All Questions
Tagged with fa.functional-analysis sp.spectral-theory
385 questions
7
votes
1
answer
245
views
Lower estimate of the minimal eigenvalue of a Hamiltonian
Consider a linear operator $H\colon L^2(\mathbb{R}^3)\to L^2(\mathbb{R}^3)$ given by
$$H(\psi)(x):=-\Delta\psi(x)+V(x)\cdot \psi(x),$$
where $V\colon \mathbb{R}^3\to \mathbb{R}$ is a continuous (or ...
- 21.8k
2
votes
0
answers
150
views
Limit circle/point of an ODE with finite eigenvalues
Consider the following Sturm–Liouville (SL) eigenvalue problem defined in $(-\infty,0]$ or $[0,\infty)$ or $(-\infty,+\infty)$ $$(py')'-qy=-\lambda^2wy,$$ in which $p(x)=x^2$, $w(x)=1$, and $q(x)=(x/2+...
- 155
2
votes
0
answers
306
views
Interesting examples of spectral decompositions of BOUNDED operators with both continuous and discrete spectrum
I would like to have a few basic examples of bounded self-adjoint operators $T$ (more generally bounded normal would be fine) on a Hilbert space $(H,\langle,\rangle)$ for which the following criteria ...
- 7,609
8
votes
2
answers
323
views
Matrix rescaling increases lowest eigenvalue?
Consider the set $\mathbf{N}:=\left\{1,2,....,N \right\}$ and let $$\mathbf M:=\left\{ M_i; M_i \subset \mathbf N \text{ such that } \left\lvert M_i \right\rvert=2 \text{ or }\left\lvert M_i \right\...
- 225
4
votes
0
answers
86
views
Characterizing the separability of the Gelfand space of a semisimple commutative Banach algebra
Problem. Is the separability of the Gelfand space of a semi-simple commutative Banach algebra $A$ equivalent to the existence of a countable family $\{\varphi_n\}_{n\in\omega}$ of multiplicative ...
- 4,535
4
votes
0
answers
410
views
Spectral Gap of Elliptic Operator
Under what conditions on $a(x)$ and domain $D$, the spectral gap of the elliptic operator $ \nabla \cdot(a(x)\cdot \nabla)$ defined on $D$, can be controlled?
The boundary condition is that the ...
- 325
3
votes
1
answer
214
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 ...
- 969
5
votes
1
answer
379
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 $...
- 5,529
5
votes
0
answers
242
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$. ...
- 3,669
4
votes
1
answer
350
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 ...
- 25.8k
2
votes
0
answers
190
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$$
where $x_0$ is an arbitrary but fixed ...
17
votes
3
answers
770
views
Does a spectral gap lift to covering spaces?
Let $M$ be a complete Riemannian manifold. Denote $\Delta_M\ge0$ the unique self-adjoint extension of the Laplace-Beltrami operator in $L^2(M)$ and $\sigma(\Delta_M)\subset [0,\infty)$ its spectrum. ...
- 779
16
votes
1
answer
656
views
Approximate eigenvectors for a set of non-commuting self-adjoint operators
This problem is motivated by finding the right mathematical setting for expressing the compatibility of classical physics with quantum mechanics.
Let $\mathcal H$ be a Hilbert space and $S$ a ...
- 161
2
votes
2
answers
242
views
iid random operator and its spectrum
consider an insteresting question:
given Banach Space $ \mathcal{B}$, independent identical distribution random operator on $ \mathcal{B}$: $ (T_i)_{i \ge 1} $, where operator space is endowed with ...
- 553
5
votes
1
answer
606
views
Spectrum of the product of operators
Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space $F$.
Let $A,B\in \mathcal{B}(F)^+:=\left\{T\in \mathcal{B}(F);\,\langle Tx, x\...
- 724
3
votes
1
answer
215
views
Discrete spectrum of Schrodinger operator
Assume $\Omega$ is a non-compact region or manifold with dimension $\geq4$. Let $H=-\Delta+V$ be Schrodinger operator. Here $V$ is a (smooth)function.
I know that if $V\geq c>0$ or $V\to c>0$,...
- 1,915
2
votes
0
answers
213
views
Density in the graph norm
Let $T\in B(H)$, where $H$ is a (separable) complex Hilbert space. Let $A$ be an unbounded self-adjoint operator acting in $H$. We denote by $D(A)$ its domain. We endow it with the graph norm, e.g., $\...
- 71
14
votes
1
answer
514
views
Generalizing the Fourier isomorphism between Sobolev spaces and weighted $L^2$ spaces to (locally) compact groups?
Motivating examples:
Let $V$ be a real vector space with Haar measure $dv$. The fourier transform induces the following topological isomorphism: $$H^s(V,dv) \cong L^2(V^*,(1+|v^*|^2)^sdv^*)$$
The ...
