DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 621–629

GEOMETRIC QUANTIZATION OF THE LANDAU PROBLEM ON HYPERBOLIC RIEMANN SURFACES

ANTONIO LÓPEZ ALMOROX, CARLOS TEJERO PRIETO

Abstract. The techniques of symplectic reduction are applied to the study of the Landau problem on the Poincaré plane and compact Riemann surfaces of genus g > 1. This allows us to study the manifold of orbits of constant energy. Once this is done we proceed to study the geometric quantization of this problem showing its relationship, in the compact case, with Maass automorphic forms and operators.

1. Introduction

The quantum Hall effect deals with electrons moving in a two dimensional conductor sample under the influence of a magnetic field orthogonal to its surface [1, 2]. One uses the Hamiltonian for a single electron moving in this magnetic field, with a potential which represents the action of the periodic ionic lattice and usually one assumes this lattice to be an euclidean one (torus) [3, 4]. However, the first stage in the treatment of the Hall effect requires the study of the so called Landau problem; that is, one neglects the above mentioned potential term [5].

A generalization of this set-up involving a hyperbolic lattice ( a lattice generated under the action of a Fuchsian group on the hyperbolic plane or a Riemann surface of genus g > 1) has been considered recently and some quantum aspects of this problem have been analyzed in [6, 7, 8, 9, 10].

In this paper we study the symplectic aspects and the geometric quantization of the hyperbolic Landau problem. These techniques allow us to prove that the prequantization line bundles on the Riemann surfaces are precisely the line bundles of automorphic forms of a given weight which depends on the genus of the Riemann surface and the magnetic field. In particular, using the Blattner-Kostant-Sternberg quantization method we stablish the relationship between the quantum Hamiltonian and the Maass operator which is used in the Physics literature.

1991 Mathematics Subject Classification. 58F05, 58F06, 81S10, 11F03,11F37.

Key words and phrases. Geometric quantization, Landau problem, Negative curvature, Automorphic forms.

Each author was partially supported by a D.G.E.S research project PB-96-1305 and by a Junta de Castilla y León research project SA-27/98.

This paper is in final form and no version of it will be submitted for publication elsewhere.

2. Symplectic aspects of the hyperbolic Landau problem

The Landau problem is the study of the motion of a charged particle on an oriented Riemannian surface (X, T_{2}) under the action of a covariantly constant magnetic field F = BΩ_{2} where B ∈ ℝ and Ω_{2} is the area element associated with the metric tensor T_{2} (see [11]).

As is well known [12], a symplectic formulation of this problem can be given on the tangent bundle T X ⟶^{τ} X with the symplectic 2-form ω_{2} = mdθ + eτ^{∗}F where θ is the pullback to T X of the canonical 1-form of T^{∗}X under the isomorphism T X ⟶^{♭} T^{∗}X determined by the metric, and the constants m and e are the mass and the charge of the particle respectively. The Hamiltonian is H(D_{x}) = \frac{1}{2}mT_{2}(D_{x}, D_{x}) with D_{x} ∈ T X.

In this paper (X, T_{2}) will be the hyperbolic plane or a compact Riemann surface.

2.1. The Landau problem on the hyperbolic plane. The configuration space is the hyperbolic plane ℍ = {z ∈ ℂ/Imz > 0} with the metric T_{2} = \frac{R^{2}}{y^{2}} (dx^{2} + dy^{2}) of Gaussian negative curvature K = −\frac{1}{R^{2}} . We have the following explicit expressions:

ω_{2} = \frac{mR^{2}}{y^{2}} (dx ∧ dẋ + dy ∧ dẏ) + \frac{eBR^{2}}{y^{2}} dx ∧ dy

eF = \frac{eBR^{2}}{y^{2}} dx ∧ dy

H = \frac{mR^{2}}{2y^{2}} (ẋ^{2} + ẏ^{2})

