On spectral minimal partitions II, the case of the rectangle

TL;DR

This paper investigates spectral minimal 3-partitions of rectangles, analyzing the transition from nodal to non-nodal partitions as the aspect ratio varies, and explores related isospectrality questions using advanced quantum Hamiltonian models.

Contribution

It describes the transition mechanism between nodal and non-nodal minimal partitions for rectangles and introduces new approaches involving Aharonov-Bohm Hamiltonians for isospectrality analysis.

Findings

Identification of the critical aspect ratio for partition transition

Description of the transition mechanism between different minimal partitions

Introduction of Aharonov-Bohm Hamiltonians to solve isospectrality questions

Abstract

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.