On the spectrum of a differential operator on a Hilbert-Pólya space

In this paper we study the spectrum of a fundamental differential operator on a Hilbert-Pólya space. A number is an eigenvalue of this differential operator if and only if it is a nontrivial zero of the Riemann zeta function. An explicit formula is given for the eigenfunction associated with each nontrivial zero of the zeta function. Every eigenfunction is characterized via the Poisson summation formula by a sequence of mysterious functions whose explicit formulas are given.