Mathematics – Classical Analysis and ODEs
Scientific paper
2006-11-21
Math. Nach. 282 (2009), No. 12, 1623--1636.
Mathematics
Classical Analysis and ODEs
15 pages, small corrections, accepted for publication in Math. Nach
Scientific paper
We study the regularity of the roots of complex monic polynomials $P(t)$ of fixed degree depending smoothly on a real parameter $t$. We prove that each continuous parameterization of the roots of a generic $C^\infty$ curve $P(t)$ (which always exists) is locally absolutely continuous. Generic means that no two of the continuously chosen roots meet of infinite order of flatness. Simple examples show that one cannot expect a better regularity than absolute continuity. This result will follow from the proposition that for any $t_0$ there exists a positive integer $N$ such that $t \mapsto P(t_0\pm (t-t_0)^N)$ admits smooth parameterizations of its roots near $t_0$. We show that $C^n$ curves $P(t)$ (where $n = \deg P$) admit differentiable roots if and only if the order of contact of the roots is $\ge 1$. We give applications to the perturbation theory of normal matrices and unbounded normal operators with compact resolvents and common domain of definition: The eigenvalues and eigenvectors of a generic $C^\infty$ curve of such operators can be arranged locally in an absolutely continuous way.
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