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Chapter 1 Introduction 1.1 Physical background 1.2 Related results of ordinary differential operators 1.3 Structure of the book Chapter 2 Approximations of eigenvalues and eigenfunctions 2.1 Notation and theoretic results 2.2 Main ideas of the algorithms 2.3 General methods for constructing examples 2.4 Examples with 2-independent BCs 2.5 Examples with 2-dependent BCs 2.6 Oscillations of eigenfunctions for discontinuous Sturm-Liouville problems Chapter 3 Computing the indices of eigenvalues 3.1 Notation and theoretic results 3.2 Algorithm and implementation 3.3 Examples with a positive f 3.4 Examples with an indefinite f 3.5 Examples about ��*(��) and ��(��) Chapter 4 Relations among eigenvalues of Sturm-Liouville problems 4.1 Notation and basic results 4.2 Geometric characterization of ��n 4.3 Interlacing relations among eigenvalues Chapter 5 Third-order eigenparameter dependent differential operators 5.1 Preliminaries 5.2 The Banach space 5.3 Derivative formulas of eigenvalues Chapter 6 Application of Sturm-Liouville problems 6.1 Construction and stability of Riesz bases 6.2 Eigenvalue problems of internal solitary waves Appendix A Fundamentals Sturm-Liouville problems A.1 Classes of Sturm-Liouville problems A.2 Characteristic function Appendix B Thomson-Haskell method Appendix C First-order linear differential equations C.1 Existence and uniqueness of a solution C.2 Rank of a solution and variation of parameters C.3 Continuous dependence of solution on the problem References