Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Localization properties of high energy eigenfunctions on flat tori   We consider the question of when the Laplace eigenfunctions on an arbitrary
flat torus $\mathbf{T}_\Gamma:=\mathbf{R}^d/\Gamma$ are flexible enough to
approximate, over the natural length scale of order $1/\sqrt\lambda$, where
$\lambda\gg1$ is the eigenvalue, an arbitary solution of the Helmholtz equation
$\Delta h + h=0$ on $\mathbf{R}^d$. This problem is motivated by the fact that,
by the asymptotics for the local Weyl law, "approximate Laplace eigenfunctions"
do have this approximation property on any compact Riemannian manifold. What we
find is that the answer depends solely on the arithmetic properties of the
spectrum. Specifically, recall that the eigenvalues of $\mathbf{T}_\Gamma$ are
of the form $\lambda_k=Q_\Gamma(k)$, where $Q_\Gamma$ is a quadratic form and
$k \in \mathbf{Z}^d$. Our main result is that the eigenfunctions of
$\mathbf{T}_\Gamma$ have the desired approximation property if and only
$Q_\Gamma$ is a multiple of a quadratic form with integer coefficients. In
particular, the set of lattices $\Gamma$ for which this approximation property
holds has measure zero but includes all rational lattices. A consequence of
this fact is that when $Q_\Gamma$ is a multiple of a quadratic form with
integer coefficients, Laplace eigenfunctions exhibit an extremely flexible
behavior over scales of order $1/\sqrt\lambda$. In particular, there are
eigenfunctions of arbitrarily high energy that exhibit nodal components
diffeomorphic to any compact hypersurface of diameter $O(1/\sqrt\lambda)$.

29
The categories of this paper are: funct-an
10
The categories of this paper are: q-bio
36
The categories of this paper are: cmp-lg
31
The categories of this paper are: ao-sci
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
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The categories of this paper are: nucl-th
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
34
The categories of this paper are: chem-ph
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The categories of this paper are: q-alg
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The categories of this paper are: bayes-an
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The categories of this paper are: atom-ph
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The categories of this paper are: eess