Construction of concrete orthonormal basis for (L^2,\Gamma,\chi)-theta functions associated to discrete subgroups of rank one in (C,+)

TL;DR

This paper constructs a complete orthonormal basis for (L^2, Gamma, chi)-theta functions associated with rank one discrete subgroups of (C,+), revealing their structure and spectral decomposition related to the Landau operator.

Contribution

It introduces a new orthonormal basis for these theta functions and demonstrates their spectral decomposition involving Landau operator eigenspaces.

Findings

Established a Hilbertian orthogonal decomposition involving Landau eigenspaces.

Constructed explicit orthonormal bases for entire theta functions.

Expressed the reproducing kernel in terms of generalized theta functions.

Abstract

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].