Landau levels on the hyperbolic plane in the presence of Aharonov Bohm fields

Transcription

1 Available online at Journal of Functional Analysis ) Landau levels on the hyperbolic plane in the presence of Aharonov Bohm fields Takuya Mine a,, Yuji Nomura b a Graduate School of Science and Technology, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto , Japan b Department of Computer Science, Graduate School of Science and Engineering, Ehime University, 3 Bunkyo-cho, Matsuyama, Ehime , Japan Received 21 February 212; accepted 3 June 212 Available online 14 June 212 Communicated by B. Driver Abstract We consider the magnetic Schrödinger operators on the Poincaré upper half plane with constant Gaussian curvature 1. We assume the magnetic field is given by the sum of a constant field and the Dirac δ measures placed on some lattice. We give a sufficient condition for each Landau level to be an infinitely degenerated eigenvalue. We also prove the lowest Landau level is not an eigenvalue if the above condition fails. In particular, the infinite degeneracy of the lowest Landau level is equivalent to the infiniteness of the zeromodes of the two-dimensional Pauli operator. 212 Elsevier Inc. All rights reserved. Keywords: Magnetic Schrödinger operator; Landau level; Zero-mode; Pauli operator; Hyperbolic plane; Aharonov Bohm effect; Aharonov Casher theorem 1. Introduction 1.1. Motivation The Landau levels hω c n + 1/2) n =, 1, 2,..., ω c = eb/mc)) are the infinitely degenerated eigenvalues of the Schrödinger operator in a homogeneous magnetic field of intensity B on * Corresponding author. Fax: addresses: mine@kit.ac.jp T. Mine), nomura@cs.ehime-u.ac.jp Y. Nomura) /$ see front matter 212 Elsevier Inc. All rights reserved.

2 172 T. Mine, Y. Nomura / Journal of Functional Analysis ) Fig. 1. Tessellation by the modular group SL 2 Z). Each hyperbolic triangle is a fundamental cell of this group. All these triangles are non-compact. the Euclidean plane. The discrete structure of the Landau levels is connected with the representation of the Heisenberg group, and plays a crucial role in the study of the quantum Hall effect, De Haas van Alphen effect, etc. We can also consider the Schrödinger operator with a homogeneous magnetic field on the hyperbolic upper half plane H, which is first studied by Maass [23] in his study of the modular functions. Then the spectrum consists of a finite number of infinitely degenerated eigenvalues in the lower part of the spectrum also called the Landau levels ), and continuous spectrum in the higher part. Comtet [6] gives a physical interpretation of the spectral structure; the low-energy classical particles in H subjected to a homogeneous magnetic field are trapped by the magnetic field, but the high-energy particles can escape to infinity due to the negative curvature of H. Moreover, Bulaev, Geyler and Margulis [5] and Lisovyy [22] show that the Landau levels are still infinitely degenerated even if we add one pointlike magnetic field or, the Aharonov Bohm magnetic field) to the homogeneous field. We can easily show that the same is true if we add a finite number of pointlike magnetic fields. 1 On the other hand, the authors [26] study the perturbation of a homogeneous magnetic field by pointlike magnetic fields on a lattice in the Euclidean plane. In this case, the Landau levels might be collapsed by the Aharonov Bohm phase shift [2] caused by the magnetic flux of the pointlike magnetic fields inside the cyclotron orbit of the electron. Nevertheless, [26] shows that the low-energy Landau levels still survive as infinitely degenerated eigenvalues. The number of such levels is determined by the magnetic flux through the fundamental cell of the period lattice. The aim of the present paper is to generalize the result of [26] on the hyperbolic plane. Especially we pay attention to i) the effect of the negative curvature, and ii) the effect of the structure of the period lattice. In contrast to the Euclidean case, the structure of a lattice in the hyperbolic plane can be much more complicated see Fig. 1). Our conclusion is that the negative curvature decreases the number of infinitely degenerated eigenvalues, but the complexity of the lattice group has no effect, though some technical difficulty occurs in the case of non-compact fundamental cell. If we take the flat-space limit, we reproduce the result of [26] at least formally. 1 This can be done by putting Φ = γ Γ z γ)and f = z + i) m in the solutions 22) and26) below.

3 T. Mine, Y. Nomura / Journal of Functional Analysis ) Moreover, we also study the number of the zero-modes the eigenfunctions with the eigenvalue ) for the Pauli operators. The Aharonov Casher theorem [3] says that the number of the zero-modes for the Pauli operator on the Euclidean plane is given by [ ] ) Φ Φ / Z, ) Φ 1 Φ =1, 2,..., Φ = ), where Φ is the total magnetic flux through the plane divided by 2π we normalize physical constants as h = e = 2m = 1), and [x] denotes the integer part of x. The A C theorem is generalized by Miller [24] and Erdös and Vugalter [12] under some weaker assumptions on the magnetic field. The A C result also suggests there are infinite zero-modes in the case Φ =. This statement is rigorously proved by Dubrovin and Novikov [9,1], Shigekawa [31], Geyler and Št ovíček [15] and Rozenblum and Shirokov [3] for the Euclidean case, and by Inahama and Shirai [18] and Geyler and Št ovíček [16] for the hyperbolic case. Actually, it is easy to deduce an estimate for the number of the zero-modes for the Pauli operator from our result for the Schrödinger operator, since the zero-modes are just the lowest Landau eigenfunctions of the Schrödinger operators see 13)) Landau levels for the Schrödinger operators The Poincaré upper half plane is given by H ={z = x + iy x R, y>}, endowed with the metric ds 2 = y 2 dx 2 + dy 2 ) and the surface form ω = y 2 dx dy.itiswell known that the Riemannian manifold H,ds 2 ) has negative constant Gaussian curvature 1, and is isomorphic to the Poincaré disc D, ds 2 ) given by D = { w = u + iv w < 1 }, ds 2 = 4du2 + dv 2 ) 1 w 2 ) 2, via the Cayley transform w = z i)/z + i). We introduce a 1-form a = a x dx + a y dy called the magnetic vector potential, and define the magnetic Schrödinger operator L on H by L = y 2 D x a x ) 2 + D y a y ) 2), where D x = i x, D y = i y.themagnetic field is the 2-form given by da, where d denotes the exterior derivative. We always assume a x,a y L 1 loc H; R), soda can be defined at least in the distributional sense. We say da is a constant magnetic field if da = Bω for some constant B R. In the present paper, we assume the magnetic field is the sum of a constant field and δ-magnetic fields on a lattice, as described below. In the sequel, we use some terminology from the theory of automorphic forms. A reader not familiar with automorphic forms can refer to Section 6. 1)

