Analytical template in the frequency domain for GW echoes from exotic compact object
Reference: Adriano Testa & Paolo Pani, An analytical template for gravitational-wave echoes: signal characterization and prospects of detection with current and future interferometers, https://arxiv.org/abs/1806.04253

Legend:
M: mass
ωR, ωI: quasinormal mode frequency (real and imaginary part) of the black hole (τ=-1/ωI)
t0p: starting time of the black-hole ringdown
ϕ: phase of the quasinormal mode
α: parameter of the Poschl-Teller potential
----
R: reflectivity coefficient at the surface of the exotic compact object
d: width of the cavity (d=xm-x0)
xm: location of the peak of the potential in tortoise coordinates (xm>0)
x0: location of the surface in tortoise coordinate (x0<0)

Quit[];

$Assumptions={ω∈ Reals, d∈ Reals,α∈ Reals,,R∈ Reals};

RBH=-(((E^(2 I xm ω)) Gamma[((α-(Sqrt[-4 V0+(α^2)])-2 I ω)/(2 α))] Gamma[((α+(Sqrt[-4 V0+(α^2)])-2 I ω)/(2 α))] Gamma[1+((I ω)/α)])/(Gamma[((α-(Sqrt[-4 V0+(α^2)]))/(2 α))] Gamma[((α+(Sqrt[-4 V0+(α^2)]))/(2 α))] Gamma[1-((I ω)/α)]));
TBH=-((I Gamma[((α-(Sqrt[-4 V0+(α^2)])-2 I ω)/(2 α))] Gamma[((α+(Sqrt[-4 V0+(α^2)])-2 I ω)/(2 α))] Gamma[1+((I ω)/α)] Sinh[((π ω)/α)])/(π Gamma[1-((I ω)/α)]));
Kappa=((TBH R Exp[-2 I ω x0])/(1-R RBH Exp[-2 I ω x0]))/.x0->xm-d;(*transfer function*)

(*Single-mode ringdown template:*)
hp =HeavisideTheta[t -t0p]Ap Exp[-t/τ] Cos[ωR t+ϕ]/.τ ->-1/ωI//Simplify;

hpp =FourierTransform[hp,t,ω]//Simplify

Final form, source near the surface:

Zgeneral =Simplify[ hpp + Kappa(Exp[2 I ω x0](((1+RBH Exp[-2 I ω x0]))/TBH))hpp//.x0->xm-d//.V0->(1/4) ((α^2)+4 ((ωR)^2)),ωR>0]//FullSimplify

Source localized in a generic location:

Δ=xs-xm (location of the source relative to the peak of the potential)

γ=((Gamma[1+((I ω)/α)])/(Gamma[1-((I ω)/α)])) Gamma[(1/2)-((I (ω-ωR))/α)] Gamma[(1/2)-((I (ω+ωR))/α)];
Zgeneral=hpp(1-(((E^(2 I d ω)) 2 π R Sinh[((π ω)/α)]γ LegendreP[-(1/2)+((I ωR)/α),((I ω)/α),-Tanh[α Δ]])/(2 (π +(E^(2 I d ω)) R Cosh[((π ωR)/α)] γ) (-2 I Cosh[((π (ω+ωR))/α)] LegendreQ[-(1/2)+((I ωR)/α),((I ω)/α),-Tanh[α Δ]]+π LegendreP[-(1/2)+((I ωR)/α),((I ω)/α),-Tanh[α Δ]] Sinh[((π (ω+ωR))/α)]))));