δ[λ_]:= ψ[30]/.(NDSolve[{ψ''[x]== -λ ψ[x]+((-5 x^2 +1)/(x^4 +1)) ψ[x],ψ[-20]== 0.001,ψ'[-20]==0.001},ψ,{x,-20,30}][[1]])

Plot[((-5 x^2 +1)/(x^4 +1)),{x,-10,10
},PlotRange->{-3,1.5}]

V[x_]:=((-5 x^2 +1)/(x^4 +1))

λ =-0.93285

λ =-0.932879;
 s=NDSolve[{ψ''[x]==  -λ ψ[x]+((-5 x^2 +1)/(x^4 +1))ψ[x],ψ[-20]== 0.001,ψ'[-20]==0.001 λ},ψ,{x,-20,30}];Plot[Evaluate[ψ[x]/.s],{x,-20,20}, PlotRange->{-200,200}]

δ[λ_]:= ψ[30]/.(NDSolve[{ψ''[x]== λ ψ[x]+((-5 x^2 +1)/(x^4 +1))ψ[x],ψ[-20]== 0.001,ψ'[-20]==0.001 },ψ,{x,-20,30}][[1]])

Plot[δ[λ],{λ,0.4,0.935},PlotRange->{-10000,10000}]

Plot[Evaluate[ψ[x]/.s],{x,-20,20}]