TRACE FORMULAS AND CONSERVATION-LAWS FOR NONLINEAR EVOLUTION-EQUATIONS

New trace formulas for linear operators associated with Lax pairs or zero-curvature representations of completely integrable nonlinear evolution equations and their relation to (polynomial) conservation laws are established. We particularly study the Korteweg-de Vries equation, the nonlinear Schrodinger equation, the sine-Gordon equation, and the infinite Toda lattice though our methods apply to any element of the AKNS-ZS class. In the KdV context, we especially extend the range of validity of the infinite sequence of conservation laws to certain long-range situations in which the underlying one-dimensional Schrodinger operator has infinitely many (negative) eigenvalues accumulating at zero. We also generalize inequalities on moments of the eigenvalues of Schrodinger operators to this long-range setting. Moreover, our contour integration approach naturally leads to higher-order Levinson-type theorems for Schrodinger operators on the line.