Case 1 a,b,α, β α α + β β α = ua + vb β = sa + tb α α + β β = (ua + vb (ua + vb + (sa + tb (sa + tb = (u a a + uva b + uvb a + v b b + (s a a + sta b

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Bogoliubov H = a a + aa < 0 H 0 >, < 1 H 1 >, < H > < H 0 > a H = α α α α, α = 1, α, α = α, α = 0 α α α H = α α Postulate( a,b α, β a, a = b, b = 1, a, b = b, a = 0 α, α = β, β = 1, α, β = β, α = 0 α, β a,b H = α α + β β 1

Case 1 a,b,α, β α α + β β α = ua + vb β = sa + tb α α + β β = (ua + vb (ua + vb + (sa + tb (sa + tb = (u a a + uva b + uvb a + v b b + (s a a + sta b + stb a + t b b = (u a a + uva b + v b b + (s a a + sta b + t b b = (u + s a a + (v + t b b + (uv + sta b E 1 α α + E β β = E 1 (ua + vb (ua + vb + E (sa + tb (sa + tb = E 1 (u a a + uva b + uvb a + v b b + E (s a a + sta b + stb a + t b b = E 1 (u a a + uva b + v b b + E (s a a + sta b + t b b = (u E 1 + s E a a + (v E 1 + t E b b + (uve 1 + ste a b (u,v,s,t α, α = αα α α = (ua + vb(ua + vb (ua + vb (ua + vb = (u aa + uvbb + uvba + v bb (u a a + uva b + uvb a + v b b = (u (a a + 1 + v (b b + 1 (u a a + v b b = u + v β, β = s + t α, β = 0 β, α = 0 α, α = β, β = 1 u + v = s + t = 1 u + v = 1,s + t = 1 (u,v,s,t (u, v, s, t = ( 1, 1, 1, 1 a a + b b + a b = α α + β β

Case a,b,α, β α = ua + vb β = ua + vb α α + β β = (ua + vb(ua + vb + (ua + vb (ua + vb = (u a a + uva b + uvba + v bb + (u aa + uvab + uvb a + v b b = (u a a + uva b + uvab + v (1 + b b + (u (1 + a a + uvab + uva b + v b b = u a a + v b b + uva b + uvab + (u + v E 1 α α + E β β = E 1 (ua + vb(ua + vb + E (ua + vb (ua + vb = E 1 (u a a + uva b + uvba + v bb + E (u aa + uvab + uvb a + v b b = E 1 (u a a + uva b + uvab + v (1 + b b + E (u (1 + a a + uvab + uva b + v b b = u (E 1 + E a a + v (E 1 + E b b + uv(e 1 + E a b + uv(e 1 + E ab + (u E 1 + v E α, α = u v β, β = v u α, β = 0 β, α = 0 3

Case 3 a,b,α, β α = ua + vb β = va + ub α α + β β = (ua + vb(ua + vb + (va + ub (va + ub = (u a a + uva b + uvba + v bb + (v aa + uvab + uvb a + u b b = (u a a + uva b + uvab + v (1 + b b + (v (1 + a a + uvab + uva b + u b b = (u + v a a + (u + v b b + uva b + uvab + (u + v E 1 α α + E β β = E 1 (ua + vb(ua + vb + E (va + ub (va + ub = E 1 (u a a + uva b + uvba + v bb + E (v aa + uvab + uvb a + u b b = E 1 (u a a + uva b + uvab + v (1 + b b + E (v (1 + a a + uvab + uva b + u b b = (u E 1 + v E a a + (u E + v E 1 b b + uv(e 1 + E a b + uv(e 1 + E ab + (u E 1 + v E α, α = u v β, β = u v α, β = 0 β, α = 0 Case u v = v u = 1 Case3 u v = 1 Case3 u,v 4