Let us note that (ℍ, eF) is a symplectic manifold and that the group ℙSL(2, ℝ) acts transitive and symplectically on ℍ by isometries as follows

\begin{matrix}ℙSL(2, ℝ) & × & ℍ & ⟶ & ℍ \\ (γ = ( \begin{matrix}a_{γ} & b_{γ} \\ c_{γ} & d_{γ} \end{matrix} ) & , & z) & ↦ & γ · z = R(\frac{a_{γ}z + b_{γ}R}{c_{γ}z + d_{γ}R}) \end{matrix}

This action on (ℍ, eF) admits an equivariant moment map J: ℍ → 𝔰𝔩(2, ℝ)^{∗} with respect to the coadjoint action, which is given by

J_{1} = J^{∗}A_{1} = −\frac{eBR}{2y}(x^{2} + y^{2} − R^{2})
J_{2} = J^{∗}A_{2} = \frac{eBR^{2}}{y} x
J_{3} = J^{∗}A_{3} = −\frac{eBR}{2y}(x^{2} + y^{2} + R^{2})

where < A_{1}, A_{2}, A_{3} >is a basis of 𝔰𝔩(2, ℝ).

GEOMETRIC QUANTIZATION OF THE LANDAU PROBLEM ON HYPERBOLIC … 623

The previous action of ℙSL(2, ℝ) on ℍ induces a second symplectic action on (T ℍ, ω_{2}) without fixed points that preserves the Hamiltonian H and admits an equivariant moment map J̄: T ℍ → 𝔰𝔩(2, ℝ)^{∗}, which is called hyperbolic angular momentum. One has the following expressions for the hyperbolic angular momenta [13]:

J̄_{1} = J̄^{∗}A_{1} = −\frac{mR}{2y^{2}} ((x^{2} − y^{2} − R^{2})ẋ + 2xyẏ) − \frac{eBR}{2y}(x^{2} + y^{2} − R^{2})
J̄_{2} = J̄^{∗}A_{2} = \frac{mR^{2}}{y^{2}} (xẋ + yẏ) + \frac{eBR^{2}}{y} x
J̄_{3} = J̄^{∗}A_{3} = −\frac{mR}{2y^{2}} ((x^{2} − y^{2} + R^{2})ẋ + 2xyẏ) − \frac{eBR}{2y}(x^{2} + y^{2} + R^{2})

It is important to remark that the Hamiltonian can be expressed in terms of the hyperbolic angular momenta as

H = \frac{1}{2mR^{2}} (J̄_{1}^{2} + J̄_{2}^{2} − J̄_{3}^{2} + e^{2}B^{2}R^{4})

In order to study the submanifold of constant energy of T ℍ, let us recall that

Proposition. The coadjoint orbits of the group ℙSL(2, ℝ) are:

(1) O_{0} = (0, 0, 0).

(2) O_{k^{2},+}^{hyp} = {α / α · α = −k^{2} , α_{3} > 0} , ∀ k^{2} ∈ ℝ^{+}.

(3) O_{k^{2},−}^{hyp} = {α / α · α = −k^{2} , α_{3} < 0} , ∀ k^{2} ∈ ℝ^{+}.

(4) O_{+}^{cone} = {α ≠ (0, 0, 0) / α · α = 0 , α_{3} > 0}.

(5) O_{−}^{cone} = {α ≠ (0, 0, 0) / α · α = 0 , α_{3} < 0}.

(6) O_{k^{2}}^{hyp 1 sheet} = {α / α · α = k^{2}} , ∀ k^{2} ∈ ℝ^{+}. where α = (α_{1}, α_{2}, α_{3}) ∈ 𝔰𝔩(2, ℝ)^{∗} and α · α = α_{1}^{2} + α_{2}^{2} − α_{3}^{2}.

Proposition. If G_{α} denotes the isotropy subgroup of a point α ∈ 𝔰𝔩(2, ℝ)^{∗} with respect to the coadjoint action of ℙSL(2, ℝ), then one has that:

(1) If α ∈ O_{0}, then G_{α} = ℙSL(2, ℝ).

