Odd-order differential expressions with positive supporting coefficients

Abstract

SynopsisThe deficiency indices (mean deficiency index) and the essential spectrum for a class of odd order ordinary differential expressions are determined. The considered expressions are relatively bounded or relatively compact perturbations of symmetric expressions with odd order terms having as coefficients real powers of the independent variable.

A fourth order limit-3 expression with nonempty essential spectrum

“…Consider first the expression Py = ix?y' for which it is known that o e (P) = U (see e.g. [6,Corollary 7.3]). Rota's spectral mapping theorem [4] implies that o e (P 2 ) = [0, »), P 2 is explicitly given as P 2 y = -xy"~\y'.…”

A construction of symmetric differential expressions with non-empty essential spectrum

“…This question has been made known to the author by Professor M. S. P. Eastham and Professor W. N. Everitt. It has been inspired by the known criteria linking nonminimal equal deficiency indices with a discrete spectrum (see [5], [7]). The negative answer to the above question would generalise the well-known theorem for expressions in the limit-circle case to this larger class of expressions.…”

“…Here McLeod's example [4] provides such an expression. Its square is a symmetric expression of order 8 with deficiency indices (5,5), (6,6) or (7,7) but it also involves nonreal coefficients.…”