Initialisation

(Subscript[Y, ℓ_,m_])[θ_,ϕ_]=SphericalHarmonicY[ℓ,m,θ,ϕ];

(Subscript[Y, ℓ_])[θ_]=SphericalHarmonicY[ℓ,0,θ,ϕ];

Needs["MaTeX`"]
SetOptions[MaTeX,"Preamble"->{"\\usepackage{amssymb,amsmath,latexsym,amsfonts,amsthm,xcolor,bm,mhchem}","\\definecolor{darkgreen}{RGB}{0, 180, 0}","\\definecolor{darkcyan}{RGB}{0, 180, 180}","\\definecolor{darkmagenta}{RGB}{180,0, 180}"}];

Radius of convergence

(Subscript[ψ, i_])[θ_]=(Subscript[Y, i])[θ];

(Subscript[ϕ, i_])=(Subscript[ψ, i])[θ1](Subscript[ψ, i])[θ2];

(Subscript[P, l_])[θ_]=((Sqrt[2l+1])/(4 π )) LegendreP[l,Cos[θ]];

(Subscript[ϵ, l_])=((l (l+1))/(2 (R^2))) +(2/R)-(1/((2l+1)R));

DiagonalMatrix[Table[2(Subscript[ϵ, l]),{l,0,8}]];

HnRMP[R_]=(({{(2/R), 0, 0, 0, 0, 0, 0, 0, 0}, {0, 2 ((1/(R^2))+(5/(3 R))), 0, 0, 0, 0, 0, 0, 0}, {0, 0, 2 ((3/(R^2))+(9/(5 R))), 0, 0, 0, 0, 0, 0}, {0, 0, 0, 2 ((6/(R^2))+(13/(7 R))), 0, 0, 0, 0, 0}, {0, 0, 0, 0, 2 ((10/(R^2))+(17/(9 R))), 0, 0, 0, 0}, {0, 0, 0, 0, 0, 2 ((15/(R^2))+(21/(11 R))), 0, 0, 0}, {0, 0, 0, 0, 0, 0, 2 ((21/(R^2))+(25/(13 R))), 0, 0}, {0, 0, 0, 0, 0, 0, 0, 2 ((28/(R^2))+(29/(15 R))), 0}, {0, 0, 0, 0, 0, 0, 0, 0, 2 ((36/(R^2))+(33/(17 R)))}}));

T[R_]=(1/(R^2))Table[KroneckerDelta[i,j]i(i+1),{i,0,8},{j,0,8}];

(({{0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, (2/(R^2)), 0, 0, 0, 0, 0, 0, 0}, {0, 0, (6/(R^2)), 0, 0, 0, 0, 0, 0}, {0, 0, 0, (12/(R^2)), 0, 0, 0, 0, 0}, {0, 0, 0, 0, (20/(R^2)), 0, 0, 0, 0}, {0, 0, 0, 0, 0, (30/(R^2)), 0, 0, 0}, {0, 0, 0, 0, 0, 0, (42/(R^2)), 0, 0}, {0, 0, 0, 0, 0, 0, 0, (56/(R^2)), 0}, {0, 0, 0, 0, 0, 0, 0, 0, (72/(R^2))}}))

vsh[i_,j_,n_]:=(1/R)  (∑_(l=0)^(2n))(2i+1)(2j+1)((ThreeJSymbol[{i,0},{j,0},{l,0}])^4)

