Massive scalar bound states of a Kerr black hole
References: Sam Dolan “Instability of the massive Klein-Gordon field on the Kerr spacetime” (2007);  Richard Brito, Vitor Cardoso and Paolo Pani, “Black hole superradiance: from fundamental physics to astrophysics” (2015)

We first set the parameters of the Kerr spacetime, the field and the mode we’re interested in:

LEAVER[w_,ar_,mu_,l_,m_,NITMAX_,NITMAX2_]:=(
br=Sqrt[1-ar^2];
rminus=(1-br);
rplus=1+br;
s=0;
k1=1/2*Abs[m-s];
k2=1/2*Abs[m+s];
Alm=l*(l+1)-s*(s+1);
OmegaH=ar/(2*M*rplus);
(**************************************)
(******)
k1=1/2*Abs[m-s];
k2=1/2*Abs[m+s];
(*Alm=l*(l+1)-s*(s+1);*)
wang=(Sqrt[(w^2)-(mu^2)]);

γang=Function[n,2*ar*wang*(n+k1+k2+s)];βang=Function[n,n*(n-1)+2*n*(k1+k2+1-2*ar*wang)-(2*ar*wang*(2*k1+s+1)-(k1+k2)*(k1+k2+1))-(ar^2*wang^2+s*(s+1)+Sep)];
αang=Function[n,-2*(n+1)*(n+2*k1+1)];
Leaver31Ang[Sep_]:=Module[{Rn},
For[{n=NITMAX2;Rn=-1.0;},
n>0,
{
Rn=((γang[n])/(βang[n]-αang[n]*Rn));n--;}
];Rn];
Leaver33Ang[Sep_]:=((βang[0])/(αang[0]))-Leaver31Ang[Sep];
Alm=FindRoot[Leaver33Ang[Sep]==0,{Sep,Alm}][[1]][[2]];
(**************************************)
q=-Sqrt[mu^2-w^2];c0=1-2*I*w-2*I/br*(w-ar*m/2);c1=-4+4*I*(w-I*q*(1+br))+4*I/br*(w-ar*m/2)-2*(w^2+q^2)/q;c2=3-2*I*w-2*I/br*(w-ar*m/2)-2*(q^2-w^2)/q;c3=2*I*(w-I*q)^3/q+2*(w-I*q)^2*br+q^2*ar^2+2*I*q*ar*m-Alm-1-(w-I*q)^2/q+2*q*br+2*I/br*((w-I*q)^2/q+1)*(w-ar*m/2);c4=(w-I*q)^4/q^2+2*I*w*(w-I*q)^2/q-2*I/br*(w-I*q)^2/q*(w-ar*m/2);γ=Function[n,(n^2)+(c2-3)*n+c4+(-c2+2)*0];
β=Function[n,-2*n^2+(c1+2)*n+c3];α=Function[n,(n^2)+(c0+1)*n+c0];
Leaver31=Module[{Rn},For[{n=NITMAX;Rn=-1.0;},n>0,{Rn=((γ[n])/(β[n]-α[n]*Rn));n--;}
];Rn];
Leaver33=((β[0])/(α[0]))-Leaver31);

LEAVER[(18/100),(9/10),(2/10),1,1,100,100]

fun[x_?NumberQ]:=LEAVER[x,0.9,0.2,1,1,200,100];
FindRoot[fun[x],{x,0.199+(10^(-8))I}]