Polyanalytic reproducing Kernels on the quantized annulus

Abstract

While dealing with the constant-strength magnetic Laplacian on the annulus, we complete Peetre’s work. In particular, the eigenspaces associated with its discrete spectrum true turns out to be polyanalytic spaces with respect to the invariant Cauchy–Riemann operator, and we write down explicit formulas for their reproducing kernels. When the magnetic field strength is an integer, we compute the limits of the obtained kernels when the outer radius of the annulus tends to infinity and express them by means of the fourth Jacobi theta function and of its logarithmic derivatives. Under the same quantization condition, we also derive their transformation rule under the action of the automorphism group of the annulus.