Table[vsh[i,j,i+j],{i,0,8},{j,0,8}]//N

Vsh[R_]={{(1/R),((0.3333333333333333`)/R),((0.2`)/R),((0.14285714285714285`)/R),((0.1111111111111111`)/R),((0.09090909090909091`)/R),((0.07692307692307693`)/R),((0.06666666666666667`)/R),((0.058823529411764705`)/R)},{((0.3333333333333333`)/R),((1.16`)/R),((0.3768707482993197`)/R),((0.23894179894179896`)/R),((0.17771406083094396`)/R),((0.14227083457852688`)/R),((0.11891502436956983`)/R),((0.10227882311991975`)/R),((0.08979197219053826`)/R)},{((0.2`)/R),((0.3768707482993197`)/R),((1.163265306122449`)/R),((0.3735274826183917`)/R),((0.2394342476260558`)/R),((0.18040103974169908`)/R),((0.146121949770896`)/R),((0.12336704694085163`)/R),((0.10701846146841107`)/R)},{((0.14285714285714285`)/R),((0.23894179894179896`)/R),((0.3735274826183917`)/R),((1.1585046810221635`)/R),((0.36821375982215143`)/R),((0.23639936203151535`)/R),((0.1789388409525929`)/R),((0.1457035485478321`)/R),((0.123649435931834`)/R)},{((0.1111111111111111`)/R),((0.17771406083094396`)/R),((0.2394342476260558`)/R),((0.36821375982215143`)/R),((1.1538451523898603`)/R),((0.36372465712226587`)/R),((0.23326087447106705`)/R),((0.17682041129967005`)/R),((0.14432821252077524`)/R)},{((0.09090909090909091`)/R),((0.14227083457852688`)/R),((0.18040103974169908`)/R),((0.23639936203151535`)/R),((0.36372465712226587`)/R),((1.150007604086675`)/R),((0.36012786983290884`)/R),((0.230517596686319`)/R),((0.17475900884511636`)/R)},{((0.07692307692307693`)/R),((0.11891502436956983`)/R),((0.146121949770896`)/R),((0.1789388409525929`)/R),((0.23326087447106705`)/R),((0.36012786983290884`)/R),((1.1469018642211581`)/R),((0.35723120605820047`)/R),((0.2281924994972908`)/R)},{((0.06666666666666667`)/R),((0.10227882311991975`)/R),((0.12336704694085163`)/R),((0.1457035485478321`)/R),((0.17682041129967005`)/R),((0.230517596686319`)/R),((0.35723120605820047`)/R),((1.1443650135634595`)/R),((0.35486187367716066`)/R)},{((0.058823529411764705`)/R),((0.08979197219053826`)/R),((0.10701846146841107`)/R),((0.123649435931834`)/R),((0.14432821252077524`)/R),((0.17475900884511636`)/R),((0.2281924994972908`)/R),((0.35486187367716066`)/R),((1.1422625061276128`)/R)}};

Table[((8(π^2))/R)(∫_0^π)(Subscript[P, l])[θ](1/(2Sin[θ/2]))(Subscript[P, m])[θ]Sin[θ]\[DifferentialD]θ,{l,0,8},{m,0,8}]//N

Vcsf[R_]={{(1/R),((0.5773502691896258`)/R),((0.4472135954999579`)/R),((0.3779644730092272`)/R),((0.3333333333333333`)/R),((0.30151134457776363`)/R),((0.2773500981126146`)/R),((0.2581988897471611`)/R),((0.24253562503633297`)/R)},{((0.5773502691896258`)/R),((1.4`)/R),((0.848367780597815`)/R),((0.6837493894061095`)/R),((0.592346380077668`)/R),((0.5311600271468729`)/R),((0.48620730337181`)/R),((0.45126077735968606`)/R),((0.4230319832800408`)/R)},{((0.4472135954999579`)/R),((0.848367780597815`)/R),((1.5714285714285714`)/R),((0.9885743706824466`)/R),((0.8114774831432803`)/R),((0.7117377302633934`)/R),((0.6440520175728418`)/R),((0.5936999228322687`)/R),((0.5541199346189097`)/R)},{((0.3779644730092272`)/R),((0.6837493894061095`)/R),((0.9885743706824466`)/R),((1.6815850815850817`)/R),((1.0845553992409294`)/R),((0.900512006141845`)/R),((0.7959982094351871`)/R),((0.7245684949157892`)/R),((0.6710809207892108`)/R)},{((0.3333333333333333`)/R),((0.592346380077668`)/R),((0.8114774831432803`)/R),((1.0845553992409294`)/R),((1.7630017041781747`)/R),((1.1578091708471627`)/R),((0.9692291776523858`)/R),((0.8615000962930917`)/R),((0.787532983589977`)/R)},{((0.30151134457776363`)/R),((0.5311600271468729`)/R),((0.7117377302633934`)/R),((0.900512006141845`)/R),((1.1578091708471627`)/R),((1.82765737874097`)/R),((1.2171341675960432`)/R),((1.0253250611469975`)/R),((0.9152400219292804`)/R)},{((0.2773500981126146`)/R),((0.48620730337181`)/R),((0.6440520175728418`)/R),((0.7959982094351871`)/R),((0.9692291776523858`)/R),((1.2171341675960432`)/R),((1.8813088755369007`)/R),((1.2670212868739044`)/R),((1.072783723443656`)/R)},{((0.2581988897471611`)/R),((0.45126077735968606`)/R),((0.5936999228322687`)/R),((0.7245684949157892`)/R),((0.8615000962930917`)/R),((1.0253250611469975`)/R),((1.2670212868739044`)/R),((1.9271727209854121`)/R),((1.3100807176325626`)/R)},{((0.24253562503633297`)/R),((0.4230319832800408`)/R),((0.5541199346189097`)/R),((0.6710809207892108`)/R),((0.787532983589977`)/R),((0.9152400219292804`)/R),((1.072783723443656`)/R),((1.3100807176325626`)/R),((1.9672311134494134`)/R)}};

