Sturm-Liouville Equations: Eigenvalues, Eigenfunctions, and Approximation Methods

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Dive into Sturm-Liouville equations, exploring eigenvalues, eigenfunctions, and approximation techniques such as Rayleigh-Ritz and Green's function solutions. Learn about the properties, orthogonality, and completeness of eigenfunctions. Discover various boundary conditions and the formal completeness statement in this comprehensive lecture review.

  • Sturm-Liouville Equations
  • Eigenvalues
  • Eigenfunctions
  • Approximation Methods
  • Mathematical Methods
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  1. PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF in Olin 103 Notes on Lecture 22 Chap. 7 (F&W) Sturm-Liouville equations 1. Eigenvalues and eigenfunctions review 2. Rayleigh-Ritz approximation method 3. Green s function solution methods based on eigenfunction expansions 4. Green s function solution methods based on solutions of the homogeneous equations 10/14/2024 PHY 711 Fall 2024-- Lecture 22 1

  2. 10/14/2024 PHY 711 Fall 2024-- Lecture 22 2

  3. 10/14/2024 PHY 711 Fall 2024-- Lecture 22 3

  4. 10/14/2024 PHY 711 Fall 2024-- Lecture 22 4

  5. Review Sturm-Liouville equations defined over a range of x. d dx d dx + = Homogenous problem: ( ) x ( ) v x ( ) x ( ) x 0 0 d dx d dx + = Inhomogenous problem: ( ) x ( ) v x ( ) x ( ) x ( ) F x Eigenfunctions: d dx d dx + = ( ) x ( ) v x ( ) ( ) x f x ( ) f x n n n Note that, because Sturm-Liouville operator is Hermitian, the eigenvalues are real and the eigenfunctions are orthogonal. In the last lecture, we argued that the eigenfunctions form a complete set over the range of x defined for the particular system. 10/14/2024 PHY 711 Fall 2024-- Lecture 22 5

  6. Eigenvalues and eigenfunctions of Sturm-Liouville equations In the domain : a x b d d x v x f x dx dx Alternative boundary conditions; 1. ( ) ) ( ) or 2. x dx + = ( ) ( ) ( ) ( ) x f x ( ) n n n = = ( ) b x 0 f a f m m ( ( ) dx df dx df x df = = ( ) x 0 m m a b ( ) a ( ) x d f b = = or 3. ( ) ( ) nd b a m m d f a f m m Properties: Eigenvalues are real n b ( ) x f x f = Eigenfunctions are orthogonal: ( ) ( ) x dx , N n m nm n a b 2 where ( )( x ( )) f x . N dx n n a 10/14/2024 PHY 711 Fall 2024-- Lecture 22 6

  7. Formal statement of completeness of eigenfunctions: Completeness of eigenfunctions: ( ) ( ') n n f x f x N ( ) x ( ) = ' x x n n 10/14/2024 PHY 711 Fall 2024-- Lecture 22 7

  8. Variation approximation to lowest eigenvalue In general, there are several techniques to determine the eigenvalues nand eigenfunctions fn(x). When it is not possible to find the ``exact'' functions, there are several powerful approximation techniques. For example, the lowest eigenvalue can be approximated by minimizing the function , h h d dx d dx h S h + ( ) S x ( ) x ( ) v x 0 ( ) h x where is a variable function which satisfies the correct boundary values. The ``proof'' of this inequality is based on the notion that can in principle be expanded in terms of the (unknown) exact eigenfunctions fn(x): where the coefficients Cn can be ( ) ( ), n n n ( ) h x = h x C f x assumed to be real. 10/14/2024 PHY 711 Fall 2024-- Lecture 22 8

