Trace Formulae for Fourth Order Differential Operators and their Applications

Abstract: Paper I. Trace Formulae and Spectral Properties of FourthOrder Differential Operators.We derive trace formulae forfourth order differential operators in dimension one anddiscuss their connection with sharp Lieb-Thirring inequalitiesfor the Riesz means of negative eigenvalues of order γ≥ 7/4. We also construct reflectionless potentials forfourth order differential operators.Paper II. Lieb-Thirring Inequalities for Higher OrderDifferential Operators.We derive Lieb-Thirringinequalities for the Riesz means of eigenvalues of order γ≥ 3=4 for a fourth order operator in arbitrarydimensions. We also consider some extensions to polyharmonicoperators, and to systems of such operators, in dimensionsgreaterthan one.Paper III. Follytons and the Removal of Eigenvalues forFourth Order Differential Operators.(Joint with J. Hoppeand A. Laptev). A non-linear functional Q[u, v]is given that governs the loss, respectively gain,of (doubly degenerate) eigenvalues of fourth order differentialoperators L = ∂4+ ∂ u ∂ + v on the line. Apart fromfactorizing L as A'A + E0, providing several explicit examples, and derivingvarious relations betweenu, vand eigenfunctions of L, we finduandvsuch that L is isospectral to the free operator L0= ∂4 up to one (multiplicity 2) eigenvalueE0<0. Not unexpectedly, this choice ofu, vleads to exact solutions of the correspondingtime-dependent PDE’s.