The UV prolate spectrum matches the zeros of zeta

We describe a remarkable property of the self-adjoint extension of the prolate spheroidal operator introduced in 1998 by A.C. The restriction of this operator to the interval whose characteristic function commutes with it is well known, has a discrete positive spectrum, and is well understood. What we have discovered is that the restriction of the prolate differential operator to the complement of the finite interval admits (besides a replica of the above positive spectrum) negative eigenvalues whose ultraviolet (UV) behavior reproduces that of the squares of zeros of the Riemann zeta function. Moreover, we show that their corresponding eigenfunctions belong to the Sonin space. This feature fits with the proof (by A.C. and C. Consani) of Weil’s positivity at the Archimedean place, which uses the compression of the scaling action to the Sonin space. Furthermore, we construct an isospectral family of Dirac operators whose spectra have the same UV behavior as the zeros of the Riemann zeta function.