Roehner–Valent conjecture on eigenvalues of birth-and-death operators

Let $Q$ be the birth-and-death operator associated with birth rates $(\lambda_n)$ and death rates $(\mu_n)$, and let $\sigma_p(Q)$ denote its point spectrum. Assume

$$\lim_{n\to\infty}\frac{\mu_n}{\lambda_n}=1,\qquad \lim_{n\to\infty}\frac{\lambda_n}{n^\alpha}=a,$$

for constants $a>0$ and $0<\alpha\leq2$.

Roehner–Valent conjecture. The operator has no eigenvalues:

$$\sigma_p(Q)=\varnothing.$$

The conjecture was proposed for operators arising from birth-and-death processes. The paper states that Chihara later showed it is false without additional assumptions, so this claim is refuted.