1
vote
0answers
62 views

Recovering Spherical Harmonics from Discrete Samples

Consider a collection of $N$ points on the 2-sphere chosen uniformly at random. Let's say that there's an edge between two such vertices if their geodesic distance is less than $r_N$. The resulting ...
2
votes
0answers
89 views

Laplacian on manifolds with corners

So far I've studied smooth riemannian manifolds and the Laplace-Beltrami operator associated to it. Therefore I know basic theorems like Stoke's theorem, divergence theorem, green's identities etc. ...
9
votes
1answer
311 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 ...
15
votes
1answer
253 views

Avoiding integers in the spectrum of the Laplacian of a Riemann surface

Let $\Sigma^2$ be an orientable compact surface of genus $gen(\Sigma)\geq2$, and denote by $\mathcal M(\Sigma)$ the moduli space of hyperbolic metrics on $\Sigma$, i.e., Riemannian metrics of constant ...
1
vote
1answer
272 views

Asymptotic of the heat kernel

This is the same question I asked in stackexchange: http://math.stackexchange.com/questions/519152/asymptotic-of-the-heat-kernel I read Proposition 3.23(page 101) in Rosenberg's book "The Laplacian ...
2
votes
2answers
91 views

Eigenfunctions of Schrödinger Operators on the boundary

Hello, let's consider a compact and connected Riemannian manifold with the Schrödinger Operator $L=-\Delta +V:dom(H)\subset L^2(M)\rightarrow L^2(M)$ whereas $dom(L):=\lbrace f\in ...
3
votes
1answer
243 views

The first eigenvalue of the Schrödinger operator is simple.

Hello, let $(M,g)$ be a compact and connected Riemannian manifold (possibly with $\partial M\neq \emptyset$). We consider the Friedrichs extension of $L=-\Delta +V: C^{\infty}(M,\mathbb{R})\subset ...
1
vote
1answer
109 views

regularity of eigenfunctions of Schrödinger Operator

Hello, I consider a compact and connected (smooth) Riemannian manofold $(M,g)$. I'm interested in the eigenfunctions of the Schrödinger Operator $L=-\Delta+ V$ acting on (smooth) functions. Do you ...
2
votes
2answers
251 views

Gap between first two nonzero Laplacian eigenvalues on closed compact surface?

Much has been said about bounds on Laplacian eigenvalues, and the literature can be tough to sort through! I am specifically interested in the case where the domain is a closed compact surface, and am ...
3
votes
1answer
235 views

First eigenvalue of $\Delta$ on Kaehler manifold with $Ricci\ge k$.

Let $M$ be a Kaehler manifold of complex dimension $n$. Let $\Delta$ be the real Laplacian of the underline Riemannian manifold. Let's assume the Ricci curvature of $M$ satisfies $\text {Ric}\ge ...
7
votes
1answer
230 views

Eigenfunctions restricted on closed geodesics

Consider the flat torus $T^2=\frac{\mathbb{R}^2}{l_1\mathbb{Z}\oplus l_2\mathbb{Z}}$. It is easy to see that the eigenvalues of the Laplacian on torus, $-\frac{\partial^2}{\partial ...
1
vote
0answers
130 views

Distance between probability amplitude functions

Suppose we have two probability measures $P_1$ and $P_2$ on some Riemannian manifold $(\Sigma,g)$. There are many potential distance measures between $P_1$ and $P_2$: The Wasserstein distance For ...
11
votes
1answer
935 views

Relationship between Green's function and geodesic distance?

I am interested in showing that a certain Green's function can be used to approximate the distance function on a Riemannian manifold in the following sense. Let $(M,g)$ be a Riemannian manifold and ...
2
votes
2answers
548 views

Exotic spectrum of Laplace operator

Given a closed Riemannian manifold and a generalized Laplace $\Delta$ operator, it is well known that $\Delta$ has discrete spectrum $(\lambda_n)_n$ (arranged in a increasing way, not counting ...
2
votes
1answer
452 views

Global Lichnerowicz Formula Proof (in the Kahler case)?