(2) If α ∈ O_{k^{2},±}^{hyp}, then G_{α} ≃ SO(2).

(3) If α ∈ O_{±}^{cone}, then G_{α} ≃ E(1).

(4) If α ∈ O_{k^{2}}^{hyp 1 sheet} then G_{α} ≃ SO(1, 1)^{o}.

Let us denote by T^{ϵ}ℍ = {D_{x} ∈ T ℍ/H(D_{x}) = ϵ} the submanifold of T ℍ of constant energy ϵ and assume that eB < 0 (similar results hold for eB > 0).

Theorem. Taking into account that ϵ = \frac{1}{2mR^{2}} (J · J + e^{2}B^{2}R^{4}) with J · J = J̄_{1}^{2} + J̄_{2}^{2} − J̄_{3}^{2}, one has that

(1) T^{0}ℍ = J̄^{−1}(O_{k^{2},+}^{hyp}) for ϵ = 0.

(2) T^{ϵ}ℍ = J̄^{−1}(O_{k^{2},+}^{hyp}) for 0 < ϵ < \frac{e^{2}B^{2}R^{2}}{2m}.

(3) T^{ϵ}ℍ = J̄^{−1}(O_{+}^{cone}) for ϵ = \frac{e^{2}B^{2}R^{2}}{2m}.

(4) T^{ϵ}ℍ = J̄^{−1}(O_{k^{2}}^{hyp 1 sheet}) for ϵ > \frac{e^{2}B^{2}R^{2}}{2m}.

Corollary. Let H = ϵ be the energy of the particle, then: (a) If ϵ = 0, then the trajectory on ℍ is a point. (b) If 0 < ϵ < \frac{e^{2}B^{2}R^{2}}{2m}, then the trajectory on ℍ is closed (periodic). (c) If ϵ ≥ \frac{e^{2}B^{2}R^{2}}{2m}, then the trajectory on ℍ is open.

For ϵ = 0, the manifold of orbits of zero energy T^{0}ℍ is identified with the zero section of the tangent bundle T ℍ, and ω_{2}_{|T^{0}ℍ} is symplectic.

For ϵ ≠ 0, T^{ϵ}ℍ is a three dimensional submanifold of T ℍ. Let ω_{2}^{ϵ} = ω_{2}_{|T^{ϵ}ℍ} and let us denote by kerω_{2}^{ϵ} its null foliation. One has kerω_{2}^{ϵ} =< D_{H}^{ϵ} > where D_{H}^{ϵ} is the restriction to T^{ϵ}ℍ of the Hamiltonian vector field D_{H} associated to the Hamiltonian H , that is i_{D_{H}} ω_{2} = −dH.

Definition. The manifold of orbits of constant energy (when it exists) is the symplectic manifold (\overline{ℍ}^{ϵ} ≡ T^{ϵ}ℍ/kerω_{2}^{ϵ}, \overline{ω}_{2}^{ϵ}) and ϕ^{∗}\overline{ω}_{2}^{ϵ} = ω_{2}^{ϵ} where ϕ : T^{ϵ}ℍ ⟶ \overline{ℍ}^{ϵ} is the natural projection.

As a consequence of the Kazhdan-Kostant-Sternberg theorem [12] one has:

Theorem. [13] The moment map J̄ induces symplectic diffeomorphisms (1) Ĵ: \overline{ℍ}^{0} → O_{e^{2}B^{2}R^{2},+}^{hyp} for ϵ = 0.

(2) Ĵ: \overline{ℍ}^{ϵ} → O_{k^{2},+}^{hyp} for 0 < ϵ < \frac{e^{2}B^{2}R^{2}}{2m}.

(3) Ĵ: \overline{ℍ}^{ϵ} → O_{+}^{cone} for ϵ = \frac{e^{2}B^{2}R^{2}}{2m}.

(4) Ĵ: \overline{ℍ}^{ϵ} → O_{k^{2}}^{hyp 1 sheet} for ϵ > \frac{e^{2}B^{2}R^{2}}{2m} where every coadjoint orbit is endowed with the Kostant-Kirillov-Souriau symplectic form.

