On the domain of convergence of spherical harmonic expansions

Math Physics Seminar
Thursday, April 27, 2023 - 12:00pm for 1 hour (actually 50 minutes)
Skiles 005 and online at https://gatech.zoom.us/j/94065877775
Ovidiu Costin – Ohio State University – costin@math.ohio-state.edu
Federico Bonetto
We settle a 60 year old question in mathematical physics, namely finding the exact domain of convergence of the spherical harmonic expansions (SHE, expansions at infinity in Legendre polynomials) of the gravitational potential of a planet. These expansions are the main tool in processing satellite data to find information about planet Earth in locations that are inaccessible, as well as the subsurface mass distribution and other quantities, with innumerable practical applications.
Despite many decades of investigation it was not known whether SHE converge all the way to the topography or only in the complement of the so called Brillouin sphere, the smallest sphere enclosing our planet. We show that regardless of the smoothness of the density and topography, short of outright analyticity, the spherical harmonic expansion of the gravitational potential converges exactly in the closure of the exterior of the Brillouin sphere, and convergence below the Brillouin sphere occurs with probability zero. We go further by finding a necessary and sufficient condition for convergence below the Brillouin sphere, which requires a form of analyticity at the highest peak on the planet, which would not hold for any realistic celestial body. Due to power-law corrections to the geometric growth of the coefficients, that we calculate for the first time in this paper, there is some amount of compensation of this divergence. However, with the increased accuracy of modern measurements divergence is bound to result in unacceptably large errors. The SHE can be made convergent though, and used optimally.
These questions turn out to be very delicate and challenging asymptotic analysis ones, which we solve using asymptotic techniques combined with elements of microlocal analysis and resurgence.
Work in collaboration with R.D. Costin, C. Ogle and M. Bevis