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1 207 : msjmeeting-207sep-07i00 ( ) Abstract 989 Korotyaev Schrödinger Gérard Laba Multiparticle quantum scattering in constant magnetic fields - propagator ( ). ( ) 20 Sigal-Soffer [22] 987 Gérard- Laba [9] [0] Adachi-Tamura [4] [5] Gérard- Laba Adachi [] Skibsted [2] 0 2 Møller [8] Adachi [2] 0 2 Yajima [23] 3 Nakamura [20] Møller-Skibsted [9] (Mourre ) Nakamura Korotyaev [5] Mathematics Subject Classification: 8Q05. Keywords: Magnetic fields, scattering theory, time-periodic quantum system. mkawa@rs.tus.ac.jp

2 2. R 2 B(t) = (0, 0, B(t)) x = (x, x 2 ) R 2, p = i = ( i, i 2 ) m > 0, q R\{0} H 0 (t) H 0 (t) = 2m (p qa(t, x))2, A(t, x) = B(t) 2 ( x 2, x ). L 2 (R 2 ) A(t, x) V H(t) = H 0 (t) + V V V (x) V (x) (V) : (V) V C(R 2 ) σ > 0 C > 0 V (x) C( + x ) σ H 0 (t) propagator U 0 (t, 0), H(t) propagator U(t, 0) U(t, 0) U 0 (t, 0) B(t) B(t) { B t I B, B(t) = I B := [nt, nt + T B ), I 0 := + T B, (n + )T ), 0 t I 0, n Z n Z[nT B > 0, 0 < T B < T t I B B = (0, 0, B) ( on ) t I 0 ( off ) B(t + T ) = B(t), t on, off H B, Schrödinger H 0 H B := 2m (p qa(x))2, A(x) = B 2 ( x 2, x ), H 0 := p2 2m H 0 (t) H 0 (t) = { H B t I B, H 0 t I 0

3 U 0 (t, 0) { e i(t nt )H B U 0 (nt, 0) t [nt, nt + T B ), U 0 (t, 0) = e i(t nt T B)H 0 e it BH B U 0 (nt, 0) t [nt + T B, (n + )T ). off T 0 := T T B U 0 (T, 0) = e it 0H 0 e it BH B, U 0 (nt, 0) = (U 0 (T, 0)) n U 0 (t, 0) U 0 (T, 0) ω = qb m, ω = ω 2, ω = ω 2 = ω 4. ω Larmor Lemma 2.. t 0 S 0 0(t; x) S B 0 (t; x, y) S0(t; 0 x) := /(2t) 2πit eix2, S0 B m ω (t; x, y) := eim ω cot( ω t)x2 /2 e im ω ( ˆR( ωt)x) y/ sin( ω t) e im ω cot( ω t)y2 /2 2πi sin( ω t) e ith 0 ψ = S0(t; 0 x y)ψ(y)dy, R 2 e ith B ψ = S0 B (t; x, y)ψ(y)dy R 2 ( cos θ sin θ ˆR(θ) = sin θ cos θ H B Avron-Herbst-Simon [6], Adachi-Kawamoto [3] U 0 (T, 0) S 0 (T ; x, y) S 0 (T ; x, y) = 2πic θ e ix2 /(2θ ) e i( ˆR(ϕ )x) y/(c θ ) e iσ y2 /(2θ ), c, σ, θ, ϕ (2) L, L 2, L 2, L 22 θ = L 2 L 22, c = L 22, σ = L L 22, ϕ = ωt B. )

4 U 0 (nt, 0) = (U 0 (T, 0)) n S 0 (nt ; x, y) S 0 (nt ; x, y) = 2πic n θ n e ix2 /(2θ n) e i( ˆR(ϕ n)x) y/(c nθ n) e iσny2 /(2θ n). () c n, σ n θ n, ϕ n ( = θ n+ c 2 σ = c n+ θ n+ ( σ n+ = σ n θ n+ c 2 n ϕ n+ = ϕ + ϕ n. ) θ + c σ c n (θ /σ + θ n ), ) + θ n (c σ ) 2 (θ /σ + θ n ), c 2 n(θ /σ + θ n ), ϕ n, θ n ϕ n = n ωt B θ n θ n θ n+ = L 2θ n + L 22 L θ n + L 2, L = cos( ω T B ) ω T 0 sin( ω T B ), L 2 = m ω ( ω T 0 cos( ω T B ) + sin( ω T B )), (2) L 2 = m ω sin( ω T B ), L 22 = cos( ω T B ) α ± α = L α + L 2 L 2 α + L 22 D D = λ 2 0, λ 0 := L + L 22 2 = cos( ω T B ) 2 ω T 0 sin( ω T B ), λ ± := λ 0 ± λ 2 0, D > 0 µ n := λn + λ n λ + λ = λn + λ n 2 λ 2 0 (3) L 2 µ n θ n = L 22 µ n µ n