2.2. The Landau problem on compact Riemann surfaces. Let S be a compact Riemann surface of genus g > 1 with a metric T_{2} of constant curvature K = −\frac{1}{R^{2}} . Riemann’s uniformization theorem tells us that S ≃ ℍ/Γ for some Fuchsian group Γ containing only hyperbolic elements. Moreover ℍ ⟶^{p} ℍ/Γ is the universal Riemannian covering [14].

Let ω̃_{2} be the symplectic form and H̃ the Hamiltonian on T S. Since p^{∗}F̃ = F, p^{∗}ω̃_{2} = ω_{2} and p^{∗}H̃ = H, the local expressions for ω̃_{2}, F̃ and H̃ are the same as in ℍ. But we have

Proposition. [13] (T S, ω̃_{2}, H̃) does not admit Hamiltonian vector fields coming from S and leaving H̃ invariant (that is, there are no infinitesimal symmetries).

However, Γ acts on T ℍ and T S = T (ℍ/Γ) ≃ T ℍ/Γ. Let us denote by T^{ϵ}S the submanifold of T S of constant energy. Then T^{ϵ}S ≃ T^{ϵ}ℍ/Γ. One also has that kerω_{2}^{ϵ} is Γ-invariant, therefore the manifold of orbits of constant energy \overline{S}^{ϵ} is identified with \overline{ℍ}^{ϵ}/Γ ≃ O/Γ. But the action Γ × O ⟶^{Φ} O can have fixed points. In fact, given α ∈ O, its isotropy group for the action Φ is Γ_{α} = Γ ∩ G_{α}.

Since all the elements of Γ are hyperbolic, G_{α} is trivial except in the case G_{α} ≃ SO(1, 1)^{o} and this happens for ϵ > \frac{e^{2}B^{2}R^{2}}{2m}.

Theorem. We have symplectic diffeomorphisms

(1) J̃: \overline{S}^{0} → O_{e^{2}B^{2}R^{2},+}^{hyp}/Γ for ϵ = 0 (2) J̃: \overline{S}^{ϵ} → O_{k^{2},+}^{hyp}/Γ for 0 < ϵ < \frac{e^{2}B^{2}R^{2}}{2m},

(3) J̃: \overline{S}^{ϵ} → O_{+}^{cone}/Γ for ϵ = \frac{e^{2}B^{2}R^{2}}{2m}

Thus we obtain another proof of the following result given in [15]

Corollary. (1) For 0 < ϵ < \frac{e^{2}B^{2}R^{2}}{2m} every trajectory on S is closed and con-

tractible.

(2) For ϵ = \frac{e^{2}B^{2}R^{2}}{2m} every trajectory on S is open.

(3) For ϵ > \frac{e^{2}B^{2}R^{2}}{2m} there is a non contractible closed trajectory representing every homotopy class in π_{1}(S) ≃ Γ.

3. Geometric Quantization of the Landau problem

Our configuration space X is an oriented Riemannian surface and the phase space is (T X, ω_{2} = mdθ + eτ^{∗}F). We assume that [\frac{1}{h}eF] ∈ H^{2}(X, ℤ). Then the prequantization condition [\frac{1}{h}ω_{2}] ∈ H^{2}(T X, ℤ) is fulfilled.

Let π̄ : (L̄, \overline{∇}, ⟨ , ⟩_{T X}) → (T X, ω_{2}) be a prequantization line bundle [16, 17] for (T X, ω_{2}); that is, a hermitian line bundle with a connection such that:

a) iCurv(\overline{∇}) = \frac{1}{ℏ}ω_{2}

b) D⟨s_{1}, s_{2}⟩_{T X} = ⟨\overline{∇}_{D}s_{1}, s_{2}⟩_{T X} + ⟨s_{1}, \overline{∇}_{D}s_{2}⟩_{T X} Therefore the prequantization operator Q̄(f̄) : Γ(T X, L̄) ⟶ Γ(T X, L̄) corre-

sponding to f̄ ∈ C^{∞}(T X) is given by

Q̄(f̄) = −iℏ\overline{∇}_{Df̄} + f̄

As we have pointed out, the magnetic field defines a symplectic structure eF on the configuration space X. Since [\frac{1}{h}ω_{2}] ∈ H^{2}(T X, ℤ) if and only if [\frac{1}{h}eF] ∈ H^{2}(X, ℤ) one has the following

