All Questions
Tagged with sp.spectral-theory operator-theory
152 questions
74
votes
2
answers
3k
views
Can you hear the shape of a drum by choosing where to drum it?
I find the problem of hearing the shape of a drum fascinating. Specifically, given two connected subsets of $\mathbb R^2$ with piecewise-smooth boundaries (or a suitable generalization to a riemannian ...
- 2,358
27
votes
0
answers
1k
views
Unital $C^{*}$ algebras whose all elements have path connected spectrum
A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$.
What is an example of a non commutative ...
- 356
22
votes
1
answer
3k
views
Why should I look at the resolvent formalism and think it is a useful tool for spectral theory?
Wikipedia calls resolvent formalism a useful tool for relating complex analysis to studying the spectra of a linear operator on a Banach space. Sure, I believe you because I've seen results that use ...
- 393
18
votes
4
answers
1k
views
Who first used the multiplication operator version of spectral theory
This is another history question.
Hilbert phrased the spectral theorem in terms of resolutions of the identity.
While this remained the form of Stone and von Neumann, they did also have the ...
- 827
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
15
votes
2
answers
819
views
An orbit of symmetric polynomials
Consider the ring of polynomials $R:=\mathbb{Z}[x_1,x_2,x_3]$. Define the operators $E, I:R\rightarrow R$ by $Ef(x_1,x_2,x_3)=f(x_1-1,x_2,x_3)$ and the identity $If=f$.
Let $\mathcal{L}:R\rightarrow R$...
- 43.2k
12
votes
1
answer
191
views
Spectra on different spaces
This is a method request: I am looking for techniques that allow me to investigate problems like this:
Let $T_1: \ell^1 \rightarrow \ell^1$ be a bounded operator with $\Re(\sigma(T_1)) \subset (-\...
- 305
11
votes
1
answer
487
views
Is the spectrum of a "self adjoint" operator real on $\ell^p$?
There might be an obvious answer to the question, but it doesn't come to mind.
Suppose we have an infinite matrix $A=(a_{ij})$, which defines a bounded linear operator on $\ell^p$, i.e. for all ...
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
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
9
votes
2
answers
778
views
Rellich's theorem from compact resolvent
On a compact Riemannian manifold, we know that the Laplacian $\Delta$ has compact resolvent. In proving this, one typical way is to use Rellich's theorem about the compact embedding of $H^1(M)$ into $...
- 91
8
votes
1
answer
844
views
A doubt about the parts of the spectrum of tensor products
Let $\mathcal{H}$ be any complex Hilbert space of infinite dimensional. By an operator $T$ I mean a linear bounded transformation from $\mathcal{H}$ into $\mathcal{H}$, i.e, $T:\mathcal{H}\rightarrow\...
- 81
7
votes
3
answers
2k
views
Why do we distinguish the continuous spectrum and the residual spectrum?
As we know, continuous spectrum and residual spectrum are two cases in the spectrum of an operator, which only appear in infinite dimension.
If $T$ is a operator from Banach space $X$ to $X$, $aI-T$ ...
- 391
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
7
votes
1
answer
375
views
Spectrum of "classical" operators
Lately, I've been reading a couple of papers from different one-dimensional PDE contexts on which operators like $\mathcal{L}:=-\partial_x^2+c_*+\Phi$ repeatedly appear. Usually, on these contexts $\...
- 395
7
votes
1
answer
414
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 ...
- 536
6
votes
3
answers
3k
views
Non-empty resolvent set, then operator closed?
On Hilbert spaces, the following is true:
Let $T$ be a densely-defined linear operator with non-empty resolvent set, then $T$ is closed.
The obvious proof I see to show this uses explicitly the ...
- 115
6
votes
1
answer
575
views
Spectrum of the complex harmonic oscilllator
Let
$$
H_\lambda=-\frac{d^2}{dx^2}+\lambda^2 x^2,\quad\lambda>0.
$$ It is known that the spectrum of $H_\lambda$ is the set $\{(2n-1)\lambda,n\in \Bbb N^*\}$. Now put
$$
(U_\mu \phi)(x)= e^{\mu\...
- 521
6
votes
1
answer
778
views
Resolvents of Schrodinger operators
In the free case one can compute the resolvents of the Laplacian $-\Delta$ in many cases explicitly, in the sense that they are given by an integral operator. Often, one uses the Hille-Yosida theorem ...
- 305
6
votes
1
answer
353
views
Domains of raising and lowering operators in QM
Let $H : \operatorname{dom}(H) \subset L^2(\Omega) \rightarrow L^2(\Omega)$, where $dom(H) \subset H^2(\Omega)$, $\Omega \subset \mathbb{R}$ should be a bounded open interval(so 1-d setting(!)) and $H$...
user54300
6
votes
1
answer
1k
views
Is the sum of spectral projections a projection?
Let $T$ be a closed operator on a Hilbert space with discrete spectrum. Then for $\{\lambda_1,...\lambda_n\}\in\sigma(T)$ one can define the spectral projections
$$P_{\{\lambda_1,...\lambda_n\}}=\frac{...
- 241
6
votes
0
answers
113
views
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
5
votes
2
answers
625
views
Reconstruction of second-order elliptic operator from spectrum
Let $M$ be a compact smooth manifold, $(\lambda_n)_{n=1}^{\infty}$ be a square-summable monotonically increasing sequence of non-negative numbers, and let $(f_k)_{k=1}^{\infty}$ be continuous ...
- 364
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
5
votes
3
answers
557
views
Show that the Laplacian operator on the Heisenberg group is negative
The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law
$$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$
For $z=x+ i y \in \mathbb C$ ...