HMPsh[λ_,R_,n_]:=Take[HnRMP[R]+λ (T [R]+ Vsh[R]-HnRMP[R]),n,n]//N;
HMPcsf[λ_,R_,n_]:=Take[HnRMP[R]+λ (T[R]+Vcsf[R]-HnRMP[R]),n,n]//N;

HWCEsh[λ_,R_,n_]:=Take[T [R]+ λ Vsh[R],n,n];
HWCEcsf[λ_,R_,n_]:=Take[T[R]+λ Vcsf[R],n,n];

HSCEsh[λ_,R_,n_]:=Take[λ T[R]+Vsh[R],n,n];
HSCEcsf[λ_,R_,n_]:=Take[λ T[R]+Vcsf[R],n,n];

HENsh[λ_,R_,n_]:=Take[DiagonalMatrix[Diagonal[T[R]+Vsh[R]]]+λ (T[R]+Vsh[R]-DiagonalMatrix[Diagonal[T[R]+Vsh[R]]]),n,n];
HENcsf[λ_,R_,n_]:=Take[DiagonalMatrix[Diagonal[T[R]+Vcsf[R]]]+λ (T[R]+Vcsf[R]-DiagonalMatrix[Diagonal[T[R]+Vcsf[R]]]),n,n];

Rcv[R_,H_,n_]:=Module[{P,dPdϵ,EP,λEP},
P[λ_,ϵ_]:=CharacteristicPolynomial[H[λ,R,n],ϵ];
dPdϵ[λ_,ϵ_]:=D[P[λ,ϵ],ϵ];
EP=NSolve[{P[λ,ϵ]==0,dPdϵ[λ,ϵ]==0},{ϵ,λ},WorkingPrecision->30];
{λEP=λ/.EP};
Abs[λEP]//Sort//First]

DomSing[R_,H_,n_]:=Module[{P,dPdϵ,EP,λEP},
P[λ_,ϵ_]:=CharacteristicPolynomial[H[λ,R,n],ϵ];
dPdϵ[λ_,ϵ_]:=D[P[λ,ϵ],ϵ];
EP=NSolve[{P[λ,ϵ]==0,dPdϵ[λ,ϵ]==0},{ϵ,λ},WorkingPrecision->30];
{λEP=λ/.EP};
SortBy[λEP,Abs,NumericalOrder]//First]

Show[LogLinearPlot[{Rcv[R,HMPsh,2],Rcv[R,HWCEsh,2],Rcv[R,HSCEsh,2],Rcv[R,HENsh,2]},{R,0.5,100},PlotRange->{{0.5,100},{0,6.25}},PlotTheme->"Scientific",FrameStyle->Directive[Thick,16],FrameLabel->{MaTeX["R",Magnification->1.5],MaTeX["R_{\\text{CV}}",Magnification->1.5]},PlotLegends->Placed[{MaTeX["\\text{MP}",Magnification->1.5],MaTeX["\\text{WCE}",Magnification->1.5],MaTeX["\\text{SCE}",Magnification->1.5],MaTeX["\\text{EN}",Magnification->1.5]},{Top,Center}]],LogLinearPlot[1,{R,0.5,500},PlotStyle->{Black,Dotted}],ImageSize->Large]

Show[LogLinearPlot[{Rcv[R,HMPcsf,2],Rcv[R,HWCEcsf,2],Rcv[R,HSCEcsf,2],Rcv[R,HENcsf,2]},{R,0.5,100},PlotRange->{{0.5,50},{0,5}},PlotTheme->"Scientific",FrameStyle->Directive[Thick,16],FrameLabel->{MaTeX["R",Magnification->1.5],MaTeX["R_{\\text{CV}}",Magnification->1.5]},PlotLegends->Placed[{MaTeX["\\text{MP}",Magnification->1.5],MaTeX["\\text{WCE}",Magnification->1.5],MaTeX["\\text{SCE}",Magnification->1.5],MaTeX["\\text{EN}",Magnification->1.5]},{Top,Center}]],LogLinearPlot[1,{R,0.5,500},PlotStyle->{Black,Dotted}],ImageSize->Large]

Show[LogLinearPlot[{Rcv[R,HMPcsf,3],Rcv[R,HWCEcsf,3],Rcv[R,HSCEcsf,3],Rcv[R,HENcsf,3]},{R,0.5,100},PlotRange->{{0.5,100},{0,5}},PlotTheme->"Scientific",FrameStyle->Directive[Thick,16],PlotLegends->Placed[{MaTeX["\\text{MP}",Magnification->1.5],MaTeX["\\text{WCE}",Magnification->1.5],MaTeX["\\text{SCE}",Magnification->1.5],MaTeX["\\text{EN}",Magnification->1.5]},{Top,Center}],FrameLabel->{MaTeX["R",Magnification->1.5],MaTeX["R_{\\text{CV}}",Magnification->1.5]}],LogLinearPlot[1,{R,0.5,500},PlotStyle->{Black,Dotted}],ImageSize->Large]

Show[LogLinearPlot[{Rcv[R,HMPcsf,2],Rcv[R,HMPcsf,3],Rcv[R,HMPcsf,4],Rcv[R,HMPcsf,6],Rcv[R,HMPcsf,8]},{R,0.5,100},PlotRange->{{0.5,10},{0,5}},PlotTheme->"Scientific",FrameStyle->Directive[Thick,16],FrameLabel->{MaTeX["R",Magnification->1.5],MaTeX["R_{\\text{CV}}",Magnification->1.5]},PlotLegends->Placed[{MaTeX["K=2",Magnification->1.5],MaTeX["K=3",Magnification->1.5],MaTeX["K=4",Magnification->1.5],MaTeX["K=6",Magnification->1.5],MaTeX["K=8",Magnification->1.5]},{Right,Top}]],LogLinearPlot[1,{R,0.5,500},PlotStyle->{Black,Dotted}],ImageSize->Large]

Show[LogLinearPlot[{Rcv[R,HWCEcsf,2],Rcv[R,HWCEcsf,3],Rcv[R,HWCEcsf,4],Rcv[R,HWCEcsf,6],Rcv[R,HWCEcsf,8]},{R,0.5,100},PlotRange->{{0.5,10},{0,3.4}},PlotTheme->"Scientific",FrameStyle->Directive[Thick,16],FrameLabel->{MaTeX["R",Magnification->1.5],MaTeX["R_{\\text{CV}}",Magnification->1.5]},PlotLegends->Placed[{MaTeX["K=2",Magnification->1.5],MaTeX["K=3",Magnification->1.5],MaTeX["K=4",Magnification->1.5],MaTeX["K=6",Magnification->1.5],MaTeX["K=8",Magnification->1.5]},{Right,Top}]],LogLinearPlot[1,{R,0.5,500},PlotStyle->{Black,Dotted}],ImageSize->Large]

Show[LogLinearPlot[{(Sqrt[R]) Rcv[R,HMPcsf,2],(Sqrt[R]) Rcv[R,HMPcsf,3],(Sqrt[R]) Rcv[R,HMPcsf,4]},{R,0.5,100},PlotRange->{{0.5,10},{0,5}},PlotTheme->"Scientific",FrameStyle->Directive[Thick,16],FrameLabel->{MaTeX["R",Magnification->1.5],MaTeX["R_{\\text{CV}}",Magnification->1.5]},PlotLegends->Placed[{MaTeX["K=2",Magnification->1.5],MaTeX["K=3",Magnification->1.5],MaTeX["K=4",Magnification->1.5],MaTeX["K=6",Magnification->1.5],MaTeX["K=8",Magnification->1.5]},{Right,Top}]],LogLinearPlot[1,{R,0.5,500},PlotStyle->{Black,Dotted}],ImageSize->Large]