5 D off : T 0,cr := cos( ωt B ) ω sin( ωt B ), T 0 T 0,cr λ 0 + 0, i.e. D = λ 2 0 0, 0 < T 0 < T 0,cr (λ 0 + )(λ 0 ) < 0, i.e. D = λ 2 0 < 0. T 0,cr Theorem 2.2 (Adachi- K). propagator U 0 (nt, 0) () T 0 T 0,cr θ n = L 2 µ n L 22 µ n µ n, c n = L 22 µ n µ n, σ n = L µ n µ n L 22 µ n µ n, ϕ n = n ωt B T 0 = T 0,cr θ n = nl 2 nl 22 (n )λ 0, σ n = nl (n )λ 0 nl 22 (n )λ 0, c n = λ n 0 {nl 22 (n )λ 0 }, ϕ n = n ωt B Remark 2.3. [3] t R U 0 (t, 0) t nt L L U 0 (nt, 0)ψ L (R 2 ) 2πc nθ n ψ L (R 2 ) (4) c n θ n = { L 2 µ n T 0 T 0,cr, L 2 nλ n 0 T 0 = T 0,cr T 0 < T 0,cr λ 2 0 < (3) λ ± C λ ± < (4) nt e i(nt )H B L L U 0 (T, 0) ([3]) T 0 = T 0,cr L L (4) n e i(nt )H 0 off T 0,cr off T 0,cr

6 T 0 > T 0,cr λ 0 < λ > λ + < L 2 0 n T 0 T 0,res := sin( ω T B) ω cos( ω T B ) U 0 (nt, 0)ψ L (R 2 ) 2π µ n L 2 ψ L (R 2 ) Ce νn ψ L (R 2 ), ν > 0 Proposition 2.4. T 0 > T 0,cr T 0 T 0,res θ > 0, ψ L 2 (R 2 ) C > 0 ( + x ) θ U 0 (t, 0)( + x ) θ ψ dt C ψ L 2 (R 2 ) L 2 (R 2 ) R Remark 2.3 t R t t n = nt ( + x ) θ U 0 (nt, 0)( + x ) θ ψ C ψ L 2 (R 2 ) L 2 (R 2 ) (5) n Z Riesz-Thorin p 2, q = p/(p ) U 0 (nt, 0)ψ L p (R 2 ) Ce 2νn(/2 /p) ψ L q (R 2 ). Kato [4] ( + x ) θ U 0 (nt, 0)( + x ) θ ψ L 2 (R 2 ) Ce 2νn/p ( + x ) θ 2 L p (R 2 ) ψ L 2 (R 2 ) p 2 θ > 0 pθ > 2 p (5) T 0 > T 0,cr [8], [], [2], [7], [23] K := L 2 (T; L 2 (R 2 )), T = R/T Z ψ K, ψ(t) = ψ(t; ) L σ 0, L σ (L σ 0 ψ)(t) = U 0 (t, t σ) ψ(t σ), (L σ ψ)(t) = U(t, t σ) ψ(t σ).

7 L σ 0, L σ K Stone K Ĥ0, Ĥ e iσĥ0 ψ = L σ 0 ψ, e iσĥ ψ = L σ ψ Ĥ0 U 0 (t, 0) Floquet Floquet : Lemma 2.5. [ Yajima [24]] f K Ĥf = λf f(t) L 2 (R 2 )- f(t) = e iλt U(t, 0)f(0), U(T, 0)f(0) = e iλt f(0) U(T, 0)φ = e it λ φ f(t) := e itλ U(t, 0)φ D(Ĥ) Ĥf = λf. Lemma 2.6 (Kitada-Yajima [7], Enss-Veselić [8]). L 2 (R 2 ) = L 2 c(u(t, 0)) L 2 p(u(t, 0)) L 2 c(u(t, 0)) L 2 p(u(t, 0)) U(T, 0) Lemma 2.5 Floquet Ĥ Floquet Howland-Yajima method (Howland [] [2], Yajima [23]) Lemma 2.7 (Howland [] [2], Yajima [23]). W ± iσĥ0 := s lim eiσĥe σ ± Ran(W ± ) = K ac (Ĥ) Kac(Ĥ) K Ĥ W ± := s lim t ± U(t, 0) U 0 (t, 0) Ran(W ± ) = L 2 ac(u(t, 0)) L 2 ac(u(t, 0)) L 2 (R 2 ) U(T, 0) Theorem 2.8 (Adachi- K [3]). V (V) on, off T B, T 0 (T 0) 0 < ω T B < π, (T ) T 0 > T 0,cr, (T 2) T 0 T 0,res, if π/2 < ω T B < π Ran ( W ±) = L 2 ac(u(t, 0))

