Ground-state second-order energy shift is nonpositive
= Ground-state second-order energy shift is nonpositive
For a nondegenerate lowest state, every excited-state denominator $E_0-E_k$ is negative. Since each numerator $|V_{k0}|^2$ is nonnegative, their sum is nonpositive, and strictly negative if a coupled higher state exists. The statement assumes the perturbative expansion and appropriate sum or spectral integral are defined.