Chapter 24 Sturm-Liouville problem

1
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 ?????, chapter 8, lecture
note of Ordinary Differential equation

2
Existence and uniqueness 1
on interval
Sturm-Liouville equation
Assumptions
1
, and
is continuous differentiable on closed interval
, or say
, 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
Let
be continuous space equipped with norm
3
Existence and uniqueness 2
is complete under norm
1
by
define a mapping
2
extract supnorm
Existence and uniqueness
is a contraction mapping if
4
Fundamental matrix
on interval
Sturm-Liouville equation
Transform to ODE system by setting
has two fundamental solutions
satisfying
is
Solution of initial value problem
Definition fundamental matrix of
is
satisfying
Question Can we expect that we have two linear
independent fundamental solutions, say
5
Abels formula 1
Consider general ODE system of dimension two
with
( from product rule)
First order ODE
implies
are linearly independent
6
Abels formula 2
with fundamental matrix
Abels formula
since
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?
7
Dirichlet eigenvalue problem 1
Dirichlet eigenvalue problem
Observation
1
,in fact,
If
is eigen-pair of Dirichlet eigenvalue problem,
then
is continuous on closed interval
Exercise
2
Under assumption
, and
, we can define inner-product
where
is complex conjugate of
1
If w is not positive, then such definition is not
an inner-product
2
is Dirac Notation, different from conventional
form used by Mathematician
Thesis of Veerle Ledoux
Matrix computation
Dirac Notation
8
Dirichlet eigenvalue problem 2
3
Define differential operator
, then it is linear and
4
Greens identity
The same
hence
Dirichlet boundary condition
9
Dirichlet eigenvalue problem 3
5
Definition operator L is called self-adjoint on
inner-product space
If Greens identity is zero, say
Matrix computation (finite dimension)
Functional analysis (infinite dimension)
Matrix A is Hermitian (self-adjoint)
operator L is self-adjoint
Matrix A is Hermitian, then
1
A is diagonalizable and has real eigenvalue
operator L is Hermitian, then
eigenvectors are orthogonal
2
1
L is diagonalizable and has real eigenvalue
eigenvectors are orthogonal
2
orthonormal
10
Dirichlet eigenvalue problem 4
and
are two eigen-pair of
Suppose
and
are two eigen-pair
Theorem 1 all eigenvalue of Sturm-Liouville
Dirichlet problem are real
ltproofgt
Hence
is eigen-pair if and only if
is eigen-pair
11
Dirichlet eigenvalue problem 5
and
are two eigen-pair of
Theorem 2 if
If eigenvalue
, then
are orthogonal in
ltproofgt from Theorem 1, we know E1 and E2 are
real
Theorem 3 (unique eigen-function) eigenfunction
of Sturm-Liouville Dirichlet problem is unique,
in other words, eignevalue is
simple.
ltproofgt
Abels formula
suppose
and
Let
, then
for some constant c
since
12
Dirichlet eigenvalue problem 6
Theorem 4 (eigen-function is real) eigenfunction
of Sturm-Liouville Dirichlet problem can be
chosen as real function.
ltproofgt
suppose
is eigen-function with eigen-value
, satisfying
and
and
are both eigen-pair, from uniqueness of
eigen-function, we have
Hence
we can choose real function u as eigenfunction
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
Exercise (failure of uniqueness) consider
, find eigen-pair and
show eigenvalue is not simple, can you explain
this? (compare it with Theorem 3)
13
Dirichlet eigenvalue problem 7
Theorem 5 (Sturms Comparison theorem) let
be eigen-pair of Sturm-Liouville Dirichlet
problem.
. Precisely speaking
suppose
, then
is more oscillatory than
Between any consecutive two zeros of
, there is at least one zero of
ltproofgt
Let
are consecutive zeros of
and
on
as left figure
suppose
on
are eigen-pair, then
define
IVT
14
Dirichlet eigenvalue problem 8
on
on
and
on
implies
Question why
, any physical interpretation?
is more oscillatory than
Sturm-Liouville equation
Time-independent Schrodinger equation
Definition average quantity of operator
over interval
by
15
Dirichlet eigenvalue problem 9
Express operator L as
where
where
neglect boundary term
where
and
Heuristic argument for Theorem 5
Average value of
Average value of
is more oscillatory than
16
Prufer method 1
Sturm-Liouville Dirichlet eigenvalue problem
Idea introduce polar coordinate
in the phase plane
Exercise check it
Objective find eigenvalue E such that
Objective find eigenvalue E such that
Advantage of Prufer method we only need to solve