8 3. Howland-Yajima method Kato s smooth perturbation method (Kato [4])., i.e., V (Ĥ + i) K 2., i.e., ρ = V /2 sign(v ), ρ 2 = V /2, ψ K Γ R, sup λ Γ, ±µ>0 ρ (Ĥ λ iµ) ρ 2 ψ K C ψ K 3. W ± 3 Cook-Kuroda method W ± ψ C 0 (R 2 ) 0 V U 0 (t, 0)ψ L 2 (R 2 ) dt C 2.4 θ = σ 0 C C V U 0 (t, 0)ψ L 2 (R 2 ) dt 0 ( + x ) σ U 0 (t, 0)( + x ) σ ( + B(L 2 (R 2 )) x )σ ψ L 2 (R 2 ) dt σ > 0 2. Ĥ0 Kato-Kuroda [6] Ĥ H 0 Imz > 0 Floquet ( Imz < 0 ) ρ (Ĥ0 z) ρ 2 ψ = i { T n= 0 t + 0 e i(t+nt s)z ρ U 0 (t + nt, s)(ρ 2 ψ)(s)ds e i(t s)z ρ U 0 (t, s)(ρ 2 ψ)(s)ds } 2.4. U 0 (t, 0)

9 4. Remarks Remark 4.. Hill f (t) + ( ) 2 qb(t) f(t) = 0 2m Remark 4.2. Bony-Carles-Häfner-Michel [7] V (x) C(log( + x )) ε Ishida [3] log Remark 4.3. T 0 = T 0,cr 3 [5] 2 [5], [3] References [] Adachi, T.: On spectral and scattering theory for N-body Schrödinger operators in a constant magnetic field, Rev. in Math. Phys., 2, (2002) [2] Adachi, T.: Asymptotic completeness for N-body quantum systems with longrange interactions in a time-periodic electric field, Comm. Math. Phys., 275, (2007) [3] Adachi, T., Kawamoto, M.: Quantum scattering in a periodically pulsed magnetic field, A. H. P., 7, , (206) [4] Adachi, T., Tamura, H.: Asymptotic completeness for long-range many particle systems with Stark effect, J. Math. Sci. Univ. Tokyo, 2, 77-6 (995) [5] Adachi, T., Tamura, H.: Asymptotic completeness for long-range many particle systems with Stark effect, II, Comm. Math. Phys. 74, (996) [6] Avron, J.E., Herbst, I.W., Simon, B.: Schrödinger operators with magnetic fields. I. General interactions. Duke Math. J. 45, (978) [7] Bony, J. F., Carles, R., Häfner, D., Michel, L.: Scattering theory for the Schrödinger equation with repulsive potential, J. Math. Pures Appl. 84, (2005) [8] Enss, V., Veselić, K.: Bound states and propagtiong states for time-dependent hamiltonians, Ann. de l I. H. P., sect. A 39, 59-9 (983) [9] Gérard, C., Laba, I.: Scattering theory for N-particle systems in constant magnetic fields, Duke Math. J., 76, (994) [0] Gérard, C., Laba, I.: Scattering theory for N-particle systems in constant magnetic fields, II, Comm. in P.D.E., 20, (996) [] Howland, J.S.: Scattering theory for time-dependent Hamiltonians. Math. Ann. 207, (974)

10 [2] Howland, J.S.: Scattering theory for Hamiltonians periodic in time. Indiana Univ. Math. J. 28, (979) [3] Ishida, A.: The borderline of the short-range condition for the repulsive Hamiltonian, J. Math. Anal. and Appl., 438, (206) [4] Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 62, (966) [5] Korotyaev, E. L.: On scattering in an external, homogeneous, time-periodic magnetic field, Math. USSR-Sb. 66, (990) [6] Kato, T., Kuroda, S. T.: The abstract theory of scattering. Rocky Mt. J. Math., 27-7 (97) [7] Kitada, H., Yajima, K.: Bound states and scattering states for time periodic hamiltonians, Ann. de l I. H. P., sect. A 39, (983) [8] Møller, J. S.: Two-body short-range systems in a time-periodic electric field, Duke Math. J., 05, (2000) [9] Møller, J. S., Skibsted, E.: Spectral theory of time-periodic many-body systems, Adv. in Math., 88, (2004) [20] Nakamura, S.: Asymptotic completeness for three-body Schrödinger equations with time-periodic potentials, J. Fac. Sci. Univ. Tokyo. Sect. IA, Math., 33, (986) [2] Skibsted, E.: Asymptotic completeness for particles in combined constant electric and magnetic fields, II, Duke Math. J., 89, (997) [22] Sigal, I. M., Soffer, A.: The N-particle scattering problem: asymptotic completeness for short-range quantum system, Ann. of Math. 25, (987) [23] Yajima, K.: Schrödinger equations with potentials periodic in time. J. Math. Soc. Jpn. 87, (977) [24] Yajima, K.: Resonances for the AC-Stark effect, Comm. Math. Phys., 87, (982)

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