12 The Sturm-Liouville Eigenvalue problem

TABLE OF CONTENTS

1. The boundary value problem

2. The Sturm-Liouville eigenvalue problem

3. Application of the Sturmian theory

3.1 Reality of eigenvalues

3.2 Orthogonality of eigenfunctions

3.2.1 Normalization

3.3 Nature of eigenvalues

4. Expansion in terms of eigenfunctions

5. An illustrative Example

LEARNING OBJECTIVES

1. Two point boundary conditions are introduced and conditions for existence of a unique solution to inhomogeneous equations or inhomogeneous boundary conditions determined.

2. The Sturm-Liouville eigenvalue problem is defined, theorems regarding reality of eigenvalues and orthogonality of eigenfunctions proved, and the normalization of the eigenfunctions discussed.

3. It is proved that eigenvalues with regular boundary conditions are simple.

4. Expansion of a function in terms of eigenfunctions of Sturm-Liouville eigenvalue problem is obtained.

5. A detailed example to illustrate all the aspects of eigenvalues and eigenfunctions discussed above is provided.

T h e  S t u r m – L i o u v i l l e  e i g e n v a l u e  p r o b l e m

1. The boundary value problem

Before we take up the Sturm-Liouville eigenvalue problem, we wish to make some general remarks about two point boundary conditions. In module DE-2 we had considered linear second order equations and the corresponding initial value problem. When we tackle two point boundary conditions, though the differential equation remains the same the problem of existence or nonexistence of a solution or the uniqueness of the solution becomes quite different and leads to what is called the eigenvalue problem.

We wish to find a solution of the equation

We first state certain theorems regarding the uniqueness of the solution of the boundary value problem where the right hand side of the equations (1) and (2) are nonzero.

Theorem-1

The boundary value problem for the equation

with boundary conditions (2) has a unique solution if, and only if, the solution with the homogeneous boundary conditions

has only the trivial solution.

Proof

Then

Hence following the same procedure as above, we have

But the coefficient matrix is non-singular , so we have

Conversely if the system given by equations (3) and (4) has only the trivial solution, then from equation (8) it follows that equation (7) holds from which it follows that equation (3) has a unique solution.              QED

Theorem-2

The next theorem is about the inhomogeneous equation (1). It states that the boundary value problem (1) together with (2):

has a unique solution if the homogeneous system, equations (3) and (4), has only the trivial solution.

Proof

Or

Since the homogeneous problem has only the trivial solution, the matrix on the left can be inverted, so that

Example

Consider the equation

This has the general solution

The boundary value problem

must have a unique solution.  The unique solution is

On the other hand, for the boundary value problem

2.  The Sturm-Liouville eigenvalue problem

We now wish to consider eigenvalue problem of the form

Here λ is a parameter which is quite often the energy or frequency. We consider boundary conditions of the regular type

as well as of the periodic type

As we have already seen with two point boundary conditions, the problem may or may not have a non-trivial solution. Very often a nontrivial solution may exist for some specific values of the parameter λ. In that case λ is called an eigenvalue and the corresponding solution an eigenfunction associated with the eigenvalue λ. Our aim now is to find the eigenvalues and corresponding eigenfunctions and study their general properties.

Example

Consider the boundary value problem

The characteristic equation is

There are three cases to be considered:

3.  Application of the Sturmian theory

As we have already seen in the last module, second order linear homogeneous equation can always be put in self-adjoint form. So let us look at the equation

We will now prove certain theorems regarding the eigenvalues and eigenfunctions of the Sturm-Liouville system. But first we prove the following theorem for a self-adjoint system.

Theorem-1

If

Proof

3.1 Reality of eigenvalues

Theorem-2

All eigenvalues of the Sturm-Liouville problem are real.

