Polyanalytic Reproducing Kernels on the Quantized Annulus

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.

normalising constant is corrected, the Bergman kernel of the punctured disc is retrieved

References in corpus (3)

• Romanovski polynomials in selected physics problems
• Zeros of the i.i.d. Gaussian Laurent series on an annulus: weighted Szegő kernels and permanental-determinantal point processes
• Josephson critical currents in annular superconductors with Pearl vortices