Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Inverse spectral problems on a closed manifold   In this paper we consider two inverse problems on a closed connected
Riemannian manifold $(M,g)$. The first one is a direct analog of the Gel'fand
inverse boundary spectral problem. To formulate it, assume that $M$ is divided
by a hypersurface $\Sigma$ into two components and we know the eigenvalues
$\lambda_j$ of the Laplace operator on $(M,g)$ and also the Cauchy data, on
$\Sigma$, of the corresponding eigenfunctions $\phi_j$, i.e.
$\phi_j|_{\Sigma},\partial_\nu\phi_j|_{\Sigma}$, where $\nu$ is the normal to
$\Sigma$. We prove that these data determine $(M,g)$ uniquely, i.e. up to an
isometry. In the second problem we are given much less data, namely,
$\lambda_j$ and $\phi_j|_{\Sigma}$ only. However, if $\Sigma$ consists of at
least two components, $\Sigma_1, \Sigma_2$, we are still able to determine
$(M,g)$ assuming some conditions on $M$ and $\Sigma$. These conditions are
formulated in terms of the spectra of the manifolds with boundary obtained by
cutting $M$ along $\Sigma_i$, $i=1,2$, and are of a generic nature. We consider
also some other inverse problems on $M$ related to the above with data which is
easier to obtain from measurements than the spectral data described.

9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
21
The categories of this paper are: adap-org
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
6
The categories of this paper are: astro-ph
34
The categories of this paper are: chem-ph
13
The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
11
The categories of this paper are: hep-lat
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat