Schrodinger Operators, Integrable Systems, and Other Simple Models in Mathematical Physics

点击翻译按钮获取中文摘要

Proposal DMS-0401277PI: Rowan Killip, UCLATitle: Schroedinger operators, integrable systems, and othersimple models in mathematical physicsABSTRACTThe project is devoted to the furtherance of the mathematicalunderstanding of certain simple physical models: (a) The long-timeasymptotics of the KdV equation will be studied for slowly decreasinginitial data via the inverse scattering/spectral method, with recentdevelopments in the spectral theory of such operators with merelysquare-integrable potentials leading the way. Of particular interest isthe question of what behaviours can be attributed to embedded singularspectrum in the way that solitons are related to isolated eigenvalues. (b)The Schrodinger equation with random potentials (or Anderson model) andits connections to unique continuation and thence to the uncertaintyprinciple (particularly in the form advocated by Fefferman). This alsomakes links to symplectic geometry. (c) The classical Coulomb gas at alltemperatures, or equivalently, random matrices at general $\beta$. Thiswill be pursued through the study of orthogonal polynomials with randomrecurrence coefficients as pioneered by Dumitriu and Edelman. (d) Thestability of the absolutely continuous spectrum of general Schrodingeroperators under rough long-range (say square-integrable) perturbation.By studying simple physical models, it is possible to concentrate onessential difficulties, without being waylaid by technicalities. Themethods and perhaps more importantly, perspectives that developed forthese simple models then inform those working closer to applications.Three examples taken from this project are the following: (a) By studyingrandom matrices at general inverse temperature, beta, one hopes to betterunderstand the most interesting case: when beta equals two. This case isso interesting because of its (currently mostly empirical) connection tothe zeros of the Riemann zeta function. Of course, analytic number theoryhas much to offer society at the present particularly in terms ofcryptography; while this project does not address these questionsdirectly, one must be careful to remember the many tributaries that make amighty river. (b) While integrable Hamiltonian PDEs have receivedintensive study in recent decades, attention has mostly been directed tothe cases of periodic or rapidly-decreasing initial data. This side-stepsthe very natural question of what behaviours are attributable to theexistence of embedded singular spectrum for the Lax operator. As is wellunderstood, solitons are a consequence of isolated eigenvalues. Apotential implication of this work is the prediction of new quasi-particlemodes in non-linear media. (c) The better understanding of inversescattering found from the study of the one-dimensional Schrodingerequation with rough and slowly decaying potentials may lead toimprovements in remote sensing technologies.