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2 votes
0 answers
67 views

Regularity and decay of Fourier-like series on a manifold

Let $D$ be a first-order self-adjoint elliptic differential operator acting on sections of a vector bundle $S$ over a closed manifold $M$. Then it is well-known that the various eigenspaces $E_\lambda$...
geometricK's user avatar
  • 1,903
3 votes
1 answer
214 views

Convergence of spectrum

Let $M$ be a compact manifold and $g_k$ be a sequence of Riemannain metrics smoothly converging to another Riemannian metric $g$. Let $\{\lambda^k_j\}$ be the spectrum of the Laplacian of the ...
Hammerhead's user avatar
  • 1,211
4 votes
0 answers
199 views

Spectral problems with the wrong sign on the Poincaré disk

Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$ equipped with the Poincaré metric $g$ of negative scalar curvature $-1$. Denote by $\Delta_g = \mathrm{Tr}_g(\nabla^g d) = - d^{\ast_g} d$ ...
Bilateral's user avatar
  • 2,816
5 votes
1 answer
224 views

Spectral theory of infinite volume hyperbolic manifolds

I have a question about the discrete spectrum of the Laplace operator on hyperbolic manifolds with infinite volume. I understand the case of infinite area surfaces: see Chapter 7 (Sections 1 and 2) of ...
SMS's user avatar
  • 1,407
14 votes
1 answer
668 views

Why are we interested in spectral gaps for Laplacian operators

Let $M$ be a Riemannian manifold and let $\Delta$ be its Laplacian operator. There is a large literature on a spectral gap for such a $\Delta$, that is, finding an interval $(0,c)$ which does not ...
Sven Mortenson's user avatar
1 vote
0 answers
100 views

Question about Dirac operator

Let $D$ be a generalized Dirac operator on a complete Riemannian manifold. I'm a little confused to prove that there exists a constant $c>0$ such that $$\|D\sigma\|^2\geq c^2\|\sigma\|^2$$ for $\...
Radeha Longa's user avatar
2 votes
0 answers
53 views

A question about the choice of a special harmonc spinor

Let $X$ be a complete Riemannian manifold and $H$ be the kernel of generalized Dirac operator $D$ on $L(S)$, where $S$ is the Dirac bundle. Let $K$ be a compact subset of $X$ and $K\subset \Omega$ be ...
Radeha Longa's user avatar
3 votes
1 answer
190 views

Laplace eigenfunction on a polygonal domain symmetric about an axis

Consider a polygon $\Omega \subseteq \mathbb{R}^2$, and let us consider the usual Laplacian operator $\Delta = \partial_x^2 + \partial_y^2$, with Dirichlet boundary conditions. My question comes from ...
user170399's user avatar
1 vote
0 answers
184 views

One question about Schrodinger Semigroups-(B. Simon)

This question comes from the paper: B. Simon, Schrodinger Semigroups, Bull. A.M.S., (1982) Vol. 7 (3). On the Theorem C.3.4(subsolution estimate) of the paper, it says that: Let $Hu=Eu$ and $u\in L^...
DLIN's user avatar
  • 1,915
17 votes
3 answers
770 views

Does a spectral gap lift to covering spaces?

Let $M$ be a complete Riemannian manifold. Denote $\Delta_M\ge0$ the unique self-adjoint extension of the Laplace-Beltrami operator in $L^2(M)$ and $\sigma(\Delta_M)\subset [0,\infty)$ its spectrum. ...
Jan Bohr's user avatar
  • 779
7 votes
1 answer
281 views

Harmonic functions on $(M,g)$ closed, induce an embedding in Euclidean space

In Hajime Urakawa's monograph The Spectral Geometry of the Laplacian on page 41, we make an assumption that I can't quite justify on my own. The following is our setup: Let $(M^n,g)$ be a closed ...
Dominic Wynter's user avatar
1 vote
1 answer
242 views

Is the trace of the heat kernel always finite?

consider any smooth Riemannian manifold $(N,g)$, an open subset $U\subset N$ and the Dirichlet heat kernel $p(t;x,y)$ for $U$. I am wondering, if it is true that $\int_U p(t;x,x)dx <\infty$ for any ...
Denilson Orr's user avatar
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
7 votes
2 answers
920 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 ...
Alex's user avatar
  • 101
7 votes
1 answer
1k 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 ...