Trace formulas for some singular differential operators and applications

We characterize the convergence of the series Sigma lambda(n)(-1), where la mbda(n) are the non-zero eigenvalues of some boundary value problems for de generate second order ordinary differential operators and we prove a formul a for the above sum when the coefficient of the zero-order term vanishes. W e study these operators both in weighted Hilbert spaces and in spaces of co ntinuous functions. After investigating the boundary behaviour of the eigen functions, we give applications to the regularity of the generated semigrou ps.