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By Luecke W.
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87) ∞ = 2π 2 rν+µ+1 e−r dr ei(ν−µ)ϕ dϕ 0 0 = µ † · = Ω dx dy e(x+iy)ˆa Ω | a ˆ† Ω | e(x+iy)ˆa Ω ν Ω 1 ∞ (ν+µ)/2 −ξ ξ e dξ 2 0 π ν! δνµ . 88), every vector in Haˆ way be written as a continuous superposition of coˆ aˆ (x + iy) Ω , z ∈ C . But, of course, this representation is not herent states D unique:39 ˆ aˆ (x + iy) Ω : z ∈ C is an over-complete subset of Haˆ . 1)) even shows that n f (x2 + y 2 ) (x + iy) ˆ aˆ (x + iy) Ω dx dy = 0 D holds for every function f and every n ∈ N for which the integral exists.
Z2n that converges for all (z1 , . . , z2n ) ∈ C2n and vanishes if zν = zn+ν ∀ ν ∈ {1, . . , n} . Then P is identically zero. Outline of proof: Obviously, def Pˇ (z1 , . . , z2n ) = P (z1 − i zn+1 , . . , zn − i zn+n , z1 + i zn+1 , . . , zn + i zn+n ) is a power series of the complex variables z1 , . . , z2n that vanishes on R2n . Therefore, it vanishes identically. This suggests that ρˆ may be represented in the form ρˆ = s- lim n→∞ ρn (α1 , . . ,αn | d2 α1 d2 αn ··· π π with suitably chosen functions ρn (α1 , .
E k1 (x, ω1 ) . . E kν (x, ων ) δ (ω1 + . . + ων − ω) dω1 . . dων . kν(ω1 , . . 9 For the tensor χjk (ω) of the linear susceptibility first order perturbation theory is sufficient. 2 Various Electric Polarization Effects See (R¨omer, 1994, Chapter 5). 10) ↔ Draft, January 2, 2008 7 For a listing of the dependence of the lowest nonlinear susceptibilities on the symmetry classes for crystals see (Brunner and Junge, 1982, Sect. 1). kν(ω, . . , ω) = 0 for some ν > 1 . , 1962a). 64 CHAPTER 3.