- 7,789
2
votes
0
answers
343
views
Spectrum of Laplacian depending on boundary conditions [closed]
Consider a compact domain $\Omega \subset \mathbb{R}^n$ with smooth boundary for simplicity. Consider the Laplacian operator with zero Dirichlet boundary conditions on $\Omega$. It is well-known that $...
- 1,407
7
votes
1
answer
281
views
Harmonic functions on $(M,g)$ closed, induce an embedding in Euclidean space
In Hajime Urakawa's monograph The Spectral Geometry of the Laplacian on page 41, we make an assumption that I can't quite justify on my own. The following is our setup:
Let $(M^n,g)$ be a closed ...
- 1,247
5
votes
1
answer
1k
views
Reference request: The resolvent is analytic in the resolvent set
I am busy reading through Taylor's paper Spectral Theory of Closed Distributive Operators.
On page 192, he defines the resolvent and spectrum of $T$:
Later on in the paragraph, he then proceeds by ...
- 289
0
votes
1
answer
181
views
Does asymptotic behavior guarantee uniqueness?
Suppose $w$ is a solution of
$$\frac{d^2}{dx^2}w+\{u(x)+k^2\}w=0$$
with asymptotic condition
$$\lim_{x\rightarrow \infty}w(x)e^{ikx}=1$$
and $u\in L^1_1(\mathbb{R})=\{f:\int_\mathbb{R}(1+|x|)|f|dx<...
- 145
0
votes
1
answer
198
views
Doubt in proof of $\lim_{n \to \infty} [\lambda R(\lambda, A)]^n = P.$
I have a doubt in proof of Lemma $4.7$ of this paper.
Lemma: Let $A$ be a closed operator on a complex Banach space $E$ and assume that $0$ is an eigenvalue of $A$ and a pole of the resolvent $R(\...
- 343
2
votes
1
answer
135
views
Confusion in definition of peripheral spectrum
I had asked this question on Mathematics Stack Exchange, $2$ days ago but it got no response so I'm asking here.
If $A$ is a closed operator, then the peripheral spectrum of $A$ is defined to be all ...
- 343
6
votes
0
answers
281
views
Spectral properties of Non-local Differential operators on real line
I am encountering non-local (and nonlinear) PDEs in my work. To compute stability, I am trying to numerically estimate the spectrum of linearized-but-nonlocal version of the said PDEs.
Definition: A ...
- 318
9
votes
0
answers
230
views
Using Property (T) to approximate invertible matrices
In the wikipedia article for Kazhdan's Property (T), there's an intriguing application:
Similarly, groups with property (T) can be used to construct finite sets of invertible matrices which can ...
2
votes
1
answer
497
views
Spectrum of magnetic Laplacian
Consider the discrete magnetic Laplacian on $\mathbb Z^2.$
$$(\Delta_{\alpha,\lambda}\psi)(n_1,n_2) = e^{-i \pi \alpha n_2} \psi(n_1+1,n_2) + e^{i\pi \alpha n_2} \psi(n_1-1,n_2) + \lambda \left(e^{i ...
- 21
2
votes
0
answers
58
views
Absolute continuity of DOS measure for Schrödinger operators
Kotani theory gives roughly that for ergodic operators there is a certain equivalence between absolutely continuous spectrum and an absolutely continuous density of states measure.
I would like to ...
- 21
2
votes
1
answer
555
views
Resolvent of the Laplacian as a pseudodifferential operator and its single layer potential
In M.Taylor's book "Partial differential equations II" it is shown that the fundamental solution $E(x,y)$ of the Laplacian equation gives rise to an elliptic pseudodifferential operator $S$ on the ...
- 1,329
1
vote
2
answers
219
views
Ground state has always constant sign?
Question: What hypothesis on the potential $V$ are required such that the ground state $\phi_0$ has constant sign?
Consider the Schrödinger operator in 1 dimension with potential $V$:
$$\mathcal{H}=-...
- 129
2
votes
1
answer
135
views
Pseudo-polynomial potentials for Schrödinger operators
Consider the one dimensional Schrödinger hamiltonian $\mathcal{H}=-\frac{\hbar^2}{2} \frac{d^2}{dx^2} + V(x)$.
Suppose that $V:\mathbb{R} \rightarrow \mathbb{R}^+$ is a continuous and confining ...
- 129
2
votes
0
answers
78
views
Generalization of supersymmetry to dimension 3
in two dimensions there is a simple trick to study the spectrum of operators of the form
$$\textbf{A}:=\left( \begin{matrix}0 && A^* \\ A && 0 \end{matrix}\right)$$
The trick is to ...
- 167
3
votes
0
answers
198
views
Asymptotic stability of eigenvalues by compact perturbations
I need some references concerning the asymptotic stability of eigenvalues by compact perturbations. In [T. Kato, Perturbation theory for linear operators] there are some results concerning stability ...
- 1,329
0
votes
1
answer
268
views
Spectral Theorem, $AB = BA \implies B\Phi(f) = \Phi(f)B$
I'm studying the spectral theorem as appears in Reed and Simon's Functional Analysis.
