Isospectral Families of High-order Systems∗

Earlier work of the authors concerning the generation of isospectral families of second order (vibrating) systems is generalized to higher-order systems (with no spectrum at infinity). Results and techniques are developed first for systems without symmetries, then with Hermitian symmetry and, finally, with palindromic symmetry. The construction of linearizations which retain such symmetries is discussed. In both cases, the notion of strictly isospectral families of systems is introduced implying that properties of both the spectrum and the sign-characteristic are preserved. Open questions remain in the case of strictly isospectral families of palindromic systems. Intimate connections between Hermitian and unitary systems are discussed in an Appendix. ∗The authors are grateful to an anonymous reviewer for perceptive comments which led to significant improvements in exposition. †The research of Peter Lancaster was partly supported by a grant from the Natural Sciences and Engineering Research Council of Canada