Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Critical partitions and nodal deficiency of billiard eigenfunctions   The paper addresses the the number of nodal domains for eigenfunctions of
Schr\"{o}dinger operators with Dirichlet boundary conditions in bounded
domains. In dimension one, the $n$th eigenfunction has $n$ nodal domains. The
Courant Theorem claims that in any dimension, the number of nodal domains of
the $n$th eigenfunction cannot exceed $n$. However, in dimensions higher than 1
the equality can hold for only finitely many eigenfunctions. Thus, a "nodal
deficiency" arises. Examples are known of eigenfunctions with arbitrarily large
index $n$ that have just two nodal domains.
  It was suggested in the recent years to look at the partitions of the domain,
rather than eigenfunctions. It was shown in a recent paper by Helffer,
Hoffmann-Ostenhof and Terracini that (under some natural conditions) bipartite
partitions minimizing the maximum of the ground-state energies in sub-domains
of the partition, correspond to the "Courant sharp" eigenfunctions, i.e. to
those with zero nodal deficiency.
  In this paper, the authors show, under some genericity conditions, among the
bipartite equipartitions, the nodal ones correspond exactly to the critical
points of an analogous functional, with the nodal deficiency being equal to the
Morse index at this point. This explains, in particular, why all the minimal
partitions must be Courant sharp.

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The categories of this paper are: q-alg
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The categories of this paper are: comp-gas
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
22
The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: q-fin
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The categories of this paper are: q-bio
7
The categories of this paper are: hep-th
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The categories of this paper are: eess
0
The categories of this paper are: hep-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: econ
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The categories of this paper are: solv-int