g[ϵ_, ξ0_, ρ_, f_]:=Evaluate[{ (ψ[-ξ0] +  ψ'[-ξ0]/(I (Sqrt[ ϵ])))/(2  (E^(-I (Sqrt[ϵ]) ξ0))),(ψ[-ξ0] -  ψ'[-ξ0]/(I (Sqrt[ ϵ])))/(2 (E^(I (Sqrt[ ϵ]) ξ0)))}/.NDSolve[{-ψ''[ξ] + (ρ^2) 
                        f[ξ] ψ[ξ] == ϵ ψ[ξ], ψ[ξ0] == (E^(I (Sqrt[ ϵ]) ξ0)), ψ'[ξ0] == I (Sqrt[ ϵ]) ψ[ξ0]}, ψ, {ξ, -ξ0, ξ0}]] [[1]]


t[ϵ_, ξ0_, ρ_, f_]:=(((Abs[g[ϵ, ξ0, ρ, f][[1]]]))^(-2))
r[ϵ_, ξ0_, ρ_, f_]:=(((Abs[g[ϵ, ξ0, ρ, f][[2]]/g[ϵ, ξ0, ρ, f][[1]]]))^2)

f[x_] = (E^(-(x^2)))

t[1, 4, 1, f]
r[1, 4, 1, f]
t[1, 4, 1, f] + r[1, 4, 1, f]

Plot[t[ϵ, 4, 1, f], {ϵ, 0, 3}]

Plot[t[ϵ, 4, 10, f], {ϵ, 0, 150}, 
      PlotRange -> {0, 1}]

g[x_] = (1/(1 + (x^10)))

Plot[t[ϵ, 4, 1, g], {ϵ, 0, 10}, 
      PlotRange -> {0, 1}]

Plot[t[ϵ, 4, 5, g], {ϵ, 0, 60}, 
      PlotRange -> {0, 1}]