Sturm-Liouville问题的几何结构 英文版
作者：傅守忠，王忠，吴宏友 著
出版时间：2019年版
内容简介
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。
内页插图
目录
Contents
Chapter 1 Basic on Sturm-Liouville Problems 1
1.1 Classes of Sturm-Liouville Problems 1
1.2 Characteristic Function 9
1.3 Equations with Piece-Wise Constant Coeffcients 14
1.4 Sturm's Comparison Theorem and Prufer Transformation 19
Chapter 2 Di.erentiable Manifolds and Lie Groups 31
2.1 Differentiable Manifolds 31
2.2 Differentiable Maps and Tangent Vectors 41
2.3 Complex Manifolds 45
2.4 Lie Groups 48
Chapter 3 Geometric Structures on Spaces of Boundary Conditions 51
3.1 Spaces of Boundary Conditions 52
3.2 Characteristic Curve and Surfaces 59
3.3 Di.erentiability of Continuous Eigenvalue Branches 71
3.4 Analyticity of Continuous Eigenvalue Branches 78
Chapter 4 Inequalities among Eigenvalues 84
4.1 More on Characteristic Function 85
4.2 Asymptotic Analysis of Fundamental Matrix 93
4.3 Inequalities among Eigenvalues for any coupled self-adjoint Boundary Condition 96
4.4 Ranges of on BR and BC 102
4.5 The Relationship among Three Multiplicities of a Di.erential
Operator's Eigenvalue 104
Chapter 5 Dependence of the n-th Eigenvalue on the Sturm-Liouville Problem 118
5.1 Continuity Principle 120
5.2 Continuous Dependence of on the Differential Equation 120
5.3 Discontinuity of λn 122
5.4 Comments on Di.erentiability of λn 131
5.5 The Index Problem for Eigenvalues for Coupled Boundary Conditions 135
Appendix A Fist-Order Linear Di.erential Equations 150
A.1 Existence and Uniqueness of Solution 150
A.2 Rank of a Solution and Variation of Parameters 155
A.3 Continuous Dependence of Solution on Problem 158
References 161
Index 170
