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d2 - + ( V (x) + W (x) ) = E, x R. dx2 E V 1 L2 (R) W S = { C : Y1 < Im < Y2} W ( ) W W ( + 2) = W () /2 1/ cos(x) N N N 11 366 43 88 Hp = d2 - + V (x) L2(R) V dx2 1 L2 (R) Hp [E1, E2] [E3, E4],... [E2n+1, E2n+2],... E1 < E2 E3 <... E2n E2n+1 < E2n+2 ... En + n + [E2n-1, E2n] n 1 (E2n, E2n+1) n 1 E2n < E2n+1 n -(x)+V (x)(x) = E(x) (x + 1) = (x) C x (0, E) = 1 S C (E) = exp(ik (E)) k E k S S C d2 - (x) + (V (x) + W (x + )) (x) = E (x), x R. dx2 (x + 1, ) = (x, + ). 0 d2 - (x, ) + V (x) (x, ) = E()(x, ), E() = E - W (), dx2 k Hp () = k (E()) E() = Ej W Im d d d > 0 ( - ) d < 0, y = Im .

dy dy K K 1 2 Im 1 < Im 2 K 1 2 D E E0 (x, ) = q(E) e (x, E), q(E) = k(E), E = E(), E0 = E(0), 0 D E0 q D E = E0 K f K i f = exp d ((x, ) + o (1)), 0. 0 E = E0 D S(D) {Y1 < Im < Y2} D D D () +(, ) D f i f exp( d) + V0 E0 f (x, , E) R S(D) V0 x f S(D) E V0 K D V V0 E0 (x, , E) [-X, X] K V X > 0 f x Im d Im ( - ) d W () = cos C1 W () = cos m < M R Im = m Im = M 1 2 1 2 = 1 2 E = E0 R f i f exp( d) + 1 R 1 2 Im < 0 f R f 2 Im < 0 S {C1 Im C2} U S f f 0 W (0) = 0 0 Im (() 0 (0))d = 0 0 2/3 1 2 3 1 0 1 1 0 0 E() V 1 1 2 3 V S1 S2 S3 S1 1 2 S2 2 3 S1 1 S1 1 Im () > 0 i f f exp d + S1 (2 V ) S2 f \ 1 V 1 S1 1 d2 - (x) + ( V (x - z) + W (x) ), (x) = E(x), x R, dx2 0 z < 2 V W V, W : R R W 2 0 < < 2 2/ z z s ac 0 x 1,2(x, z) 0 z < 1 0 z < 1 1,2 z z x 1,2(x+2/, z+2/) x 1,2(x, z) 1(x, z) (x + 2/, z + 2/) = M(z)(x, z), (x, z) =, 2(x, z) M(z) 2 2 x (1, 2) z det M(z) 1 M(z+1) = M(z) h = {2/} M (1,2) (m + 1) = M(z + mh)(m), m Z, : Z C2, h m x 1,2 M z R f x R m Z f(x + 2m/, z) 0 -1 1 2 = (x, z - mh) -m, =.







1 2 fx(x + 2m/, z) 1 0 x x f J E J E - W (R) n Hp > 0 D (0, 1) mes (D (0, )) = + o e-/, 0;

D B J mes (B) = O(/2) mes (J) J \ B E J \ B (x, E) (x) = eip(E)x P(x - z, x, E), p E J P P- = P+ (x, , E) P+(x, , E) 1 x R 2 R H2 x E J \ B J R n N [E2n-1, E2n] E J E - W (R) Hp W [0, 2) 0 < < 2 W J J J 1 (E) = (Sn-1(E)+Sn(E))+o(1), Sj = i ()d, j = n, n-1. 4 j j 2 W E J 0 2 j j = n-1, n E - W (j ) = Ej 0 Re 2 j j E J Sj j Sj Sj E J W () = cos > 0 J E J E - < E1, E1 < E + < E2. E - W (R) W E J 0 Re 2 j j = 1, 2 E - W (j) = Ej j j = 1, 2, 3 = d, Sh = i d, Sv = i d. 2 1 3 E J Sh Sv J Sh Sv J J (E) > 0 J E(l) J l Z 1 (E(l)) = /2 + l, l Z. 1/ J Il Il o () E(l) 2 v h |Il| e-S (E(l))/ + e-S (E(l))/ (E(l)) Il 2 Il Il S(E) = Sv(E) - Sh(E) J- = {E J : S(E) < 0} J+ = {E J : S(E) > 0}.