Proposition. For every prequantization line bundle (L̄, \overline{∇}, ⟨ , ⟩_{T X}) ⟶^{π̄} (T X, ω_{2}) there exists a unique (up to equivalence) prequantization line bundle (L, ∇, ⟨ , ⟩_{X}) ⟶^{π} (X, eF) such that:

(1) τ^{∗}L = L̄. Then, if V is the vertical polarization of the tangent bundle and Γ(T X, L̄)^{V} denotes the set of vertical polarized sections, there exists an isomorphism

\begin{matrix}Γ(X, L) & ⟶^{τ^{∗}} & Γ(T X, L̄)^{V} \\ s & ⟼ & s^{v} \end{matrix}

(2) ∀ s_{1} s_{2} ∈ Γ(X, L) , ⟨s_{1}, s_{2}⟩_{X} = ⟨s_{1}^{v}, s_{2}^{v}⟩_{T X}

(3) ∀ s ∈ Γ(X, L), ∀D ∈ 𝔛(X) , (∇_{D}s)^{v} = \overline{∇}D̄s^{v} + \frac{im}{ℏ}θ(D̄)s^{v} where D̄ is the canonical lifting of D ∈ 𝔛(X) to T X.

For every f ∈ C^{∞}(X) , let D_{f} be the Hamiltonian vector field associated to f with respect to the symplectic structure eF (that is i_{D_{f}} eF = −df). Then the prequantization operator Q(f) : Γ(X, L) ⟶ Γ(X, L) is given by

Q(f) = −iℏ∇_{D_{f}} + f

If we consider the following map

\begin{matrix}C^{∞}(X) & ⟶^{∧} & C^{∞}(T X) \\ f & ⟼ & f̂ = τ^{∗}(f) + mθ(D̄_{f}) \end{matrix}

then we have

Proposition. ∀ f ∈ C^{∞}(X), there is a commutative diagram

Γ(X, L) ⟶^{τ^{∗}} Γ(T X, L̄)^{V}
Q(f)↓  ↓^{Q̄(f̂)}
Γ(X, L) ⟶^{τ^{∗}} Γ(T X, L̄)^{V}

Remark. In the case X = ℍ, we can see that Ĵ_{i}, ∀1 ≤ i ≤ 3, are precisely the hyperbolic angular momenta J̄_{i}. The prequantization of the hyperbolic angular momenta Q̄(J̄_{i}) acting on Γ(T X, L̄)^{V} coincides, via τ^{∗}, with the prequantization operators Q(J_{i}) acting on Γ(X, L).

Since X is oriented, its orientation induces a metaplectic structure on (T X, ω_{2}) which leads to a trivial half-form bundle for the vertical polarization. Therefore

Proposition. [16, 18] The Blattner-Kostant-Sternberg quantization method for the vertical polarization allows us to quantize the Hamiltonian H giving

Q_{H}^{BKS} = \frac{ℏ^{2}}{2m}(∇^{∗}∇ + \frac{s}{6})

where ∇^{∗}∇ is the Bochner Laplacian of (L, ∇) and s is the scalar curvature of the metric. Therefore Q_{H}^{BKS} acts on L^{2}(X, L).

3.1. The Hyperbolic plane ℍ. On the hyperbolic plane ℍ the prequantization line bundle L ⟶^{π} ℍ is trivial. We can take a global section s_{0} such that ∇s_{0} = −\frac{i}{ℏ}A ⊗ s_{0} where A = eBR^{2} \frac{dx}{y} and therefore we can take ⟨s_{0}, s_{0}⟩_{ℍ} = 1.

