Higher Derivatives of Spectral Functions Associated with One-dimensional Schrodinger Operators

We investigate the existence and asymptotic behaviour of higher derivatives of the spectral function, rho(lambda), on the positive real axis, in the context of one-dimensional Schrodinger operators on the half-line with integrable potentials. In particular, we identify sufficient conditions on the potential for the existence and continuity of the nth derivative, rho((n))(lambda), and outline a systematic procedure for estimating numerical upper bounds for the turning points of such derivatives. The potential relevance of our results to some topical issues in spectral theory is discussed.