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1 vote
1 answer
309 views

Eigenvalues of a Schrödinger operator

I'm interested in the existence of eigenfunctions and finding eigenvalues of the following operator $$L(\varphi) = \varphi_{rr} - \frac{1}{r} \varphi_r - [V + \frac{m}{r^2}] \varphi$$ $$\varphi(0) = \...
JMK's user avatar
  • 337
14 votes
1 answer
830 views

Spectrum of matrix involving quantum harmonic oscillator

The quantum harmonic oscillator relies on two classical objects, the so-called creation and annihilation operator $$a ^* = x- \partial_x \text{ and }a = x+\partial_x.$$ Fix two numbers $\alpha,\beta \...
Kung Yao's user avatar
  • 192
14 votes
1 answer
1k views

Computing spectra without solving eigenvalue problems

There is a rather remarkable conjecture formulated in this paper, "Computing spectra without solving eigenvalue problems," https://arxiv.org/pdf/1711.04888.pdf and in this talk by Svitlana Mayboroda ...
Victor Galitski's user avatar
3 votes
1 answer
212 views

Eigenvalue estimates for operator perturbations

I edited the question to a general mathematical question, since I found the answer in Carlo Beenakker's reference and think that my initial question was mathematically misleading. What was behind ...
Sascha's user avatar
  • 536
2 votes
0 answers
149 views

Off-diagonal estimates for Poisson kernels on manifolds

Let $(M,g)$ be a complete Riemannian manifold, $\Delta$ its Laplace-Beltrami operator and $T_t = (e^{t \Delta})_{t \geq 0}$ the associated heat semigroup. We can define the subordinated Poisson ...
Adrián González Pérez's user avatar
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\...
Bazin's user avatar
  • 16.2k
2 votes
3 answers
1k views

Sobolev norms of eigenfunctions

Let D be a domain in R^n, and let f be an eigenfunction of the Laplacian with Dirichlet boundary condition with eigenvalue $\lambda$. Assume that f has L^2 norm 1. I want to know if I can say anything ...
Debraj Chakrabarti's user avatar
0 votes
0 answers
27 views

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 ...
Student's user avatar
  • 537