Case 4 a,b,α, β α = ua + v b β = va + u b α α + β β = (u a + vb(ua + v b + (v a + ub (va + u b = ( u a a + u v a b + uvba + v bb + ( v aa + u v ab + uvb a + u b b = ( u a a + u v a b + uvab + v (1 + b b + ( v (1 + a a + u v ab + uva b + u b b = ( u + v a a + ( u + v b b + (u v + uva b + (u v + uvab + ( u + v E 1 α α + E β β = E 1 (u a + vb(ua + v b + E (v a + ub (va + u b = E 1 ( u a a + u v a b + uvba + v bb + E ( v aa + u v ab + uvb a + u b b = E 1 ( u a a + u v a b + uvab + v (1 + b b + E ( v (1 + a a + u v ab + uva b + u b b = (E 1 u + E v a a + (E u + E 1 v b b +(E 1 u v + E uva b + (E u v + E 1 uvab + (E 1 u + E v α, α = u v β, β = u v α, β = 0 β, α = 0 5

Bogoliubov a, b α, β a, b α, β Case1 α = ua + vb β = sa + tb α α + β β = (u + s a a + (v + t b b + (uv + sta b a = uα + vβ b = sα + tβ a a + b b = (u + s α α + (v + t β β + (uv + stα β H = a a + b b + a b + a b + ab a = uα + vβ b = sα + tβ H = ( α α + ( β β + ( α β + ( α β +... α α, β β u,v,s,t α, α β, β u,v,s,t Bogoliubov Case1 4 =0 4 6

Example H = gn V + P m a P a P + 1 gn ( a P a P + a P a P + a P a P + mgn P a P = u P b P + v P b P a P = αb P + βb P H 1 = gn V + P = gn V = gn V = gn V = gn V m a P a P + 1 gn ( a P a P + mgn P + ( P m + gn a P a P + 1 gn ( mgn P + 1 gn ( mgn P + ( P ( m + gn u P b P + v P b P (u P b P + v P b P + 1 gn ( mgn P + ( P ( m + gn u P b P b P + u P v P b P b P + u P v P b P b P + v P b P b P ( mgn P + 1 gn + ( P ( m + gn u P b P b P + v P b P b P + u P v P b P b P + u P v P b P b P + v P (a P a P + a P a P H = 1 gn = 1 gn ( ( ( (u P b P + v P b P α b P + β b P + u P b P + v P b P αb P + βb P = 1 gn u P α b P b P + u P β b P b P + v P α b P b P + v P β b P b P +u P αb P b P + u P βb P b P + v P αb P b P + v P βb P b P = 1 gn (u P α + u P α b P b P + ( v P β + v P β b P b P + ( u P β + v P α b P b P + (u P β + v P α b P b P + (u P α + v P β 7

H α, β H (α,β b P b P b P b P b P b P b P b P a(u P,v P u P + u P v P P u P v P + u P v P u P v P + u P v P u P + v P b(u P,v P u P v P u P v P u P v P u P + v P c(u P,v P u P u P + u P u P v P v P + v P v P u P v P + u P v P u P v P + u P v P u P u P + v P v P d(u P,v P u P u P + u P u P v P v P + v P v P u P v P + u P v P u P v P + u P v P u P u P + v P v P e(v P,u P u P v P + u P v P u P v P + u P v P u P u P + v P v P u P u P + v P v P u P v P + u P v P f(v P,u P u P v P + u P v P u P v P + u P v P u P u P + v P v P u P u P + v P v P u P v P + u P v P g(v P,u P u P v P + u P v P u P v P + u P v P u p + v P u p + v P u P v P h(v P,u P u P v P + u P v P u P v P + u P v P u p + v P u p + v P u P v P + u P v P b P b P b P b P a g u,v H 1 = gn V + 1 gn ( mgn P + ( P ( ( m + gn u P b P b P + v P b P b P + u P v P b P b P + b P b P + v P H = 1 gn (u P α b P b P + (v P β b P b P + (u P β + v P α (b P b P + b P b P + (u P α + v P β H b P (α,β b P b P b P b P b P + b P b P a(u P,v P u P v P u P v P u P + v P b(u P,v P u P v P u P v P u P + v P c(u P,v P u P u P v P v P u P v P + u P v P u P u P + v P v P d(u P,v P u P u P v P v P u P v P + u P v P u P u P + v P v P e(v P,u P u P v P u P v P u P u P + v P v P u P v P + u P v P f(v P,u P u P v P u P v P u P u P + v P v P u P v P + u P v P g(v P,u P u P v P u P v P u p + v P u P v P h(v P,u P u P v P u P v P u p + v P u P v P 8

