Zeta determinants aligned with ZetaOberseminar Global Analysis and Operator Algebras

by Henri Moscovici (Ohio State University)

Friday Jul 3, 2026, 11:00 AM → 12:00 PM Europe/Berlin

Lipschitzsaal (Mathezentrum)

The prolate spheroidal wave operator, whose underlying ODE goes back to classical
work on heat conduction in ellipsoids through separation of variables in prolate
spheroidal coordinates for the Helmholtz equation, has played a surprisingly rich and
unexpected role across several fields. After gaining considerable visibility through
its ”lucky accident” role in the 1960s solution by Slepian, Landau, and Pollak of
the time- and band-limiting problem for signals, it reappeared in the late 1990s as a
cutoff mechanism in Connes’ trace-formula framework recovering the Riemann–Weil
explicit formula in number theory. Connes also observed that, when extended to
the whole real line, the prolate wave operator admits a unique self-adjoint extension
commuting both with the Fourier transform and with its truncation to the finite
time interval. This extension turned out to be unexpectedly significant: in 2022, Connes and myself discovered that it possesses a purely discrete negative spectrum
confined to the Sonin subspace, whose eigenvalues display a striking resemblance
to the zeros of the Riemann zeta function. Recent work of Ramis, Richard–Jung,
and Thomann (2025) shows that the Sonin space is precisely the repository of this
negative spectrum. The aim of this talk is to show that the restrictions of the prolate
operators to the cor- responding Sonin spaces possess zeta determinants naturally
aligned with the Riemann zeta function. Moreover, they also admit semi-adelic
extensions whose zeta determinants exhibit the same phenomenon. Interestingly,
the Sonin prolate operators are closely related to a class of Sturm–Liouville operators
studied extensively by Matthias Lesch and his collaborators.