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A differentiability property and its application to the spectral theory of differential operators

  • Robert M. Kauffman (a1)


For an ordinary differential operation Lλ of order 2N which depends differentiably on a parameter λ, we study the differentiability with respect to λ of all solutions to Lλf = 0 which are in L2[a,∞). Applications to spectral theory are given, including a formula for the rate of change with respect to the end-point a of the spectrum of the weighted eigenvalue problem Lf = λwf, f∈L2[a,∞), f[i](a) = 0 for i ≦N − 1. The weight w may be a function or an operator. The formula seems new even when w = 1.



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