Massive Scalar on AdS

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$Assumptions=r∈Positive&&n∈Integers

<<EDCRGTCcode.m;
xIN={t,r,ϕ};
g=DiagonalMatrix[{-((r^2)+1),(1/((r^2)+1)),(r^2) }];
RGtensors[g,xIN];

Eq=multiDot[gUU,covD[covD[f[t,r,ϕ]]],{1,1},{2,2}]-(mass^2)f[t,r,ϕ]

Eqr=Eq/.f->Function[{t,r,ϕ},Exp[-I ω t+I nn ϕ]Q[r]]

Eqrfin=FullSimplify[Numerator[FullSimplify[Eqr]]/((E^(I (nn ϕ))))]

Refine[FullSimplify[Together[DSolve[Eqrfin==0,Q[r],r]]][[1]][[1]][[2]]]

Clearly the branch multiplying C[2] diverges as r aproaches the origin. So we keep only the second branch

S[r_]=%7/.C[2]->0

FullSimplify[Series[S[r],{r,Infinity,0}]]

We can see that there are two branches approaching infinity as (r^(-Δ+)) and (r^(-Δ-)). For positive squared mass we have that the second branch diverges, and therefore we must impose the following conditions

Solve[(1/2) (1+(Sqrt[1+(mass^2)])+nn-ω)==-p,ω]

Which is the spectrum for the AdS3 Oscillons. Notice that p=0,1,2,3,... is the overtone number and nn is the angular momentum of the scalar field. Notice that the spectrum is equispaced. Let’s now see some plots of the normal mode functions

S[r]/.ω->1+(Sqrt[1+(mass^2)])+nn+2 p//FullSimplify

{CN=1,mass=1}

NM[r_,p_,nn_]=((((((r^2)))^((1+nn)/2)) (((1+(r^2)))^((1/2) (1+(Sqrt[1+(mass^2)])+nn+2 p))) CN Hypergeometric2F1[1+nn+p,1+(Sqrt[1+(mass^2)])+nn+p,1+nn,-(r^2)])/r)

NM[r,0,0]

Fundamental mode

Plot[Simplify[NM[r,0,0]],{r,0,10},PlotRange->All]

First harmonic

Plot[Simplify[NM[r,1,0]],{r,0,10},PlotRange->All]