A$. Consider a finite rank perturbation of $A$ of the form $B=UV^{T}$ where $U$ and $V$ are $n\times ...
5
votes
1answer
166 views
Eigenvalue and eigenvector of ergodic Markov operator for continuous space Markov chain
As we know that the transition matrix $P$ of a Markov chain with finite space is a stochastic matrix, and from Perron-Frobenius Theorem, we know that the spectral radius of the matrix $P$ is $1$, and ...
3
votes
0answers
82 views
Is the square root of curl^2-1/2 a natural (Dirac-)operator?
In current computations on a particular $3$-dimensional Riemannian manifold, a first order differential operator $D:\Gamma^\infty(TM,M)\to \Gamma^\infty(TM,M)$ acting on vector fiels shows up, with ...
14
votes
1answer
233 views
Generalizing the Fourier isomorphism between Sobolev spaces and weighted $L^2$ spaces to (locally) compact groups?
Motivating examples:
Let $V$ be a real vector space with Haar measure $dv$. The fourier transform induces the following topological isomorphism: $$H^s(V,dv) \cong L^2(V^*,(1+|v^*|^2)^sdv^*)$$
The ...
3
votes
1answer
183 views
Proof of Jacobson's lemma in operators theory
Let $A$ and $B$ be two bounded linear operators on an infinite-dimensional complex Hilbert space $H$.
Why the non-zero points of the spectrum of $AB$ coincide with those of the spectrum of $BA$?
2
votes
0answers
115 views
Spectrum of Laplacian depending on boundary conditions [closed]
Consider a compact domain $\Omega \subset \mathbb{R}^n$ with smooth boundary for simplicity. Consider the Laplacian operator with zero Dirichlet boundary conditions on $\Omega$. It is well-known that $...
7
votes
1answer
207 views
Harmonic functions on $(M,g)$ closed, induce an embedding in Euclidean space
In Hajime Urakawa's monograph The Spectral Geometry of the Laplacian on page 41, we make an assumption that I can't quite justify on my own. The following is our setup:
Let $(M^n,g)$ be a closed ...
2
votes
0answers
39 views
Exponential decay for wave equation in even dimensions
Consider the wave equation
$$
u_{tt} = \Delta_x u - q(x)u, \quad x \in\mathbb R^d, \; t > 0,\tag{1}\\
u(0,x) = u_0(x) \in H^1_\text{comp}(\mathbb R^d),\\
u_t(0,x) = u_1(x) \in L^2_\text{comp}(...
1
vote
1answer
96 views
Lower bound of the spectrum of a Schrodinger operator on a bounded domain
I am trying to look for references on estimate of the lower bound of the spectrum of a Schrodinger operator $-\Delta + V$ on a bounded domain in three-dimensional space. For simplicity, we can take ...
3
votes
1answer
207 views
Reference request: The resolvent is analytic in the resolvent set
I am busy reading through Taylor's paper Spectral Theory of Closed Distributive Operators.
On page 192, he defines the resolvent and spectrum of $T$:
Later on in the paragraph, he then proceeds by ...
0
votes
0answers
66 views
Stability of Schrodinger operators on bounded domains
Consider a Schrodinger operator $H=H_0+V$, where $H_0$ is a $L_2$ realization of the negative Laplace operator $-\Delta$ with homogeneous Neumann boundary condition on a bounded, smooth domain $\Omega\...
1
vote
0answers
47 views
Deriving the time evolution of the reflection coefficient for 1d cubic NLS
Update: I have found that the detailed answer to my questions is contained in the book "Solitons: an introduction" by P.G. Drazin and R.S. Johnson. Generally speaking, this seems to be a great book ...
0
votes
1answer
151 views
Does asymptotic behavior guarantee uniqueness?
Suppose $w$ is a solution of
$$\frac{d^2}{dx^2}w+\{u(x)+k^2\}w=0$$
with asymptotic condition
$$\lim_{x\rightarrow \infty}w(x)e^{ikx}=1$$
and $u\in L^1_1(\mathbb{R})=\{f:\int_\mathbb{R}(1+|x|)|f|dx<...
0
votes
1answer
143 views
Doubt in proof of $\lim_{n \to \infty} [\lambda R(\lambda, A)]^n = P.$
I have a doubt in proof of Lemma $4.7$ of this paper.
Lemma: Let $A$ be a closed operator on a complex Banach space $E$ and assume that $0$ is an eigenvalue of $A$ and a pole of the resolvent $R(\...