when solution
is found with condition
then
17
Prufer method 2
Observation
1
Suppose
has unique solution
with initial condition
fixed
is a strictly increasing function of variable
2
is Hooks law
has general solution
and
implies
18
Prufer method 3
numerically
Question Given energy E, how can we solve
Forward Euler method consider first order ODE
Uniformly partition domain
as
1
2
Let
and approximate
by one-side finite difference
continuous equation
discrete equation
Exercise use forward Euler method to solve
angle equation
for different E 1, 4, 9, 16 as right figure
It is clear that
is a strictly increasing function of variable
19
Prufer method 4
3
Although
has a unique solution
. But we want to find energy E
such that
, that is
is constraint.
Example for model problem
20
Prufer method 5
4
never decrease in a point where
number of zeros of
in
number of multiples of
in
Ground state
has no zeros except end points.
21
Prufer method 6
First excited state
has one zero in
model problem
second excited state
has two zeros in
conjecture k-th excited state
has k zeros in
22
Prufer method 7
Theorem 6 consider Prufer equations of regular
Sturm-Liouville Dirichlet problem
and
Let boundary values satisfy following
normalization
Then the kth eigenvalue
satisfies
Moreover the kth eigen-function
has k zeros in
Remark for detailed description, see Theorem
2.1 in
Veerle Ledoux, Study of Special Algorithms for
solving Sturm-Liouville and Schrodinger Equations.
23
Prufer method 8
Scaled Prufer transformation a generalization
of simple Prufer method
scaling function, continuous differentiable
where
1
2
Theorem 6 holds for scaled Prufer equations
Exercise use Symbolic toolbox in MATLAB to check
scaled Prufer transformation.
24
Prufer method 9
Scaled Prufer transformation
Simple Prufer transformation
Recall time-independent Schrodingers equation
dimensionless form
reduce to 1D Dirichlet problem
25
Exercise 1 1
Consider 1D Schrodinger equation with Dirichlet
boundary condition
use standard 3-poiint centered difference method
to find eigenvalue
1
number of grids 50, (n 50)
1
all eigenvalues ae positive, can you explain
this?
2
Ground state
has no zeros in
1st excited state
has 1 zero in
2nd excited state
has 2 zero in
Check if k-th eigen-function has k zeros in
26
Exercise 1 2
solve simple Prufer equation for first 10
eigenvalues
2
Forward Euler method N 200
is consistent with that in theorem 6
1
increases gradually staircase shape with
2
plateaus at
and steep slope around
Question what is disadvantage of this
staircase shape?
27
Exercise 1 3
Forward Euler method N 100
Forward Euler method N 200
under Forward Euler method with number of grids,
N 100, this is wrong!
Question Can you explain why
is not good?
hint see page 21 in reference
Veerle Ledoux, Study of Special Algorithms for
solving Sturm-Liouville and Schrodinger Equations.
28
Exercise 2 1
Scaled Prufer transformation
1D Schrodinger
where
is continuous
Suppose we choose
Question function f is continuous but not
differentiable at x 1. How can we obtain
and
Although we have known
on open set
on open set
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Exercise 2 2
solve simple Prufer equation for first 10
eigenvalues
Choose scaling function
Forward Euler method N 100
Forward Euler method N 100
scaling function S
NO scaling function S
Compare both figures and interpret why
staircase disappear when using scaling function