On the Ground Level of Purely Magnetic Algebro-Geometric 2D Pauli Operator (spin 1/2)

TL;DR

This paper investigates the complex Bloch-Floquet eigenfunctions of the ground level of purely magnetic 2D Pauli operators, revealing new algebro-geometric solutions and their connection to nonlinear hierarchies, with implications for magnetic field configurations.

Contribution

It introduces the first known algebro-geometric operators for the 2D Pauli operator with nonzero flux and explores their spectral properties and ground states.

Findings

Found algebro-geometric operators for zero and nonzero flux cases.

Discovered periodic operators with zero flux and Bohm-Aharonov phenomena.

Identified ground state properties for genus zero and higher cases.

Abstract

Full manifold of the complex Bloch-Floquet eigenfunctions is investigated for the ground level of the purely magnetic 2D Pauli operators (equal to zero because of supersymmetry). Deep connection of it with the 2D analog of the "Burgers Nonlinear Hierarchy" plays fundamental role here. Everything is completely calculated for the broad class of Algebro-Geometric operators found in this work for this case. For the case of nonzero flux the ground states were found by Aharonov-Casher (1979) for the rapidly decreasing fields, and by Dubrovin-Novikov (1980) for the periodic fields. No Algebro-Geometric operators where known in the case of nonzero flux. For genus $g=1$ we found periodic operators with zero flux, singular magnetic fields and Bohm-Aharonov phenomenon. Our arguments imply that the delta-term really does not affect seriously the spectrum nearby of the ground state. For $g>1$ our theory requires to use only algebraic curves with selected point leading to the solutions elliptic in the variable $x$ for KdV and KP in order to get periodic magnetic fields. The algebro-geometric case of genus zero leads, in particular, to the slowly decreasing lump-like magnetic fields with especially interesting variety of ground states in the Hilbert Space $\cL_2(\bR^2)$.