Show[LogLinearPlot[{Rcv[R,HENcsf,2],Rcv[R,HENcsf,3],Rcv[R,HENcsf,4],Rcv[R,HENcsf,5]},{R,0.5,100},PlotRange->{{0.5,100},{0,6.25}},PlotTheme->"Scientific",FrameStyle->Directive[Thick,16],FrameLabel->{MaTeX["R",Magnification->1.5],MaTeX["R_{\\text{CV}}",Magnification->1.5]},PlotLegends->Placed[{MaTeX["2",Magnification->1.5],MaTeX["3",Magnification->1.5],MaTeX["4",Magnification->1.5],MaTeX["5",Magnification->1.5]},{Right,Top}]],LogLinearPlot[1,{R,0.5,500},PlotStyle->{Black,Dotted}],ImageSize->Large]

Show[LogLinearPlot[{Rcv[R,HENsh,2],Rcv[R,HENsh,3],Rcv[R,HENsh,4],Rcv[R,HENsh,5]},{R,0.5,100},PlotRange->{{0.5,100},{0,6.25}},PlotTheme->"Scientific",FrameStyle->Directive[Thick,16],FrameLabel->{MaTeX["R",Magnification->1.5],MaTeX["R_{\\text{CV}}",Magnification->1.5]},PlotLegends->Placed[{MaTeX["2",Magnification->1.5],MaTeX["3",Magnification->1.5],MaTeX["4",Magnification->1.5],MaTeX["5",Magnification->1.5]},{Right,Top}]],LogLinearPlot[1,{R,0.5,500},PlotStyle->{Black,Dotted}],ImageSize->Large]

Show[LogLinearPlot[{Rcv[R,HMPsh,2],Rcv[R,HMPcsf,2],Rcv[R,HWCEsh,2],Rcv[R,HWCEcsf,2]},{R,0.5,100},PlotRange->{{0.5,100},{0,6.25}},PlotTheme->"Scientific",FrameStyle->Directive[Thick,16],FrameLabel->{MaTeX["R",Magnification->1.5],MaTeX["R_{\\text{CV}}",Magnification->1.5]},PlotLegends->Placed[{MaTeX["\\text{MP SH}",Magnification->1.5],MaTeX["\\text{MP CSF}",Magnification->1.5],MaTeX["\\text{WCE SH}",Magnification->1.5],MaTeX["\\text{WCE CSF}",Magnification->1.5]},{Right,Top}]],LogLinearPlot[1,{R,0.5,500},PlotStyle->{Black,Dotted}],ImageSize->Large]

Show[LogLinearPlot[{Rcv[R,HMPsh,3],Rcv[R,HMPcsf,3],Rcv[R,HWCEsh,3],Rcv[R,HWCEcsf,3]},{R,0.5,100},PlotRange->{{0.5,100},{0,6.25}},PlotTheme->"Scientific",FrameStyle->Directive[Thick,16],FrameLabel->{MaTeX["R",Magnification->1.5],MaTeX["R_{\\text{CV}}",Magnification->1.5]},PlotLegends->Placed[{MaTeX["\\text{MP SH}",Magnification->1.5],MaTeX["\\text{MP CSF}",Magnification->1.5],MaTeX["\\text{WCE SH}",Magnification->1.5],MaTeX["\\text{WCE CSF}",Magnification->1.5]},{Right,Top}]],LogLinearPlot[1,{R,0.5,500},PlotStyle->{Black,Dotted}],ImageSize->Large]

Show[LogLinearPlot[{Rcv[R,HMPsh,4],Rcv[R,HMPcsf,4],Rcv[R,HWCEsh,4],Rcv[R,HWCEcsf,4]},{R,0.5,100},PlotRange->{{0.5,100},{0,6.25}},PlotTheme->"Scientific",FrameStyle->Directive[Thick,16],FrameLabel->{MaTeX["R",Magnification->1.5],MaTeX["R_{\\text{CV}}",Magnification->1.5]},PlotLegends->Placed[{MaTeX["\\text{MP SH}",Magnification->1.5],MaTeX["\\text{MP CSF}",Magnification->1.5],MaTeX["\\text{WCE SH}",Magnification->1.5],MaTeX["\\text{WCE CSF}",Magnification->1.5]},{Right,Top}]],LogLinearPlot[1,{R,0.5,500},PlotStyle->{Black,Dotted}],ImageSize->Large]

Table[N[DomSing[R,HMPcsf,8],3],{R,{0.1,1,2,3,5,10,100,1000}}]

Table[N[DomSing[R,HWCEcsf,8],3],{R,{0.1,1,2,3,5,10,100,1000}}]