4 174 T. Mine, Y. Nomura / Journal of Functional Analysis ) Assumption 1.1. There exist real constants B and α with <α<1, a Fuchsian group G of the first kind, and a discrete subset Γ of H the union of H and the cusps; see Section 6.2) invariant under the action of G, such that da = Bω + γ Γ 2παδ γ, 2) where δ γ is the Dirac δ measure at the point γ. In Section 2, we shall construct a x,a y C H \ Γ ; R) L 1 loc H; R) satisfying 2). The restriction <α<1 loses no generality, because the integral difference of α can be gauged away see e.g. Geyler and Št ovíček [15, Section 6]). When we need to indicate the value B or α) explicitly, we write L B or L B,α )forl,etc. As in the Euclidean case Adami and Teta [1] or Dabrowski and Št ovíček [7]), we can prove that the operator H B, = L C H\Γ) is not essentially self-adjoint if Γ. We choose several self-adjoint extensions of H B, as follows. Let a = a x dx + a y dy satisfying 2). We define operators Π x, Π y, A B and A B by Π x = yd x a x ), Π y = yd y a y ), A B = Π x + iπ y, A B = Π x iπ y + 1. Notice that A B is the formal adjoint of A B in the sense A B u, v) = u, A B v) for every u, v C H \ Γ), where u, v) = H uvω = H uvy 2 dxdy. The following relation is called the shape invariance see Benedict and Molnár [4], Molnár, Benedict and Bertrand [27] or Inahama and Shirai [19]): L B = A B A B + B = A B+1 A B+1 B on H \ Γ. 3) We define two self-adjoint operators H B,max ± quadratic forms h ± B,max given by as the self-adjoint operators associated to the h + B,max [u]= A Bu 2 + B u 2, Q h + { B,max) = u L 2 H; ω) A B u L 2 H; ω) }, h B,max [u]= A B+1 u 2 B u 2, Q h { B,max) = u L 2 H; ω) A B+1 u L2 H; ω) }, where u 2 = u, u) and the derivatives A B u and A Bu are defined as distributions on H \ Γ. By 3), H B,max ± are two self-adjoint extensions of H B,. We also define operators H ± B,min as the self-adjoint operators associated to the quadratic forms h ± B,min given by

5 T. Mine, Y. Nomura / Journal of Functional Analysis ) h + B,min [u]= A Bu 2 + B u 2, Q h + B,min) = C H \ Γ), h B,min [u]= A B+1 u 2 B u 2, Q h B,min) = C H \ Γ), where the overline denotes the form closure. However, both operators H ± B,min coincide with the Friedrichs extension of H B,, so we usually omit ± and write simply H ± B,min = H B,min. The difference between three operators is, roughly speaking, the boundary conditions at the lattice points. The domains DH B,max ± ) contain some functions singular at γ Γ, while all the elements of DH B,min ) satisfy some repulsive conditions, that is, lim z γ uz) = for u DH B,min). When Γ = da is a constant magnetic field), it is known that H B, is essentially selfadjoint see e.g. Shubin [34]), so we denote the operator H ± B,max = H B,min = H B, simply by H B. The spectrum σh B ) of H B is well known see e.g. Roelcke [29], Elstrodt [11], Comtet[6] or Inahama and Shirai [19]): σh B ) = { NB) n= {E B,n} [B 2 + 1/4, ) B > 1/2), [B 2 + 1/4, ) B 1/2), 4) where NB) is the largest integer less than B 1/2, and E B,n = 2n + 1) B nn + 1). The eigenvalues {E B,n } NB) n= are called the Landau levels. When B > 1/2, all the eigenvalues E B,n are infinitely degenerated, due to the invariance of the magnetic field under the action of SL 2 R). We study the degeneracy of E B,n for the operators H B,# ± # = max or min, as in the sequel), when Γ is a lattice. We require some number-theoretical assumptions for the lattice Γ. Assumption 1.2. For some m N positive integers) and some even integer k, there exists an automorphic form Ψ of weight k on G satisfying the following conditions: i) All the zeros of Ψ are of order m. ii) Γ coincides with the set of the zeros of Ψ in H. iii) Ψ is not zero at every cusp of G. We denote Φ = Ψ 1/m. Notice that Φ is a single-valued holomorphic function on H having only first order zeros on Γ. When G has cusps, we additionally assume the following. Assumption 1.3. There exists a cusp form which has no zeros in H. In Section 6, we see that there are many examples of the groups G and the lattices Γ satisfying the above conditions. Let ι be the projection from SL 2 R) to PSL 2 R) = SL 2 R)/{±1}.Forz H, we denote G z ={g G gz = z} and e z = #ιg z ). We call z a fixed point if e z > 1.

6 176 T. Mine, Y. Nomura / Journal of Functional Analysis ) Definition 1.4. Let G and Γ be as in Assumption 1.1.LetD be a fundamental domain of G.We choose a complete representatives {γ k } K k=1 of Γ such that γ k Γ D, Γ = K Gγ k, Gγ k Gγ k = for k k. k=1 We define the number N of points of Γ in D by N = K k=1 e 1 γ k. If Γ includes no fixed points and no boundary points of D, then N is just the number of the points of Γ in D in the usual sense. When Γ includes fixed points, N may take a fractional value. First we state our result for the case G is co-compact. We denote multh ; E) = dim KerH E) for a self-adjoint operator H. Theorem 1.5. Suppose B, α, G and Γ satisfy Assumptions 1.1, 1.2, and G is co-compact. Let D and N be as in Definition 1.4, and D the hyperbolic area of D, that is, D = D ω. Then the following holds. i) Let Φ + = B + 2παN/ D. Then mult { H B,max + if ; B) Φ+ > 1/2, = if Φ + 1/2, { if Φ+ >1/2) + 2πN/ D, multh B,min ; B) = if Φ + 1/2) + 2πN/ D. 5) 6) ii) Let Φ = B + 2π1 α)n/ D. Then, mult { HB,max if ; B) Φ > 1/2, = if Φ 1/2, { if Φ >1/2) + 2πN/ D, multh B,min ; B) = if Φ 1/2) + 2πN/ D. 7) 8) In i), the value Φ + is the average of the magnetic flux per unit area. So the results read the lowest Landau level is infinitely degenerated if the magnetic field is sufficiently strong compared with the curvature see also 17) below). The additional term 2πN/ D for H B,min is due to the repulsive boundary conditions at γ Γ. The result ii) is derived from i), the complex conjugation symmetry and the gauge invariance CH ± B,α,# C = H B, α,# H B,1 α,#, 9) where C denotes the complex conjugation map Cu = u, and means the both sides are unitarily equivalent. When G has cusps, we have the following.

7 T. Mine, Y. Nomura / Journal of Functional Analysis ) Theorem 1.6. Suppose B, α, G and Γ satisfy Assumptions 1.1, 1.2. Suppose additionally Γ has cusps, and Assumption 1.3 holds. Then, i) the statements 6) and 8) hold without any change; ii) the statement 5) holds when Φ + 1/2, or when Φ + = 1/2 and B ; iii) the statement 7) holds when Φ 1/2, or when Φ = 1/2 and B. The restriction B in ii) above or B in iii)) is necessary for some technical reasons, and not yet removed at present. Next we state the result for the higher Landau levels of H B,min. Theorem 1.7. Suppose B, α and Γ satisfy Assumptions 1.1 and 1.2. IfΓ has cusps, suppose additionally Assumption 1.3 holds. Let n be a positive integer, and assume B + 2παN ) 1 > D 2 + n + 2πN n + 1), 1) D or B + Then multh B,min ; E B,n ) =. 2π1 α)n D ) 1 > 2 + n + 2πN n + 1). D Theorem 1.7 also gives the degeneracy of the higher Landau levels of H B,max ±, because of the following relations. For n 1, we have { Ker H + B,max E B,n ) KerHB 1,min E B 1,n 1 ) B >3/2), Ker H B,max + E B,n 1) KerHB 1,min E B 1,n ) B < 1/2), { KerHB,min E B,n ) Ker H B 1,max E B 1,n 1) B > 3/2), KerH B,min E B,n 1 ) Ker H B 1,max E ) B 1,n B < 1/2), 11) 12) where means the two subspaces are isomorphic via some unitary operator and have the same dimensions). These relations are derived from the supersymmetric property of the Pauli operator for our magnetic field, as defined below Zero-modes for the Pauli operators In Inahama and Shirai [18], they define the Dirac operator D, the Pauli operator P and its diagonal components P B ± by2 A ) D = B A, P = D 2 = B A ) B P + ) A B A B A = B P. B B 2 Geyler and Št ovíček [16] adapt another definition, that is, P ± B,max = H ± B,max B. Here we adapt the above definition from the viewpoint of the supersymmetry 16). In the Euclidean case, Persson [28] discusses the definition of the Pauli operator with the Aharonov Bohm magnetic fields in detail.

