A note on Sturm-Liouville problems whose spectrum is the set of prime numbers

We show that there is no classical regular Sturm-Liouville problem on a finite interval whose spectrum consists of infinitely many distinct primes numbers. In particular, this answers in the negative a question raised by Zettl in his book on Sturm-Liouville theory. We also show that there {it may} exist such a problem if the parameter dependence is nonlinear.