Spectral theory is a study of generalized notions of eigenvalues and eigenvectors.
3
votes
1answer
89 views
Do the maximum and minimum values of a Laplacian eigenfunction have the same magnitude?
Let $\Delta$ be the scalar Laplace-Beltrami operator on a compact, connected, orientable 2-manifold without boundary smoothly embedded in $\mathbb{R}^3$ and let $\phi$ be one of its eigenfunctions, ...
3
votes
0answers
111 views
Construct a multiplication operator which has dense point spectrum
By a multiplication operator here we mean an operator
$$Af(t)=m(t)f(t), \qquad f \in D(A)=\{x \in L^2(\mathbb{R} \mid m(t)f(t) \in L^2(\mathbb{R})\}$$
where $m$ is a Borel measurable function on ...
3
votes
0answers
395 views
Continuous spectrum can shrink to an isolated point
Let $A$ be a bounded linear operator in a Hilbert space $H$.
I had the misconception that the continuous spectrum of $A$ would necessarily have some "continuous" appearance: an interval, a union of ...
1
vote
0answers
93 views
Does a projection valued measure (PVM) induce a PVM on a generic subspace of the Hilbert space?
Let $E:{\cal B}(X) \to Pr({\cal H})$ be a projection valued measure (PVM), where ${\cal B}(X)$ is the Borel $\sigma$-algebra of a suitable topological space $X$ and $Pr({\cal H})$ is the set of ...
2
votes
1answer
181 views
How to characterize self-adjoint operators in terms of orthogonal diagonalizability
Have a look at the following excerpt of Tosio Kato (taken from Zeidler Applied functional analysis vol. I):
The fundamental quality required of operators representing physical quantities in ...
4
votes
2answers
140 views
Changing the manifold, preserving the discrete spectrum
On a Riemannian manifold $M$, the Laplace operator $L$ is uniquely defined.
If $M$ is not compact, then $L$ admits a continuous spectrum.
Is there a way of "changing" $M$ and/or $L$ in say $M'$ ...
4
votes
0answers
133 views
When functions, analytically continued, carry over certain properties
Let $ \Omega $ be a sufficiently smooth planar region in $ \mathbb{R}^2 $ with spectrum $ \Gamma $ (the set of eigenvalues of the Laplace operator on functions which vanish on the boundary $ \partial ...
7
votes
1answer
218 views
Does there exist a self-adjoint operator whose spectrum consists wholly of prime numbers?
The zeros of the canonical Riemann zeta function have been compared to the prime numbers, and they have a number of special, definite connections. The infamous zeros have also been conjectured to be ...
5
votes
1answer
163 views
Convergence of spectra under strong convergence of operators
Say $\left\{A_n\right\}$ is a sequence of bounded self-adjoint operators on a separable Hilbert space, converging in strong operator topology to a (bounded, self-adjoint) operator $A$. Denote the ...
0
votes
1answer
537 views
spectrum of right shift operator on $\ell^2(\mathbb{Z})$
Consider the right shift operator on $\ell^2(\mathbb{Z})$. Is there a way of calculating (well, showing what it is since I already know it's $z$ s.t $|z| = 1$) its spectrum without reference to it ...
1
vote
1answer
296 views
Spectrum of sum of operators on Banach spaces
Let $A$ and $B$ be two operators on a Banach space $X$. I am interested in the relationship between the spectra of $A$, $B$ and $A+B$. In particular, are there any set theoretic inclusions or ...
1
vote
2answers
292 views
How do the solutions to the wave and heat equations converge in general?
I would like to check my understanding with someone if possible.
When we cover the heat and wave equations, for instance, in "methods" courses at university, they normally restrict the initial ...
40
votes
1answer
1k views
Example of a compact set that isn't the spectrum of an operator
This question is a follow-up to this recent question and related to that one.
Is there an easy example of an (infinite-dimensional) Banach space $X$ and a non-empty compact set $K \subset ...
14
votes
3answers
620 views
Can spectrum “specify” an operator?
Given a bounded operator $A$ on a Banach space $X$, one may find the spectrum $\sigma(A)\subset{\bf C}$.
Here are my questions:
Given some set in the complex plane, say, $S\subset{\bf C}$, ...
0
votes
0answers
121 views
Uniqueness of solution to a differential equation
Consider the second-order ODE
$-(py')'+(q-\lambda$$w)y=wf$ $(1)$
with $y$ an $L^2$ complex-valued function on $[a,b]$ subject to the boundary conditions:
...
2
votes
1answer
150 views
Schrödinger operator: where is the generator to be defined?
The theory as I know it
Let $\mathcal{H}$ be a Hilbert space and $(A, D(A))$ a self-adjoint operator acting on it. The Spectral Theorem (cfr. Reed & Simon Methods of modern mathematical physics, ...
6
votes
3answers
756 views
Spectral decomposition of a normal matrix
I'd like to find the spectral decomposition of $A$:
$$A = \begin{pmatrix}
2-i & -1 & 0\\
-1 & 1-i & 1\\
0 & 1 & 2-i
\end{pmatrix}$$
i.e. $A=\sum_{i}\lambda_i P_i$ where ...
1
vote
3answers
204 views
references for the spectral theorem
Recently, I am thinking about the question in spectral theory. And I finally found that I need help with the properties of unitary operator. Its a consequence of the spectral theorem for the normal ...
5
votes
1answer
435 views
spectrum of the “discrete Laplacian operator”
In numerical analysis, the discrete Laplacian operator $\triangle$ on $\ell^2({\bf Z})$ can be written in terms of the shift operator
$\triangle=S+S^*-2I$
where $S$ is the right shift operator. ...
2
votes
2answers
1k views
spectrum of right shift operator
Here is the question:
Considering the right shift operator $S$ on $l^2({\bf Z})$, what can one know about ran$(S-\lambda)$?
Here is what I thought:
If one wants to prove that the operator ...
4
votes
2answers
395 views
Matrix Differential Equation with a Skew-Symmetric Matrix
From a bank of masters exams:
Say the position of a particle moving
in $\mathbb{R}^n$ is given by a smooth
vector-valued function $\vec{x}(t)$.
Suppose that $\vec{x}(t)$ satisfies a
...
12
votes
1answer
621 views
Spectrum of a linear operator
Let $\ell^2 =\ell^2(\mathbb{Z})$. Choose $\theta \in ]0,1[$ and set:
$$Tx=(\theta x_{n-1} +(1-\theta)x_{n+1})_{n\in \mathbb{z}}$$
for each $x=(x_n)_{n\in \mathbb{Z}}\in \ell^2$ (thus $T$ is a convex ...
1
vote
2answers
315 views
Basic Spectral Theory Problem: Finding the Point/Continuous Spectrum of an Operator
I have the following problem:
Determine the point spectrum and the continuous spectrum of the operator $$(A\psi )(x)=\theta (x)(\cos x)\psi (x)$$ on $L_2(\mathbb R,dx)$, where $\theta(x)=0$ for ...
5
votes
1answer
874 views
What's the connection between the Laplace transform and the Fourier transform?
Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a ...
