gMuNuGeneric={{-f[x], 0 ,0,0}, {0, 1/f[x], 0, 0}, {0, 0, C[x]^2, 0}, {0, 0, 0,C[x]^2}};
MatrixForm[gMuNuGeneric];
GMuNuGeneric=Inverse[gMuNuGeneric];
coord2D={t, x, y, z};
sqrtngGeneric=Simplify[Sqrt[-Det[gMuNuGeneric]], {C[x]>0}];
box4DGeneric=0;box4D2Generic=0;
Do[ {box4D2Generic=1/sqrtngGeneric*(D[sqrtngGeneric*GMuNuGeneric[[i,j]]*D[Theta[t,x]/C[x],coord2D[[j]]] ,coord2D[[i]]]),box4DGeneric=box4DGeneric+box4D2Generic}, {i, 1, 2}, {j, 1, 2}];
box4DGenericDimple=Collect[Simplify[box4DGeneric], {Derivative[0,1][Theta][t,x], Derivative[2,0][Theta][t,x], Derivative[0,2][Theta][t,x],Theta[t,x]}];
box4DGenericDimpleCorrected=Simplify[box4DGenericDimple*C[x]];

The Potential contains already contains an overal negative sign because it is found through the Mode equation.

PotentialGeneric=Coefficient[box4DGenericDimpleCorrected,Theta[t,x]];
EffectivePotentialGeneric=PotentialGeneric*f[x];
A2[x_]=n[x]*(c[x]^2-v[x]^2)/c[x];
DA2[x_]=D[A2[x], x];
B2[x_]=(Sqrt[n[x]/c[x]]);
DB2[x_]=D[B2[x], x];
DDB2[x_]=D[DB2[x], x];
EffectivePotentialGeneric2019=Expand[(EffectivePotentialGeneric/.{f[x]->A2[x], f'[x]->DA2[x], C[x]->B2[x],C'[x]->DB2[x] ,C''[x]->DDB2[x] })];

rho1plus=264/100;
rho1minus=11;
XXX=27/10*10^(-7);
TTT=10^(-4);
Mplus=rho1minus/rho1plus;
uNISt=166053906660/100000000000*10^-27;(*1.66053906660*10^-27*);
hbarNIST=1054571817/1000000000*10^-34;(*1.054571817*10^-34*);
MRB87NISt=86909187/1000000*uNISt;(*86.909187*uNISt*);
CInt2016=25*10^-5;
CExt2016=57*10^-5;
XiInt=hbarNIST/(MRB87NISt*CInt2016);
XiExt=hbarNIST/(MRB87NISt*CExt2016);
Xi=Sqrt[XiInt*XiExt]
sigma=Sqrt[Mplus-1]/(Xi);
nZero[x1_]=(1/XXX)*rho1plus*(Mplus+(1-Mplus)/(Cosh[sigma*(xstart1)]^2));
vZero=1/(TTT*nZero[xstart1]);
cZero=(XXX/TTT)*Sqrt[(XXX*nZero[xstart1])/(rho1plus*rho1minus^2)];

N[Xi]

xstartSolver=NSolve[vZero==cZero, xstart1, Reals, WorkingPrecision->80];

n2[x_]=(1/XXX)*rho1plus*(Mplus+(1-Mplus)/(Cosh[sigma*(x+xstart1/.xstartSolver[[1]])]^2));(*This contains approximately the location of c[x]=v[x]*)
dn2=D[n2[x], x]/n2[x];
ddn2=D[dn2, x]/n2[x];
v2=1/(TTT*n2[x]);
dv2=D[v2, x]/n2[x];
ddv2=D[dv2, x]/n2[x];
c2=(XXX/TTT)*Sqrt[(XXX*n2[x])/(rho1plus*rho1minus^2)]
dc2=D[c2, x]/n2[x];
ddc2=D[dc2,x]/n2[x];
N[(c2-v2)/.x->-10000*Xi]

Thcalc1=N[(2*N[v2/.x->0, 20]*Sqrt[N[c2/.x->-1000*Xi, 20]*N[v2/.x->-1000*Xi, 20]])/((N[c2/.x->-1000*Xi, 20]*N[v2/.x->-1000*Xi, 20])+N[v2/.x->0]^2),20]

Rhcalc1=N[(N[v2/.x->-1000*Xi, 20]*N[c2/.x->-1000*Xi, 20]-N[v2/.x->-0, 20]^2)/(N[v2/.x->-1000*Xi, 20]*N[c2/.x->-1000*Xi, 20]+N[v2/.x->-0, 20]^2),20]

N[(c2-v2)/.x->0]

NSolve[v2==c2, x, Reals]

ncvPlot=Plot[{ n2[x]*XXX*TTT, c2, v2}, {x, -2*Xi, 0}, PlotLegends->{ "n[x]*X*T","c[x]", "v[x]"}, ImageSize->Large]

Export["D:\\Dropbox\\BH\\BEC\\nofxcofxvofx\\CorrectCVN\\2020Files\\Plots\\ncvPlot.jpg",ncvPlot, ImageResolution->200]

EffectivePotential2019Genericv1Values=EffectivePotentialGeneric2019/.{c[x]->c2, v[x]->v2,n[x]->n2[x],v'[x]->dv2,c'[x]->dc2,n'[x]->dn2,c''[x]->ddc2, v''[x]->ddv2  ,  n''[x]->ddn2};
EffectivePotential2019PlotExterior=Plot[-EffectivePotential2019Genericv1Values, {x, -2*Xi, 0}, ImageSize->Large, ImageSize->Large, PlotRange->All, PlotLegends->{ "V_Eff"}]

