TRACE FORMULAS AND CONSERVATION-LAWS FOR NONLINEAR EVOLUTION-EQUATIONS

New trace formulas for linear operators associated with Lax pairs or z ero-curvature representations of completely integrable nonlinear evolu tion equations and their relation to (polynomial) conservation laws ar e established. We particularly study the Korteweg-de Vries equation, t he nonlinear Schrodinger equation, the sine-Gordon equation, and the i nfinite Toda lattice though our methods apply to any element of the AK NS-ZS class. In the KdV context, we especially extend the range of val idity of the infinite sequence of conservation laws to certain long-ra nge situations in which the underlying one-dimensional Schrodinger ope rator has infinitely many (negative) eigenvalues accumulating at zero. We also generalize inequalities on moments of the eigenvalues of Schr odinger operators to this long-range setting. Moreover, our contour in tegration approach naturally leads to higher-order Levinson-type theor ems for Schrodinger operators on the line.