Theory and Related Topics) 数理解析研究所講究録別冊 = RIMS Kokyuroku Bessa (2010), B16:

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RIMS Kokyuroku essa (2010), 16: 135-140 Issue Date 2010-04 URL http://hdl.handle.

1 The Spectrum of Schrodinger TitleAharonov-ohm magnetic fields operato (Spec Theory and Related Topics) Author(s) MINE, Takuya; NOMURA, Yuji Citation 数理解析研究所講究録別冊 = RIMS Kokyuroku essa (2010), 16: Issue Date URL Right Type Departmental ulletin Paper Textversion publisher Kyoto University

2 RIMS Kôkyûroku essatsu 16 (2010), The spectrum of Schrödinger operators with periodic Aharonov-ohm magnetic fields y Takuya MINE and Yuji NOMURA Abstract We shall consider the magnetic Schrödinger operators on R 2. The magnetic field is the sum of a non-zero uniform magnetic field and periodic pointlike magnetic field on a lattice. In [1], we gave a sufficient condition for each Landau level to be a infinitely degenerated eigenvalue. This condition is also necessary for the lowest Landau level. Moreover, in the threshold case, the spectrum near the lowest Landau level is purely absolutely continuous. In this paper, we shall characterize the eigenfunctions corresponding to the second Landau level and show the absolutely continuity of the spectrum near the second Landau level in the threshold case. 1. Introduction We identify a vector z = (x, y) R 2 with a complex number z = x + iy in the sequel. We consider a magnetic Schrödinger operator on R 2 L = ( ) 2 1 i + a, where a = (a x, a y ) is the magnetic vector potential. rot a = x a y y a x satisfies We assume the magnetic field (1.1) rot a(z) = + γ Γ 2παδ γ (z) Received April 1, Revised July 23, Mathematics Subject Classification(s): 81Q10, 35P15, 35Q40, 47F05, 47N50 Key Words: Schrödinger operator; periodic magnetic field; Aharonov ohm magnetic field; Landau level Supported by JSPS grant Wakate and JSPS grant Kiban C Department of Comprehensive Sciences, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto , Japan. mine@kit.ac.jp Department of Computer Science, Graduate School of Science and Engineering, Ehime University, 3 unkyo-cho, Matsuyama, Ehime , Japan. nomura@cs.ehime-u.ac.jp c 2010 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

3 136 T. Mine and Y. Nomura in the distribution sense, where is a positive constant, α is a real number satisfying 0 < α < 1, Γ is a lattice of rank 2 in R 2 and δ γ is the Dirac measure concentrated at γ Γ. The vector potential a satisfying (1.1) is constructed as follows: a(z) = (Im ϕ(z), Re ϕ(z)), (1.2) ϕ(z) = 2 z + αζ(z), where ζ(z) is the Weierstrass ζ function associated with Γ: ζ(z) = 1 z + ( 1 z γ + 1 γ + z ) γ 2. We define a operator L by γ Γ\{0} Lu = Lu, u D(L) = C 0 (R 2 \ Γ). L is a positive symmetric operator on L 2 (R 2 ). We denote the Friedrichs extention of L by H. It is shown that (1.3) D(H) = {u L 2 (R 2 ) H 2 loc(r 2 \ Γ) Lu L 2 (R 2 ), lim u(z) = 0 for any γ Γ}. z γ From the inequality H, it follows that σ(h) [, ), where σ(h) is the spectrum of H. We prepare some notations. Let H 0 be the Schrödinger operator with constant magnetic field. It is well-known that where σ(h 0 ) = {E n n = 1, 2, }, (1.4) E n = (2n 1) which is an infinitely degenerated eigenvalue and is called the nth Landau level. Let ω 1, ω 2 R 2 be a basis of the lattice Γ, that is Γ = ω 1 Z ω 2 Z and we assume Im(ω 2 /ω 1 ) > 0. The set Ω denotes a fundamental domain of Γ defined by { Ω = z = sω 1 + tω s < 1 2, 1 2 t < 1 }. 2 The condition (1.5) 2π Ω + α Q

4 The spectrum of Schrödinger operators with periodic Aharonov-ohm magnetic fields 137 is called the rational flux condition. The number of the left-hand side is the magnetic flux in a fundamental domain divided by 2π. For an interval I R, P I (H) denotes the spectral projection of H corresponding to I. We define an entire function σ by (( σ(z) = z 1 z ) ) e z γ + z 2 2γ 2, γ which satisfies the equation γ Γ\{0} (1.6) σ (z) = ζ(z)σ(z). Our results in [1] are the followings. Theorem 1.1 ([1]). Assume > 0 and 0 < α < 1. (i) If 2π Ω + α > 1, then E 1 = is an infinitely degenerated eigenvalue of H, and the eigenspace of H associated with the eigenvalue E 1 is { } (1.7) e 4 z 2 σ(z) α σ(z)f(z) L 2 (R 2 ) f is an entire function. (ii) If 2π Ω + α < 1, then E 1 is not an eigenvalue of H. If we additionally assume the rational flux condition (1.5), then there exists a positive number ε such that σ(h) [ + ε, ). (iii) If 2π Ω + α = 1, then E 1 is not an eigenvalue of H, and E 1 is the edge of the purely absolutely continuous spectrum, i.e. there exists a constant E such that < E 3, [, E) σ(h) and Ran P [,E) (H) H ac, where H ac denotes the absolutely continuous subspace for the operator H. Theorem 1.2 ([1]). Assume > 0 and 0 < α < 1. If Ω + α > n for some 2π positive integer n, then E n is an infinitely degenerated eigenvalue of H. Following theorem is our main result in this paper. Theorem 1.3. Assume > 0 and 0 < α < 1. (i) If 2π Ω + α > 2, then E 2 is an infinitely degenerated eigenvalue of H, and the eigenspace of H associated with the eigenvalue E 2 is { (1.8) A ( ) } e 4 z 2 σ(z) α σ(z) 2 f(z) L 2 (R 2 ) f is an entire function, where (ii) If A = 2 z + ϕ(z). 2π Ω + α = 2, then E 2 is not an eigenvalue of H and E 2 σ ac (H), where σ ac (H) is the absolutely continuous spectrum of H.

