Polyanalytic Reproducing Kernels on the Quantized Annulus

TL;DR

This paper characterizes eigenspaces of the magnetic Laplacian on the annulus as polyanalytic spaces, providing explicit formulas for their reproducing kernels involving Jacobi theta functions, especially under quantized magnetic field strengths.

Contribution

It completes Peetre's work by explicitly describing the polyanalytic eigenspaces and their kernels, including transformation rules under automorphisms for quantized magnetic fields.

Findings

Reproducing kernels expressed via Jacobi theta functions.

Eigenspaces are true-polyanalytic spaces.

Transformation rules under automorphisms derived for integer magnetic fields.

Abstract

While dealing with the constant-strength magnetic Laplacian on the annulus, we complete J. Peetre's work. In particular, the eigenspaces associated with its discrete spectrum are true-polyanalytic spaces with respect to the invariant Cauchy-Riemann operator, and we write down explicit formulas for their reproducing kernels. The latter are expressed by means of the fourth Jacobi theta function and of its logarithmic derivatives when the magnetic field strength is an integer. Under this quantization condition, we also derive the transformation rule satisfied by the reproducing kernel under the automorphism group of the annulus.