Proof

Let

Equate real and imaginary parts of the above equation and we get

Now multiply the first equation by v, the second by u and subtract the resulting first equation from the second:

Since the solution satisfies the boundary conditions

Further since the boundary conditions are linear this implies

So u and v satisfy the conditions of theorem-1; as a result the first integral in equation (17) vanishes and

3.2 Orthogonality of eigenfunctions

Theorem-3

Proof

so the above equation becomes

Since the two eigenvalues are distinct, it follows that

3.2.1 Normalization

We know that if y is a solution of a linear homogeneous differential equation then so is any constant multiple of y. The same of course is true for eigenfunctions as well. We can always use this arbitrariness to “normalize” the eigenfunction in any suitable manner. Usually the eigenfunction is normalized so that

If a given eigenfunction y(x) is not normalized, let φ(x) = cy(x).  Then φ(x) is properly normalized provided

3.3 Nature of eigenvalues

Theorem-4

The eigenvalues of the regular Sturm-Liouville eigenvalue problem are simple. An eigenvalue is said to be simple if there is only one linearly independent eigenfunction corresponding to it. In other words, if u and v are two eigenfunctions with eigenvalue λ, then = for some constant c.

Proof

Let u and v be eigenfunctions of regular Sturm-Liouville eigenvalue problem with eigenvalue λ.  Then

Theorem-5

4. Expansion in terms of eigenfunctions

Consider a function f(x) defined on the interval [a, b]. Let the function be “reasonably well behaved”. Most functions we have to deal with satisfy the criterion of “reasonableness”. Even discontinuities are allowed but non-integrable divergences are definitely a problem. Then such a function can be expanded in a series of eigenfunctions of the Sturm-Liouville problem.

5. An illustrative Example

Let us look at the familiar and the simplest equation, the harmonic oscillator, under various types of boundary conditions:

The general solution of this equation is

Case-1

The boundary conditions are

For λ ≤ 0, the boundary conditions demand a = b = 0, so we have only the null solution.

For λ > 0, the boundary conditions demand b = 0 and for a nontrivial solution

Thus the only eigenvalues and corresponding eigenfunctions are

As expected, all eigenvalues are real and simple and eigenfunctions orthogonal.

Case-2

The boundary conditions are

For λ ≤ 0, the boundary conditions demand a = b = 0, so we have only the null solution.

For λ > 0, the boundary conditions demand b = 0 and for a nontrivial solution

Thus this time the eigenvalues and corresponding eigenfunctions are

For the same equation the eigenvalues and eigenvectors change with the boundary conditions.

Case-3

This time the boundary conditions are

For λ = 0, the boundary conditions are satisfied with y = a. Thus the constant function y = a is an eigenfunction with eigenvalue λ = 0. As before, in this case also for λ< 0, boundary conditions allow only the null solution.

For λ > 0, the boundary conditions demand a = 0 and for a nontrivial solution

Case-4

Now we take the boundary conditions to be slightly more complicated:

Hence λ = 0 is not an eigenvalue.

Boundary conditions demand the following:

For a nontrivial solution we obtain the condition

On applying the boundary conditions we get

For a nontrivial solution we obtain the condition

Case-5

All the four boundary conditions that we have considered so far were regular boundary conditions. We now look at the periodic boundary condition:

On solving these two equations for a and b we have

Or

For a nontrivial solution the determinant of the coefficient matrix must be zero.  This leads to the condition

Hence the eigenvalues are

However both the coefficients, a and b, are indeterminate. Hence for each eigenvalue there are two linearly independent eigenfunctions:

SUMMARY

• We introduce two point boundary conditions and prove certain theorems regarding the conditions for existence of a unique solution to inhomogeneous equations or inhomogeneous boundary conditions.
• We introduce the Sturm-Liouville eigenvalue problem and prove theorems regarding reality of eigenvalues and orthogonality of eigenfunctions. Normalization of the eigenfunctions is also discussed.
• We next prove that the eigenvalues of the Sturm-Liouville eigenvalue problem are simple for regular boundary conditions but may not be so for periodic boundary conditions.
• We next obtain expansion of a function in terms of eigenfunctions of Sturm-Liouville eigenvalue problem.
• Finally we provide a detailed example to illustrate all the aspects of eigenvalues and eigenfunctions discussed above.