Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On spectral minimal partitions II, the case of the rectangle   In continuation of \cite{HHOT}, we discuss the question of spectral minimal
3-partitions for the rectangle $]-\frac a2,\frac a2[\times ] -\frac b2,\frac
b2[ $, with $0< a\leq b$. It has been observed in \cite{HHOT} that when
$0<\frac ab < \sqrt{\frac 38}$ the minimal 3-partition is obtained by the three
nodal domains of the third eigenfunction corresponding to the three rectangles
$]-\frac a2,\frac a2[\times ] -\frac b2,-\frac b6[$, $]-\frac a2,\frac
a2[\times ] -\frac b6,\frac b6[$ and $]-\frac a2,\frac a2[\times ] \frac b6,
\frac b2[$. We will describe a possible mechanism of transition for increasing
$\frac ab$ between these nodal minimal 3-partitions and non nodal minimal
3-partitions at the value $ \sqrt{\frac 38}$ and discuss the existence of
symmetric candidates for giving minimal 3-partitions when $ \sqrt{\frac
38}<\frac ab \leq 1$. Numerical analysis leads very naturally to nice questions
of isospectrality which are solved by introducing Aharonov-Bohm Hamiltonians or
by going on the double covering of the punctured rectangle.

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The categories of this paper are: supr-con
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The categories of this paper are: stat
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-fin
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The categories of this paper are: funct-an
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The categories of this paper are: solv-int
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The categories of this paper are: eess
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The categories of this paper are: econ
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The categories of this paper are: acc-phys
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The categories of this paper are: adap-org
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The categories of this paper are: atom-ph
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The categories of this paper are: q-alg
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The categories of this paper are: math-ph
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The categories of this paper are: hep-th
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The categories of this paper are: ao-sci
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The categories of this paper are: mtrl-th