Theorem: 2x2 Van Vleck block has trace E1+E2, det E1*E2 - V^2, second-order shift V^2/(E1-E2)

Theorem (Van Vleck 2x2 quasi-degenerate canonical form): for the real-symmetric Hamiltonian H = [[E_1, V], [V, E_2]] the linear-algebra invariants are: trace(H) = E_1 + E_2 (level-sum invariant under any unitary); det(H) = E_1*E_2 - V^2…