Assume we have constructed the continuous functional calculus for a self adjoint bounded operator $A$ on a ...
- 181
2
votes
1
answer
244
views
Smoothness of distributions defined by oscillation integrals
In M.A. Shubin's book Pseudodifferential Operators and Spectral Theory, we have the following statement.
Let $X\subset\mathbb{R}^n$ be an open set, and fix a symbol $a\in S_{\varrho,\delta}^m(X\...
- 1,247
5
votes
1
answer
1k
views
The spectrum of the discrete Laplacian
Consider a connected (we define connected components by defining the set of vertices where every vertex has one neighbour) sublattice $V$ of the square lattice $V \subset\mathbb{Z}^2.$
On this we ...
- 53
0
votes
1
answer
185
views
Meromorphic solutions to Legendre's equation
I just saw the following question that was asked yesterday on math overflow on meromorphic solutions to ODEs
Although, I understand the answers and comments to the questions, I did not understand how ...
- 501
4
votes
1
answer
366
views
Dissipative operator on Banach spaces
An operator $A$ is called dissipative if for all $x \in D(A)$ and $\lambda >0$
$$ \left\lVert (A-\lambda)x \right\rVert \ge \lambda \left\lVert x \right\rVert.$$
On a Hilbert space this is ...
- 501
6
votes
4
answers
1k
views
Resource recommendation: Spectral theory and $C^*$ algebras
I have formally studied functional analysis, both as university courses, and by myself, but this is one area of mathematics I find so huge and complicated, I have a hard time properly getting into it.
...
- 2,792
2
votes
0
answers
142
views
Self-adjointness on Banach spaces
Let $A \in L(X,Y)$ be a bounded operator between Banach spaces. Then its dual operator $A' \in L(Y',X')$ has the same spectrum as $A$ by the closed range theorem.
Now, if we have an unbounded ...
- 501
1
vote
0
answers
130
views
Spectrum of an operator
Let $L$ an operator self-adjoint acting on $L^2(\Bbb{R}^{2})$ such that :
$L(\phi_{\alpha,\beta})=(|\alpha|-|\beta|)(\phi_{\alpha,\beta})$ where $(\phi_{\alpha,\beta})$ is an orthonormal basis for ...
- 11
10
votes
2
answers
1k
views
Harmonic oscillator discrete spectrum
Let us act intentionally stupid and assume we do not know that we can solve for the spectrum of the harmonic oscillator
$$-\frac{d^2}{dx^2}+x^2$$
explicitly.
Is there an abstract argument why the ...
- 501
5
votes
1
answer
2k
views
Commuting with self-adjoint operator
Let $T$ be an (unbounded) self-adjoint operator. Assume that there is a bounded operator $S$ such that $TS=ST.$ For which kind of $f$ do we have that $f(T)S=Sf(T)?$
My thought was that using a ...
- 501
5
votes
0
answers
122
views
How to solve this operator equation numerically?
I would like to know how one solves Sturm-Liouville problems on $(0,\infty)$ NUMERICALLY for the eigenvalues that are of the form
$$-f''(x)+\frac{1}{\sinh(x)^2}f(x)=\lambda f(x).$$
So even if there ...
- 501
1
vote
0
answers
259
views
An estimate for the solution of an elliptic PDE depending on a parameter
Let $\Omega\subset\mathbb R^n$ be a bounded domain with a sufficiently smooth boundary $\partial\Omega$.
We assume $\lambda_1\in\mathbb R$ is the principle eigenvalue of the operator
$$
-\Delta:\ H^...
- 375
4
votes
1
answer
725
views
Eigenfunction of Laplacian
On $L^2(\mathbb{R}^n)$ it is true that $\Delta$ has $\sigma(\Delta)=(-\infty,0].$ Also, there are no eigenfunction. Yet, even if one would not know this, negativity $\langle \Delta u,u \rangle \le 0$ ...
- 329
10
votes
3
answers
1k
views
References: spectral analysis of the Laplacian operator
I'm looking for several references on the spectral analysis of the Laplacian operator. It is such a well-known topic, but I'm a bit struggling to locate modern systematic expositions in the literature....
user89890
10
votes
1
answer
3k
views
Trace of integral trace-class operator
I have seen many answers to the converse question (which seems to be difficult in general), but I would like to ask the following:
Let $T: L^2 \rightarrow L^2$ be a trace-class operator that is also ...
user69109
1
vote
0
answers
76
views
Which sets support which spectra?
I know (and this is of course rather elementary) that an isolated point in the spectrum of a self-adjoint operator $T$ always belongs to the point-spectrum.
I would like to ask: Are there similar ...
- 173
3
votes
1
answer
529
views
Spectrum of self-adjoint operator
As a non functional analyst, I stumbled over the following question:
Given a self-adjoint Operator $T:D(T) \subset H \rightarrow H.$ Assume we know that $T$ has some eigenvalue $\lambda$ which is ...
- 173