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Chapter 1 Basic on Sturm-Liouville Problems 1.1 Classes of Sturm-Liouville Problems 1.2 Characteristic Function 1.3 Equations with Piece-Wise Constant Coeffcients 1.4 Sturm's Comparison Theorem and Prufer Transformation Chapter 2 Differentiable Manifolds and Lie Groups 2.1 Differentiable Manifolds 2.2 Differentiable Maps and Tangent Vectors 2.3 Complex Manifolds 2.4 Lie Groups Chapter 3 Geometric Structures on Spaces of Boundary Conditions 3.1 Spaces of Boundary Conditions 3.2 Characteristic Curve and Surfaces 3.3 Di.erentiability of Continuous Eigenvalue Branches 3.4 Analyticity of Continuous Eigenvalue Branches Chapter 4 Inequalities among Eigenvalues 4.1 More on Characteristic Function 4.2 Asymptotic Analysis of Fundamental Matrix 4.3 Inequalities among Eigenvalues for any coupled self-adjoint Boundary Condition 4.4 Ranges of ¦Ën on BR and BC 4.5 The Relationship among Three Multiplicities of a Differential Operator's Eigenvalue Chapter 5 Dependence of the n-th Eigenvalue on the Sturm-Liouville Problem 5.1 Continuity Principle 5.2 Continuous Dependence of on the Differential Equation 5.3 Discontinuity of ¦Ën 5.4 Comments on Di.erentiability of ¦Ën 5.5 The Index Problem for Eigenvalues for Coupled Boundary Conditions Appendix A Fist-Order Linear Differential Equations A.1 Existence and Uniqueness of Solution A.2 Rank of a Solution and Variation of Parameters A.3 Continuous Dependence of Solution on Problem References Index