《Sturm-Liouville問題的幾何結構（英文版）》是2019年科學出版社出版的圖書，作者是傅守忠、王忠、吳宏友。
基本介紹
- 中文名：Sturm-Liouville問題的幾何結構（英文版）
- 作者：傅守忠、王忠、吳宏友
- 出版社：科學出版社
- 出版時間：2019年1月
- 頁數：180 頁
- 定價：88 元
- 開本：16 開
- 裝幀：平裝
- ISBN：9787030619853
內容簡介,目錄,
內容簡介
本書從幾何的角度去研究Sturm-Liouville問題的譜，將邊界條件可化為解析流形（如正則自伴邊條件形成一個4維的緊解析流形），將所有是Sturm-Liouville問題放在一起形成一個空間。在該空間賦予一定的拓撲，成功解決了譜對問題的連續依賴性，揭示出Sturm-Liouville問題特徵值的許多新的性質，如連續特徵值分支在上述結構下的可微性，特徵值的解析重數、代數重數和幾何重數之間的關係等。
目錄
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
