Abstract. i x + a x +. a = (a x, a y ) z γ + 1 γ + z )

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1 Abstract R, Aharonov-Bohm Schrödinger Landau level Aharonov-Bohm Schrödinger 1 Aharonov-Bohm R Schrödinger ( ) ( ) 1 1 L a = i + a = i x + a x + ( ) 1 i y + a y (1). a = (a x, a y ) rot a = ( x a y y a x )(z) = B + γ Γ παδ(z γ), z R () 0 B, 0 α < 1, Γ R., z = x + iy C (x, y) R Γ = Zω 1 Zω, Im (ω /ω 1 ) > 0 Γ Weierstrass ζ ζ(z) = 1 z + ( 1 z γ + 1 γ + z ) (3) γ γ Γ\{0} a = (a x, a y ) = (Im φ(z), Re φ(z)), φ(z) = B z + αζ(z) (4), a ( 0 < α < 1 ), L a L a u = L a u, Dom(L a ) = C 0 (R \ Γ)

2 L a (L a u, u) B u L a Friedrichs H a H a Aharonov-Bohm Schrödinger H a H a u = L a u in D (R \ Γ), Dom(H a ) = {u L (R ) La u L (R ), lim z γ u(z) = 0 for any γ Γ}. (5) Schrödinger 1). (4) α = 0, rot a = B L a C 0 (R ) H B Schrödinger H B σ(h B ) σ(h B ) = {E n = (n 1)B n = 1,, } E n n Landau level L n E n L 1 ( Landau level B ) = {u L (R ) u = e B 4 z h(z) for some entire function h} ) j+1e φ j (z) = B 4 z z j {φ j } j=0,1,, L 1 1 j!π ( B L 1 P 1 K 1 K 1 (z, w) = B π e B 4 ( z + w zw) = B π e B 4 ( z w iim(zw)) A B = z + B z, A B = z + B z L n+1 = A B L n = (A B )n L 1 A B L n+1 = L n { L n (B) n 1 } 1 (A B )n 1 φ j (n 1)! L n P n K n K n (z, w) = K 1 (z, w)l n 1 ( B z w ) j=0,1,, l n (x) = n k=0 nc k ( x) k k! Laguerre

3 ) + 1 δ ( [Na], Exner-Št ovíček-vytřas [E S V]) rot a = B + παδ(z) {E n, E n + αb} n=1 E n, E n + αb n 3) Aharonov-Bohm[A B] rot a = παδ(z) Alexandrova- [A-T], - [I T] δ δ,. Ω Γ. Theorem 1. ([M N1]) 0 < B, 0 < α < 1.. (1)(i) B π Ω + α > 1 E 1 = B { e B 4 z σ(z) α σ(z)f(z) L (R ) f : entirefunction }. σ(z) = z γ Γ\{0} (( 1 z ) ) e z γ + z γ γ (ii) B Ω + α = 1 B < E 3B E, σ(h) [B, 3B) = π [B, E] [B, 3B),. (iii) B π Ω + α < 1 E 1 = B, B Ω + α π, B < inf σ(h).. () n B π Ω + α > n, E n (3)R = min γ Γ\{0} γ n 0 N R 0 > 0, c > 0 R > R 0 σ(h) [B, E n0 +1) = n 0 n=1 ({E n } S n ), S n [E n + αb e cr, E n + αb + e cr ] ρ(s n ) = n Ω, B π n Ω ρ(e n) B π n 1 Ω

4 n = 1,, n 0 ρ Landau level Theorem. 0 < B, 0 < α < 1.. (1) B π Ω + α > E = 3B ( ) {A e B 4 z σ(z) α σ(z) f(z) L (R ) } f : entirefunction. A = z + φ () B π Ω + α =, E = 3B Aharonov-Bohm Anderson type Aharonov-Bohm Schrödinger ( ) 1 H ω = i + a ω, rot a ω = B + πα γ (ω)δ(z γ). α γ (ω) [0, 1), {α γ (ω)} γ Γ µ, (4) ζ ζ ω (z) = α 0(ω) z + γ Γ\{0} γ Γ ( 1 α γ (ω) z γ + 1 γ + z ) γ Σ R σ(h ω ) = Σ a.s. Theorem 3.([M N]) (1) (i) supp µ ({0} {1}) σ(h B ) Σ B = 0 Σ = [0, ) (ii) supp µ ({0} {1}) = B = 0 0 < inf Σ. () n N (i) B π Ω + E[α γ] > np(α γ 0) E n a.s. (ii) B π Ω + E[α γ] < P(α γ 0) E 1 a.s. (6)

