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
69 questions from the last 365 days
1
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0
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70
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Eigenfunctions of the Laplacian on $\Bbb R^d$ in Fourier space
Let $\Delta$ be the Laplacian on $\mathbb R^d$. There are no eigenfunctions of the Laplacian in $L^2(\mathbb R^d)$, but $e^{ik\cdot x}$ is an eigenfunction since
$$
\Delta e^{ik\cdot x} = -|k|^2 e^{-...
- 291
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votes
0
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53
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Spectral theory of compact operator for quasi-Banach spaces
Let $X$ be a Banach space and let $Y\subset X$ be a quasi-Banach space (with compact inclusion). Suppose $T:X\to X$ is a compact operator such that $1$ is not its eigenvalue and $T|_{Y}:Y\to Y$ is ...
- 938
0
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0
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55
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reference request: conditions for pointwise and operator-norm convergence of kernel projections
At a very high level, I’m interested in the following question. Suppose $X$ is a (separable) Hilbert space, and $T_n : X \rightarrow X$ is a sequence of finite rank self-adjoint maps that converges (...
- 101
2
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0
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70
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Laplace spectrum on $U(n)$
Consider $\psi:SU(n)\times U(1)\to U(n)$, $(w,z)\mapsto \bar{z}\cdot w$. One can show that $\psi$ serves as a projection and $SU(n)\times U(1)$ is a principal $\mathbb Z_n$-bundle over $U(n)$.
Suppose ...
3
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1
answer
136
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$L^\infty$-bound on Laplace-eigenfunctions
Suppose we are on a closed Riemannian manifold $M$. Any function $f\in C^\infty(M)$ may be decomposed as $$f = \sum_{j = 0}^\infty f_j\phi_j,$$ where $\phi_j\in C^\infty(M)$ are the Laplace ...
1
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0
answers
127
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Trace type convergence of the Laplacian on the box to the Laplacian on $\mathbb R^d$
Let $-\Delta \colon H^2(\mathbb R^d) \to \mathbb R^d$ be the (negative) Laplacian on the full space and $-\Delta_L$ the Laplacian acting on $L^2([-L,L]^d)$ with some boundary conditions making it self-...
- 61
4
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1
answer
195
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Asymptotic spectrum of a complex Sturm-Liouville differential operator
Let $\varepsilon > 0$ and consider the (complex) Sturm–Liouville differential operator on $[0,1]$ given by
$$
\mathcal{L}_\varepsilon f(x) = \varepsilon^2 f''(x) + i V(x) f(x),
$$
with Neumann ...
- 333
1
vote
1
answer
144
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An application of min-max characterization of eigenvalues
Let $(M,g_0)$ be a $n$-dimensional closed Riemannian manifold with a Riemannian covering $(\widetilde{M},\widetilde{g}_0)$. Let
$$
\mathcal{V}_{ab}=\{g\colon a^2 g_0\leq g\leq b^2 g_0\}, \quad \text{...
- 563
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0
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40
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In what sense is a change of boundary conditions a finite rank perturbation?
Also asked on MSE (https://math.stackexchange.com/questions/4987654/in-what-sense-is-a-change-of-boundary-conditions-a-finite-rank-perturbation and https://math.stackexchange.com/questions/4875398/how-...
- 101
2
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1
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124
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Choice of the eigenbasis for the Dirac operator on $S^d$
This question is a simplified version of my previous one. I think that adding a gauge potential complicates the problem too much.
Let us consider the Dirac operator $D$ on the $d$-sphere $S^d$ with ...
- 3,477
4
votes
1
answer
318
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Functoriality of infinite suspension spectrum functor on infinity groupoids!
Consider the functor $F: C \rightarrow D $ of $\infty$-groupoids. Is there any explicit proof somewhere in the literature that $\Sigma^{\infty}$ construction is functorial? I mean how do we define $\...
- 167
1
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2
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85
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Bound on norm of difference of powers of self-adjoint operators
I found the following result without any proof. I would be very grateful for any suggestion how to start here.