Export["E:\\Dropbox\\BH\\BEC\\nofxcofxvofx\\CorrectCVN\\2020Files\\Potentialvariations\\Plots\\M5p00EffectivePotential2019PlotExterior.jpg",EffectivePotential2019PlotExterior, ImageResolution->200]

ncvPlotInt=Plot[{ n2[x]*XXX*TTT, c2, v2}, {x, -2*Xi, -xstart1/.xstartSolver[[1]]}, PlotLegends->{ "n[x]*X*T","c[x]", "v[x]"}, ImageSize->Large]

Export["E:\\Dropbox\\BH\\BEC\\nofxcofxvofx\\CorrectCVN\\2020Files\\Potentialvariations\\Plots\\M5p00ncvPlotInt.jpg",ncvPlotInt, ImageResolution->200]

EffectivePotential2019PlotInterior=Plot[-EffectivePotential2019Genericv1Values, {x, -2*Xi, -xstart1/.xstartSolver[[1]]}, ImageSize->Large, ImageSize->Large, PlotRange->All, PlotLegends->{ "V_Eff"}]

Export["E:\\Dropbox\\BH\\BEC\\nofxcofxvofx\\CorrectCVN\\2020Files\\Potentialvariations\\Plots\\M4p17EffectivePotential2019PlotInterior.jpg",EffectivePotential2019PlotInterior, ImageResolution->200]

x1=-10^-22;
x3=x1;
x2=-x1;
x4=x2;
h=FullSimplify[-c2/(c2^2-v2^2)];
Dh=FullSimplify[D[h, x]];
xstar[y_]:=NIntegrate[(h), {x,  x3,y}, WorkingPrecision->70, PrecisionGoal->50, MaxRecursion->50]
xinitial1=N[-5*10^-6, 70];
xfinal=N[-10^-22,70];
xfinal2=N[-10^-22,70];
Vcheck1=EffectivePotential2019Genericv1Values/.x->xinitial1
Vcheck2=EffectivePotential2019Genericv1Values/.x->xfinal
Vcheck2=N[EffectivePotential2019Genericv1Values/.x->xfinal2]
Dh/.x->-20*10^-6

TIRIofOmegaVeff=Table[{(10^(i/11)*10^(-6)),
s1Veff=NDSolve[{1/h^2*y1Veff''[x]-Dh/h^3*y1Veff'[x]-(-((10^(i/11)*10^(-6)))^2-EffectivePotential2019Genericv1Values)*y1Veff[x]==0,y1Veff[xinitial1]==(Exp[-I*(10^(i/11)*10^(-6))*xstar[xinitial1]]),y1Veff'[xinitial1]==((-I*(10^(i/11)*10^(-6))*h*Exp[-I*(10^(i/11)*10^(-6))*xstar[xinitial1]])/.x->xinitial1)},y1Veff,{x,xinitial1,xfinal2}, WorkingPrecision->60, PrecisionGoal->18, AccuracyGoal->18, MaxSteps->1000000,InterpolationOrder->All];
1/(Evaluate[Exp[I*(10^(i/11)*10^(-6))*xstar[xfinal2]]]/2*(Evaluate[y1Veff[xfinal2]/.s1Veff]+(I/((10^(i/11)*10^(-6))*h)/.x->xfinal2)*Evaluate[y1Veff'[xfinal2]/.s1Veff])),
(Evaluate[Exp[-I*(10^(i/11)*10^(-6))*xstar[xfinal2]]]/2*(Evaluate[y1Veff[xfinal2]/.s1Veff]-(I/((10^(i/11)*10^(-6))*h)/.x->xfinal2)*Evaluate[y1Veff'[xfinal2]/.s1Veff])) / (Evaluate[Exp[I*(10^(i/11)*10^(-6))*xstar[xfinal2]]]/2*(Evaluate[y1Veff[xfinal2]/.s1Veff]+(I/((10^(i/11)*10^(-6))*h)/.x->xfinal2)*Evaluate[y1Veff'[xfinal2]/.s1Veff])) ,Evaluate[y1Veff[xfinal2]/.s1Veff]}, {i,101,110}];

Evaluate[y1Veff[xfinal2]/.s1Veff]

Omega1Veff=Flatten[TIRIofOmegaVeff[[All, 1]]];
TI1Veff=Flatten[Map[Abs,TIRIofOmegaVeff[[All, 2]]]];
RI1Veff=Flatten[Map[Abs,TIRIofOmegaVeff[[All, 3]]]];
dataTIVeff=Transpose[{Omega1Veff, TI1Veff}];
dataRIVeff=Transpose[{Omega1Veff, RI1Veff}];

ThPlotVeff=ListLogLinearPlot[dataTIVeff,PlotStyle->Automatic,AxesLabel->{"ω"}, PlotLegends->{"|T_H|"}, PlotRange->{0.95, 1}]

Export["E:\\Dropbox\\BH\\BEC\\nofxcofxvofx\\CorrectCVN\\2020Files\\Potentialvariations\\Plots\\M4p17ThPlotVeff50p6.jpg",ThPlotVeff,ImageResolution->200]

RhPlotVeff=ListLogLinearPlot[dataRIVeff,PlotStyle->Automatic,AxesLabel->{"ω"}, PlotLegends->{"|R_H|"}, PlotRange->All]

Export["E:\\Dropbox\\BH\\BEC\\nofxcofxvofx\\CorrectCVN\\2020Files\\Potentialvariations\\Plots\\M4p17RhPlotVeff50p6.jpg",RhPlotVeff,ImageResolution->200]

Export["E:\\Dropbox\\BH\\BEC\\nofxcofxvofx\\CorrectCVN\\2020Files\\Potentialvariations\\RawData\\M4p17THRHTableDatap6.mx",TIRIofOmegaVeff]

TIRIofOmegaVeff[[10, 4]]

Beep[]