The project is devoted to the furtherance of the mathematical understanding of certain simple physical models: (a) The long-time asymptotics of the KdV equation will be studied for slowly decreasing initial data via the inverse scattering/spectral method, with recent developments in the spectral theory of such operators with merely square-integrable potentials leading the way. Of particular interest is the question of what behaviours can be attributed to embedded singular spectrum in the way that solitons are related to isolated eigenvalues. (b) The Schrodinger equation with random potentials (or Anderson model) and its connections to unique continuation and thence to the uncertainty principle (particularly in the form advocated by Fefferman). This also makes links to symplectic geometry. (c) The classical Coulomb gas at all temperatures, or equivalently, random matrices at general $eta$. This will be pursued through the study of orthogonal polynomials with random recurrence coefficients as pioneered by Dumitriu and Edelman. (d) The stability of the absolutely continuous spectrum of general Schrodinger operators under rough long-range (say square-integrable) perturbation.
By studying simple physical models, it is possible to concentrate on essential difficulties, without being waylaid by technicalities. The methods and perhaps more importantly, perspectives that developed for these simple models then inform those working closer to applications. Three examples taken from this project are the following: (a) By studying random matrices at general inverse temperature, beta, one hopes to better understand the most interesting case: when beta equals two. This case is so interesting because of its (currently mostly empirical) connection to the zeros of the Riemann zeta function. Of course, analytic number theory has much to offer society at the present particularly in terms of cryptography; while this project does not address these questions directly, one must be careful to remember the many tributaries that make a mighty river. (b) While integrable Hamiltonian PDEs have received intensive study in recent decades, attention has mostly been directed to the cases of periodic or rapidly-decreasing initial data. This side-steps the very natural question of what behaviours are attributable to the existence of embedded singular spectrum for the Lax operator. As is well understood, solitons are a consequence of isolated eigenvalues. A potential implication of this work is the prediction of new quasi-particle modes in non-linear media. (c) The better understanding of inverse scattering found from the study of the one-dimensional Schrodinger equation with rough and slowly decaying potentials may lead to improvements in remote sensing technologies.
