Nodal sets of magnetic Schroedinger operators of Aharonov-Bohm type and energy minimizing partitions

TL;DR

This paper studies the nodal sets of magnetic Schrödinger operators with Aharonov-Bohm flux, revealing conditions under which the domain is partitioned into three minimal energy regions and linking critical magnetic configurations to nodal line clustering.

Contribution

It establishes a connection between magnetic energy criticality and three-part minimal domain partitions for Aharonov-Bohm type operators, with uniqueness and continuous dependence results.

Findings

Magnetic energy is critical if and only if the domain is partitioned into three parts by nodal lines.

The three-part partition is minimal, unique, and depends continuously on the data.

Critical points of the Rayleigh quotient correspond to clustering of nodal lines.

Abstract

In this paper we consider a stationary Schroedinger operator in the plane, in presence of a magnetic field of Aharonov-Bohm type with semi-integer circulation. We analyze the nodal regions for a class of solutions such that the nodal set consists of regular arcs, connecting the singular points with the boundary. In case of one magnetic pole, which is free to move in the domain, the nodal lines may cluster dissecting the domain in three parts. Our main result states that the magnetic energy is critical (with respect to the magnetic pole) if and only if such a configuration occurs. Moreover the nodal regions form a minimal 3-partition of the domain (with respect to the real energy associated to the equation), the configuration is unique and depends continuously on the data. The analysis performed is related to the notion of spectral minimal partition introduced in [20]. As it concerns eigenfunctions, we similarly show that critical points of the Rayleigh quotient correspond to multiple clustering of the nodal lines.