SturmLiouvilleproblem.pptVIP

Chapter 24 Sturm-Liouville problem Speaker: Lung-Sheng Chien Reference: [1] Veerle Ledoux, Study of Special Algorithms for solving Sturm-Liouville and  Schrodinger Equations. [2] 王信華教授, chapter 8, lecture note of Ordinary Differential equation Sturm-Liouville equation: Assumptions: 1 on interval is continuous differentiable on closed interval , or say , and , and , and 2 is continuous on closed interval , or say 3 , and Proposition 1.1: Sturm-Liouville initial value problem is unique on under three assumptions. proof: as before, we re-formulate it as integral equation and apply “contraction mapping principle” Existence and uniqueness [1] Let be continuous space equipped with norm Existence and uniqueness [2] is complete under norm define a mapping by 1 2 extract supnorm is a contraction mapping if Existence and uniqueness Fundamental matrix Sturm-Liouville equation: on interval Transform to ODE system by setting has two fundamental solutions satisfying Solution of initial value problem is Definition: fundamental matrix of is satisfying Question: Can we expect that we have two linear independent fundamental solutions, say Abel’s formula [1] Consider general ODE system of dimension two: with ( from product rule) First order ODE implies are linearly independent Abel’s formula [2] since with fundamental matrix Abel’s formula: Definition: Wronskian , then fundamental matrix of Sturm-Liouville equation can be expressed as In our time-independent Schrodinger equation, we focus on eigenvalue problem Definition: boundary condition is called Dirichlet boundary condition Question: What is “solvability condition” of Sturm-Liouville Dirichlet eigenvalue problem? Dirichlet eigenvalue problem [1] Dirichlet eigenvalue problem: Observation: 1 If is eigen-pair of Dirichlet eigenvalue problem, then ,in