fi = Exp[-Pi*lam/4]*ParabolicCylinderD[-1/2+I*lam,-Exp[-I*Pi/4]*u]

fis = Exp[-Pi*lam/4]*ParabolicCylinderD[-1/2-I*lam,-Exp[I*Pi/4]*u]

fideriv= D[fi,u]

fisderiv= D[fis,u]

W= Sqrt[u^2/4+lam]

V=-u/(u^2+4*lam)

beta2 =(1/(2*W))*(fideriv+(I*W-V/2)*fi)*(fisderiv+(-I*W-V/2)*fis)

W2 = W  - (0.5)* (u^2+4*lam)^{-1.5} +(1.25)*u^2*(u^2+4*lam)^{-2.5}

beta2W2 =(1/(2*Abs[W2]))*(fideriv+(I*W2-V/2)*fi)*(fisderiv+(-I*W2-V/2)*fis)

Plot[beta2W2/.{lam->2},{u,-20,20}]

u0 = -20

delu = 10^(-2)

Data = Table[{N[u0+i*delu],Part[Re[beta2W2/.{lam->1,u->u0+i*delu}],1],Part[Re[beta2W2/.{lam->2,u->u0+i*delu}],1],Part[Re[beta2W2/.{lam->3,u->u0+i*delu}],1]},{i,0,4000}];

(*  Export["D:\\Userdata\\olduserdata\\Research\\de Sitter States\\Beta2Efield.txt",Data,"Table"]  *)

Data2 = Table[{N[u0+i*delu],Re[N[beta2/.{lam->1,u->u0+i*delu}]],Part[Re[N[beta2W2/.{lam->1,u->u0+i*delu}]],1]},{i,0,4001}];

(*

Export["D:\\Userdata\\olduserdata\\Research\\de Sitter States\\Beta2Efield2.txt",Data2,"Table"]  *)

Part[Re[N[beta2W2/.{lam->1,u->u0}]],1]

N[beta2W2/.{lam->1,u->u0}]

(*  Now check the Wronskian condition. *)
Wron = Expand[I*fis*fideriv-I*fi*fisderiv]

Wron/.{lam->1,u->0}

N[Wron/.{lam->1,u->10}]

N[Wron/.{lam->1,u->0}]

N[Wron/.{lam->1,u->-10}]

N[Wron/.{lam->2,u->-10}]

N[Wron/.{lam->2,u->0}]

(*  Now compute alpha2 and check the difference. *)
alpha2 =(1/(2*W))*(fideriv+(-I*W-V/2)*fi)*(fisderiv+(I*W-V/2)*fis);

diff = Expand[alpha2-beta2]

N[diff/.{lam->1,u->-10}]

N[diff/.{lam->1,u->0}]

N[diff/.{lam->1,u->10}]

N[diff/.{lam->2,u->10}]

N[diff/.{lam->2,u->0}]

N[diff/.{lam->3,u->0}]

N[diff/.{lam->3,u->10}]

alpha2W2 =(1/(2*Abs[W2]))*(fideriv+(-I*W2-V/2)*fi)*(fisderiv+(I*W2-V/2)*fis);

diffW2 = Expand[alpha2W2-beta2W2];

N[diffW2/.{lam->1,u->-10}]

N[diffW2/.{lam->1,u->0}]

N[diffW2/.{lam->2,u->-10}]

N[diffW2/.{lam->2,u->10}]

N[diffW2/.{lam->3,u->10}]

N[diffW2/.{lam->3,u->0}]

N[diffW2/.{lam->3,u->-10}]

N[beta2/.{lam->1,u->100}]

N[beta2W2/.{lam->1,u->100}]

N[beta2/.{lam->1,u->1000}]

N[beta2W2/.{lam->1,u->1000}]

{0.0018674427315578987` +0.` I}

N[E^(-2*Pi)]

N[beta2W2/.{lam->2,u->1000}]

N[E^(-4*Pi)]

N[beta2W2/.{lam->3,u->1000}]

N[E^(-6*Pi)]