  9. Estimation of the lowest eigenvalue continued: From the eigenfunction equation, we know that ( ) ( ) ( ) n It follows that: ( ) ( ) ( ) a It also follows that : = = ( ) ( ) x f x ( ). S x h x S x C f x C n n n n n n b = = N 2 | C | . h S h h x S x h x dx n n n n b = = ( ) ( ) ( ) h x 2 | C | , h h x h x dx N n n a n 2 | C | N h S h n n n = Therefore . n 0 2 | C | N h h n n n 10/14/2024 PHY 711 Fall 2024-- Lecture 22 9

  10. 2 | C | N h S h n n n Some additional comments -- = . n 0 2 | C | N h h n n n 2 h S h C N = n n N w her e and =1 f f f n n n n 2 h h | C = = 0 0 n n m m m For the case of only two non-trivial e ige val es: n u h S h ( ) = + 1 1 f = + f f 0 0 0 1 0 1 h h 0 1 10/14/2024 PHY 711 Fall 2024-- Lecture 22 10

  11. Rayleigh-Ritz method of estimating the lowest eigenvalue h S h 2 ( ) xample: n n x f f dx , 0 h h d = = = E ( ) with (0) x ( ) 0 f f a n n n 2 n x a = = Exact eigenfunctions: ( ) sin 1,2,3.. .. f x n n 2 2 9.869604 40 4 n = = = Exact e igenvalue : s 1,2,3.... n n 2 2 a a a = Trial function ( ) x ( ) f x x a trial 2 d dx ( ) ( ) x a x x a x 10 a 2 = Raleigh-Ritz estimate: ( ) ( x x a ) 2 x a x 10/14/2024 PHY 711 Fall 2024-- Lecture 22 11

  12. f1 exact f ftrial x 10/14/2024 PHY 711 Fall 2024-- Lecture 22 12

  13. Rayleigh-Ritz method of estimating the lowest eigenvalue h S h 2 2 2 ( ) Example: n x G x d x Another example this time with a variable parameter , 0 h d f h ( ) x + = = ( ) = ( ) with ( x ) 0 n x f f f f n n n n 2 = gx trial function ( ) f e trial f f S f G trial trial = + Raleigh-Ritz estimate: ( ) g g trial 4 g f trial trial trial( ) G g / g G Note that for differential equation of the Schoedinger equation of the harmonic oscillator: 2 G= = 1 2 = = ( ) g G g G m mE 0 trial 0 = E trial 0 0 2 2 10/14/2024 PHY 711 Fall 2024-- Lecture 22 13

  14. Recap -- Rayleigh-Ritz method of estimating the lowest eigenvalue Example from Schroedinger eq ( ) ( ) 2 2 ( x e f = uation f r one-dimensional o harmonic oscillator: 2 2 1 d f x + = = = 2 2 ( ) with ( x ) ( ) 0 m x f x E f f f n x n n n n n 2 m d 2 gx Trial functi o n ) trial f f S f 2 2 2 2 / m trial trial = + Raleigh- Ritz estima e: t ( ) g g E trial 2 4 m g f trial 1 2 trial m Exact answer = = ( ) g E g 0 trial 0 Do you think that there is a reason for getting the correct answer from this method? a. Chance only b. Skill 10/14/2024 PHY 711 Fall 2024-- Lecture 22 14

  15. 10/14/2024 PHY 711 Fall 2024-- Lecture 22 15

  16. Solution to inhomogeneous problem by using Greens functions Inhomogenous problem: d d x v x dx dx : function s Green' d x dx Formal solution: + = ( ) ( ) ( ) x ( ) x ( ) F x d ( ) ' x + = ( ) ( ) ( ) ( , ) ' x v x x G x x dx b + ( ) x = ( ) x ( , ') ( ') G x x F x dx ' 0 a Solution to homogeneous problem 10/14/2024 PHY 711 Fall 2024-- Lecture 22 16

  17. Formal solution: b + ( ) x = ( ) x ( , ') ( ') G x x F x dx ' 0 a Solution to homogeneous problem What is the homogeneous equation psi_0(x)? Homogenous problem: d d x dx dx + = ( ) ( ) v x ( ) x ( ) x 0 0 In this lecture, we will discuss several methods of finding this Green s function. This topic will also appear in PHY 712 10/14/2024 PHY 711 Fall 2024-- Lecture 22 17