8 178 T. Mine, Y. Nomura / Journal of Functional Analysis ) We define self-adjoint realizations P B,max ±, P ± B,min of P± B to the quadratic forms p B,max ±, p± B,min given by as the self-adjoint operators associated p + B,max [u]= A Bu 2, Q p + B,max) = { u L 2 H; ω) AB u L 2 H; ω) }, pb,max [u]= A B u 2, Q pb,max ) { = u L 2 H; ω) A B u L2 H; ω) }, p + B,min [u]= A Bu 2, Q p + B,min) = C H \ Γ), p B,min [u]= A B u 2, Q p B,min) = C H \ Γ). Clearly we have H B,# + = P B,# + + B, H B,# = P B+1,# B. 13) The relation 13) and Theorem 1.5 tell us the following corollary for the degeneracy of the zeromodes of the Pauli operators. Corollary 1.8. Under the same assumptions of Theorem 1.5, we have the following. i) Let Φ + = B + 2παN/ D. Then mult { P B,max + if ; ) Φ+ > 1/2, = if Φ + 1/2, mult { P + if B,min ; ) Φ+ >1/2) + 2πN/ D, = if Φ + 1/2) + 2πN/ D. 14) ii) Let Φ = B + 2π1 α)n/ D. Then, mult { PB,max if ; ) Φ > 1/2, = if Φ 1/2, mult { P if B,min ; ) Φ > 1/2) + 2πN/ D, = if Φ 1/2) + 2πN/ D. 15) Especially 14) and 15) are quite similar to [18, Theorem 1.6], but a little different because of the δ-magnetic fields. Of course, Theorem 1.6 and 13) give us another corollary for the cusp case. As is usual in the theory of Pauli operators, the spin-up component P B,# + and the spin-down component PB,# satisfy the following supersymmetric relations see Proposition 3.1 below): P B,max ± Ker P ± B,max ) P B,min Ker P B,min ). 16) The relations 16) together with 13)imply11) and 12).

9 T. Mine, Y. Nomura / Journal of Functional Analysis ) To see the effect of the curvature more explicitly, let us change the Gaussian curvature from 1 to 1/A for some constant A>. We replace ds 2 by ds A ) 2 = Ay 2 dx 2 + dy 2 ), and consequently ω by ω A = Ay 2 dx dy, L B by L A B = 1/A)L AB, E B,n by EB,n A = 2n + 1) B nn + 1)/A, etc. Then, the statement 14) is changed into mult { P A,+ if B,max ; ) Φ+ > 1/2A), = if Φ + 1/2A). 17) If we formally take the flat-space limit A, we obtain the classical Aharonov Casher result 1) in the case Φ =. Similarly, 1) is changed into B + 2παN D > 1 A ) n + 2πN n + 1). 18) D Taking the limit A again, we obtain the corresponding result in the Euclidean case [26, Theorem 1.1]. 3 Our analysis relies on the theory of the automorphic forms. However, in the non-critical case, there is a possibility of another proof using the entire function theory for the asymptotic behavior of the canonical product, as in the Euclidean case. In fact, the canonical product on the unit disc is introduced by Tsuji [36], and its asymptotics is studied by Girnyk [17] and Sons [35] though Girnyk says the angular density of zeros in the unit disc does not uniquely determine the asymptotics of the canonical product). We hope to study this direction in the future work. The rest of the paper is organized as follows. In Section 2 we shall construct the vector potentials satisfying 2). In Section 3, we shall write down the eigenfunctions corresponding to the Landau level E B,n explicitly. In Section 4, we shall prove main theorems in the case G is co-compact. In Section 5, we shall prove main theorems in the case G has cusps. In Section 6, we shall review some definitions and fundamental facts about the automorphic forms, and give some examples of groups and automorphic forms satisfying our assumptions. All the figures and the numerical values are obtained by using Mathematica Vector potentials In this section, we shall construct the vector potential a satisfying 2). Similar constructions are found in [15,16]. Let Φ be the function given after Assumption 1.2. The function Φ is a single-valued holomorphic function on H having only first order zeros at every γ Γ. Put φ = log Φ) = Φ /Φ, then φ is a meromorphic function on H having only first order poles with residue 1 at every γ Γ. Define a 1-form a by a = B y dx + α Imφ dz), 19) 3 In the paper [26], the index of the Landau level starts from n = 1. So we should adjust the index n when we compare our result with [26, Theorem 1.1].

10 171 T. Mine, Y. Nomura / Journal of Functional Analysis ) where dz = dx + idy. Then, by the Cauchy Riemann relation da = B y 2 dy dx + α Im zφdz dz) = Bω for z H \ Γ.Forz near γ Γ,wehave ) dz d Imφ dz) = d Im = d y log z γ dx + x log z γ dy ) z γ = log z γ dx dy = 2πδ γ, where = 2 x + 2 y and we used the distributional equality log z dx dy = 2πδ. Thus a satisfies the equality 2). By the gauge invariance, we can assume a is given by 19) without loss of generality. 3. Eigenfunctions for Landau levels In this section, we shall construct the eigenfunctions of H B,# + # = max, min) for the eigenvalue B, and the eigenfunctions of H B,min for the higher Landau levels. First we prove the supersymmetric relation in the introduction. Proposition 3.1. The equivalence relation 16) holds. Proof. The proof is quite similar to that of [25, 8)], so we give only the sketch. We define two linear operators A B u = A B u, DA B ) = C H \ Γ), A B u = A B u, D A B) = C H \ Γ), where the overline denotes the closure with respect to the operator norm. Since A B is the formal adjoint of A B,wehave A B u = A B u, A B u = A Bu, D A ) { B = u L 2 H; ω) A B u L2 H; ω) }, D A ) { B = u L 2 H; ω) A B u L 2 H; ω) }, where we regard A B and A B as operators on the Schwartz distribution space D H \ Γ). Then we have P B,max + = A B A ), B P B,min = A ) A B B, 2) PB,max = A BA B ), P + B,min = A B) A B, 21) since the form domains of the both sides of each equality are equal. Then the relation X X Ker X) XX Ker X ) see e.g. Deift [8, Theorem 3]) implies the conclusion.

