Anderson 2008 12 ( ) Anderson 2008 12 1 / 14
Anderson ( ) Anderson 2008 12 2 / 14
Anderson P.W.Anderson 1958 ( ) Anderson 2008 12 3 / 14
Anderson tight binding Anderson tight binding Z d u (x) = V i u t (u(x) u(y)) y: y x =1 = Hu = u + λvu ( ) Anderson 2008 12 4 / 14
Anderson tight binding Anderson tight binding Z d u (x) = V i u t V (u(x) u(y)) y: y x =1 = Hu = u + λvu {V ω (x)} x Z d V ω (x) g(v) 0 g(v) M ( ) Anderson 2008 12 4 / 14
Anderson tight binding Anderson tight binding Z d u (x) = V i u t V (u(x) u(y)) y: y x =1 = Hu = u + λvu {V ω (x)} x Z d V ω (x) g(v) 0 g(v) M R d ( ) Anderson 2008 12 4 / 14
Dynamical localization (mean square displacement) r(t) 2 = x Z d x 2 u (t, x) 2 ( ) Anderson 2008 12 5 / 14
Dynamical localization (mean square displacement) V = 0 r(t) 2 = x Z d x 2 u (t, x) 2 r(t) 2 Ct 2 as t ( ) Anderson 2008 12 5 / 14
Dynamical localization (mean square displacement) V = 0 r(t) 2 = x Z d x 2 u (t, x) 2 r(t) 2 Ct 2 as t Dynamical localization r(t) 2 C as t ( ) Anderson 2008 12 5 / 14
Dynamical localization (mean square displacement) V = 0 r(t) 2 = x Z d x 2 u (t, x) 2 r(t) 2 Ct 2 as t Dynamical localization r(t) 2 C as t Brown motion r(t) 2 Ct as t ( ) Anderson 2008 12 5 / 14
H = H = E 0 E Σ (de ), (Σ (de ) ) ( ) Anderson 2008 12 6 / 14
H = H = E Σ (de ), (Σ (de ) ) E 0 Schrödinger ) ( ) u(t, ) = (e ith u 0 ( ) = e ite Σ (de ) u 0 ( ) E 0 ( ) Anderson 2008 12 6 / 14
H = H = E Σ (de ), (Σ (de ) ) E 0 Schrödinger ) ( ) u(t, ) = (e ith u 0 ( ) = e ite Σ (de ) u 0 ( ) E 0 Σ (de ) E point mass H u 0 E φ u(t, ) = e ite φ ( ) = t u(t, ) ( ) Anderson 2008 12 6 / 14
H = H = E Σ (de ), (Σ (de ) ) E 0 Schrödinger ) ( ) u(t, ) = (e ith u 0 ( ) = e ite Σ (de ) u 0 ( ) E 0 Σ (de ) E point mass H u 0 E φ u(t, ) = e ite φ ( ) = t u(t, ) Σ (de ) u(t, x) t u (t, x) 2 = u 0 (x) 2 = u(t, ) t x Z d x Z d ( ) Anderson 2008 12 6 / 14
H = H = E Σ (de ), (Σ (de ) ) E 0 Schrödinger ) ( ) u(t, ) = (e ith u 0 ( ) = e ite Σ (de ) u 0 ( ) E 0 Σ (de ) E point mass H u 0 E φ u(t, ) = e ite φ ( ) = t u(t, ) Σ (de ) u(t, x) t u (t, x) 2 = u 0 (x) 2 = u(t, ) t x Z d x Z d Σ (de ) Σ (de ) = Σ ac (de ) Σ sc (de ) Σ p (de ) Σ (de ) = Σ p (de ) Σ (de ) = Σ ac (de ) (extended states) ( ) Anderson 2008 12 6 / 14
1D, 2D 3D ( ) Anderson 2008 12 7 / 14
1D, 2D 3D 3D ( ) Anderson 2008 12 7 / 14
K.Ishii: Localization of eigenstates and transport phenomena in 1-dim. disordered systems, Prog.Theor.Phys.Suppl. 53(1973) Σ ω ac (de ) = 0 ( ) Anderson 2008 12 8 / 14
K.Ishii: Localization of eigenstates and transport phenomena in 1-dim. disordered systems, Prog.Theor.Phys.Suppl. 53(1973) Σ ω ac (de ) = 0 I.Goldseid,S.Molchanov,L.Pastur: A pure point spectrum of the stochastic 1-dimensional Schröinger equation. Funct. Anal. Appl. 11(1977) Σ ω (de ) = Σ ω p (de ) ( ) Anderson 2008 12 8 / 14
