Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fourier coefficients of restrictions of eigenfunctions   Let $\{e_j\}$ be an orthonormal basis of Laplace eigenfunctions of a compact
Riemannian manifold $(M,g)$. Let $H \subset M$ be a submanifold and let
$\{\psi_k\}$ be an orthonormal basis of Laplace eigenfunctions of $H$ with the
induced metric. We obtain joint asymptotics for the Fourier coefficients \[
  \langle \gamma_H e_j, \psi_k \rangle_{L^2(H)} = \int_H e_j \overline \psi_k
\, dV_H, \] of restrictions $\gamma_H e_j$ of $e_j$ to $H$. In particular, we
obtain asymptotics for the sums of the norm-squares of the Fourier coefficients
over the joint spectrum $\{(\mu_k, \lambda_j)\}_{j,k - 0}^{\infty}$ of the
(square roots of the) Laplacian $\Delta_M$ on $M$ and the Laplacian $\Delta_H$
on $H$ in a family of suitably `thick' regions in $\mathbb R^2$. Thick regions
include (1) the truncated cone $\mu_k/\lambda_j \in [a,b] \subset (0,1)$ and
$\lambda_j \leq \lambda$, and (2) the slowly thickening strip $|\mu_k -
c\lambda_j| \leq w(\lambda)$ and $\lambda_j \leq \lambda$, where $w(\lambda)$
is monotonic and $1 \ll w(\lambda) \lesssim \lambda^{1 - 1/n}$. Key tools for
obtaining these asymptotics include the composition calculus of Fourier
integral operators and a new multidimensional Tauberian theorem.

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The categories of this paper are: stat
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The categories of this paper are: math-ph
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The categories of this paper are: solv-int
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: patt-sol
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The categories of this paper are: nlin
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The categories of this paper are: hep-th
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The categories of this paper are: acc-phys
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The categories of this paper are: supr-con
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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th