Entire functions in weighted $L_2$ and zero modes of the Pauli operator with non-signdefinite magnetic field

TL;DR

This paper studies the space of entire functions square integrable with specific weights related to a non-signdefinite magnetic field, exploring conditions for infinite or zero dimensions and implications for zero modes of the Pauli operator.

Contribution

It extends known results for sign-definite magnetic fields to non-signdefinite cases, providing methods to construct entire functions with prescribed asymptotic behavior.

Findings

Both infinite and zero-dimensional spaces can occur for non-signdefinite B

Methods are developed to construct entire functions with specific growth properties

Results relate to the dimension of zero energy subspaces of the Pauli operator

Abstract

For a real non-signdefinite function $B(z)$, $z\in \C$, we investigate the dimension of the space of entire analytical functions square integrable with weight $e^{\pm 2F}$, where the function $F(z)=F(x_1,x_2)$ satisfies the Poisson equation $\D F=B$. The answer is known for the function $B$ with constant sign. We discuss some classes of non-signdefinite positively homogeneous functions $B$, where both infinite and zero dimension may occur. In the former case we present a method of constructing entire functions with prescribed behavior at infinity in different directions. The topic is closely related with the question of the dimension of the zero energy subspace (zero modes) for the Pauli operator.