K.Ishii: Localization of eigenstates and transport phenomena in 1-dim. disordered systems, Prog.Theor.Phys.Suppl. 53(1973) Σ ω ac (de ) = 0 I.Goldseid,S.Molchanov,L.Pastur: A pure point spectrum of the stochastic 1-dimensional Schröinger equation. Funct. Anal. Appl. 11(1977) Σ ω (de ) = Σ ω p (de ) coupling λ ( ) Anderson 2008 12 8 / 14
J.Frölich, T.Spencer: Absence of diffusion in the Anderson tight-binding model for large disorder or low energy, CMP. 88(1983) λ 1 = λ = 0 = ( ) Anderson 2008 12 9 / 14
J.Frölich, T.Spencer: Absence of diffusion in the Anderson tight-binding model for large disorder or low energy, CMP. 88(1983) λ 1 = λ = 0 = (Green G E = (H E ) 1 ( ) Anderson 2008 12 9 / 14
J.Frölich, T.Spencer: Absence of diffusion in the Anderson tight-binding model for large disorder or low energy, CMP. 88(1983) λ 1 = λ = 0 = (Green G E = (H E ) 1 Wegner Λ Z d H Λ = H Λ Prob (dist (E, sph Λ ) δ) n λ (E )δ Λ, n λ (E ) = ( ) Anderson 2008 12 9 / 14
J.Frölich, T.Spencer: Absence of diffusion in the Anderson tight-binding model for large disorder or low energy, CMP. 88(1983) λ 1 = λ = 0 = (Green G E = (H E ) 1 Wegner Λ Z d H Λ = H Λ Prob (dist (E, sph Λ ) δ) n λ (E )δ Λ, n λ (E ) = Combes-Thomas dist (E, sph Λ ) = κ (E C) G Λ,E (x, y) κ 1 e cκ x y, c ( ) Anderson 2008 12 9 / 14
J.Frölich, T.Spencer: Absence of diffusion in the Anderson tight-binding model for large disorder or low energy, CMP. 88(1983) λ 1 = λ = 0 = (Green G E = (H E ) 1 Wegner Λ Z d H Λ = H Λ Prob (dist (E, sph Λ ) δ) n λ (E )δ Λ, n λ (E ) = Combes-Thomas dist (E, sph Λ ) = κ (E C) G Λ,E (x, y) κ 1 e cκ x y, c Resolvent H = H 0 + H 1 = G E = G 0,E G 0,E H 1 G E ( ) Anderson 2008 12 9 / 14
J.Frölich, T.Spencer: Absence of diffusion in the Anderson tight-binding model for large disorder or low energy, CMP. 88(1983) λ 1 = λ = 0 = (Green G E = (H E ) 1 Wegner Λ Z d H Λ = H Λ Prob (dist (E, sph Λ ) δ) n λ (E )δ Λ, Combes-Thomas n λ (E ) = dist (E, sph Λ ) = κ (E C) G Λ,E (x, y) κ 1 e cκ x y, c Resolvent H = H 0 + H 1 = G E = G 0,E G 0,E H 1 G E Multiscale analysis: H 0 = H Λ H Λ c { 1 if x y = 1, x Λ, y Λ H 1 (x, y) = c or vise versa 0 otherwise Resolvent ( ) Anderson 2008 12 9 / 14
J.Frölich, T.Spencer: Absence of diffusion in the Anderson tight-binding model for large disorder or low energy, CMP. 88(1983) λ 1 = λ = 0 = (Green G E = (H E ) 1 Wegner Λ Z d H Λ = H Λ Prob (dist (E, sph Λ ) δ) n λ (E )δ Λ, Combes-Thomas n λ (E ) = dist (E, sph Λ ) = κ (E C) G Λ,E (x, y) κ 1 e cκ x y, c Resolvent H = H 0 + H 1 = G E = G 0,E G 0,E H 1 G E Multiscale analysis: H 0 = H Λ H Λ c { 1 if x y = 1, x Λ, y Λ H 1 (x, y) = c or vise versa 0 otherwise Resolvent Green GE ω (0, y) c ω e c y ( ) Anderson 2008 12 9 / 14
Erdös, Lázló- Salmhofer, Manfred- Yau, Horng-Tzer: Towards the quantum Brownian motion. Mathematical physics of quantum mechanics, 233 257, Lecture Notes in Phys., 690, Springer, Berlin, 2006 3 d, 1 λ = r(t) 2 Dt if λ 2 t λ 2 ε for ε > 0 r(t) 2 Ct 2 if t λ 2 ( ) Anderson 2008 12 10 / 14