5 3 Aharonov-Bohm ds = y (dx + dy ) H = {z C Imz > 0} L (H, ω) Schrödinger ω = y dx dy 1-form) a = a x dx + a y dy L a = y { (D x + a x ) + (D y + a y ) } Schrödinger D x = 1 i x, D y = 1 i y (-form) da = ( x a y y a x )dx dy a = B dx (B R : ) da = Bω y L B = y { ( D x + B y ) + D y H B L B C 0 (H) H B Maass Laplacian (e.g. Maass[M], Roelcke[R], Elstrodt[E], - [I S]). { [B σ(h B ) =, ) ( B 1), N(B) n=0 {E n} [B + 1, ) ( B > 1) (7) 4 N(B) B 1 E n = (n + 1) B n(n + 1) H Γ = SL (Z) i weight 6 Eisenstein G 6 (z) = (m,n) Z \(0,0) } 1 (mz + n) 6 ϕ(z) = α G ( ) 6(z) B G 6(z), a = y + Im ϕ dx + (Re ϕ)dy (8) Aharonov-Bohm da = Bω + γ Γ παδ γ dx dy

6 SL (Z) Schrödinger H a D SL (Z) Theorem 5. ([M N3]) B 0, 0 < α < 1 N = 1 (1) n N B + παn > 1 + n + πn (n + 1) E n H a () B + παn E 0 = B H a 1 + πn B + παn N w H Γ = SL (Z) w (8) ϕ ϕ(z) = (z)(j(z) j(w)), j(z) = (60G 4(z)) 3 (z) ( Γ = SL (Z) i, Γ = SL (Z) ρ ϕ G 6 (z), G 4 (z) ρ = 1+ 3i ). (z) (The discriminant modular form) j(z) j-invariant SL (R) G (i) G \ H (ii)g = Γ 0 (11) ( 11 References [A B] Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev., 115 (1959), [E] J. Elstrodt, Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, I-III, Math. Ann. 03 (1973), ; Math. Z. 13 (1973), ; Math. Ann. 08, (1974), [E S V] P. Exner and P. Št ovíček and P. Vytřas, Generalized boundary conditions for the Aharonov-Bohm effect combined with a homogeneous magnetic field, J. Math. Phys., 43 (00), no. 5,

7 [I S] Y. Inahama and S. Shirai, The essential spectrum of Schrodinger operators with asymptotically constant magnetic fields on the Poincare upper-half plane, J. Math. Phys., 44 (003), no. 1, [I T] H.T. Ito and H. Tamura, Semiclassical analysis for magnetic scattering by two solenoidal fields, J. London Math. Soc., () 74 (006), no. 3, [L] B. J. Levin, Distribution of zeros of entire functions, American Mathematical Society, [M] H. Maass, Die Differentialgleichungen in der Theorie der elliptischen Modulfunktionen, Math. Ann., 15 (1953), [M N1] T. Mine and Y. Nomura, Periodic Aharonov-Bohm solenoids in a constant magnetic field, Rev. Math. Phys., 18 (006), no. 8, [M N] T. Mine and Y. Nomura, The spectrum of Schrödinger operators with random δ magnetic fields, Ann. Inst. Fourier, Grenoble, 59, (009) [M N3] T. Mine and Y. Nomura, Landau levels on the hyperbolic plane in the presence of Aharonov-Bohm fields, submitted [Na] Y. Nambu, The Aharonov-Bohm problem revisited, Nuclear Phys. B, 579 (3) (000), [A-T] I. Alexandrova and H. Tamura, Aharonov-Bohm effect in resonances of magnetic Schrodinger operators with potentials with supports at large separation, Ann. Henri Poincare, 1 (011), no. 4, [R] W. Roelcke, Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I, II, Math. Ann., 167 (1966),

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