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

TL;DR

This paper proves that no classical regular Sturm-Liouville problem on a finite interval can have a spectrum consisting solely of infinitely many prime numbers, answering a question by Zettl, but nonlinear cases remain open.

Contribution

It demonstrates the non-existence of classical regular Sturm-Liouville problems with prime spectra and explores possibilities for nonlinear parameter dependence.

Findings

No classical regular Sturm-Liouville problem has prime spectrum

The question by Zettl is answered negatively

Nonlinear parameter dependence might allow such problems

Abstract

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.