A Maple assisted study of a Schroedinger Newton, a k a Schroedinger Poisson, a k a Choquard, a k a

Michael Kiessling speaks to the Experimental Mathematics Seminar.

Abstract: In several different contexts (mathematical) physicists have proposed a nonlinear system of PDEs which can be recast into a single Schroedinger equation with a Schroedinger potential that is the solution to a Poisson equation with the square of the solution of the Schroedinger equation as source term; as such it is known under a variety of names (see above). For instance, Roger Penrose proposed it in his theory of gravity-induced quantum-mechanical wave function collapse, but also down-to-earth condensed matter theories, by Pekar and later by Choquard, produce this equation without invoking gravity. Interestingly, the question of the asymptotic large-distance behavior has received several plausible but conflicting answers which cannot all be true simultaneously. Recently I managed to settle the issue of the leading order term --- partly rigorously, partly with the help of Maple. Subsequently Andrey Yudin from Moskow managed to tickle Maple to produce an intriguing formula for all the putative asymptotic correction terms to the leading order term which are of power law type. Numerically this formula seems pretty accurate, but it has yet to be proved; moreover, there are correction terms beyond all orders of powers, and a formula for these terms has yet to be found.

About 33 1/2 minutes into the recording a slip of tongue happens; namely, Prof. Kiessling says: "The L-two norm of the gradient of psi is controlled by the L-six norm of psi itself," but of course he meant to say: "The L-two norm of the gradient of psi controls the L-six norm of psi itself."