On a problem in eigenvalue perturbation theory

We consider additive perturbations of the type Ht=H0+tVHt=H0+tV, t∈[0,1]t∈[0,1], where H0H0 and V   are self-adjoint operators in a separable Hilbert space HH and V is bounded. In addition, we assume that the range of V   is a generating (i.e., cyclic) subspace for H0H0. If λ0λ0 is an eigenvalue of H0H0, then under the additional assumption that V   is nonnegative, the Lebesgue measure of the set of all t∈[0,1]t∈[0,1] for which λ0λ0 is an eigenvalue of HtHt is known to be zero. We recall this result with its proof and show by explicit counterexample that the nonnegativity assumption V≥0V≥0 cannot be removed.