Construction of concrete orthonormal basis for (L^2,Γ,χ)-theta functions associated to discrete subgroups of rank one in (C,+)
arXiv:1212.1870 · doi:10.1063/1.4811463
Abstract
Let χbe a character on a discrete subgroup Γof rank one of the additive group (C,+). We construct a complete orthonormal basis of the Hilbert space of (L^2,Γ,χ)-theta functions. Furthermore, we show that it possesses a Hilbertian orthogonal decomposition involving the L^2-eigenspaces of the Landau operator Δ_ν; ν>0, associated to the eigenvalues νm. For m=0, the associated L^2-eigenspace is the Hilbert subspace of entire (L^2,Γ,χ)-theta functions. Corresponding orthonormal basis are constructed and the corresponding reproducing kernel can be expressed in terms of the generalized theta function of characteristic [α,0].
arXiv comment17 pages