For every f ∈ C^{∞}(ℍ) we have

Q(f) = −\frac{iℏy^{2}}{eBR^{2}} (−\frac{∂f}{∂y} \frac{∂}{∂x} + \frac{∂f}{∂x} \frac{∂}{∂y} ) + y \frac{∂f}{∂y} + f

In particular we have for J_{1}, J_{2}, J_{3}

Q(J_{1}) = −\frac{iℏ}{2R} ((R^{2} − x^{2} + y^{2})\frac{∂}{∂x} − 2xy \frac{∂}{∂y} ) − eBRy

Q(J_{2}) = −iℏ (x\frac{∂}{∂x} + y \frac{∂}{∂y} )

Q(J_{3}) = \frac{iℏ}{2R} ((R^{2} + x^{2} − y^{2})\frac{∂}{∂x} + 2xy \frac{∂}{∂y} ) − eBRy

Proposition. The Casimir operator of the action of ℙSL(2, ℝ) on Γ(X, L) is

Q(J_{1})^{2} + Q(J_{2})^{2} − Q(J_{3})^{2} = −ℏ^{2} (y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) − 2i\frac{eBR^{2}}{ℏ} y \frac{∂}{∂x})

Proposition.

∇^{∗}∇ = −\frac{1}{R^{2}} (y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) − 2i\frac{eBR^{2}}{ℏ} y \frac{∂}{∂x} − \frac{e^{2}B^{2}R^{4}}{ℏ^{2}} )

Corollary. The Hamiltonian of the quantum Landau problem on the hyperbolic plane is

Q_{H}^{BKS} = \frac{1}{2mR^{2}} (Q(J_{1})^{2} + Q(J_{2})^{2} − Q(J_{3})^{2} + e^{2}B^{2}R^{4}) − \frac{ℏ^{2}}{6mR^{2}}

Remark. This expression is equal to the classical one except for the term −\frac{ℏ^{2}}{6mR^{2}}

3.2. Compact Riemann surfaces S of genus g > 1. Let S = ℍ/Γ and ℍ ⟶^{p} S = ℍ/Γ be the canonical projection. The symplectic form on S is given by eF̃ = eBΩ̃_{2} where Ω̃_{2} is the area element with p^{∗}Ω̃_{2} = Ω_{2}.

Proposition. There exists a prequantization line bundle (L̃, ˜∇, ⟨ , ⟩_{S}) ⟶^{π̃} (S, eF̃) if and only if

\frac{eB}{h} Ω̃_{2} ∈ H^{2}(S, ℤ) ⇔ −\frac{eBR^{2}}{ℏ} χ(S) = c_{1}(L̃) ∈ ℤ

Theorem. Let (L, ∇, ⟨ , ⟩_{ℍ}) ⟶^{π} (ℍ, eF) be a prequantization line bundle. If −\frac{eBR^{2}}{ℏ} ∈ \frac{4π}{sμ(F)}ℤ = \frac{1}{ḩi(S)}ℤ then there exists an action of Γ on L ⟶^{π} ℍ covering the action of Γ on ℍ such that:

1. It leaves ∇ and ⟨ , ⟩_{ℍ} invariant.

2. (L/Γ, ∇/Γ, ⟨ , ⟩_{ℍ}/Γ) ⟶^{π̃} (S, eF̃) is a prequantization line bundle. Proof. Let Γ̄ be the group in SL(2, ℝ) covering Γ. Taking a global section s_{0} of L ⟶^{π} ℍ one has that L ≃ ℍ × ℂ and the action is given by an automorphy factor

\begin{matrix}Γ̄ & × & L & ⟶ & L \\ (γ & , & (z, α)) & ↦ & (γ(z), e^{iξ(γ,z)}α) \end{matrix}

where

\begin{aligned}ξ: & Γ̄ & × & ℍ & ⟶ & ℝ \\ & (γ & , & z) & ↦ & ξ(γ, z) \end{aligned}

verifies ξ(γ_{1}γ_{2}, z) − ξ(γ_{1}, γ_{2}(z)) − ξ(γ_{2}, z) ∈ 2πℤ. The connection 1-form of ∇ is ω = −\frac{eBR^{2}}{ℏ} \frac{dx}{y} + dσ with dσ being the Maurer-Cartan 1-form of U(1) and the condition γ^{∗}ω = ω ∀ γ ∈ Γ̄, implies that