11 T. Mine, Y. Nomura / Journal of Functional Analysis ) Proposition 3.2. For # = max or min, a function u belongs to KerH B,# + B) = Ker P B,# + if and only if there exists a holomorphic function f on H such that { y uz) = B Φz) α fz) # = max), y B Φz) α Φz)f z) # = min), 22) and u L 2 H; ω). Proof. By 2), we have u Ker P B,max + u Ker A B) A B u =, u L 2 H; ω). 23) By 21), we have u Ker P + B,min u Ker A B A B u =, u L 2 H; ω), Put z = x + i y )/2. Then the operator A B is written as lim uz) = for every γ Γ. 24) z γ A B = iy 2 z + By ) i αφz) = iy B+1 Φ α 2 z ) Φ α y B. Thus the solution to the equation A B u = is given by u = Φ α y B gz), where g is a holomorphic function on H \ Γ.Ifu DP B,max + ),wehaveu L2 H; ω), and theconverseistrueby23). If u DP + B,min ), u must satisfy the boundary condition in 24). Since Φ α = O z γ α ) as z γ and <α<1, the function gz) must be factorized as gz) = Φz)f z), where f is a holomorphic function on H notice that Φz) has only first order zeros). Next, let us consider the higher Landau levels. Proposition 3.3. Let B>1/2. Suppose u C H \ Γ) satisfies L B u = E B,n u for some n =, 1, 2,...Then Proof. By 3), we have L B+1 A B+1 u = E B+1,n+1A B+1u. 25) If L B u = E B,n u,wehave L B+1 = A B+1 A B+1 + B + 1, L B = A B+1 A B+1 B.

12 1712 T. Mine, Y. Nomura / Journal of Functional Analysis ) L B+1 A B+1 u = A B+1 A B+1A B+1 u + B + 1)A B+1 u = A B+1 L B + 2B + 1)u = E B,n + 2B + 1)A B+1 u = E B+1,n+1A B+1 u. Proposition 3.4. Suppose B>1/2. Let n = 1, 2, 3,...and let f be a holomorphic function on H. Put u = A B+n A B+1 y B Φz) α Φz) n+1 fz) ). 26) If u L 2 H; ω), then u DH B+n,min ) and H B+n,min u = E B+n,n u. Proof. Let f be a holomorphic function on H and put v = y B Φz) α Φz) n+1 fz). Then we have L B v = Bv = E B, v by Proposition 3.2. Since u = A B+n A B+1 v,wehave L B+n u = E B+n,n u by Proposition 3.3. Since u L 2 H; ω),wehavel B+n u L 2 H; ω). Moreover, by the explicit form of v and A B,weseeuz) = O z γ 1 α ) as z γ. Thus the boundary conditions lim z γ uz) = hold and u DH B+n,min ). 4. Co-compact case In this section, we assume G is co-compact i.e. G has no cusps) and prove the statements for H B,# + in Theorems 1.5 and 1.7. Then, those for H B,# hold because of the complex conjugation symmetry 9). The basic idea is based on the proof of [16, Theorem 8] Infiniteness of the lowest Landau eigenfunctions First we assume { Φ+ > 1/2 if#= max, Φ + > 1/2 + 2πN/ D if # = min, 27) where Φ + = B + 2παN/ D, and prove multh B,# + ; B) =. By Theorem 6.1 below, 27) is equivalent to { 2B + kα/m > 1 # = max), 2B + kα 1)/m > 1 # = min), 28) where m, k are the numbers given in Assumption 1.2. Let u be the function 22) with f = z+i) j, where j is a positive integer. By Proposition 3.2, it suffices to prove u L 2 H; ω) for sufficiently large j. Since Ψ is an automorphic form of weight k,wehave Ψgz)= cz + d) k Ψz)

13 T. Mine, Y. Nomura / Journal of Functional Analysis ) for every z H and g = ) ab cd G. By this equality and we see that the function Im gz = Im z cz + d 2, ρz) = y k/2m) Φz) is periodic with respect to G-action and has zeros only on Γ. Consider the case # = min. Since G is co-compact, the periodicity of ρ implies there exists a constant M> such that for every z H. Thus we have H u 2 ω M Φz) 21 α) My kα 1)/m H y 2B 2+kα 1)/m z + i 2j dxdy. By 28), the last integral converges if we take j sufficiently large. When # = max, we have to take care of the singularity of Φ α on Γ. Take sufficiently small ɛ> and put U ={z H dist ds z, Γ ) < ɛ}. By the periodicity of ρ, there exists M> such that for every z in H \ U. Thus we have H u 2 ω M H\U Φz) 2α My kα/m y 2B 2+kα/m z + i 2j dxdy + U y 2B Φz) 2α z + i 2j ω. 29) By 28), the first term in RHS of 29) converges for sufficiently large j. The second term is bounded by M g G sup y 2B+kα/m z + i 2j 3) z U gd where M = U D ρz) 2α ω. To see 3) is finite, we need the following lemma. In the sequel, we denote for z H and ɛ>. B ɛ z) = { z H distds z,z ) <ɛ },

14 1714 T. Mine, Y. Nomura / Journal of Functional Analysis ) Lemma 4.1. For every ɛ>, z H and z, z B ɛ z ), we have e 2ɛ < Im z Im z <e2ɛ, 31) e 2ɛ < z + i z + i <e2ɛ. 32) Proof. Let z = x + iy H. Then, the set B ɛ z ) is an open disc in H and the diameter of B ɛ z ) parallel to the imaginary axis is the segment from x + ie ɛ y to x + ie ɛ y. The first statement 31) follows from this fact. Next, consider the line l passing through the two points z and i, and let z 1 and z 2 be the two intersection points of l and B ɛ z ), with Im z 1 < Im z 2. Then we have for z, z B ɛ z ) z + i z + i < z 2 + i z 1 + i = Im z Im z < Im z 2 e 2ɛ. Im z 1 Taking the reciprocal of the both sides, we have 32). Notice that there are only finite points of Γ in D, since D is compact and Γ is discrete. By Lemma 4.1,thesumin3) is bounded by e 2ɛ M 1 y 2B+kα/m z + i 2j ω U D g G U gd e 2ɛ M 1 y 2B+kα/m 2 z + i 2j dxdy < U D for sufficiently large j, because of 28). Thus we prove u L 2 H; ω) Non-existence of the lowest Landau eigenfunctions H Next we prove the non-existence part of Theorem 1.5. We need two lemmas. Lemma 4.2. Let p 1. Then, we have for any holomorphic function f on H y p fz) 2 ω =. 33) Proof. Let σ be the inverse Cayley transform from D to H, that is, Then we have H z = σw= i 1 + w 1 w. 34) y = Im z = 1 w 2 1 w 2. 35)

15 T. Mine, Y. Nomura / Journal of Functional Analysis ) Since σ is an isometry, the left-hand side of 33) is written as 1 w 2 ) p fσw) 2 ω. 36) D 1 w 2 Put gw) = 1 w) p fσw), then g is holomorphic on D. Consider the Taylor expansion of g Then the integral 36) equals 2π a n 2 n= gw) = 1 a n w n. n= 4r 2n+1 1 r 2) p 2 dr, since ω = 41 r 2 ) 2 rdr dθ in the polar coordinate. Since p 2 1, the integral diverges for every n. Since f, we have g, therefore at least one coefficient a n is non-zero. Thus we have the conclusion. Lemma 4.3. Let p R. Then, for sufficiently small ɛ>, there exist ɛ = ɛ ɛ) > ɛ and C = Cɛ,p) > such that, y p fz) 2 ω C y p fz) 2 ω B ɛ z )\B ɛ z ) B ɛ z ) for any z H and any holomorphic function f on B ɛ z ). Moreover, ɛ as ɛ. Proof. By Lemma 4.1, sup y p e 2ɛ p inf z B ɛ z ) z B ɛ z ) yp 37) for any ɛ >, any z H and any z B ɛ z ). Thus it is sufficient to prove the case p =. Since ω is invariant under the action of SL 2 R), we can assume z = i.againby37), it is sufficient to show that fz) 2 dxdy fz) 2 dxdy 38) B ɛ i)\b ɛ i) B ɛ i) for some ɛ = ɛ ɛ) with ɛ asɛ. For <ɛ<log4/3), put ɛ = e ɛ 1 and ɛ = log4 3e ɛ ). Since the diameter of B ɛ i) parallel to the imaginary axis is the segment from e ɛ i to e ɛ i,wehave B ɛ i) { z z i <ɛ } { z z i < 3ɛ } B ɛ i).

