Construction of concrete orthonormal basis for (L²,Gamma,chi)-theta functions associated to discrete subgroups of rank one in (C,+)

Let \chi be a character on a discrete subgroup \Gamma of rank one of the additive group (C,+). We construct a complete orthonormal basis of the Hilbert space of (L^2,\Gamma,\chi)-theta functions. Furthermore, we show that it possesses a Hilbertian orthogonal decomposition involving the L^2-eigenspaces of the Landau operator \Delta_\nu; \nu>0, associated to the eigenvalues \nu m. For m=0, the associated L^2-eigenspace is the Hilbert subspace of entire (L^2,\Gamma,\chi)-theta functions. Corresponding orthonormal basis are constructed and the corresponding reproducing kernel can be expressed in terms of the generalized theta function of characteristic [\alpha,0].