  18. How do we arrive at the formal solution? Formal solution: b + ( ) x = ( ) x ( , ') ( ') G x x F x dx ' 0 a Note that this form satisfies the inhomogenous equation d d S x x dx dx + Define ( ) ( ) ( ) v x ( ) x b ( ) x = + ( ) S x ( ) S x ( ) x ( ) S x ( , ') ( ') G x x F x dx ' 0 a b ( ) x = = ( ) S x 0 + ( ') ( ') x F x dx ' ( ) x F x a 10/14/2024 PHY 711 Fall 2024-- Lecture 22 18

  19. Using complete set of eigenfunctions to form Greens function -- Suppose that we can find a Green's function defined as follows: d d x v x x G x x dx dx ( ) + = ( ) ( ) ( ) ( , ') ' x x Completeness of eigenfunctions: ( ) ( ') n n f x f x N In terms of eigenfunctions: d d x v x dx dx f x f x G x x = Recall: ( ) x ( ) = ' x x n n ( ) ( ') f x f x N ( ) x + = ( ) ( ) ( ) x ( , ') G x x n n n n ( ) ( ')/ N By construction ( , ') n n n n n 10/14/2024 PHY 711 Fall 2024-- Lecture 22 19

  20. Example Sturm-Liouville problem: = 1; ( ) = = = = Example: ( ) 1; ( ) 0; 0 and x x v x a b L x = = 1; ( ) sin F x F 0 L Inhomogenous equation: 2 d dx x = 1 ( ) x sin F 0 2 L 10/14/2024 PHY 711 Fall 2024-- Lecture 22 20

  21. Eigenvalue dx equation : 2 d = ( ) ( ) f x f x n n n 2 Eigenfunct ions Eigenvalue : s 2 2 L n x n = = ( ) sin f x n n L L Completeness of eigenfunctions: ( ) n n f x f x N ( ') ( ) x ( ) = ' x x n n = 2 L ' n x L n x L ( ) In this example: sin sin ' x x = 1 n 10/14/2024 PHY 711 Fall 2024-- Lecture 22 21

  22. = 2 L ' n x L n x L N ( ) In reality, for finite summation sin sin ' x x = 1 n x=1/2, L=1 N=100 N=10 x 10/14/2024 PHY 711 Fall 2024-- Lecture 22 22

  23. Green' function s d x : d ( ) ' x + = ( ) ( ) ( ) ( , ) ' x v x x G x x dx dx Green' function s for the example : ' n x n x sin sin ( ) ( / ) ' 2 f x f x N L L n n = = n n n G(x,x') 2 L n n 1 L 10/14/2024 PHY 711 Fall 2024-- Lecture 22 23

  24. Using Green's function to solve inhomogenous equation: = 2 d dx x 1 ( ) x sin with boundary values (0)= ( )=0 F L 0 2 L L ' x = + ( ) x ( ) x ( , ') G x x F sin ' dx 0 0 L 0 n x L 2 sin L 2 L ' ' n x L x = + ( ) x ( ) x sin sin ' F dx 0 0 L n n 0 1 L F x ( ) = + ( ) x sin 0 2 x 0 L 1 L 10/14/2024 PHY 711 Fall 2024-- Lecture 22 24

  25. Another method of constructing Greens functions -- using two solutions to the homogeneous problem : function s Green' d x dx T wo homogeneous solutions d d x v x x dx dx d ( ) ' x + = ( ) ( ) ( ) ( , ) ' x v x x G x x dx + = = ( ) ( ) ( ) ( ) 0 for , g x i a b i 1 = Let ( , ') ( ) ( ) G x x g x g x a b W d d ( ') x where ( ') x ( ') ( ') x ( ' ) W g g x g g x a b b a ' ' dx dx 10/14/2024 PHY 711 Fall 2024-- Lecture 22 25