16 1716 T. Mine, Y. Nomura / Journal of Functional Analysis ) By the mean value theorem, we have fz)= 1 2π 2π f z + 2ɛ e iθ) dθ for z B ɛ i). Notice that z + 2ɛ e iθ B ɛ i) \ B ɛ i). By the Schwarz inequality and the Fubini theorem, we have B ɛ i) fz) 2 dxdy 1 2π 2π B ɛ i) B ɛ i)\b ɛ i) f z + 2ɛ e iθ) 2 dxdydθ f z + 2ɛ e iθ) 2 dxdy. Thus 38) holds. Suppose the contrary of 27) holds. Then we have { 2B + kα/m 1 # = max), 2B + kα 1)/m 1 # = min). 39) Let u be the function 22) for a holomorphic function f. By Proposition 3.2, it suffices to prove u/ L 2 H; ω). By definition, we have where H u 2 ω = H y β ρz) fz) 2 ω, 4) { 2B + kα/m # = max), β = 2B + kα 1)/m # = min), { ρz) 2α # = max), ρz) = # = min). ρz) 2α 1) In both cases, we have β 1by39), and ρz) is periodic with respect to G-action. When # = max, ρ is bounded since G is co-compact, so we have inf z H ρz) >. Thus the integral 4) diverges by Lemma 4.2. When # = min, the function ρ has zeros on Γ, so we need a little modification. For sufficiently small ɛ>, let C, ɛ be the constants given in Lemma 4.3 with p = β.wetakeɛ and ɛ so small that {B ɛ γ )} γ Γ are mutually disjoint. Put Ω = γ Γ B ɛγ ). Since ρz) is G-periodic and has zeros only on Γ,wehave

17 T. Mine, Y. Nomura / Journal of Functional Analysis ) by the compactness of D. Suppose the integral 4) converges. By 41), we have H\Ω inf ρz) > 41) z H\Ω y β fz) 2 ω<. 42) Since β 1, we have by Lemma 4.2 Ω y β fz) 2 ω =. 43) However, Lemma 4.3 and 42) imply the left-hand side of 43) converges. This is a contradiction. Therefore i) of Theorem 1.5 is proved Infiniteness of the higher Landau eigenfunctions Lastly, we shall consider the case n 1, and prove multh B,min ; E B,n ) = under the assumption 1). Actually we prove an equivalent statement as follows. We assume B + 2παN D > πN n + 1), 44) D and prove that E B+n,n is an infinitely degenerated eigenvalue of H B+n,min. By Theorem 6.1,44) is equivalent to 2B kn + 1 α)/m > 1. 45) By Proposition 3.4, it suffices to prove the function u given by 26) belongs to L 2 H; ω) for infinitely many independent holomorphic functions f on H. Let us write down u more explicitly. Put v = y B Φz) α Φz) n+1 fz). 46) Then u = A B+n A B+1 v. The operator A B+1 is written as A B+1 = 2iy z B + iαyφ, where z = x i y )/2. Since φ = Φ /Φ = log Φ),wehave A B+1 v = 2iy n + 1 α)log Φ) + log f) ) v 2Bv = 2iyη 2B)v,

18 1718 T. Mine, Y. Nomura / Journal of Functional Analysis ) where η = n + 1 α)log Φ) + log f). By induction using the equality A B+j = A B+1 j 1), we can prove the function u = A B+n A B+1v is a finite linear combination of v and the terms of the form y j 1 j 1 1 z η ) y j l j l 1 z η ) v, 1 j 1 j l, 1 j 1 + +j l n. 47) We choose f = z + i) p for sufficiently large p. Then y j j 1 z η = n + 1 α)y j j z log Φ + y j j z logz + i) p. 48) For the second term of the right-hand side of 48), we have y j j z logz + i) p = pj 1)!y j z + i j pj 1)!. 49) In order to estimate y j j z log Φ = y j j z log Ψ /m, we prepare some lemmas. Notice that we do not use the assumption G is co-compact in the following lemmas. Lemma 4.4. Let G be a Fuchsian group of the first kind and Ψ an automorphic form of weight k. Then, for g = ) ab j cd G, z H, and j = 1, 2,...,the function z log Ψ )gz) is the sum of j 1)!k ccz + d) ) j and a finite linear combination of the terms of the form ) j lcz ccz + d) + d) 2l z l log Ψ )z), l = 1,...,j. Proof. Since Ψgz)= cz + d) k Ψz),wehave Since we have by differentiating the both sides of 5) log Ψgz)= k logcz + d)+ log Ψz). 5) z gz = z az + b cz + d = 1 cz + d) 2, 1 z log Ψ )gz) cz + d) 2 = kc cz + d + z log Ψz) z log Ψ )gz) = kccz + d)+ cz + d) 2 z log Ψz). This equality implies the assertion is true for j = 1. Then we can prove the assertion for j 2 by induction.

19 T. Mine, Y. Nomura / Journal of Functional Analysis ) Lemma 4.5. Suppose the same assumptions as in Lemma 4.4 hold. Then, for any j = 1, 2,..., there exists a constant C> independent of g G and z H such that z j log Ψ ) gz) 1 Im gz) j j 1)!k + C ) j Im z) l z l log Ψz) l=1 51) for any g G and z H. Proof. By the equality Im gz = Im z/ cz + d 2,wehave ) Im z 1/2 cz + d = 52) Im gz and ccz + d) = Imcz + d) cz + d 2 Im z cz + d Im z = 1 Im gz. 53) Then the conclusion follows immediately from 52), 53), and Lemma 4.4. For z H, we write z = gz g G, z D). Then we have by the G-periodicity of ρ vz) = y B kn+1 α)/2m) ρ z ) n+1 α z + i p. 54) By Lemma 4.5,wehave y j j z log Ψ ) z) j 1)!k + C j y l z l log Ψ z ), 55) where y = Im z.by54) and 55), the absolute value of 47) is bounded by a linear combination of v and the terms of the form l=1 y l 1+ +l p l 1 z log Ψ z ) l p z log Ψ z ) ρ z ) n+1 α y B kn+1 α)/2m) z + i p, 1 l 1 l p, 1 l 1 + +l p n. 56) Since Ψ has an m-th order zero at γ Γ, the function z l log Ψ has a pole of order l at γ. Since ρz ) n+1 α = O z γ n+1 α ) near z = γ, the first line of 56) converges to as z γ, and is bounded on D by the compactness of D. Thus we have u 2 Cy 2B kn+1 α)/m z + i 2p 57) for some positive constant C. By45), the right-hand side of 57) belongs to L 2 H; ω) for sufficiently large p, and the proof is completed.

