Gibbs measures of nonlinear Schrödinger equations (NLS) are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. The NLS can be viewed a classical limit of many-body quantum theory. We are interested in the problem of the derivation of Gibbs measures as mean-field limits of Gibbs states in many-body quantum mechanics.
We consider dimension d=2 and interaction given by the delta function, which corresponds to the Euclidean $\Phi^4_2$ theory.
We analyse the regime when the density is large and the range of the interaction is small.
Our proof is based on two main ingredients.
(a) An infinite-dimensional stationary phase argument, based on a functional integral representation.
(b) A Nelson-type estimate for a nonlocal field theory in two dimensions.
This is joint work with J.Fröhlich, A. Knowles, and B. Schlein.
Vedran Sohinger is an Associate Professor of Mathematics at the University of Warwick. His research interests are nonlinear dispersive PDEs and many-body quantum problems. Vedran received his BA from the University of California, Berkeley in 2006 and his PhD from the Massachusetts Institute of Technology in 2011, where his PhD advisor was Gigliola Staffilani. He was a Simons Postdoctoral Fellow at the University of Pennsylvania 2011-2014, a Postdoctoral Fellow at ETH Zurich 2014-2016, and a Postdoc at the University of Zurich 2016-2017. Since 2017, he has been at the University of Warwick. In 2020, Vedran was awarded the EPSRC New Investigator Award.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai