H = {{(Subscript[ϵ, 1]), α}, {α, (Subscript[ϵ, 2])}};
MatrixForm[H]

eigenvalores = Eigenvalues[H]

(Subscript[ℰ, 1]) = 
      FullSimplify[(eigenvalores[[1]] /. {α -> 0}) + (D[
                  eigenvalores[[1]], {α, 1}] /. {α -> 
                    0}) α + (D[
                  eigenvalores[[1]], {α, 2}] /. {α -> 
                    0}) (α^2)]
(Subscript[ℰ, 2]) = 
      FullSimplify[(eigenvalores[[1]] /. {α -> 0}) + (D[
                  eigenvalores[[2]], {α, 1}] /. {α -> 
                    0}) α + (D[
                  eigenvalores[[2]], {α, 2}] /. {α -> 
                    0}) (α^2)]