Asymptotic of complex hyperbolic geometry and L2-spectral analysis of Landau-like Hamiltonians

In this paper we show that the flat Hermitian complex geometry of Cn, n⩾1, is approximated by the complex hyperbolic geometry of the Bergman complex balls Bρn⊂Cn of radius ρ>0. Furthermore, it will be shown that some elements of the L2-spectral analysis, such as the spectrum, the L2-eigenprojector and the resolvent kernels, associated to the so-called Landau-like Hamiltonian HB,ρ on Bρn give rise to their analogous of the Landau-like Hamiltonian HB,∞ on Cn by letting ρ tend to infinity.