On a conjecture concerning the means of the eigenvalues of random Sturm-Liouville boundary value problems

It has been known for several years that the expected value $\left \langle {{\lambda _1}} \right \rangle$ of the smallest eigenvalue of a self-adjoint positive definite random Sturm-Liouville boundary value problem satisfies the relation $\left \langle {{\lambda _1}} \right \rangle \le {\mu _1}$ where ${\mu _1}$ is the smallest eigenvalue of the corresponding deterministic problem obtained by replacing each random coefficient by its mean. It has been an open question whether similar inequalities are valid for the higher eigenvalues. The answer is negative, as shown by the counterexample given in this note.