J I J+ > 0 D (0, 1) 0 mes (D (0, )) = + O(e-/) D Il I mes (Il ac) = mes Il (1 + o(1)) ac I J- Il I S(E) = Sv(E) - Sh(E) 1 > 0 > 0 < 0 S(E) > 0 > 1 S(E) < 0 0 < < 1 S 0 < < 1 S W () = cos > 0 Hp J E J E2n-1 < E - < E2n, E2n+1 < E + < E2n+2, n N E - W (R) E - W (R) n (n + 1) Sv Sh j = n, n + 1 j Sv,j Sh,j j E J (E) < 0, (E) > 0. n+1 n Sh(E) = Sh,n+1(E) + Sh,n(E) t,j(E) = exp(-S,j(E)/) {v, h} j {n, n + 1} th(E) = exp(-Sh(E)/) n (n + 1) 2 min{Im 2n-2(E), Im 2n+3(E)} > max{Sh(E), Sv,n(E), Sv,n+1(E)}, E J.

j {0 Re , Im 0} E() = Ej 1 0 := minEJ min{Sh(E), Sv,n(E), Sv,n+1(E)} 2 E0 J V0 C E0 0 J V0 e- / (l) {Ej }l J V0 j = n, n + 1 1 (l) j(Ej ) = + l, l Z, 2 j : V0 C V0 R 0 j(E) = j(E) + o() j = n, n + 1 (l) {Ej }l 0 e- / (l) {En+1}l (l) (l) {En }l En+1 E(n + 1) E(n + 1) (l) {En }l (l) E(n + 1) {En }l E(n+1) (l) 0 dist E(n + 1), {En }l > 2e- / dist (E, A) E A 0 (e- /) E(n + 1) I(n + 1) n(V ) E(n + 1) + 2 (E)th(E) tg n(E)/ n+1 E=E(n+1) -1 cos th(E) n(E)/ + tv,n+1(E) n(V ) E=E(n+1) V n I(n + 1) 2 n(V ) 1 n(V ) > 1 (l) E(n) {En }l E(n + 1) I(n + 1) E(n + 1) E(n) I(n + 1) E(n + 1) E(n) E(n + 1) E(n) I(n + 1) S (l) Sn+1(E) = Sv,n+1(E) - Sh(E) - ln dist E, {E0 }.

l I(n + 1) 1 = max {-Sn+1(E(n + 1)), 0} + o(1).

2 I(n + 1) E(n) c > 0 Sn+1(E(n + 1)) < -c I(n + 1) c > 0 > 0 D (0, 1) mes (D (0, )) = + O e-/ 0 D Sn+1(E(n + 1)) > c mes (I(n + 1) ac) = mes (I(n + 1)) (1 + o(1)) ac E(n) E(n + 1) I(n) E(n) I(n + 1) c > 0 D I(n + 1) I(n) N N N Sh(E) - Sv,n+1(E) Sh(E)-Sv,n(E) J J 0 I(n + 1) I(n) N exp (-0/) Sn+1 I(n + 1) I(n + 1) I(n) J V (l) dist E(n + 1), {En }l 0 0 2e- / e- / E(n) E(n + 1) tv,n tv,n+1 max(tv,n,tv,n+1) = 2 = 2.

th th = (E(n) + E(n + 1))/2 0 1, 1 1, 1.

1 1 1 0 0 E(n) E(n+1) e- / 1 j {n, n + 1} I(j) tv,j E(j) E(n) E(n + 1) I(n) I(n + 1) /2 1 E(n) E(n + 1) / I(n) I(n + 1) 1 I(n) I(n + 1) I(n) I(n + 1) E(n) E(n + 1) I(n) I(n + 1) 1 n(V ) > 1 I(n) I(n + 1) /2 1 1 I(n) I(n + 1) E(n) E(n + 1) I(n) I(n + 1) th th E(n) E(n + 1) I(n) I(n + 1) th |E(n + 1) - E(n)| |E(n + 1) - E(n)| tv,n+1() tv,n+1() 1 1 E(n) E(n+1) th E(n) E(n + 1) E(n + 1) E(n) 1 1 1 I(n) I(n + 1) tv,n+1 tv,n E(n) E(n+1) I(n) I(n + 1) E(n) E(n + 1) |E(n)-E(n+1)| |E(n) - E(n + 1)| tv,n+1 |E(n) - E(n + 1)| tv,n+1 E(n) E(n+1) I(n) I(n + 1) tv,n+1 E(n) th/tv,n+1 E(n) E(n+1) I(n) I(n + 1) I(n) I(n + 1)

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