IUMJ

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

Authors: Benedetta Noris and Susanna Terracini

Issue: Volume 59 (2010), Issue 4, 1361-1404

Abstract:

In this paper we consider a stationary Schr\"odinger 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 [B. Helffer, T. Hoffmann-Ostenhof, S. Terracini, \emph{Nodal domains and spectral minimal partitions}, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire \textbf{26} (2009), 101-138}. As it concerns eigenfunctions, we similarly show that critical points of the Rayleigh quotient correspond to multiple clustering of the nodal lines.