a0=1000000; 
a1=-10000000;
a2=-5000000;

bd1[x_,t_,eta_]:=1+(-9/10)eta^2t x^(-2/3)+(81/280)eta^4t^2 x^(-4/3)+((1/6)eta^2-(27t^3/560)eta^6)x^(-2)+((-9/88)((-20eta^2+16eta^4)/15)t+(9 27t^4/(88 560))eta^8)x^(-8/3);
bd2[x_,t_,eta_]:=(-2/3)(-9/10)eta^2t x^(-5/3)+(-4/3)(81/280)eta^4t^2 x^(-7/3)+(-2)((1/6)eta^2-(27t^3/560)eta^6)x^(-3)+(-8/3)((-9/88)((-20eta^2+16eta^4)/15)t+(9 27t^4/(88 560))eta^8)x^(-11/3);


examineft3d1b[h_,eta_,rmin_,rmax_,rmin2_,rmax2_]:=(
Print["Re-version. list01b is the list of {Re part, Im part} of lambda of zero-pt."];
Print["Re-part of lambda is ",h rmin, " to ", h rmax, "."];
Print["Im-part of lambda is ",h rmin2, " to ", h rmax2, "."];
Do[Do[y31[j,k]=NDSolve[{y''[x]+eta^2(-(1+x^2)^(-2)+(h j+I h k)(1+x^2)^(-4/3))y[x]==0,y[a0]==bd1[a0,h j+I h k,eta],y'[a0]==bd2[a0,h j+I h k,eta]},y,{x,a1,a0}]; 
y32[j,k]=y[a1]/.y31[j,k]; 
y33[j,k]=y[a2]/.y31[j,k]; 
y34[j,k]=Re[y33[j,k][[1]]-y32[j,k][[1]]]; 
,{j,rmin,rmax}],{k,rmin2,rmax2}];
list011={};
list012=Table[{h j,h k,y34[j,k]},{j,rmin,rmax},{k,rmin2,rmax2}];
Do[Do[AppendTo[list011,list012[[j]][[k]]],{j,1,rmax-rmin+1}],{k,1,rmax2-rmin2+1}];
checker[h,list012];
list01b=listint;
ListPlot3D[list011, Mesh->All])


examineft3d2b[h_,eta_,rmin_,rmax_,rmin2_,rmax2_]:=(
Print["Im-version. list02b is the list of {Re part, Im part} of lambda of zero-pt."];
Print["Re-part of lambda is ",h rmin, " to ", h rmax, "."];
Print["Im-part of lambda is ",h rmin2, " to ", h rmax2, "."];
Do[Do[y35[j,k]=NDSolve[{y''[x]+eta^2(-(1+x^2)^(-2)+(h j+I h k)(1+x^2)^(-4/3))y[x]==0,y[a0]==bd1[a0,h j+I h k,eta],y'[a0]==bd2[a0,h j+I h k,eta]},y,{x,a1,a0}]; 
y36[j,k]=y[a1]/.y35[j,k]; 
y37[j,k]=y[a2]/.y35[j,k]; 
y38[j,k]=Im[y37[j,k][[1]]-y36[j,k][[1]]]; 
,{j,rmin,rmax}],{k,rmin2,rmax2}];
list015={};
list016=Table[{h j,h k,y38[j,k]},{j,rmin,rmax},{k,rmin2,rmax2}];
Do[Do[AppendTo[list015,list016[[j]][[k]]],{j,1,rmax-rmin+1}],{k,1,rmax2-rmin2+1}];
checker[h,list016];
list02b=listint;
ListPlot3D[list015, Mesh->All])

checker[h_,list00b_]:=(
listint={};
If[Length[list00b]≥2,Do[Do[If[list00b[[j]][[k]][[3]] list00b[[j+1]][[k]][[3]]≤0,
If[Abs[list00b[[j]][[k]][[3]]]≤Abs[list00b[[j+1]][[k]][[3]]],AppendTo[listint,{list00b[[j]][[k]][[1]],list00b[[j]][[k]][[2]]}],AppendTo[listint,{list00b[[j+1]][[k]][[1]],list00b[[j+1]][[k]][[2]]}]];(*Print["lambda= ",list00b[[j]][[k]][[1]]," + i ",list00b[[j]][[k]][[2]]," , the value= ",list00b[[j]][[k]][[3]]]; Print["lambda= ",list00b[[j+1]][[k]][[1]]," + i ",list00b[[j+1]][[k]][[2]]," , the value= ",list00b[[j+1]][[k]][[3]]];
Print[""]*)
],{k,1,Length[list00b[[j]]]}],{j,1,Length[list00b]-1}]
];
Do[If[Length[list00b[[j]]]≥2,Do[If[list00b[[j]][[k]][[3]] list00b[[j]][[k+1]][[3]]≤0,If[Abs[list00b[[j]][[k]][[3]]]≤Abs[list00b[[j]][[k+1]][[3]]],AppendTo[listint,{list00b[[j]][[k]][[1]],list00b[[j]][[k]][[2]]}],AppendTo[listint,{list00b[[j]][[k+1]][[1]],list00b[[j]][[k+1]][[2]]}]];
(*Print["lambda= ",list00b[[j]][[k]][[1]]," + i ",list00b[[j]][[k]][[2]]," , the value= ",list00b[[j]][[k]][[3]]]; Print["lambda= ",list00b[[j]][[k+1]][[1]]," + i ",list00b[[j]][[k+1]][[2]]," , the value= ",list00b[[j]][[k+1]][[3]]];Print[" "]*)
],{k,1,Length[list00b[[j]]]-1}]
],{j,1,Length[list00b]}])

examineft3dnewb[h_,eta_,rmin_,rmax_,rmin2_,rmax2_]:=(
examineft3d1b[h,eta,rmin,rmax,rmin2,rmax2];
examineft3d2b[h,eta,rmin,rmax,rmin2,rmax2];
ListPlot[{list01b,list02b},PlotStyle->{Blue,Orange}]
)

(* plot satisfying points for the range h*rmin ≤ Re(lambda) h*rmax 
and h*rmin2 ≤ Im(lambda) ≤ h*rmax2*)
examineft3dnewb[0.1,Exp[3Pi I/8],-100,0,-100 ,0]