Let $B_{1}$, $B_{2}$ be two non-negative self-adjoint operators on some Hilbert space ...
- 51
2
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0
answers
102
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Semiclassical limit of spectral gap of Schrödinger operators with nonsmooth potential
Let $\Omega$ be a connected compact subset of $\mathbb{R}^d$. It is well known that for a smooth potential $V:\Omega \to \mathbb{R}$ that has a unique nondegenerate minimum $V(0) = 0$, the operator $H ...
- 1,551
6
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0
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113
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Schwartz kernel of spectral projection of Laplacian and integrated density of states
I'm reposting here a question I asked on MSE which did not receive an answer.
I am considering the Dirichlet Laplacian $\Delta$ on some smooth domain $U$. For now assume that $U$ is bounded, and later ...
- 251
0
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0
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60
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Spectral analysis of Dirac operators coupled to gauge potential on $\mathbb{R}^n$
Dirac operators on compact manifolds seem to have been studied well, such as in this book and also this one.
However, I cannot easily find comprehensive treatment of Dirac operators coupled to gauge ...
- 3,477
0
votes
1
answer
118
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Nodal domain theorem for clamped plate equation
Let $\lambda_k$ and $\varphi_k$ be the $k$-th eigenvalue and a corresponding eigenfunction of the clamped plate equation in a bounded domain $\Omega \subset \mathbb{R}^N$, $N \geq 2$.
That is, $\...
- 87
3
votes
1
answer
67
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Infinite direct sum decomposition of the heat semigroup on $\mathbb R$
This question is based on a very similar question posted by me yesterday. A very nice solution was provided by Aleksei Kulikov. Here I modify my question slightly.
Let $Q_t$ be the heat semigroup on $...
- 407
2
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0
answers
331
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What is the spectrum of this differential operator?
My self-adjount differential operator $L$ is defined by $$L f(x) \equiv u(x) \frac{\partial^2}{\partial x^2} \left( u(x) f(x) \right)$$ where $u(x)$ is a known but arbitrary smooth function that ...
- 151
0
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0
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49
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On the $L^p$ estimate and Weyl's law of Eigenfunctions in Sogge's Book
I have recently started to study the book "Fourier integrals in classical analysis " by Sogge mainly oscillatory integral decay methods. I have a question from the chapters 4 and 5. Mainly ...
- 11
1
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0
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76
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A representation of positive matrix
Let $\mathcal H$ be a Hilbert space. Let $-\frac{1}{2}<r<0.$ Denote $c_p:=\int_{0}^\infty\frac{t^r}{1+t}dt.$ Suppose $A$ be a positive invertible operator in $B(\mathcal H).$ Is it true that $A^...
- 1,879
12
votes
3
answers
2k
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Why is resonance such a widespread phenomenon?
It is easy to mathematically describe the motion of a mass which is attached to a spring and also pushed around by a sinusoidal force. We get a differential equation of the form:
$$\frac{\mathrm{d}^2x}...
- 1,433
0
votes
1
answer
73
views
Computing spectrum of very simple Schrödinger operator
I asked this question recently on a thread in math stack exchange, but with no real answers suggested. I think this is a relatively simple variation on the classical free Laplacian spectrum, so I ...
- 633
2
votes
0
answers
65
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Generalized Fourier transforms associated to Schroedinger operators
Let $n\geq 1$. Let $q\in C^{\infty}_0(\mathbb R^n)$ be compactly supported and consider the operator $P= -\Delta+q(x)$ on $\mathbb R^n$. We will assume that $q$ is sufficiently small so that the ...
- 4,135
0
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0
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40
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Is the principal eigenvalue of the clamped plate minimized by the equilateral triangle among all triangles of a given area?
Let $\Omega$ be a ``nice'' domain in $\mathbb{R}^2$
(bounded and with piecewise smooth boundary). Consider the clamped plate eigenvalue problem, given by
$$\Delta^2 u = \lambda u $$
$$ u|_{\partial \...
