Title: The Van Vleck Formula on Ehrenfest time scales and stationary phase asymptotics for frequency-dependent phases (Part I)

Abstract: The Van Vleck formula is a semiclassical approximation to the integral kernel of the propagator associated to a time-dependent Schrödinger equation. Under suitable hypotheses, we present a rigorous treatment of this approximation which is valid on “Ehrenfest” time scales, which are time scales whose length grows as Planck’s constant tends to 0. Our derivation is based on an approximation to the integral kernel often called the “Herman-Kluk” approximation, which expresses the kernel as an integral superposition of Gaussians parameterized by points in phase space, and has been rigorously justified on Ehrenfest time scales. In order to derive the Van Vleck approximation from the Herman-Kluk approximation, we are led to develop stationary phase asymptotics for oscillatory integrals where the phase functions depend on the frequency parameter in a nontrivial way.

About the speaker: Matt Blair got his PhD in 2005 from University of Washington under the supervision of Hart Smith (Matt is one of Elias Stein's academic grandchildren). He was a Postdoctoral Fellow at MSRI at Berkeley, and at John Hopkins University, and a Visiting Assistant Professor at Rochester before joining UNM as an Assistant Professor in 2008. Since 2018 he is a Professor at UNM. Professor Blair's  research concerns harmonic analysis and its interactions with partial differential equations of wave and Schrödinger type. Much of his work involves finding effective ways to represent solutions to these equations and understanding the oscillatory integrals that appear as a result. He has publish high quality articles  in top mathematics journals such as Inventiones and the  American Journal of Mathematics. He  has received four consecutive NSF research grants covering the years 2008-2020. Professor Blair is an excellent teacher and mentor.