Landau automorphic functions on Cn of magnitude ν

Abstract

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 (∗).

Construction of concrete orthonormal basis for (L2, Γ, χ)-theta functions associated to discrete subgroups of rank one in $({\mathbb {C}},+)$(C,+)

“…In this case, a systematic study of F 2,ν L ,χ (C) is provided in Ref. 7. It is proved there that its description is closely related to the L 2 -spectral theory of the Landau Laplacian…”

Spectral analysis on planar mixed automorphic forms

“…The following result is an immediate consequence of (4.3) above and the dimensional formula established in [13]. Namely, we have …”

“…the usual Lebesgue measure dλ, is well known (see for example [6,17,4,7,12]). A systematic study of L ν acting on the space of Landau automorphic functions of magnitude ν is presented in [13].…”

On Concrete Spectral Properties of a Twisted Laplacian Associated with a Central Extension of the Real Heisenberg Group

“…Thus, a detailed description of the spectral properties of Δ ], when acting on biweighted automorphic functions with respect to any discrete subgroup of (C , +) (not necessary of full-rank) is of great interest. In this context, the particular case ] = 0 and Γ = C ⋊ {1}, these functions reduce further the classical one studied in [11]. We hope to focus on this in a near future.…”

“…The same observation holds when dealing with the so-called holomorphic automorphic functions. In advantage, such Landau Hamiltonian leaves invariant such space [11]. Accordingly, the first main object was the introduction of a special magnetic Schrödinger operator satisfying the following conditions:…”