  26. Some details: For 0: + + ' ' x x d dx d dx ( ) + = ( ) x ( ) v x ( ) x ( , ') ' dx G x x dx x x ' ' x x + ' x 1 d dx d dx W = ( ) ( ) ( ) 1 dx x g x g x a b ' x + ' x ( ) x W ( ') x W d dx d d x = ( ) ( ) ( ') x ( ') ( ) x ' ( ') g x g x g g x g g x a b a b b a ' ' dx d ' x d d = ( ') x ( ') x ( ') x ( ') x ( ' ) W g (Wronskian) is constant, since g g g x a b b a ' ' dx dx dW dx = Note -- 0. W ' Useful Green's function construction in one dimension: 1 ( , ' ( ) ( ) ) a b G x x g x x g W = 10/14/2024 PHY 711 Fall 2024-- Lecture 22 26

  27. d dx d dx + = ( ) x ( ) v x ( ) x ( ) x ( ) F x Green's function solution: b ( ) x = + ( ) x ( , ') ( ') G x x F x dx ' 0 a x b ( ) W ( ) W g x g x + + = ( ) x ( ') ( ') ' ( ') ( ) ' ' g x F x dx g x F x dx b a 0 a b a x Note that the integral has to be performed in two parts. While the eigenfunction expansion method can be generalized to 2 and 3 dimensions, this method only works for one dimension. 10/14/2024 PHY 711 Fall 2024-- Lecture 22 27

  28. Example from previous discussion: = 2 d dx x 1 ( ) x sin with boundary values (0)= ( )=0 F L 0 2 L 1 ( ) ( ) = Using: ( , ') for 0 G x x g x g x x L a b W 2 d dx ( ) x ( ) x = = = 1 ( ) 0 sin( ); x sin( ); g x g g L x i a b 2 ( ) x dx ( ) x dx dg dg ( ) ( ) L ( ) x ( ) ( ) x ( ) x ( ) = = + a b sin cos sin cos W g x g L x L x b a = sin x sin( sin( ) ) ' L x x = + ( ) x ( ) x sin( ') sin ' x F dx 0 0 L L 0 L sin( ) sin( ) ' x L x + sin( ') sin ' L x F dx 0 L x F x (Actually the algebra is painful). But, hurray! Same result as before. = + ( ) x ( ) x sin 0 2 0 L 1 10/14/2024 PHY 711 Fall 2024-- Lecture 22 28 L

  29. Another example -- 2 d dx = ( )/ x ( ) x electrostatic potential for charge density ( ) x 0 2 Homogeneous equation: 2 d dx = ( ) x 0 g , a b 2 = = Let Wronskian: ( ) ( ) 1 g x x g x a b ( ) dx ( ) x dg x dg x d = = ( ) x ( ) 1 W g g x b a a b Green ( , ') G x x 's function: = x x 1 x = + + ' ( ( ) x ( ) x ' ' ( ) dx x ) x dx x 0 0 0 x 10/14/2024 PHY 711 Fall 2024-- Lecture 22 29

  30. Example -- continued 2 d dx = ( ) / x ( ) x electrostatic potential for cha rge density ( ) x 0 2 x 1 x = + + ( ) x ( ) ' ' ( ) x ' ( ) x x dx x d x 0 0 0 x 0 x a x = Suppose ( ) x / x a a a 0 0 x a 0 x a 3 2 3 a 3 xa x = + + ( ) x ( ) x 0 a a x a 0 2 6 0 2 2 a x a 0 3 0 10/14/2024 PHY 711 Fall 2024-- Lecture 22 30

  31. 0 x a 3 2 3 a xa x = + ( ) x 0 a a x a 3 2 6 0 2 2 a x a 0 3 0 10/14/2024 PHY 711 Fall 2024-- Lecture 22 31

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