Erdös, Lázló- Salmhofer, Manfred- Yau, Horng-Tzer: Towards the quantum Brownian motion. Mathematical physics of quantum mechanics, 233 257, Lecture Notes in Phys., 690, Springer, Berlin, 2006 3 d, 1 λ = r(t) 2 Dt if λ 2 t λ 2 ε for ε > 0 r(t) 2 Ct 2 if t λ 2 3D 1 λ r(t) 2 Dt as t ( ) Anderson 2008 12 10 / 14
Erdös, Lázló- Salmhofer, Manfred- Yau, Horng-Tzer: Towards the quantum Brownian motion. Mathematical physics of quantum mechanics, 233 257, Lecture Notes in Phys., 690, Springer, Berlin, 2006 3 d, 1 λ = r(t) 2 Dt if λ 2 t λ 2 ε for ε > 0 r(t) 2 Ct 2 if t λ 2 3D 1 λ r(t) 2 Dt as t A.Klein: Extended states in the Anderson model on the Bethe lattice. Adv. Math. 133(1998) Bethe ( ) Anderson 2008 12 10 / 14
( ) Anderson 2008 12 11 / 14
IDS 1 N(E ) = lim Λ Z Λ # { HΛ ω E } ( ) Anderson 2008 12 11 / 14
IDS Lyapounov 1 γ (E ) = lim x U ω (x + 1, E ) = 1 N(E ) = lim Λ Z Λ # { HΛ ω E } x log Uω (x, E ) 0, ( 0 1 1 2 + E V ω (x + 1) ) U ω (x, E ) ( ) Anderson 2008 12 11 / 14
IDS Lyapounov 1 γ (E ) = lim x U ω (x + 1, E ) = 1 N(E ) = lim Λ Z Λ # { HΛ ω E } x log Uω (x, E ) 0, ( 0 1 1 2 + E V ω (x + 1) ) U ω (x, E ) ( ) Anderson 2008 12 11 / 14
IDS Lyapounov 1 γ (E ) = lim x U ω (x + 1, E ) = 1 N(E ) = lim Λ Z Λ # { HΛ ω E } x log Uω (x, E ) 0, ( 0 1 1 2 + E V ω (x + 1) Σ ω (= suppσ ω (de )) = suppdn(e ) ) U ω (x, E ) ( ) Anderson 2008 12 11 / 14
IDS Lyapounov 1 γ (E ) = lim x U ω (x + 1, E ) = 1 N(E ) = lim Λ Z Λ # { HΛ ω E } x log Uω (x, E ) 0, ( 0 1 1 2 + E V ω (x + 1) ) U ω (x, E ) Σ ω (= suppσ ω (de )) = suppdn(e ) Σac ω (= suppσac ω (de )) = {E; γ (E ) = 0} ess ( ) Anderson 2008 12 11 / 14
IDS Lyapounov 1 γ (E ) = lim x U ω (x + 1, E ) = 1 N(E ) = lim Λ Z Λ # { HΛ ω E } x log Uω (x, E ) 0, ( 0 1 1 2 + E V ω (x + 1) ) U ω (x, E ) Σ ω (= suppσ ω (de )) = suppdn(e ) Σac ω (= suppσac ω (de )) = {E; γ (E ) = 0} ess m± ω Weyl m+(e ω + i0) = m ω (E + i0) a.e. {E; γ (E ) = 0} ( ) Anderson 2008 12 11 / 14
S R V ω (x) S V ω (x) ( ) Anderson 2008 12 12 / 14
S R V ω (x) S V ω (x) Mathieu (Hu) x = u x+1 + u x 1 + 2λ cos 2π (αx + ω) u x α : λ > 1 = λ = 1 = λ < 1 = André-Aubry, Avila, Jitomirskya, Last, Simon, ( ) Anderson 2008 12 12 / 14
S R V ω (x) S V ω (x) Mathieu (Hu) x = u x+1 + u x 1 + 2λ cos 2π (αx + ω) u x α : λ > 1 = λ = 1 = λ < 1 = André-Aubry, Avila, Jitomirskya, Last, Simon, ( ) Anderson 2008 12 12 / 14
( ) Anderson 2008 12 13 / 14
( ) Anderson 2008 12 13 / 14
e.g.:penrose ( ) Anderson 2008 12 13 / 14
e.g.:penrose ( ) Anderson 2008 12 13 / 14
e.g.:penrose ( ) Anderson 2008 12 13 / 14
e.g.:penrose ( ) Anderson 2008 12 13 / 14
e.g.:penrose ( ) Anderson 2008 12 13 / 14
e.g.:penrose d 2 x ω (t) dt 2 = grad V ω (x ω (t)) = {x ω (t)} = ( ) Anderson 2008 12 13 / 14
e.g.:penrose d 2 x ω (t) dt 2 = grad V ω (x ω (t)) = {x ω (t)} = e.g.: ( ) Anderson 2008 12 13 / 14
Isaac Newton Institute for Mathematical Sciences Mathematics and Physics of Anderson localization : 50 Years After 14 July - 19 December 2008 (http://www.newton.ac.uk/programmes/mpa/) Spencer, T (IAS, Princeton) Anderson localisation: phenomenology and mathematics ( ) Anderson 2008 12 14 / 14