100x W=0.1 W=

1 2 abstract: 1 [1] 1970 Furstenberg [2] 1 [3] 1979 Anderson [4] 1 2 1 H = NX n=1 jn >V n <nj + NX n (jn ><n+1j + jn +1>< nj); fv n g HjΨ >= EΨ > (ffi n =<njψ >), ffi n+1 + ffi n 1 + V n ffi n = Effi n, M n ψ! ψ! ψ ffin +1 =Π N n=1 ffi M ffi1 E Vn 1 n ; M n = N ffi 0 1 0 1 2002 12 11 2 E-mail: hyamada@cc.niigata-u.ac.jp! 1

2.1 Harper V n V n =2 cos(2ßffn + ) (ff ; 2 Q) [5] self-duality Herbert-Jones-Thouless ( )[6] =1 [7] Harper Soukoulis-Economou [8], V n =1:9[cos(Qn)+ 1 cos(2qn)](q =0:7), 3 V n = W j cos(2ßffn)j[9] Harper V n =2 cos(ßffn ν )(0 <ν<1)[] 2.2 :C n < V n 0 V n >= ffi n 0;n (C n ο e cn ) [11] c 1 random-dimer [12] fv A ;V B g V A! V A V A ;V B! V B. [13, 14] 1 (C n ο n s (s>0)) [15, 16] fv n g S(f) ο f ff (Fourier Filtering Method) ff ff >ff c ff 2 (E =0) ff 1:9 1 1980 [17, 18] 0» ff» 1 2

Furstenberg 1» ff» 1:5 ff ' 1:5 ff 1 fv n g Furstenberg 3 X n+1 = X n +2 B 1 (1 2b)Xn B + b (0» X n < 1=2) X n 2 B 1 (1 2b)(1 X n ) B + b (1=2» X n» 1): [19] B b B 2 b = 13 i b =(B 1) 1 (2b) (1 B)=B [19]. W V n =(2X n 1)W fv n g 0» Xn < 1=2! V n = W 1=2» X n < 1! V n = W C(n) ο n B 1 2 B (n >>1) [19]. B < 2 B 2 b =0, S(f) ο f 2B 3 B 1 ff =2 B!1 B 2 2 B B 1 W! W( W! W ) m P (m) ο m B 1 B B <2 (< m><1) B 2 [20] 4 (B» 2), fl =< ln(jffi N+1j 2 +jffin j 2 ) 2N > < ::: > N!1 N [17, 18, 21] 3

Furstenberg [17] B <3=2 3=2 < B < 2 fx n g V n [22, 23] 5 (B 2) 2» B» 3 [21, 24] [25] 0x -3 7 80 6 60 5 γ ξ 4 40 3 20 2 1 W= W=0.3 0-2 -1 0 1 2 E 0-3 -2-1 0 1 2 3 E 1: Lyapunov exponents fl as a function of energy, for several potential strength W within a range [0.01, 0.8]. The system and ensemble size are 2 20 and 50 respetively. (B =2:0.) 2: Localization length ο as a function of energy, for several potential strength W = 0:1; 0:3. The system and ensemble size are 2 22 and 50 respetively. (B =3:0.) B =2:0 fl E (W << 1) (E =0) fl B = 3:0 ( ) E =0 4

-1-2 -3-4 B=1.8 2.5 3.0 W^2 9 8 7 6 5 4 3 2 γ -5-6 -7-8 -9 γ 9 8 7 6 5 4 3 2 'B=1.1' '1.8' '2.3' '3.0' W^2/3 W~1/2 0.01 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 W 0.01 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 W 3: Lyapunov exponents at band center (E = 0) as a function of potential strength W for several bifurcation parameter B in symbolic cases. 4: Lyapunov exponents at band edge (E = 2:0) as a function of potential strength W for several bifurcation parameter B in symbolic cases. (E =0) fl W fl ο W 2 [26, 27] W (E =2:0) fl W (fl ο W 2=3 ) fl ο W 1=2 ( [28, 27] ) Van-Hope [24] fl B B = 2:0 fl ( 6) fl B W fl = C(W )B + D(W )(C; D ) fl(b Λ )=0 B Λ 7 B 2 B =2 fl>0 B =2 5

20x -3 400 γ 15 5 W=0.01 0.3 0.5 x -6 300 200 0 0 0.51.01.52.02.53.0 1.5x -3 1.0 γ 0.5 W=0.01 0.3 0.5 0 0.0 1.6 1.8 2.0 2.2 B 2.4 2.6 2.8 3.0 1.6 1.8 2.0 2.2 B 2.4 2.6 2.8 3.0 5: Lyapunov exponents as a function of bifurcation parameter B, for several potential strength W in symbolic cases (E =0). The inset is expansion of a case, W =0:01. 6: Lyapunovexponents as a function of bifurcation parameter B, for several potential strength W in non-symbolic cases (E =0). 3.0 2.5 B * 2.0 1.5 1.0 0.2 0.4 W 0.6 0.8 1.0 7: Bifurcation parameter B Λ estimated by fl(b Λ )=0:0 for several potential strength W in symbolic (2) and non-symbolic cases (fl) ate =0. N W B fl N [17] [16] N, N 8 9 W N ο = N B =2:8 N ο / N ν ν<1 6

20 18 16 W=0.5 0.3 0.05 24 22 20 W=0.5 0.3 0.05 0.01 ln ξ 14 12 ln ξ 18 16 14 12 8 8 12 14 ln N 16 18 9 11 12 13 ln N 14 15 16 8: Localization length ο as a function of system size N for several potential strength W at B=2.8. The line shows ο = N. 9: Localization length ο as a function of system size N for several potential strength W at B=3.0. The line shows ο = N. 9(B =3:0) W ο / N ν (ν ο 1) (W << 1) S(f) ο f 1:5 6 B» 2:8 (W << 1) S(f) ο f ff (ff ' 1:5) ff c 2( B!1 ) [15] [16] ff c =1:9 B fl hopping( 1 (off-diagonal model) stretched fl =0 [29] hopping 7

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