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We investigate the existence and asymptotic behaviour of higher derivatives of the spectral function in the context of one-dimensional Schr¨odinger operators on the half-line with integrable potentials. In particular, we identify sufficient conditions on the potential for the existence and continuity of the n-th derivative, and outline a systematic procedure for estimating numerical upper bounds for the turning points of such derivatives. Explicit worked examples illustrate the development and application of the theory.
Gilbert, D., Harris, B., Richl, S.: Higher Derivatives of Spectral Functions Associated with One-Dimensional Schrodinger Operators. Advances and Applications, vol. 186, 2008. pp 217-228.