ϕA[ξ_,η_,R_,α_]=1/Sqrt[π]Exp[-α R / 2 (ξ+η)]
ϕB[ξ_,η_,R_,α_]=1/Sqrt[π]Exp[-α R / 2 (ξ-η)]

Vol[ξ_,η_,ϕ_,R_]=R^3/8(ξ^2-η^2);
V[ξ_,η_,R_]=-2/(R(ξ+η))-2/(R(ξ-η))+1/R;

Limit[V[ξ,η,R],R->∞]

Clear[H,S]

A=1.13275257;
B=1.61922659;

nfun=9;
wyk={A/B^2,A/B,A,A B, A B^2, A B^3, A B^4, A B^5, A B^6}

Funcs1=Table[ϕA[ξ,η,R,wyk[[i]]],{i,1,nfun}];
Funcs2=Table[ϕB[ξ,η,R,wyk[[i]]],{i,1,nfun}];
Funcs=Join[Funcs1,Funcs2];

Clear[ϕ]

TProlate[ξ_,η_,ϕ_,R_,α_]=-2/(R^2 (ξ^2-η^2))*(D[(ξ^2-1)*D[Psi[ξ,η,ϕ,R,α],{ξ,1}],{ξ,1}]+D[(1-η^2)*D[Psi[ξ,η,ϕ,R,α],{η,1}],{η,1}]);

T=Table[Integrate[-2Vol[ξ,η,ϕ,R]/(R^2 (ξ^2-η^2))*(D[(ξ^2-1)*D[Funcs[[i]],{ξ,1}],{ξ,1}]+D[(1-η^2)*D[Funcs[[i]],{η,1}],{η,1}])*Funcs[[j]],{ξ,1,∞},{η,-1,1},{ϕ,0,2 π}],{i,1,8},{j,1,8}]

HAA[R_,α_,β_]=Integrate[-2Vol[ξ,η,ϕ,R]/(R^2 (ξ^2-η^2))*(D[(ξ^2-1)*D[ϕA[ξ,η,R,α],{ξ,1}],{ξ,1}]+D[(1-η^2)*D[ϕA[ξ,η,R,α],{η,1}],{η,1}])*ϕA[ξ,η,R,β]+Vol[ξ,η,ϕ,R]ϕA[ξ,η,R,α]V[ξ,η,R]ϕA[ξ,η,R,β],{ξ,1,∞},{η,-1,1},{ϕ,0,2 π},Assumptions->{α>0,β>0,R>0}]
HAB[R_,α_,β_]=Integrate[-2Vol[ξ,η,ϕ,R]/(R^2 (ξ^2-η^2))*(D[(ξ^2-1)*D[ϕA[ξ,η,R,α],{ξ,1}],{ξ,1}]+D[(1-η^2)*D[ϕA[ξ,η,R,α],{η,1}],{η,1}])*ϕB[ξ,η,R,β]+Vol[ξ,η,ϕ,R]ϕA[ξ,η,R,α]V[ξ,η,R]ϕB[ξ,η,R,β],{ξ,1,∞},{η,-1,1},{ϕ,0,2 π},Assumptions->{α>0,β>0,R>0}]
SAA[R_,α_,β_]=Integrate[Vol[ξ,η,ϕ,R]ϕA[ξ,η,R,α]ϕA[ξ,η,R,β],{ξ,1,∞},{η,-1,1},{ϕ,0,2 π},Assumptions->{α>0,β>0,R>0}]
SAB[R_,α_,β_]=Integrate[Vol[ξ,η,ϕ,R]ϕA[ξ,η,R,α]ϕB[ξ,η,R,β],{ξ,1,∞},{η,-1,1},{ϕ,0,2 π},Assumptions->{α>0,β>0,R>0}]

Haa=Table[HAA[R,wyk[[i]],wyk[[j]]],{i,1,nfun},{j,1,nfun}];
Hab=Table[Limit[HAB[R,wyk[[i]],xx],xx->wyk[[j]]],{i,1,nfun},{j,1,nfun}];
Saa=Table[SAA[R,wyk[[i]],wyk[[j]]],{i,1,nfun},{j,1,nfun}];
Sab=Table[Limit[SAB[R,wyk[[i]],xx],xx->wyk[[j]]],{i,1,nfun},{j,1,nfun}];

H=ArrayFlatten[{{Haa,Hab},{Hab,Haa}}];
S=ArrayFlatten[{{Saa,Sab},{Sab,Saa}}];

Clear[H,S]

Eigensystem[N[{H,S}]/.R->31]//Transpose//Sort//Transpose

Krzywa1=Array[0.&,{600,2}];
Krzywa2=Array[0.&,{600,2}];
Krzywa3=Array[0.&,{600,2}];
Krzywa4=Array[0.&,{600,2}];
Krzywa5=Array[0.&,{600,2}];
Krzywa6=Array[0.&,{600,2}];
For[i=1,i≤600,i++,
RR=0.5+i*0.05;
Krzywa1[[i,1]]=RR;
Krzywa2[[i,1]]=RR;
Krzywa3[[i,1]]=RR;
Krzywa4[[i,1]]=RR;
Krzywa5[[i,1]]=RR;
Krzywa6[[i,1]]=RR;
Krzywa1[[i,2]]=Sort[Eigenvalues[N[{H,S}]/.R->RR]][[1]];
Krzywa2[[i,2]]=Sort[Eigenvalues[N[{H,S}]/.R->RR]][[2]];
Krzywa3[[i,2]]=Sort[Eigenvalues[N[{H,S}]/.R->RR]][[3]];
Krzywa4[[i,2]]=Sort[Eigenvalues[N[{H,S}]/.R->RR]][[4]];
Krzywa5[[i,2]]=Sort[Eigenvalues[N[{H,S}]/.R->RR]][[5]];
Krzywa6[[i,2]]=Sort[Eigenvalues[N[{H,S}]/.R->RR]][[6]];
]
Clear[RR];

ListLinePlot[{Krzywa1,Krzywa2,Krzywa5,Krzywa6},PlotRange->Full]