20 172 T. Mine, Y. Nomura / Journal of Functional Analysis ) Cusp case When G has cusps, the most difficulty for the proof is the unboundedness of the G- periodic function ρz) = Φz) y k/2m). For example, if is a cusp, then the non-zero limit lim y Φz) exists see the q-expansion 85) below), and then ρz) = Oy k/2m) ) as y. Moreover, the G-periodicity of ρz) implies this function is unbounded at every cusp. So some parts of the proofs in the previous section need modifications Infiniteness of the lowest Landau modes Let us consider the proof in Section 4.1. We assume 28), and we need some upper bound of the function u, where u is the function given in 22). When # = max, the divergence of ρz) at cusps causes no problem, since u =y B+kα/2m) ρz) α fz) and the exponent α is negative. So the proof in Section 4.1 is applicable without modification. When # = min, we have u =y B+kα 1)/2m) ρz) 1 α fz). So we have to control the divergence of ρz) 1 α at cusps. To this purpose, let be the cusp form giveninassumption1.3. The function is an automorphic form of weight k which has no zero in H and has zero at every cusp. For any ɛ>, the function ɛ is defined as a single-valued holomorphic function on H. By the argument in the previous section, the function y k /2 z) is periodic with respect to G. Lemma 5.1. For any ɛ>, the function is bounded on H. Fz)= ρz) 1 α y ɛk /2 z) ɛ Proof. We already know Fz)is G-periodic, so we have to show Fz)is bounded on the fundamental domain D. It suffices to show lim Fz)= 58) z c for any cusp c. We may assume c =. Since Φ = Ψ 1/m, Ψ is an automorphic form, and is a cusp form, we have q-expansions Ψz)= a n q n, n= z)= b n q n, n=1 where q = e 2πiaz for some a>. These expansions imply Φ is bounded near z = and = Oexp 2πay)) as y. Thus we have 58).

21 T. Mine, Y. Nomura / Journal of Functional Analysis ) We choose the function f in 22)as for sufficiently small ɛ> and sufficiently large p. Thus we have f = ɛ z + i) p 59) uz) Cy B k1 α)/2m) ɛk /2 z + i p, 6) where C = max Fz). Ifwetakeɛ sufficiently small, we see u L 2 H; ω) by 28) and 6) Non-existence of the lowest Landau level Next, we assume 39) and consider the proof in Section 4.2. In this case, we need some lower bound of u. When # = min, the divergence of ρz) 1 α at cusps causes no problem. When # = max, the function ρz) α tends to as z tends to cusps. So the proof needs some modification. First consider the non-critical case, that is, 2B + kα m < 1. 61) In this case, we can write down u as u =y B+kα/2m)+ɛk /2 ρz) α y k /2 ɛ fz) for sufficiently small ɛ>, where fz)= fz) ɛ is a holomorphic function on H. Since the function ɛ diverges exponentially at cusps, we can cancel the decay of ρz) α at cusps. By 61), we can prove u/ L 2 H; ω) unless f =, as in the same way in Section 4.2. Next consider the critical case, that is, 2B + kα m = 1. 62) Then u =y 1/2 ρz) α fz). In this case, the proof in the non-critical case fails, so we need more detailed analysis of the function ρ. As stated in Theorem 1.6, we need an additional assumption B. 63) We assume 62), 63) and u L 2 H; ω), and prove u =. We shall consider the problem on the Poincaré disc D. We use the notation f g z) = f gz)cz + d) k for g = ) ab cd GL2, C). Then we have f g ) g = f gg ) for any g,g GL2, C). Letσ be the inverse of the Cayley transform given by 34). Put Ψ = Ψ σ, Φ = Ψ 1/m, and G = σ 1 Gσ. Put ρw) = ρσw)= ρz), then we have ρw) = 1 w 2) k/2m) Φw). 64)

22 1722 T. Mine, Y. Nomura / Journal of Functional Analysis ) Since ρz) is G-periodic, we see ρw) is G-periodic. By 35), 64), and ω = σ ω σ denotes the pull-back operator), we have H u 2 ω = H = D = D yρz) 2α fz) 2 ω 1 w 2 1 w 2 ρw) 2α fσw) 2 ω 1 w 2 ) ρw) 2α fw) 2 ω, 65) where fw) = fσw)/1 w). Notice that f is holomorphic on D. The assumption u L 2 H; ω) implies the integral 65) converges. We shall show f =. We use the following lemma. Lemma 5.2. Let C and r be positive constants with <r < 1. Let ηw) be a non-negative continuous function on D satisfying 2π η re iθ) dθ C log1 r) 1 66) for every r with r <r<1. Let f be a holomorphic function on D satisfying Then, f =. I = r < w <1 Proof. Consider the Taylor expansion of f 1 w 2 ) ηw) 1 fw) 2 ω<. 67) fw)= a n w n. n= By the Cauchy formula 2πia n = w =r fw)/w n+1 dw,66) and the Schwarz inequality, we have 2πr n a n 2π f re iθ) dθ C log1 r) 1) 1/2 2π η re iθ) 1 f re iθ) 2 dθ ) 1/2

23 T. Mine, Y. Nomura / Journal of Functional Analysis ) for r <r<1. By 67), we have 1 4π 2 r ) 4r 2n+1 1 r 2 dr a ) log1 r) 1 n 2 C 1 r 1 w 2 ) ηw) 1 fw) 2 ω = CI <. Since the first integral diverges, we have a n = for every n, and f =. Therefore it suffices to prove η = ρ 2α satisfies 66), that is, 2π ρ re iθ) 2α dθ C log1 r). 68) In order to prove 68), we have to analyze the asymptotic behavior of ρw) as w tends to cusps. Let x 1,...,x t D the closure of D in H) be a system of the complete representatives of the cusps of G. For each x j, choose h j SL 2 R) with x j = h j and fix it hereafter. For any cusp x, there exist unique x j and not unique) g G such that x = h and h = gh j. For sufficiently small ɛ>, put V ɛ, = { z Im z > 1/ɛ }, V ɛ,x = hv ɛ,. If h = ) ab cd, we can write down Vɛ,x explicitly { V ɛ,x = z } Im z cz + a 2 > 1/ɛ if x. 69) The set V ɛ,x is independent of the choice of g G with x = gh j.ifwetakeɛ sufficiently small, we have V ɛ,x V ɛ,x = for any two different cusps x and x. By the definition of the automorphic form, the function Ψ h has the q-expansion Ψ h z ) = a n q n, q = e 2πipz, n= where p is some positive constant. The right-hand side of the q-expansion depends only on the equivalence class of the cusp x. By iii) of Assumption 1.2, wehavea, so Ψ h z ) a in V ɛ,. 7) The notation 7) means there exists some positive constant C>1 independent of x and z such that C 1 a Ψ h z ) C a for any z V ɛ,. We use this notation also in the sequel. Since z = hz V ɛ,x,wehave

