Chapter 25 Sturm-Liouville problem (II)

1. Chapter 25 Sturm-Liouville problem (II) 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

2. Prufer method Sturm-Liouville Dirichlet eigenvalue problem: Scaled Prufer transformation Simple Prufer transformation scaling function where So far we have shown that Sturm-Liouville Dirichlet problem has following properties 1 Eigenvalues are real and simple, ordered as Eigen-functions are orthogonal in with inner-product 2 Eigen-functions are real and twice differentiable 3 Moreover we have implemented (Scaled) Prufer equation with Forward Euler Method (not stable, but it can be used so far)

3. Sturm’s Comparison [1] Theorem (Sturm’s first Comparison theorem): let be eigen-pair of Sturm-Liouville problem. . Precisely speaking suppose , then is more oscillatory than Between any consecutive two zeros of , there is at least one zero of Theorem (Sturm’s second Comparison theorem): let be solutions of Sturm-Liouville problem. suppose and on (1) Between any consecutive two zeros of , there is at least one zero of (2) <proof of (1)> Simple Prufer

4. Sturm’s Comparison [2] First we consider suppose and on 1 continuity of F, G 2 Suppose , then

5. Sturm’s Comparison [3] and 1 2 Question: How to deal with the case Between any consecutive two zeros of , there is at least one zero of <proof of (2)> Suppose has consecutive zeros at Without loss of generality, we assume Moreover , we may assume and 1 2 Apply result of (1), set , then

6. Pitfall [1] Recall Sturm-Liouville Dirichlet eigenvalue problem: 1 Eigenvalues are real and simple, ordered as Question: How about asymptotic behavior of eigenvalue, say Eigen-functions are orthogonal in with inner-product 2 Question: are eigen-functions complete in is eigen-pair of Eigen-functions are real and twice differentiable 3 The more important question is Question: is operator diagonalizable in

7. Pitfall [2] Question: How about asymptotic behavior of eigenvalue, say General Sturm-Liouville problem Model problem Sturm’s second Comparison theorem (1) Between any consecutive two zeros of , there is at least one zero of (2) Hook’s Law: solution: Zeros of solution is with space Exercise: Between any consecutive two zeros of , there is at least one zero of shows

8. Pitfall [3] Question: are eigen-functions complete in General Sturm-Liouville problem Model problem Question : solution of modal problem is Is such eigenspace complete in Consider space with inner-product 1 is orthogonal in is a closed subspace 2 is unique decomposition where

9. Pitfall [4] Informally, for some to be determined Formally speaking, when we write , in mathematical sense we construct partial sum such that in L2 sense. in

10. Pitfall [5] Exercise: is the solution of

11. Pitfall [6] where and Theorem: trigonometric basis is complete in in L2 sense, where

12. Pitfall [7] Exercise: we have shown where We abbreviate f as 1 If function f is even, say , then 2 If function f is odd, say , then Modal problem has eigen-pair From above exercise, for any , we can do odd extension then . Hence Question: How about if we do even extension

13. Pitfall [8] Question: is operator diagonalizable in From Prufer transformation, we can show and 1 Eigenvalues are real and simple, ordered as Eigen-functions are orthogonal in with inner-product 2 Define domain of operator L with Dirichlet boundary condition as Clearly we have ,but we can not say is diagonalizable in Finite dimensional matrix computation infinite dimensional functional analysis Jordan form: Question: does such exists? Idea: if we can show that , then even such exists, , why? Then operator L is diagonalizable in

14. Scaled Prufer Transformation [1] Scaled Prufer transformation Time-independent Schrodinger equation where Suppose we choose Question: function f is continuous but not differentiable at x = 1. How can we obtain has jump discontinuity at

15. Scaled Prufer Transformation [2] and Observation: does not exist, we ignore it. Then fundamental Theorem of Calculus also holds, say , fundamental Theorem of Calculus holds, 1 f is continuous 2 , fundamental Theorem of Calculus holds, 3 Question: although fundamental theorem of calculus holds for function f , but if is given, How can we find f(x) numerically and have better accuracy? Reason to discussion of fundamental theorem of calculus: depends on S(x), accuracy of is equivalent to accuracy of obtaining S(x)

16. Numerical integration [1] Ignore odd power since it does not contribute to integral general form Trapzoid rule (梯形法)

17. Numerical integration [2] Example: given a partition and grid function We use Trapezoid rule to find 1 Exercise 1: let 2 Try number of grids = 10, 20, 40, 80, 160, compute and measure maximum error Plot error versus grid number, what is order of accuracy ? Exercise 2: let 3 1 If x = 1 is in the grid partition, what is order of accuracy If x = 1 is NOT in the grid partition, what is order of accuracy 2

18. Scaled Prufer Transformation [3] Question: can we modify function f slightly such that it is continuously differentiable , say and where is polynomial of degree 3, are chosen such that <sol> is achieved by following 4 conditions 1 2 3 4 where

19. Scaled Prufer Transformation [4] and but has jump discontinuity at Exercise 3: try to construct where is polynomial of degree 5 1 use Symbolic toolbox to determine coefficients 2 plot 3 use Trapezoid method to compute ,what is order of accuracy ?

20. Review Finite Difference Method Model problem: for FDM eigen-pair: solution is Question: why does error of eigenvalue increase as wave number k increases? Substitute Exercise 4: find analytic solution of where Then use FDM to solve What is order of accuracy? measure