Abstract
Given a symmetric (formally self-adjoint) ordinary linear differential expression L which is regular on the interval [0,∞) and has complex C∞ coefficients, we investigate the relationship between the deficiency indices of L and those of p(L) where p(x) is any real polynomial of degree k>1. Our main results are the inequalities: (a) For k even, say k=2m, N+(p(L)), N−(p(L))≥m[N+(L)+N−(L)] and (b) for k odd, say k=2m+1, N+(p(L))≥(m+1)N+(L)+mN−(L) and N−(p(L))≥mN+(L)+(m+1)N−(L). Here N+(M), N−(M) denote the deficiency indices of the symmetric expression M associated with the upper and lower half-planes, respectively.KeywordsLimit PointRegular PointReal CoefficientLinear Differential OperatorReal PolynomialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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