For a Kahler manifold $M$, let us denote its Dirac operator $\overline{\partial} + \overline{\partial}^\ast$, with respect to a metric $g$, by $D$. Moreover, let us dentoe the Levi-Civita connection ...
9
votes
3answers
1k views

The first eigenvalue of the laplacian for complex projective space

What is the exact value of the first eigenvalue of the laplacian for complex projective space viewed as $SU(n+1)/S(U(1)\times U(n))$?
3
votes
3answers
331 views

Estimates for the diameter of a (nice) surface?

The Question Let $M$ be a compact, connected, orientable surface without boundary of bounded genus smoothly embedded in $\mathbb{R}^3$; define the diameter $d_M$ of $M$ as the maximum minimal ...
5
votes
1answer
761 views

How to construct a scalar differential operator having the same spectrum as a non-scalar differential operator exploiting symmetries?

I am interested in eigenvalue problems for differential operators acting on one forms on closed two-dimensional manifolds and how they relate to eigenvalue problems of associated operators acting on ...
4
votes
1answer
676 views

Growth of Laplacian eigenvalues on a compact domain?

Let $\mathcal{M}$ be a compact Riemannian manifold and let $\Delta$ be the (scalar) Laplace-Beltrami operator on $\\mathcal{M}$. Then $\Delta$ has a discrete spectrum and if we order its distinct ...
12
votes
4answers
1k views

High multiplicity eigenvalue implies symmetry?

It is well known that on any compact Riemannian symmetric space $X$, the eigenvalues of the Laplacian have very high multiplicity (comparable with the Weyl bound), and the resulting actions ...
7
votes
2answers
378 views

Symmetric spaces, Horocycle spaces and intertwining operators

Let $G=KAN$ be an Iwasawa decomposition of a connected semisimple Lie group with finite center. Let us assume for simplicity that the associated symmetric space $G/K$ has rank 1. Harish-Chandras ...
3
votes
1answer
413 views

eigenvalue problem on the geodesic ball of sphere

I have a question about eigenvalue problem on the geodesic ball in $n$-dimensional sphere $\mathbb{S}^n\subset\mathbb{R}^{n+1}$. Consider the eigenvalue problem in the geodesic ball ...
3
votes
1answer
350 views

Estimating laplace-beltrami spectra for a graph surface in $R^3$

Consider a surface $\Gamma$ in $R^3$. The surface $\Gamma$ is a graph, i.e. $\Gamma = (x,y, h(x,y))$, for $x \in R^2$ and some smooth function $h$, where $h$ and all its derivatives are periodic on ...
3
votes
0answers
243 views

Controlling the Second Eigenvalue of a Schrödinger Operator

Consider a bounded domain $\Omega$ (with smooth boundary) in some Riemannian $n$-manifold $M^n$. Let $L$ be the operator $$ L=\Delta+V $$ where $\Delta$ is the Laplace-beltrami operator on $M$ (so is ...
23
votes
3answers
2k views

Trapped rays bouncing between two convex bodies

At some point during my research I was confronted with this problem, but I did not dedicate serious time to it. Anyway it stayed in the back of my mind and I'm still interested in hints for it. ...
7
votes
1answer
496 views

The smallest Laplace-Beltrami eigenvalue on hyperbolic surfaces

For $g\geq 2$, let $M_g$ be the moduli space of genus $g$ hyperbolic surfaces, and let $\lambda_1(S_x): M_g \to \mathbb{R}$ be the smallest eigenvalue of the Laplace-Beltrami operator on the surface ...
4
votes
1answer
315 views

Is there a name for this differential operator and/or its corresponding spectrum?

Let $\mathcal{M}$ be a real, compact, orientable manifold and let $X$ be a vector field on $\mathcal{M}$. Consider the functional $$E(f) = \int_{\mathcal{M}} X_p(f)^2 dV$$ where $X_p(f)$ is the ...
9
votes
3answers
2k views

Number Theory and Geometry/Several Complex Variables

This is a question for all you number theorists out there...based on my skimming of number theory textbooks and survey articles, it seems like most of the applications of geometry and complex ...
5
votes
2answers
1k views

Eigenvalues of Laplacian

What's the most natural way to establish the asymptotics of $\Delta$ on a compact Riemannian manifold $M$ of dimension $N$? The asymptotics should be $$ \#\{v < A^2\} = ...