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

  • Robert M. Kauffman (a1)

Synopsis

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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1Dunford, N. and Schwartz, J. T.. Linear Operators II: spectral theory (New York: Wiley-Interscience, 1963).
2Goldberg, S.. Unbounded linear operators (New York: McGraw-Hill, 1966).
3Kato, T.. Perturbation theory for linear operators (New York: Springer, 1966).
4Kauffman, R. M.. On the limit-n classification of ordinary differential operators with positive coefficients. Proc. London Math. Soc. 35 (1977), 496–526.
5Kauffman, R. M.. The number of Dirichlet solutions to a class of linear ordinary differential equations. J. Differential Equations 31 (1979), 117129.
6Naimark, M. A.. Linear Differential Operators, pt II (New York: Ungar, 1968).

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