ξ(γ, z) = 2\frac{eBR^{2}}{ℏ} arg (c_{γ}z + d_{γ}R) + μ_{B}(γ)

where Γ̄ ⟶^{μ_{B}} ℝ verifies μ_{B}(γ_{1}γ_{2}) − μ_{B}(γ_{1}) − μ_{B}(γ_{2}) − 4π \frac{eBR^{2}}{ℏ} w_{B}(γ_{1}, γ_{2}) ∈ 2πℤ

with

w_{B}(γ_{1}, γ_{2}) = \frac{1}{2π} \{arg (c_{γ_{1}}γ_{2}(z) + d_{γ_{1}}R) + arg (c_{γ_{2}}z + d_{γ_{2}}R) − arg (c_{γ_{3}}z + d_{γ_{3}}R)\} Finally the map

\begin{aligned}σ_{B}: & Γ̄ & × & Γ̄ & ⟶ & U(1) \\ & (γ_{1} & , & γ_{2}) & ↦ & σ_{B}(γ_{1}, γ_{2}) = e^{i4π \frac{eBR^{2}}{ℏ} w_{B}(γ_{1},γ_{2})} \end{aligned}

must define a trivial cohomology class [σ_{B}] ∈ H^{2}(Γ̄, U(1)) and [19, 20, 21] this is the case if and only if −\frac{eBR^{2}}{ℏ} ∈ \frac{4π}{sμ(F)}ℤ = \frac{1}{ḩi(S)}ℤ.

Corollary. The sections of the prequantization line bundle L̃ ⟶^{π̃} S are identified with the automorphic forms A^{k}(Γ) on ℍ of weight k = \frac{eBR^{2}}{ℏ} and multiplier ν_{B} = e^{iμ_{B}}, that is functions ψ on ℍ such that

ψ(γ(z)) = e^{iμ_{B}(γ)}e^{2i \frac{eBR^{2}}{ℏ} arg (c_{γ}z+d_{γ}R)}ψ(z) ∀ γ ∈ Γ̄

Let D_{B} = y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) − 2i\frac{eBR^{2}}{ℏ} y \frac{∂}{∂x} be the Maass Laplacian acting on au-

tomorphic forms A^{k}(Γ) [22].

Proposition. The Hamiltonian Q_{H}^{BKS} = \frac{ℏ^{2}}{2m}(∇^{∗}∇ + \frac{s}{6}) acting on L^{2}(S, L̃) is identified with the operator

Q_{H}^{BKS} = −\frac{ℏ^{2}}{2mR^{2}} D_{B} + \frac{e^{2}B^{2}R^{2}}{2m} − \frac{ℏ^{2}}{6mR^{2}}

Theorem. The discrete series of the spectrum of Q_{H}^{BKS} acting on A^{k}(Γ) is

E_{l}^{(k)} = \frac{ℏ^{2}}{2mR^{2}} [(2l + 1)|k| − l(l + 1)] − \frac{ℏ^{2}}{6mR^{2}}

with l ∈ ℤ and 0 ≤ l < |k| − \frac{1}{2}

The degeneration of each eigenvalue can be calculated using the Riemann-Roch theorem

Theorem. dim H_{l}^{(k)} = \frac{1}{2}χ(S)(2l − 2|k| + 1).

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Departamento de Matemáticas, Universidad de Salamanca Plaza de la Merced 1-4, E-37008 Salamanca, SPAIN E-mail: alm@gugu.usal.es carlost@gugu.usal.es