Dissertation
Spectral properties of differential operators with vanishing coefficients
Doctor of Philosophy (Ph.D.), Drexel University
Jun 2013
DOI:
https://doi.org/10.17918/etd-4185
Abstract
The purpose of this thesis is to ascertain whether linear differential operators with vanishing coefficients make suitable operators for Cauchy problems. Well-posedness for linear Cauchy problems - characterized by existence, uniqueness, and continuous dependence on the initial data - depends on a ray being in the spectrum of the operator and an estimate for the resolvent operator along this ray. This was originally shown by Hille and Yosida for operators when every positive real number is in the resolvent set, and later generalized by Feller, Miyadera, and Phillips. We restrict our attention to the setting where the differential operator acts on functions that depend on a spatial variable that takes values from a bounded subset of the real line. We establish ill-posedness of the Cauchy problem by analyzing the spectral properties of the differential operator and prove the spectrum is the entire complex plane for a wide variety of differential operators with vanishing coefficients. If the differential operator is the product of a polynomial in the derivative with a scaler function that has roots of finite multiplicity, we develop simple criteria for establishing ill-posedness of the Cauchy problem. We establish point spectral results when the function is real-valued with only simple roots, in particular we show the point spectrum is the entire complex plane. Much less is known when the coefficients of the differential operator depend on time. For these non-autonomous Cauchy problems (NCPs) only sufficient conditions for well-posedness are known, with necessary conditions still lacking. In this thesis we make strides with establishing necessary spectral conditions for well-posed NCPs in the case where the family of operators is continuous.
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Details
- Title
- Spectral properties of differential operators with vanishing coefficients
- Creators
- Daniel Jordon - DU
- Contributors
- Jay Douglas Wright (Advisor) - Drexel University (1970-)
- Awarding Institution
- Drexel University
- Degree Awarded
- Doctor of Philosophy (Ph.D.)
- Publisher
- Drexel University; Philadelphia, Pennsylvania
- Resource Type
- Dissertation
- Language
- English
- Academic Unit
- College of Arts and Sciences; Drexel University; Mathematics
- Other Identifier
- 4185; 991014632564604721