Second microlocalization and Fredholm theory for the three-body problem

Abstract

In this thesis, we will study the stationary quantum mechanical three-body problem at a fixed energy in a new microlocal setting. Studies along this direction have previously been considered by Vasy [Vas00], and also [Vas01] in the many-body setting. We will build upon Vasy's method by introducing second microlocalization. This will produce a new pseudodifferential calculus, which will be a refinement of Vasy's three-body calcu- lus. Using this calculus, we prove that the three-body Hamiltonian at fixed energy is a Fredholm map between suitable anisotropic Hilbert spaces.

The thesis begins with a detailed discussion of how one defines second microlocalization in the three-body setting. The main issue here is that quantization of degenerate symbols would not be sufficient to ensure the existence of a calculus. Instead, we will follow the strategy pioneered by Vasy in [Vas20a], and start by defining another calculus, which we name the 'three-cone' calculus. Such a calculus captures microlocal decay at the second microlocalized boundary face in phase space. We will then blow up the three-cone calculus to produce (in fact a slightly more refined version of) the desired second microlocalized calculus. In the process, we also discuss some familiar objects in this setting, including the principal symbol map, indicial operators, and so on.

Once we have obtained an understanding of second microlocalization, we proceed to un- derstand the three-body Hamiltonian in the new framework. We will show that there exists a suitable rescaling of the Hamiltonian dynamic that exhibits new behaviors on the blow up of the three-body phase space. In particular, its equilibria are not degenerate in any (microlocal or dynamical) sense, namely, they can all be understood as radial sets, and are either sink, source or saddle. We will prove radial point estimates in these settings. As a result, we obtain significantly more refined propagation of regularity results. In par- ticular, our method incorporates geometric diffraction [MW04, MVW08, MVW13, Hin24] into the setting of the three-body problem, which is very natural from a microlocal per- spective. This allows us to obtain semi-Fredholm maps with errors in the symbolic sense. We then improve the error term on the global faces using different methods, which still follows Vasy's work [Vas20a] closely. In the process, we will also construct explicitly the variable orders required for the aforementioned strategies to work.