Deficiency indices of polynomials in symmetric differential expressions
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 Point Regular Point Real Coefficient Linear Differential Operator Real Polynomial
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