- 3,245
5
votes
0
answers
227
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Relations between two Schwartz kernels in dimensions $n$ and $n+1$
Let $(M,g)$ be an $n$-dimensional pseudo-Riemannian manifold and $\Box_g$ be the Laplace-Beltrami operator on $M$. Consider $z \in \mathbb{C}$ such that $\mathrm{ Im}(z)>0$, and we define $P_0 := \...
- 319
2
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0
answers
41
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Why has the random Koopman matrix $ G_{xx}^{(-)} G_{yx} $ only eigenvalues on the complex unit circle?
Let U be a $\Bbb{R}^{(n+1)(n+1)} $ matrix with entries drawn from a independent normal distribution,
e.g.
$$ U_{i j} \sim N(0,1) \quad \quad i,j=1,...n+1$$
Let $ G=U U^* $ be a Gram matrix where $ U^* ...
- 253
0
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0
answers
27
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Comparison Principle for Courant Nodal Domain Theorem
Recall that Courant's Nodal domain gives an upper bound on the number of nodal domains of the $k$-th eigenfunction. I was wondering if it is possible to deduce some relation between the number of ...
- 537
0
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0
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45
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Discrete and continuous representation in Hilbert space
I’m interested in using laplacian (−Δ)
eigenfunction as a basis for H1(Rn)
. I know that in H1(Ω)
, Ω
bounded this can be done so I was wandering about H1(Rn)
.
Now let eλ
be an eigenfunction ...
0
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0
answers
14
views
A question regarding quasiperiodic and limit-periodic potentials
I am looking for specific references regarding the fact that there are quasiperiodic and, also, limit-periodic potentials $q(x)$ such that
the $L^2(\mathbb{R})$-spectrum $\sigma(H)$ of the operator
\...
0
votes
0
answers
39
views
Spectral properties of Ruelle transfer operator
Consider a compact metric space $(X,d)$, a continuous surjective map $T:X \to X$ of finite degree and the space of continuous functions $C(X)$ equipped with the supremum norm. Also, consider only (for ...
- 143
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?
22
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0
answers
869
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Can two drums almost sound the same?
Let $D\subset \mathbb R^2$ be a region and let $\Lambda=\{\lambda_1,\lambda_2,\dots\}$ be the set of eigenvalues of the Laplacian $-\Delta$ (with boundary condition $\psi=0$ on $\partial D$).
Mark Kac,...
- 3,054
1
vote
0
answers
98
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$(\lambda I-A)^{-1}-(\lambda I-B)^{-1}$ compact implies $\sigma_\text{ess}(A)=\sigma_\text{ess}(B)$
Suppose $H$ is a Hilbert space and $A$, $B$ are two adjoint operators on it (not necessarily bounded), satisfying $D(A)=D(B)$.
Question: If $\exists \lambda\in \rho(A)\cap\rho(B)$ such that $(\lambda ...
- 775
0
votes
1
answer
100
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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
0
votes
1
answer
45
views
Expectation of spectral norm of a diagonal stochastic matrix
There is a diagonal matrix $G(t)\in R^{M\times M}$ where the diagonal elements are independent Bernoulli stochastic variables, satisfying $\mathbb{E}(g_i(t) = 1) = b$, and $ \mathbb{E}(g_i(t) = 0) = 1 ...
1
vote
0
answers
35
views
Class of semi-algebraic potentials satisfying growth condition for the sublevel sets
Recently I have studied the paper: Rozenbljum, G. V. (1974). Asymptotics of the eigenvalues of the Schrödinger operator. Mathematics of the USSR-Sbornik, 22(3), 349.
In there Rozenbljum gives ...
- 1,045
4
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1
answer
108
views
Accumulation points of point spectrum of Schrödinger operator in one dimension
Consider a Schrödinger operator $H=-\partial_x^2+V(x)$, with $x\in\mathbb R$, $V(x)$ tending monotonically to $V_\pm$ as $x\to\pm\infty$, and $\min V(x)<V\pm$. Intuitively, the only accumulation ...
