Eigenvalue problem in a linearly dependent basis and the super-secular-equation

The solution of the Schrödinger equation HKΨ = EΨ is discussed in terms of a basis Φ of n functions of which only p are linearly independent. After introducing a variable ∞ and the matrix K = Φ| H - ∞ ∞ 1|Φ} = H - ∞ δ, a super-secular-equation |K - Δ. 1| = 0 is discussed. It is shown that the n-valued function Δ = Δ(∞) has (n - p) branches which are identically vanishing, and p branches which are monotonously decreasing functions of ∞; the zero-points of the latter, Δ(∞) = 0, correspond to the eigenvalues ∞ = E desired. The coefficients in the characteristic polynomial P(Z) = |K - Z. 1| are studied, and it is shown that the eigenvalue problem is equivalent to setting the sum of the principal minors of order p of |K| equal to zero. The same problem is also discussed by carrying out a canonical orthonormalization of the basis which is shown to lead to the same result. The method is applied to the problem of the group-theoretical splitting of the secular equation due to symmetry properties of the Hamiltonian. Following Byers-Brown, it is shown that a block of the secular equation corresponding to a specific irreducible representation α depends only on the characters Xα and not on the full matrices γα.