New bounds are given for the L 2-norm of the solution of the Kuramoto-Sivashinsky equation d t U(x, t) =-(d 2 x + d A x)U{x, t)-U(x, t)d x U(x, t) , for initial data which are periodic with period L. There is no requirement on the antisymmetry of the initial data. The result is limsup||£/(,ί)|| 2 < const. L 8/5 .

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Abstract. In this article, we prove and exploit a trace identity for the spectra of Schrödinger operators and similar operators. This identity leads to universal bounds on the spectra, which apply to low-lying eigenvalues, eigenvalue asymptotics, and to partition functions (traces of heat operators). In many cases they are sharp in the sense that there are specific examples for which the inequalities are saturated. Special cases corresponding to known inequalities include those of Hile and Protter and of Yang.

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