Trace Formulas for Some Singular Differential Operators and Applications

Abstract

We characterize the convergence of the series P 1 n ,w heren are the non{ zero eigenvalues of some boundary value problems for degenerate second order ordinary dierential opera- tors and we prove a formula for the above sum when the coecient of the zero{ order term vanishes. We study these operators both in weighted Hilbert spaces and in spaces of continuous functions. After investigating the boundary behaviour of the eigenfunctions, we give applications to the regularity of the generated semigroups. in the space C((a; b)), with 1 a 0 in the interior and r 2 C((a; b)), but we allow and to vanish or to have no nite limits at a, b. The asymptotic behaviour of the eigenvalues n of Sturm { Liouville problems has been extensively studied both in the non { degenerate and in the degenerate case. In particular it is well { known that jn jn 2 in the regular case or even in the singular one under the non { oscillatory limit{ circle condition, i. e., when all the solutions of the homogeneous equation uA u = 0 are square summable and keep a denite sign near the endpoints (see (27)). After assigning the boundary conditions specied below, we give here necessary and sucient conditions for the convergence of the above sum and we compute explicitly the sum in the case r 0. As regards trace formulas, we