24 1724 T. Mine, Y. Nomura / Journal of Functional Analysis ) Ψz) = Ψ h ) h 1 z) a cz + a k in V ɛ,x, ρz) = y k/2m) Ψz) 1/m a 1/m y k/2m) cz + a k/m. 71) Next, let x = σ 1 x D be a cusp for the group G. Let For z = σw,wehaveby35) U ɛ, x = σ 1 V ɛ,x. Im z cz + a 2 = 1 w 2 ci + a)w + ci + a) 2. 72) We define a new coordinate w = a + ci)w/a ci) on D since a + ci)/a ci) =1, this is just a rotation). By 69) and 72), we have w U ɛ, x 1 w 2 A 1 w 2 > 1 ɛ w U ɛ, x, 73) where A = a ci 2 = a 2 + c 2 and U ɛ, x {w = D w A A + ɛ < ɛ }. A + ɛ Notice that the value A = Ax) is dependent on the cusp x, but is independent of the choice of h with h = gh j and x = h. The relation 73) means both U ɛ, x and U ɛ, x are discs tangent to the boundary of D.By71) and 72), we have 1 w ρw) a 1/m 2 ) k/2m) A 1 w 2 in U ɛ, x. 74) Let U ɛ = x:cusp U ɛ, x, which is the union of an infinite number of disjoint discs tangent to the boundary of D see Figs. 2, 3). Notice that U ɛ is invariant under the action of G. Bythe G-periodicity, ρ is bounded outside U ɛ. Thus we have C r U c ɛ ρw) 2α dθ 2π sup w U c ɛ ρw) 2α for every <r<1, where C r ={w w =r} and c denotes the complement set. Thus it suffices to show there exist C> and <r < 1 such that for r <r<1. I r = C r U ɛ ρw) 2α dθ C log1 r) 75)

25 T. Mine, Y. Nomura / Journal of Functional Analysis ) Fig. 2. The discs U ɛ, x for G = SL 2 Z). Fig. 3. The discs U ɛ, x for G = SL 2 Z) near w =.7 +.7i. In order to estimate I r, we have to estimate the counting function of Ax), that is, Nλ)= # { x: cusp of G Ax) λ }. Lemma 5.3. There exist C> and λ > dependent only on G, such that { Cλ λ λ ), Nλ) = λ < λ ). 76) Proof. First we show Ax) has positive infimum. By 73), the radius of U ɛ, x is ɛ/a+ ɛ). Since the discs U ɛ, x are disjoint, the radii have upper bound δ<1. So ɛ A + ɛ δ A δ 1 1 ) ɛ. Put λ = δ 1 1)ɛ. The above inequality means Nλ)= forλ<λ. 77) Next, let x 1,...,x n be all the cusps satisfying Ax) λ, and we may assume Ax 1 ) Ax n ). The number n = Nλ) is actually finite, since {U ɛ, xj } n j=1 are disjoint and their radii have

26 1726 T. Mine, Y. Nomura / Journal of Functional Analysis ) lower bound ɛ/λ + ɛ). ForadiscU D tangent to D, letl r U) be the length of the arc U C r. By a simple geometric consideration, we see that l r U) is monotone non-decreasing function with respect to the radius of U. Taker<1 so that the circle C r passes through the two endpoints of a diameter of U ɛ, xn. Then we have Since {U ɛ, xj } n j=1 are disjoint, we have l r U ɛ, x1 ) l r U ɛ, xn ) 2ɛ λ + ɛ. Nλ) 2ɛ πλ 2πr 2π Nλ) λ + ɛ ɛ + π. This inequality and 77) imply the conclusion with C = πɛ 1 + λ 1 ). Let us begin the proof of 75). Put s = 1 r. By73), we have C r U ɛ, x r> A ɛ A + ɛ 1 + r 1 r ɛ>a. Thus it suffices to show Ax) 2ɛ/sC r U ɛ, x ρw) 2α dθ C log s. Since the number of equivalence classes of cusps is finite, we can ignore the term a 1/m in the asymptotics 74), and it suffices to show Ax) 2ɛ/s 1 A β C r U ɛ, x 1 w 2 ) β dθ C log s, 78) 1 w 2 where β = αk/m. Let us estimate the integral. If we write w = u + iv, then the condition w C r U ɛ, x is equivalent to u A ) 2 + v 2 < A + ɛ ɛ 2 A + ɛ) 2, u2 + v 2 = r 2. Eliminate v in this equation and put r = 1 s;wehave u>1 1 + ɛ ) s ɛ ) s 2 > ɛ ) s. A 2 A A Substituting this inequality into v 2 = 1 s) 2 u 2,wehave ) 2ɛ v 2 A + ɛ2 A 2 s 2 + 2ɛ A 2ɛ s s. 79) A

27 T. Mine, Y. Nomura / Journal of Functional Analysis ) Next, r sin θ = v implies θ = sin 1 v/r), and we have by 79) for s<1/2, where C = 1/4 ɛ/m) 1/2. Moreover, dθ dv = 1 C 8) 1 s) 2 v2 1 w 2 2s 1 w 2 s 2 + v 2. 81) By 79), 8) and 81), we see that the summand in 78) is bounded by 2 β+1 C 1 A β 2ɛs/A s ) β dv = 2 β+1 Cs 1 β 1 s 2 + v 2 A β 2ɛ/sA) By integration by parts, the LHS of 78) is bounded by a constant times ) 1 β 1 + t 2 dt. s 1 β 2ɛ/s λ 1 λ β = s 1 β N + 2ɛ/s λ + 2ɛ/s λ 2ɛ/sλ) ) 2ɛ s s 2ɛ Nλ)β ) 1 β 1 + t 2 dt dnλ) 1 λ β+1 ) 1 β 2ɛ/sλ) ) 1 β 1 + t 2 dt ) 1 β 1 + t 2 dt dλ ) 1 1 β Nλ) 2ɛ/s1/2)λ dλ) 3/2 λ β. 82) 1 + 2ɛ/sλ) By using 76) and putting λ = 2ɛ/s)k, RHSof82) is bounded by a constant times k β λ s/2ɛ) 1/k ) 1 β 1 + t 2 dt dk + 1 λ s/2ɛ) k 1/2 1 + k) β dk. The third term is bounded with respect to s. For the second term, we use 1/k ) 1 β 1 + t 2 dt C β >1/2), = log 1/k /k) β = 1/2), Ck β 1/2 β < 1/2),

28 1728 T. Mine, Y. Nomura / Journal of Functional Analysis ) and obtain 1 λ s/2ɛ) k β 1/k ) 1 β { C log s β = 1), 1 + t 2 dt dk C <β<1) as s. By the assumptions 62) and 63), we have <β= kα/m 1 this is the only part we need the assumption 63)), so we have the conclusion Infiniteness of the higher Landau modes Next we shall prove Theorem 1.7. Again we take f in 26) as59). Define v by 46) and put η = n + 1 α)log Φ) + log f) + ɛlog ). Then, u is written as a finite linear combination of the terms of the form 47), and we have instead of 48) y j j 1 z η = n + 1 α)y j j z log Φ + y j j z logz + i) p + ɛy j j z log. 83) For the first term of 83) and the second, we can still use the estimates 51) and 49). Moreover, we can apply Lemma 4.5 for the function, and obtain z j log ) gz) 1 Im gz) j k j 1)!+C ) j Im z) l z l log z). Thus the term 47) is bounded by a linear combination of v and the terms of the form l=1 y l 1+ +l p +l 1 + +l q +kn+1 α)/2m)+k ɛ/2 l 1 z log Φ z ) l p z log Φ z ) Φ z ) n+1 α l 1 z log z ) l q z log z ) z ) ɛ y B kn+1 α)/2m) ɛk /2 z + i p, 1 l 1 l p, 1 l 1 l q, 1 l 1 + +l p + l 1 + +l q n. 84) We shall show the product of the first three lines of 84) is bounded on D. Then, we have u L 2 H; ω) for sufficiently small ɛ and large p. Since the singularities come from l j z log Φz ) are canceled by Φz ) n+1 α and has no zero, it suffices to prove the product is bounded near the cusps. For the cusp x D the closure as a set in H = H R { }), take h and V ɛ,x as 69), and assume ɛ is sufficiently small so that Ψ has no zero in V ɛ,x. Put z = hz for z V ɛ,. By differentiating the equality log Ψ hz ) = k log cz + d ) + log Ψ h z ),

