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14 votes
1 answer
1k views

Spectrum of Laplacian in non-compact manifolds

What can be said about the spectrum of the Laplace-Beltrami operator on a non-compact, complete Riemannian manifold of finite volume? For example, is the point spectrum non-empty? What would be a ...
alvarezpaiva's user avatar
  • 13.5k
13 votes
1 answer
481 views

A question on a result of Colin de Verdière

Consider a compact connected surface $M$ of some genus $\gamma \geq 2$. A particular case of a famous result of Colin de Verdière (see Construction de laplaciens dont une partie finie du spectre est ...
SMS's user avatar
  • 1,407
12 votes
1 answer
1k views

Multiplicity of Laplace eigenvalues

Disclaimer: This is a very heuristic question and I will be satisfied with heuristic insights, if rigorous and precise answers are not possible. All the examples of closed surfaces (or higher ...
user82132's user avatar
  • 121
10 votes
0 answers
284 views

Comparing spectra of Laplacian and Schrödinger operator

Let $M$ be a closed (compact without boundary) Riemannian manifold. Is there a body of results that compares the eigenvalues of the Laplace-Beltrami operator with that of Schrödinger operators $-\...
noname's user avatar
  • 109
9 votes
1 answer
313 views

The scope of correspondence principle in quantum chaos

My understanding of the so-called correspondence principle in quantum chaos, is that it is a connection between the behaviour of a classical Hamiltonian system (chaotic/completely integrable) and the ...
Pig's user avatar
  • 809
8 votes
1 answer
421 views

$C^k$ one-parameter family of metrics

Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...
SMS's user avatar
  • 1,407
6 votes
0 answers
393 views

Steklov eigenvalue problem for a planar region bounded by ellipse

The Steklov problem for a compact planar region $\Omega$ is \begin{cases} \Delta u =0 &\text{in $\Omega$}, \\ \frac{\partial u}{\partial n} = \sigma u &\text{on $\partial \Omega$}, \end{...
Donghwi Seo's user avatar
5 votes
1 answer
345 views

Convergence of Riemannian metrics spectra

Consider a one-parameter real analytic family of metrics $g_t$ on a compact manifold $M$ converging to a metric $g$ in $C^k$-norm, for some $k$. It is known that the Laplace spectrum of $g_t$ will ...
student's user avatar
  • 51
3 votes
3 answers
243 views

Compact surfaces with arbitrary gaps in spectrum

Consider a sequence of positive numbers $a_n$. My question is, can we select a closed Riemann surface whose spectrum $\lambda_i$ satisfies the condition that $\lambda_{i + 1} - \lambda_i > a_i$? Of ...
anonymos's user avatar
3 votes
0 answers
112 views

Is the square root of curl^2-1/2 a natural (Dirac-)operator?

In current computations on a particular $3$-dimensional Riemannian manifold, a first order differential operator $D:\Gamma^\infty(TM,M)\to \Gamma^\infty(TM,M)$ acting on vector fiels shows up, with ...
B K's user avatar
  • 1,942
2 votes
1 answer
95 views

literature/reference request for estimates of first eigenvalue of certain Schrodinger operator on compact surfaces

On compact Riemannian surfaces (say without boundary), the Schrodinger operator I am interested in is of the form $-\Delta+2\kappa$, where $\kappa$ is the Gauss curvature. For minimal surfaces in $\...
Piojo's user avatar
  • 783
2 votes
1 answer
144 views

Spectral geometry: asymptotic sequences of subspaces of $L^2(M)$ and the geometry of $M$

Consider a closed connected Riemannian manifold $M$, together with the associated Hilbert space $L^2(M)$ defined with respect to the Riemannian volume density. Let $-\Delta$ be the positive Laplacian $...
B K's user avatar
  • 1,942