Ordinary and Partial Differential Equations pp 293-301 | Cite as

# Deficiency indices of polynomials in symmetric differential expressions

## 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.

## Keywords

Limit Point Regular Point Real Coefficient Linear Differential Operator Real Polynomial## Preview

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