Background Material and Notation
This chapter provides an overview of the background material assumed in the remainder of the book. We state major results and provide sketches of the less technical proofs, particularly where the ideas presented are instrumental in subsequent constructions. The first topic is the theory of linear systems of ordinary differential equations (ODEs). Much of this material is standard for first-year graduate courses in ODEs, such as presented in Hartman , Perko . This is followed by a review of the basic theory of functional analysis as applied to linear partial differential equations; further details can be found in the standard references Evans , Kato . We finish the general overview by discussing the point spectrum in the context of the Sturm–Liouville theory for second-order operators. These operators have a one-to-one relationship between the ordering of the eigenvalues and the number of zeros for the associated eigenfunctions, which is extremely useful in applications.
KeywordsInvariant Subspace Unstable Manifold Homoclinic Orbit Closed Operator Essential Spectrum
- L. Evans. Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998.Google Scholar
- M. Hǎrǎguş and G. Iooss. Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems. Springer, New York, 2011.Google Scholar
- W. Magnus and S. Winkler. Hill’s Equation, volume 20 of Interscience Tracts in Pure and Applied Mathematics. Interscience, New York, 1966.Google Scholar
- M. Schecter. Fredholm operators and the essential spectrum. Courant Inst. Math. Sciences, New York U., 1965. (see http://archive.org/details/fredholmoperator00sche).
- E. Titchmarsh. Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I. Oxford University Press, Oxford, UK, edition, 1946.Google Scholar
- E. Titchmarsh. Eigenfunction Expansions Associated with Second-Order Differential Equations. Part II. Oxford University Press, Oxford, UK, first edition, 1958.Google Scholar
- J. Weidmann. Spectral Theory of Ordinary Differential Operators, volume 1258 of Lecture Notes in Mathematics. Springer-Verlag, New York, 1987.Google Scholar
- J. Weidmann. Spectral theory of Sturm–Liouville operators. Approximation by regular problems. In W. Amrein, A. Hinz, and D. Pearson, editors, Sturm–Liouville Theory: Past and Present, pp. 75–98. Birkhäuser, Boston, 2005.Google Scholar