本书从几何的角度去研究Sturm-Liouville问题的谱，将边界条件可化为解析流形（如正则自伴边条件形成一个4维的紧解析流形），将所有是Sturm-Liouville问题放在一起形成一个空间。在该空间赋予一定的拓扑，成功解决了谱对问题的连续依赖性，揭示出Sturm-Liouville问题特征值的许多新的性质，如连续特征值分支在上述结构下的可微性，特征值的解析重数、代数重数和几何重数之间的关系等。

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 λn 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 λn
5.4 Comments on Di.erentiability of λn
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