The positivity conjecture for nonzero eigenvalues of $\mathfrak{L}_0$

Let $a>1$, and let $\mathfrak{L}_0$ be the linear operator arising from the $n=0$ mode of the linearized problem. Positivity conjecture for $\mathfrak{L}_0$. For all $a>1$, all nonzero eigenvalues of $\mathfrak{L}_0$ are real and positive. Numerical observations suggest that the eigenvalue $0$ has the eigenfunction $\partial_a\Psi$ up to a constant multiple and that the remaining eigenvalues are positive, but the supplied text gives no proof of the conjecture.