29 T. Mine, Y. Nomura / Journal of Functional Analysis ) we can prove an estimate like 51) Im z ) j z j log Ψ ) z ) j j 1)!k + C Im z ) l l z log Ψ h z ), and a similar estimate for. Since Im z ) l h z ) ɛ is bounded for any l, it suffices to prove j z log Ψ h z ) and j z log h z ) are bounded for z V ɛ,. The functions Ψ h and h have q-expansions n= l=1 Ψ h z) = a n q n, h z) = b n q n for z V ɛ,, where q = e 2πiaz. Since z = z q q = 2πiaq q,wehave n=1 z Ψ h = 2πia na n q n, n=1 z h = 2πia nb n q n. n=1 Thus z log Ψ h = z Ψ h /Ψ h and z log h = z h / h are holomorphic with respect to q near q =. This also implies j z log Ψ h and j z log h are holomorphic with respect to q near q =, and thus bounded in V ɛ,. This completes the proof. 6. Sufficient conditions for G to satisfy Assumptions 1.2 and 1.3 In this section, we shall review some definitions and known facts about the automorphic forms for the convenience of the readers. After that, we give some examples of Fuchsian groups satisfying Assumptions1.2 and1.3. Forthe reference, see e.g. Ford [14], Shimura [33], Shimizu [32], or Iwaniec [2] Definition of the automorphic forms Let { ) } a b SL 2 R) = a,b,c,d R, ad bc = 1. c d For g = ) ab cd SL2 R) and z H, we define the action of g on H or on H = H R { }) by the linear fractional transformation gz = az + b)/cz + d). The group SL 2 R) acts on H transitively. The Poincaré metric ds 2 = y 2 dx 2 + dy 2 ) and the surface form ω = y 2 dx dy are invariant under this action. Let ι : SL 2 R) PSL 2 R) = SL 2 R)/{±1} be the canonical projection map. An element g SL 2 R) \{±1} is called elliptic if tr g < 2, parabolic if tr g =2, hyperbolic if tr g > 2. A discrete subgroup G of SL 2 R) is called a Fuchsian group of the first kind if the quotient set G \ H has finite hyperbolic area, and co-compact if G \ H is compact. For a Fuchsian group G of the first kind, we say a closed subset D of H is a fundamental domain of G if the following i) iii) hold:

30 173 T. Mine, Y. Nomura / Journal of Functional Analysis ) i) H = g G gd. ii) The sets {g D} g G are disjoint, where D is the interior of D. iii) The boundary of D consists of a finite number of geodesics. The group G is co-compact if and only if there is a compact fundamental domain D. For z H, put G z ={g G gz = z}, e z = #ιg z ). The number e z is called the order of z with respect to G. A point z H is called a fixed point of G if e z 2. We say a fixed point z in H is called an elliptic point of G if G z contains an elliptic element. If z is an elliptic point of G, then ιg z ) is a finite cyclic group. A fixed point z is called a cusp if G z contains a parabolic element. The cusps are contained in R { }. Let G be a Fuchsian group of the first kind, m an integer. For g = ) ab cd G and a function fz)on H, put f g z) = f gz)cz + d) m. Then f g ) g z) = f gg z) holds for any g,g G. We call f a meromorphic automorphic form of weight m on the group G if the following i) iii) hold: i) f is a meromorphic function on H. ii) f g z) = fz)for every g G and z H. iii) f is meromorphic at every cusp. We denote the set of the meromorphic automorphic forms by Am, G). The meaning of iii) above is the following. For a cusp x, we can take some σ SL 2 R) such that σ =x. Then the group σ 1 G x σ fixes, so this group is generated by some element ± ) 1 r 1 for some r>. Let f be a function satisfying i) and ii). Since f σ ) σ 1gσ = f σ for every g G x,wehave f σ z) = f σ r + z) by ii). So f σ is analytic with respect to the variable q = e 2πiz/r in the annulus { < q <ɛ} for some ɛ>. Thus f σ is expressed as the q-expansion near q = : f σ z) = n= Let ν x f ) be the order of f σ with respect to q at q =, that is, ν x f ) = inf{n Z a n }. a n q n. 85) The number ν x f ) depends only on the equivalence class of x with respect to the G-action. We say f is meromorphic at x if ν x f ) >, holomorphic at x if ν x f ), and zero at x if ν x f ) >. For f Am, G), we call f an automorphic form if f is holomorphic on H and at every cusp of G. An automorphic form f is called a cusp form if f is zero at every cusp of G.We

31 T. Mine, Y. Nomura / Journal of Functional Analysis ) denote the set of the automorphic forms by Gm, G), and the set of the cusp forms by Sm,G). 4 For f Am, G) and z H, we also define ν z f ) as the order of f at z = z with respect to z. The number ν z f ) also depends only on the equivalence class of z.letz 1,...,z s D be a system of complete representatives of the zeros and the poles of f with respect to G-action), and z 1,...,z t D the closure of D in H) a system of complete representatives of the cusps. We define the number N by N = Nf)= s j=1 ν zj f ) e zj + t ν z j f ). 86) The following theorem is a direct consequence of [14, Section 49, Theorem 4] 5 and the area formula for the hyperbolic triangle with angles α, β, γ. S = π α β γ Theorem 6.1. Let G be a Fuchsian group of the first kind, D a fundamental domain of G, m an integer, f a meromorphic automorphic form of weight m, and N given by 86). Then we have 6.2. Riemann Roch theorem N = m D 4π. We shall introduce the notion of differential, then the existence of automorphic forms is equivalent to the corresponding differential. Let G be a Fuchsian group of the first kind. We denote by H the union of H and cusps of G and define a topology of H as follows: j=1 i) The topology of H as the subspace of H is the usual one. ii) For a cusp x = σ σ SL 2 R)), we take all sets of the form as an open basis around x. Ux,λ)={σz Im z>λ} {x} λ > ) We can define the complex structure on G \ H = RG) and RG) become a compact Riemann surface see e.g. Shimura [33, 1.3, 1.5]). Let π be the canonical projection map from H to RG). For P = πz) z H ) and non-zero f Am, G), we define e P = e z z H) 4 If G is co-compact, then Sm,G) = Gm, G). 5 Our weight m corresponds to the number 2m in Ford s book. See the definition of the theta function in [14, Section 45, 4)]. Essentially the same assertion is stated in the proof of [33, Theorem 2.2].