- 43
1
vote
1
answer
87
views
Where does $V$ from the spectral decomposition $A = VDV^*$ lie, if $A$ has only imaginary entries?
The spectral theorem says that for every Hermitian matrix $A \in \mathbb{C}^{n \times n}$ there is a unitary matrix $V \in U(n)$ and a diagonal matrix $D \in \mathbb{R}^{n \times n}$ such that $A = ...
- 1,295
2
votes
0
answers
127
views
Error in an argument using spectral theory
Let $Z=\sqrt{2}\mathbb Z$, and consider the sequence on $\mathbb{Z}$ $$\xi(k)= 1_{Z\cap [k,k+1]\neq\emptyset}.$$(remark that the intersection is either empty or with one point.)
Thanks to the comments,...
- 1,352
0
votes
1
answer
106
views
Spectral theory: a key to unlocking efficient insights in network datasets
In the context of directed or undirected graphs, matrices such as adjacency and Laplacian matrices are commonly used. The eigenbasis of these matrices addresses some practical implications, such as ...
- 4,058
2
votes
0
answers
67
views
Regularity and decay of Fourier-like series on a manifold
Let $D$ be a first-order self-adjoint elliptic differential operator acting on sections of a vector bundle $S$ over a closed manifold $M$. Then it is well-known that the various eigenspaces $E_\lambda$...
- 1,903
1
vote
0
answers
67
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Equidistribution on subvarieties of $\mathrm{SL}_n(\mathbb{Z})$
$\DeclareMathOperator\SL{SL}$Let $\Gamma_{\infty}$ be a maximal parabolic subgroup of $\SL_n(\mathbb{Z})$. I believe there are various equidistribution theorems, which say that Iwasawa $x$ coordinates ...
- 2,907
3
votes
1
answer
335
views
Book on Hilbert spaces, including non-separable
I am looking for a book that develops the theory of Hilbert spaces, including the spectral theorems and unitary representations, but includes non-separable Hilbert spaces in the main exposition. Any ...
1
vote
0
answers
63
views
On an estimate in the paper by Donnelly and Fefferman
I was reading the following paper by Donnelly and Fefferman https://link.springer.com/content/pdf/10.1007/BF01393691.pdf which essentially deals with the Hausdorff dimension bound of the nodal sets ...
- 41
3
votes
1
answer
251
views
Feynman–Kac formula for other operators
I recently came across the Feynman-Kac formula which states that given an open domain $\Omega\in\mathbb{R}^n$ and $f \in L^2(\Omega)$
where $x \in \Omega$ and $t > 0$, then
$e^{t\Delta_D}f(x) = ...
- 41
5
votes
2
answers
458
views
Question about Neumann eigenvalues on manifolds
Question:
Let $\Omega\subset \mathbb{S}^2_+$ be any geodesically convex subset of the hemisphere $\mathbb{S}^2_+$. Then is it true that $\mu_1(\Omega)\geq \mu_1(\mathbb{S}^2_+)$ where $\mu_1$ is the ...
- 537
2
votes
1
answer
112
views
On compactly supported functions with prescribed sparse coordinates
Let $\{\phi_n\}_{n=1}^{\infty}$ be an orthonormal basis for $L^2((0,1))$ consisting of Dirichlet eigenfunctions for the operator $-\partial^2_x + q(x)$ where $q \in C^{\infty}_c((0,1))$ is fixed. ...
- 4,135
0
votes
0
answers
79
views
Convergence of metric implies convergence of eigenvalues?
Consider the metric $g_\varepsilon=d\theta^2 + \varepsilon^2 \sin(\theta)^2 \, d\varphi^2$ on the hemisphere $S^2_+.$ I had two naive questions:
Does $g_\varepsilon$ converge to the flat metric on ...
- 537
3
votes
2
answers
392
views
Monotonicity of matrix conjugation
Let $A$ and $B$ be positive-definite matrices such that $A \le B.$
By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$
I am now curious under ...
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