g H = H 1 + H = gn V ( mgn P + 1 gn + ( P ( m + gn u P b + 1 gn P b P + v P b P b P + u P v P ( b P b P + b P b P + v P ( u P v P b P b P + b P b P + ( u P + v P ( b P b P + b P b P + (u P v P = gn V + 1 gn ( mgn P + ( P m + gn u P + (gnu P v P b P b P + ( P m + gn v P + (gnu P v P b P b P + ( P m + gn u P v P + 1 gn ( u P + v P (b P b P + b P b P + ( P m + gn v P + (gnu P v P u P, v P a P, a P a P, a P a P, a P a P, a P b P, b P b P, b P b P, b P b P, b P = u P v P = u P v P = 0 = 0 1 = u P v P 1 = u P v P = 0 = 0 9

u P v P = 1 (1 u P, v P b P b P b P b P ( P m + gn u P v P + 1 gn ( u P + v P = 0 ( (1 u P = ± x + 1, v P = ± x (3 u P = ± x + 1 x + 1, v P = ± (4 u P = ± cosh x, v P = ± sinh x (5 (4 u P, v P (4 u P, v P ( u P = v P = 0 u P, v P v P x + 1 x + 1 u P =, v P = (6 ( ( P m + gn u P v P + 1 gn ( u P + v P = 0 ( P (x + 1(x 1 m + gn + 1 ( x + 1 gn + x 1 = 0 ( P x m + gn 1 + (gnx = 0 ( P m + gn (x 1 = (gn x ( ( P ( P m + gn (gn x = m + gn ( (gn x = x = (gn 10

u P, v P x (1( u P, v P x+1 u P = ± v P = ± x+1, x = (gn, = P m + gn H = gn V + 1 gn ( mgn P + ( P m + gn u P + (gnu P v P b P b P + ( P m + gn v P + (gnu P v P b P b P + ( P m + gn u P v P + 1 gn ( u P + v P (b P b P + b P b P + ( P m + gn v P + (gnu P v P = gn V + 1 gn ( mgn P + up + (gnu P v P b P b P + v P + (gnu P v P b P b P + v P + (gnu P v P = gn V ( mgn P + 1 gn + ( up + v P + (gnu P v P b P b P + v P + (gnu P v P u P, v P x x P P P u P, v P K1 b P b P + K b P b P = (K 1 + K b P b P 11

H = gn V + 1 gn ( mgn P + ( up + v P + (gnu P v P b P b P + v P + (gnu P v P = gn V + 1 gn ( mgn P + ( x + 1 x 1 (x + 1(x 1 (gn b P b P + x 1 (x + 1(x 1 (gn = gn V + 1 gn ( mgn P x (gn x 1 + + 1 b P b P (x 1 (gn x 1 = gn V + 1 gn ( mgn P 1 x (gn x 1 b P b P + + 1 x (gn x 1 = gn V + 1 gn ( mgn P 1 + (gn (gn (gn 1 b P b P + 1 (gn (gn (gn 1 1

H = gn V + 1 gn ( mgn P 1 + (gn (gn (gn 1 b P b P + 1 (gn (gn (gn 1 = gn V + 1 gn + + 1 ( mgn P 1 (gn (gn b (gn P b P (gn (gn (gn = gn V + 1 gn ( mgn P 1 + (gn b P b P + 1 (gn = gn V + 1 ( m(gn + ɛ (P b P b P + 1 ɛ (P P 1 ( P m + gn = gn V + 1 (ɛ (P P m(gn gn + m P + ɛ (P b P b P = E 0(P + ɛ (P b P b P P P 1 13

ppendix Example a P = u P b P + v P b P a P = αb P + βb P a P = u P b P + v P b P a P = αb P + β b P b b P, b P b P α, β Takashi Inoue http://www.persianblue.net/ 14

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