0
votes
0answers
36 views

Riemann Sphere: Holomorphic Functional Calculus

Why do we consider the holomorphic functional calculus on the Riemann sphere rather than the complex plane only? Is there a serious problem? Moreover isn't any curve encircling the spectrum ones ...
0
votes
2answers
45 views

Distorted Unitary matrices

Let $U$ be an unitary and $D$ be a diagonal matrix. We know that for all vectors $v$ on the sphere $Uv$ is on the sphere and, $$\langle Uv,Uv\rangle=\langle v,v\rangle.$$ What are the vectors $v$ on ...
1
vote
2answers
63 views

invariant inner product on eigenspace

I have several questions about the following corollary: "Let G/H be a riemannian homogeneous space where G is a compact Lie group. Let $E_{\lambda}=\lbrace f\in C^{\infty}(G/H) : -\Delta f= \lambda ...
1
vote
1answer
114 views

Relation of Hodge Theorem to Eigenfunction Basis of Laplacian

The classical Hodge theorem I know of relates the de Rham cohomology groups isomorphically to the space of harmonic forms and shows that $Id=\pi+\Delta G$, where $\pi$ is the harmonic projection of ...
5
votes
4answers
187 views

Reason for Continuous Spectrum of Laplacian

For the circle $S^1$, it is well-known that the Laplace-Beltrami operator $\Delta=\text{ div grad}$ has a discrete spectrum consisting of the eigenvalues $n^2,n\in \mathbb{Z}$, as can be seen from the ...
1
vote
1answer
61 views

Asymptotic of the heat kernel

I read Proposition 3.23(page 101) in Rosenberg's book "The Laplacian on a Riemannian Manifold" and not quite clear how to get the estimate $(4\pi t)^{n/2}|Q_k * H_k|\leq C \cdot t^{k+1}$ on a compact ...
2
votes
1answer
192 views

How to convert between the hyperboloid model and the Poincare patch for $\mathbb{H}_n$?

This question is motivated by the results in this paper, http://calvino.polito.it/~camporesi/JMP94.pdf In this paper some of its most important results about the asymptotics of symmetric traceless ...
9
votes
2answers
226 views

Are there n-th roots of differential operators?

In analogy to a Dirac operator, it seems to me that formally, the equation $$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$ is solved by $$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$ Is there a ...
2
votes
0answers
179 views

Using Rayleigh Quotient to approximate the first eigenvalue of the Laplace operator on the unit disk

Let $D\subset\mathbb{R}^{2}$ unit disk, the first eigenvalue of the Laplace operator holds: $\lambda_{1}=\inf\left\{ \frac{\int_{D}\left|\triangledown ...
3
votes
1answer
131 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, ...
4
votes
2answers
155 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 $M'$ ...