Manipulate[Plot[U0 Exp[x/x0]UnitStep[-x],{x,-100, 100},PlotRange->{0,2 U0}],{{x0,5},0.1,20},{U0,1}]

Manipulate[Plot[Evaluate[Abs[ϕ[x] /. NDSolve[{(E^(x/x0)) U0 UnitStep[-x] ϕ[x]-ϕ''[x]==E0 ϕ[x],ϕ[2 4 Pi/Sqrt[E0]]==1, ϕ'[2 4 Pi/Sqrt[E0]]==I Sqrt[E0]},ϕ[x],{x,-300,300}][[1]]]],{x,-100,100},PlotRange->{0,5}],{{E0,1},0.1,2},{U0,1},{x0,0.1,20}]