Wave Propagation in Linear Media | re-examined

A.3 Solutions for the square barrier
ma = MaxRecursion/. {opts}/.Options[OscInt],
limpos, limneg},
{limneg,limpos} = GaussTruncLims[w,k,n,wp];
(If[limpos DoubleExponential,
WorkingPrecision->wp,
AccuracyGoal->ag,
PrecisionGoal->pg,
MinRecursion->mi,
MaxRecursion->ma]] +
If[limneg >= -1, 0,
NIntegrate[fneg[-t,-Pi k/Sqrt[w],-x,-Pi k,w,k,n,l][xi],
{xi,1,-limneg},
Method->DoubleExponential,
WorkingPrecision->wp,
AccuracyGoal->ag,
PrecisionGoal->pg,
MinRecursion->mi,
MaxRecursion->ma]]
) GaussConst[w,k,n]
];
GaussEvanTruncTemp[f_, x_, t_, w_, k_, n_, l_, opts___Rule] :=
Module[{limpos,limneg,
wp = WorkingPrecision/.{opts}/.Options[OscInt],
ag = AccuracyGoal/. {opts}/.Options[OscInt],
pg = PrecisionGoal/. {opts}/.Options[OscInt],
mi = MinRecursion/. {opts}/.Options[OscInt],
ma = MaxRecursion/. {opts}/.Options[OscInt]},
{limneg,limpos} = GaussTruncLims[w,k,n,wp];
limneg = Max[limneg,-1];
limpos = Min[limpos,1];
NIntegrate[f[-t,Pi k/Sqrt[w],x,-Pi k,w,k,n,l][xi],
{xi,limneg,limpos},
Method->DoubleExponential,
WorkingPrecision->wp,
AccuracyGoal->ag,
PrecisionGoal->pg,
MinRecursion->mi,
MaxRecursion->ma] GaussConst[w,k,n]
];
(*-- Driver functions for the computation of the wave integrals --*)
(* Note that for the reflected wave outside and the left-going part
inside the tunnel spatial coordinate is inverted in the function calls
because of the reverse motion of the partial waves. *)
(* Note also that the spatial coordinate for the regions inside and
behind the barrier is transformed to x-l. *)
PhiInc[x_, t_, w_, k_, n_?Positive, l_, opts___Rule] :=
GaussPassTruncTemp[GaussIncPos,GaussIncNeg,x,t,w,k,n,l,opts] +
GaussEvanTruncTemp[GaussIncPos,x,t,w,k,n,l,opts];
PhiRef[x_, t_, w_, k_, n_?Positive, l_, opts___Rule] :=
GaussPassTruncTemp[GaussRefPos,GaussRefNeg,-x,t,w,k,n,l,opts] +
GaussEvanTruncTemp[GaussRefPos,-x,t,w,k,n,l,opts];
PhiTo[x_, t_, w_, k_, n_?Positive, l_, opts___Rule] :=
GaussPassTruncTemp[GaussToPos,GaussToNeg,x-l,t,w,k,n,l,opts] +
GaussEvanTruncTemp[GaussToPos,x-l,t,w,k,n,l,opts];
PhiFro[x_, t_, w_, k_, n_?Positive, l_, opts___Rule] :=
GaussPassTruncTemp[GaussFroPos,GaussFroNeg,-x+l,t,w,k,n,l,opts] +
GaussEvanTruncTemp[GaussFroPos,-x+l,t,w,k,n,l,opts];
PhiTrans[x_, t_, w_, k_, n_?Positive, l_, opts___Rule] :=
GaussPassTruncTemp[GaussTransPos,GaussTransNeg,x-l,t,w,k,n,l,opts] +
GaussEvanTruncTemp[GaussTransPos,x-l,t,w,k,n,l,opts];
PhiTun[x_, t_, w_, k_, n_?Positive, l_, opts___Rule] :=
PhiTo[x,t,w,k,n,l,opts] + PhiFro[x,t,w,k,n,l,opts];
Phi[x_, t_, w_, k_, n_?Positive, l_, opts___Rule] :=
Which[t < 0, 0,
x >= l, PhiTrans[x,t,w,k,n,l,opts],
0 < x && x < l, PhiTun[x,t,w,k,n,l,opts],
x