4.3 Analytic properties of the correlation functions It is not possible to derive exact expressions for the functions I(k, t), G(r,t), S(k, w), except for the very simplest scattering systems. Approximations have to be made. It is therefore important to estab- lish the basic analytic properties of the functions in order to check that the approximate functions have these properties. We define an operator p(r,t) = Σ 8{r — R₁(t)}. (4.30) It gives the number density of particles at position r at time t and is known as the particle-density operator. From (A.13) we can express the operator in terms of its Fourier transform where 1 p(r,t) = Pk (t) exp(ik. r) dk, (4.31) (2πT)³ Px(t)=Σ exp{-ik. R;(t)}. (4.32) Correlation functions in nuclear scattering- 66 The integral in (4.31) is over all reciprocal space. Since R,(t) is a Hermitian operator p+(r,t) = p(r,t), (4.33) P(t)=P-(t). (4.34) From (4.2) and (4.32) N (4.35) I(k, t)=(Pk(0)p-x(t)), and from (4.17) and (4.30) G(r,t) = (p(r', 0)p(r' + r, 1)) dr'. (4.36) We may use the properties of the particle-density operator p(r,t) to prove the following results: I(k,t) = I*(K, -t), (4.37) G(r,t)=G*(-r, -1), (4.38) S(K, w)= S*(K, w), (4.39) I(K, t)=I(-K, -t+ihẞ), (4.40) G(r,t)=G(-r, -t+ihẞ), (4.41) S(K, w)= exp(hwẞ)S(-K, -w). (4.42)

4.3 Analytic properties of the correlation functions It is not possible to derive exact expressions for the functions I(k, t), G(r,t), S(k, w), except for the very simplest scattering systems. Approximations have to be made. It is therefore important to estab- lish the basic analytic properties of the functions in order to check that the approximate functions have these properties. We define an operator p(r,t) = Σ 8{r — R₁(t)}. (4.30) It gives the number density of particles at position r at time t and is known as the particle-density operator. From (A.13) we can express the operator in terms of its Fourier transform where 1 p(r,t) = Pk (t) exp(ik. r) dk, (4.31) (2πT)³ Px(t)=Σ exp{-ik. R;(t)}. (4.32) Correlation functions in nuclear scattering- 66 The integral in (4.31) is over all reciprocal space. Since R,(t) is a Hermitian operator p+(r,t) = p(r,t), (4.33) P(t)=P-(t). (4.34) From (4.2) and (4.32) N (4.35) I(k, t)=(Pk(0)p-x(t)), and from (4.17) and (4.30) G(r,t) = (p(r', 0)p(r' + r, 1)) dr'. (4.36) We may use the properties of the particle-density operator p(r,t) to prove the following results: I(k,t) = I(K, -t), (4.37) G(r,t)=G(-r, -1), (4.38) S(K, w)= S*(K, w), (4.39) I(K, t)=I(-K, -t+ihẞ), (4.40) G(r,t)=G(-r, -t+ihẞ), (4.41) S(K, w)= exp(hwẞ)S(-K, -w). (4.42)