All Questions
9 questions
15
votes
2
answers
785
views
Is there a spectral theory approach to non-explicit Plancherel-type theorems?
Teaching graduate analysis has inspired me to think about the completeness theorem for Fourier series and the more difficult Plancherel theorem for the Fourier transform on $\mathbb{R}$. There are ...
- 56.6k
12
votes
3
answers
2k
views
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:
$$\frac{\mathrm{d}^2x}...
- 1,433
6
votes
0
answers
113
views
Schwartz kernel of spectral projection of Laplacian and integrated density of states
I'm reposting here a question I asked on MSE which did not receive an answer.
I am considering the Dirichlet Laplacian $\Delta$ on some smooth domain $U$. For now assume that $U$ is bounded, and later ...
- 251
4
votes
1
answer
221
views
Non-isolated ground state of a Schrödinger operator
Question. Does there exist a dimension $d \in \mathbb{N}$ and a measurable function $V: \mathbb{R}^d \to [0,\infty)$ such that the smallest spectral value $\lambda$ of the Schrödinger operator $-\...
- 12.6k
3
votes
2
answers
642
views
Localization of Laplacian eigenfunction on the unit square?
Let A be the unit square, $\{u_k\}$ is the set of all L2-normalized Laplacian eigenfunctions with Dirichlet boundary condition. Is it true that for any open subset V, $C_V = \inf\limits_k \int\...
3
votes
0
answers
203
views
On the spectrum of Fokker–Planck with linear drift
The paper by Liberzon and Brockett, "Spectral analysis of Fokker–Planck and related operators arising from linear stochastic differential equations." SIAM Journal on Control and ...
- 116
3
votes
0
answers
102
views
Determining what happens to the spectrum of Schrödinger operator as boundary condition changes
I recently came across a problem in research, and I'm asking about it here after trying in math stack exchange with no luck.
Suppose I have a metric graph $G$ (or even a closed interval, to make ...
- 251
2
votes
1
answer
301
views
Weyl quantization and convexity
Let $C$ be a convex subset of $\mathbb R^{2n}$ and $\mathbf 1_C$ be the characteristic function of $C$. Is it true that
$$\forall u\in\mathscr S(\mathbb R^n),\quad
\langle\mathbf 1_C^{Weyl}u,u\rangle\...
- 16.2k
1
vote
0
answers
61
views
Numerical computation of spectrum for operators on real line with "confining potential"
I am looking to understand the conditions under which one can expect "reasonably" accurate solution to leading eigenvalues/eigenvectors of a second order differential operator posed on the real line.
...
- 318