- 51
5
votes
1
answer
151
views
Existence of operator with certain properties
I am curious to know the answer to the following question:
Does there exist a continuous linear operator on some Banach space $X$ such that $\Vert T \Vert=1$, and $\sigma(T)\supset \{1\}$ is isolated ...
user69109
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
5
votes
1
answer
475
views
Is irreducibility sufficient for uniqueness of invariant distribution for a Feller semigroup?
Let $(T_t)$ be a strongly continuous semigroup of positive operators on $C(K)$, where $K$ is a compact space. Assume also that $T_t1 =1 $ for every $t\geq 0$.
(This is also called a Feller semigroup.)
...
- 448
5
votes
1
answer
229
views
Canonical multiplication representation of self-adjoint operator in quantum chemistry and coding theory research
In my applied math research group, we are studying and going over functional analysis results from papers and theses from our institution to generalize their results and apply them in our discrete ...
- 620
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
4
answers
839
views
Norm bounds on spectral variation and eigenvalue variation
Let $A$ and $B$ be two matrices of eigenvalues $\lambda_i$ and $\mu_i$, respectively.
The spectral variation of $B$ w.r.t. $A$ and the eigenvalue variation of $B$ and $A$ are, respectively,
\begin{...
- 43.2k
5
votes
0
answers
227
views
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
5
votes
0
answers
515
views
Learning from eigenvalues of Hilbert-Schmidt integral operator
Do eigenvalues of the Hilbert-Schmidt integral operator determine the underlying measure up to translation, reflection and rotation?
Details: Suppose we have a measure $\mu$ on a Euclidean space $X=\...
- 903
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
4
votes
1
answer
221
views
Non-isolated ground state of a Schrödinger operator
Question. Does there exist a dimension $d \in \mathbb{N}$ and a measurable function $V: \mathbb{R}^d \to [0,\infty)$ such that the smallest spectral value $\lambda$ of the Schrödinger operator $-\...
- 12.6k
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
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
4
votes
1
answer
301
views
Trying to recover a proof of the spectral mapping theorem from old thesis/paper with continuous functional calculus
In my research group in functional analysis and operator theory (where we do physics and computer science as well), we saw in an old Russian combination paper/PhD thesis in our library a nice claim ...
- 620
4
votes
1
answer
201
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) = \...
- 536
4
votes
1
answer
353
views
When is rank-1 perturbation to a positive operator still positive?
Let $A : \mathcal{H} \to \mathcal{H}$ and $B : \mathcal{H} \to \mathcal{H}$ be trace-class (hence compact) Hermitian operators on a separable Hilbert space. Assume that $A$ is strictly positive and ...
- 695
4
votes
1
answer
213
views
Mapping properties of backward and forward heat equation
In a previous question on mathoverflow, I asked about the following:
Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions.
The ...
- 536
4
votes
1
answer
161
views
Elliptic estimates for self-adjoint operators
Let $A$ be a symmetric matrix in $\mathbb R^n$ such that $A$ is positive definite and hence satisfies $0< \lambda \le A \le \Lambda < \infty.$
Let $T$ be a densely defined and closed operator ...
- 192
4
votes
1
answer
275
views
Asymptotic behavior of Schrödinger operators
I am currently dealing with $1$ or at most $2$-dimensional Schrödinger operators on compact domains. A classical result of spectral theory is the Weyl approximations for this operator
$H = -\Delta +V$....
user54300
4
votes
1
answer
233
views
A.C. spectrum of the non additive perturbation BAB of a self-adjoint operator A where B is strictly positive
If have the following problem:
Let $A : \mathcal{H} \to \mathcal{H}$ be a bounded, self-adjoint operator on some Hilbert space $\mathcal{H}$. Let $B: \mathcal{H} \to \mathcal{H}$ be a bounded, ...
- 41
4
votes
0
answers
2k
views
Eigenvalues and spectrum of the adjoint
In a finite-dimensional Hilbert space, the eigenvalues of the adjoint $A^*$ of an operator $A$ are the complex conjugates of the eigenvalues of $A$.
But in infinite dimensions this need no longer be ...
- 3,873
4
votes
0
answers
361
views
Spectral mapping theorem
Rudin's book contains in chapter 10 a spectral mapping theorem for (self-adjoint) unbounded operators that respects the point-spectrum, in the sense that he shows $f(\sigma_p(T))=\sigma_p(f(T))$ for ...
- 305
4
votes
0
answers
171
views
quasi-nilpotent part of a dual operator
Definitions and notation.
Let $X$ be a complex Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator on $X$. We define the quasi-nilpotent part of $T$ as
\begin{equation*}H_0(T):=\left\{...
- 1,591
4
votes
0
answers
229
views
The representation-theoretic nature of an operator resolvent
Consider parameter $s$ in definition of $R(s,A)=(s I - A)^{-1}$ where $A$ is a linear operator in a vector space $X$. When $X$ is over $\mathbb{C}$, then $s$ is thought to be a complex number.
Now ...
- 580
4
votes
0
answers
454
views
Adjoint of sum of two operators. Kato-Rellich
Let $A$ be self-adjoint and $B$ be symmetric with $A$-bound less than $1$. By Kato-Rellich, I know that $(A+B)^*=A+B$. Could I also get something like $(A+iB)^*=A-iB$ or is there a counterexample to ...
- 41
3
votes
2
answers
576
views
A version of the spectral theorem for group actions
Suppose $G$ is a sufficiently nice (maybe locally compact and abelian) group which acts on the separable Hilbert space $\mathcal{H}$ by unitary transformations. Is there a generalization of the ...
- 437