Thesis

Title:

Concrete spectral analysis of twisted Laplacians on some classical and mixed automorphic functions on C n and applications.

Abstract:

We provide in this thesis some generalizations of classical automorphic functions on C^n with weight ν. We parametrize the automorphic factor in two ways, the first one is by adding a second weight µ into the automorphic factor. These are called mixed automorphic functions of first kind. The second parametrization is given by multiplying the standard factor with its copy transported using an equivariant pair ( ρ,τ ). These functions are called mixed automorphic functions of second kind. On the spaces of mixed automorphic functions of each kind, obeying a suitable growth condition, we realize the appropriate invariant Laplacians. We show that these spaces are isospectral where the spectrum reduces to discrete eigenvalues. By doing so, we provide a spectral decomposition and construct an orthogonal Hilbert basis for these spaces of mixed automorphic functions. In the last part of this thesis, we investigate some questions concerning the construction and applications of automorphic functions. More precisely, we study the kernel of Poincaré theta series operator. Last but not least, we study the reproducing kernel function of the space of holomorphic automorphic functions, which happens to be the Poincaré periodization of Fock reproducing kernel function. Thereby, we derive interesting arithmetical identities that appear in chemistry and physics. This provides a novel application of automorphic functions in mathematics as well as in other scientific fields.

Keywords:

Automorphic functions; Mixed automorphic functions; Spectral theory; Magnetic Laplacians; Poincaré Theta operator; Arithmetical identities; Lattice Sums.

Links:

Déscriptif de la thèse (in french). 

Thesis (pdf).