Mathematical Sciences: Inverse Spectral Problems and Meromorphic Solutions of Differential Equations

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ABSTRACT Proposal: DMS- 9623121 PI: Gesztesy Gesztesy proposes to study two related areas in differential equations pertaining to inverse spectral theory and meromorphic solutions of systems of ordinary differential equations related to integrable evolution equations. The first area is connected to inverse spectral theory and the characterization of isospectral sets of potential coefficients for the Schrodinger equation and representations of solutions of integrable systems such as the Korteweg-de Vries (KdV) equation. Building upon recently developed trace formulas for (multi-dimensional) Schrodinger operators in terms of appropriate Krein spectral shift functions and a general device for constructing isospectral potential coefficients, he intends to continue recent work on the inverse spectral problem for confining potentials and inverse spectral theory for short-range interactions. In connection with the second area, he proposes a strategy to solve the problem of characterizing all elliptic algebro-geometric finite-gap solutions of general matrix hierarchies of completely integrable evolution equations. In particular, Gesztesy proposes to continue work which recently led to a solution of this characterization problem for the KdV hierarchy on the basis of a newly developed approach centered around a theorem of Picard. Inverse spectral theory has significant applications in atomic, molecular, and nuclear physics and in areas as wide-ranging as computer tomography, acoustics, electromagnetics, resolution of structural/material properties of media, and aviation technology. Similarly, completely integrable equations play a vital role in nonlinear optics and especially in soliton-based optical communications systems. Gesztesy' work can be expected to have an impact on these areas.