Clear[x,A]

a=-5/128;
b=-1/128;
y=x/6;
d=x/(3Sqrt[2]);

A[x_]={{a-x/2,0,0,0,0,0,0,0},{0,a+x/2,0,0,0,0,0,0},{0,0,a-y,0,0,-d,0,0},{0,0,0,a+y,0,0,-d,0},{0,0,0,0,b-x,0,0,0},{0,0,-d,0,0,b-2y,0,0},{0,0,0,-d,0,0,b+2y,0},{0,0,0,0,0,0,0,b+x}};

A[x]//MatrixForm

Plot[Evaluate[Eigenvalues[A[x]]],{x,0,0.2},AxesLabel->{x,ΔE/α^2}]

FullSimplify[Eigenvalues[A[x]],Assumptions->Element[x,Reals]]

Eigenvectors[A[x]]

Series[FullSimplify[Eigenvalues[A[x]],Assumptions->Element[x,Reals]],{x,0,1}]

Series[FullSimplify[Eigenvalues[A[1/x]],Assumptions->Element[x,Reals]],{x,0,1}]