  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.
