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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.

We introduce a class of generalized Bargmann spaces on Cn for which we establish explicit formulas of their reproducing kernels. Some applications of the obtained formulas are given.

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We investigate the spectral theory of the invariant Landau Hamiltonian, Lν=−1/2{4∑j=1n∂2/∂zj∂z¯j+2ν∑j=1n(zj∂/∂zj−z¯j∂/∂z¯j)−ν2|z|2}, acting on the space FΓ,χν of (Γ,χ)-automorphic functions on Cn, constituted of C∞ functions satisfying the functional equation f(z+γ)=χ(γ)eiνIm⟨z,γ⟩f(z);  z∊Cn,γ∊Γ, for given real number ν>0, lattice Γ of Cn and a map χ:Γ→U(1) such that the triplet (ν,Γ,χ) satisfies a Riemann–Dirac quantization-type condition. More precisely, we show that the eigenspace EΓ,χν(λ)={f∊FΓ,χν; Lνf=ν(2λ+n)f}; λ∊C, is nontrivial if and only if λ=l=0,1,2,…,. In such case, EΓ,χν(l) is a finite dimensional vector space whose the dimension is given explicitly by dim EΓ,χν(l)=(n+l−1l)(ν/π)nvol(Cn/Γ). Furthermore, we show that the eigenspace EΓ,χν(0) associated with the lowest Landau level of Lν is isomorphic to the space, OΓ,χν(Cn), of holomorphic functions on Cn satisfying g(z+γ)=χ(γ)eν/2|γ|2+ν⟨z,γ⟩g(z), (∗) that we can realize also as the null space of the differential operator, ∑j=1n(−∂2/∂zj∂z¯j+νz¯j∂/∂z¯j) acting on C∞ functions on Cn satisfying (∗).