Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Robin Spectral Rigidity of the Ellipse   In this paper, we investigate $C^1$ isospectral deformations of the ellipse
with Robin boundary conditions, allowing both the Robin function and domain to
deform simultaneously. We prove that if the deformations preserve the
reflectional symmetries of the ellipse, then the first variation of both the
domain and Robin function must vanish. Reparameterizing allows us to show that
such smooth deformations must be flat at $\epsilon = 0$. In particular, there
exist no such analytic isospectral deformations. The key ingredients are a
version of Hadamard's variational formula for variable Robin boundary
conditions and an oscillatory integral representation of the wave trace
variation which uses action angle coordinates for the billiard map. For the
latter, we in fact construct an explicit parametrix for the wave propagator in
the interior, microlocally near orbits of rotation number $1/j$.

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The categories of this paper are: stat
9
The categories of this paper are: nlin
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The categories of this paper are: bayes-an
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The categories of this paper are: eess
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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: dg-ga
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The categories of this paper are: physics
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The categories of this paper are: cmp-lg
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-alg
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The categories of this paper are: q-fin
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The categories of this paper are: alg-geom