5 138 T. Mine and Y. Nomura 2. Outline of the proof of Theorem 1.1 Proposition 2.1. We have Hu = E 1 u for u D(H) if and only if (2.1) u(z) = e 4 z 2 σ(z) α σ(z)f(z) for some entire function f(z) and u(z) L 2 (R 2 ). Proof. We define an operator A by A = 2 z + ϕ(z), where z = 1 2 ( x i y ). From (1.6), we have (2.2) A = e 4 z 2 σ(z) α (2 z )e 4 z 2 σ(z) α. Then three operators A, A and L satisfy the following relations: L = AA + (2.3) = A A. Since H is the Friedrichs extension of L, (2.3) implies that (2.4) ((H )u, u) = (A Au, u) = Au 2 for u D(H). Therefore we have that Hu = u if and only if (2.5) Au = 0 in C \ Γ for u D(H). y (2.2), any solution of (2.5) is written as u(z) = e 4 z 2 σ(z) α h(z), where h(z) is a holomorphic function on C\Γ. From the boundary conditions lim z γ u(z) = 0 for any γ Γ, the function h(z) must be factorized as h(z) = σ(z)f(z) with an entire function f(z). We denote η j = ζ ( ωj 2 ) for j = 1, 2. Then we have the Legendre relation: (2.6) η 1 ω 2 η 2 ω 1 = 2πi,

6 The spectrum of Schrödinger operators with periodic Aharonov-ohm magnetic fields 139 and (2.7) σ(z + γ) = ( 1) m+n+mn e η(z+ γ 2 ) σ(z), where γ = mω 1 + nω 2 and η = mη 1 + nη 2 for m, n Z (see, e.g. [3]). Outline of proof of Theorem 1.1. (i) Let f(z) = P (z)e (α 1)µz2, where P (z) is an arbitrary polynomial and µ = i 4 Ω (η 1ω 2 η 2 ω 1 ). Let u be the function given by (2.1) with the above function f. y (2.6) and (2.7), we have u(z) Q(z)e d z 2, where Q(z) is some function of polynomial order and d = π(1 α) +. Since d is 4 2 Ω negative by the assumption, the solution u belongs to L 2 (R 2 ). Thus E 1 is an infinitely degenerated eigenvalue of H. (ii) Let f(z) be an arbitrary entire function which is not identically equal to 0 and let u(z) = e 4 z 2 σ(z) α σ(z)f(z). Putting g(z) = f(z)e (α 1)µz2, we have (2.8) u(z) Ce d z 2 g(z) for z satisfying dist(z, Γ) > ϵ for some ϵ > 0. Since d is positive by the assumption, the solution u dose not belong to L 2 (R 2 ) by (2.8). 3. Outline of the proof of Theorem 1.3 y using of the following Proposition 3.1 we can prove Theorem 1.3. Proposition 3.1. we have Assume 2π Ω + α 2. If Hu = E 2u for u D(H), then A 2 u = 0. For a > 0 and w C, let a (w) = {z C z w < a} and D a = C \ γ Γ a (γ).

7 140 T. Mine and Y. Nomura For sufficiently small δ > δ > 0 and ϵ > 0, we define D n δ = D δ Q n and D n δ = D δ Q n+ϵ, where Q r = {z C Re z < r + 1 2, Im z < r }. Proposition 3.1 is derived from the following Lemmas. Lemma 3.2. If Lf = E 2 f for f H 2 loc (R2 \ Γ), then we have A 2 f L2 (D n δ ) C f L2 ( D n δ ) for some C > 0 which is independent of f and n N. Lemma 3.3. Assume 2π Ω + α > 2. If Hu = E 2u and A 2 u 0, then there exists a constant c 1 > 0 such that A 2 u L2 (Dδ n) e c 1n 2 for any n N. Lemma 3.4. Assume 2π Ω + α = 2. If Hu = E 2u and A 2 u 0, then there exists a constant c 2 > 0 such that A 2 u L2 (Dδ n) c 2 n for any n N. References [1] T. Mine and Y. Nomura, Periodic Aharonov-ohm solenoids in a constant magnetic field, Rev. Math. Phys. 18, no. 8, (2006) [2] T. Mine and Y. Nomura, The spectrum of Schrödinger operators with periodic Aharonov- ohm magnetic fields, in preparation. [3] A. Hurwitz and R. Courant, Vorlesungen uber Allgemeine Funktionentheorie und Elliptische Funktionen, Springer, 1998.

数理解析研究所講究録別冊 = RIMS Kokyuroku Bessa (2011), B25:

Non-existence of certain Galois rep Titletame inertia weight : A resume (Alg Related Topics 2009) Author(s) OZEKI, Yoshiyasu Citation 数理解析研究所講究録別冊 = RIMS Kokyuroku Bessa (2011), B25: 89-92 Issue Date 2011-04