11 The Sturmian theory

TABLE OF CONTENTS

1. Introduction

2. Self-adjoint linear differential system

3. The Sturmian theory

3.1 The separation theorem

3.2 Sturm’s fundamental theorem

3.3 Condition for oscillatory solution

3.4 The first comparison theorem

3.5 The second comparison theorem

3.6 Sturm’s oscillation theorem

4. Application to the Sturm-Liouville system

LEARNING OBJECTIVES

1. In this module Sturmian theory is introduced.

2. A self adjoint linear differential system is defined. How any system can be converted into self adjoint form is explained.

3. The separation theorem and Sturm’s fundamental theorem are stated and proved.

4. Condition for the solution to be oscillatory is obtained.

5. Next the first comparison theorem, the second comparison theorem and the Sturm’s oscillation theorem are proved.

6. The theory developed above is applied to the Sturm-Liouville system.

T h e  S t u r m i a n  t h e o r y

1. Introduction

We now come back to the general theory of the ordinary second order linear differential equations. So far we have studied some general methods of solving such equations, and the related initial value problems. In module DE-2 we had seen that the initial value problem for the linear second order equation

has a unique solution provided certain conditions are satisfied. However, the existence theorem does not supply any, or very little, information on the nature of the solution. From the point of view of its applications in physics, and even otherwise, the nature of the solution is of crucial importance. In particular what interests us is the number of zeros of the solution in a given interval. This is intimately related to the question of the existence of the eigenvalues and the behaviour of the associated eigenfunctions. This is the problem that we wish to explore now. It was first studied by Sturm and is generally called the Sturmian theory.

2. Self-adjoint linear differential system

The general form of the linear second order differential expression is

The adjoint of this expression is defined as

A necessary and sufficient condition that the form be identical to its adjoint, that is, it is self-adjoint is found on equating the two expressions. Thus condition for it to be self-adjoint is

so that

Even if a given expression is not self-adjoint, it can always be made so.  This is proved by the following theorem

Theorem

Given the equation

where p, q, r are continuous and p is positive, it can always be put in the form

Proof

Divide equation (4) by p and put it in the form

3. The Sturmian theory

Since we have proved that linear second order differential equation can always be put in a self-adjoint form, we assume that the given expression is already in such a form. We now analyse the self-adjoint equation

Comparing with the usual form, equation (2)

3.1 The separation theorem

The separation theorem states that

Proof

3.2 Sturm’s fundamental theorem

Sturm’s fundamental theorem states that if the solutions of equation (4)

oscillate on the interval [a, b], they will oscillate more rapidly if p and q are diminished.

Proof

We will first prove it for the case when only q is diminished, p remaining unchanged. Let u be a solution of the equation

and v a solution of

This is a special case of what is usually called Green’s formula or theorem.

We now take up the general case which is also sometimes called the Sturm-Picone theorem. In this case the two equations are

This is known as Picone formula.

3.3 Condition for oscillatory solution

First consider the comparison equation

The solutions of equation (8) do not oscillate in [a, b] more rapidly than those of (13). But the solutions of equation (13) are known.

Now consider the other comparison equation

3.4 The first comparison theorem

Consider the systems

and

We are given

Proof

The integral on the right hand side is obviously positive. However, the quantity on the left hand side is

3.5 The second comparison theorem

Let c be any interior point of the interval [a, b] which is not a zero of either u or v. The second comparison theorem states that if c is such that u and v have the same number of zeros in the interval a<x <c , then

Proof

The required result follows immediately from this inequality.

3.6 Sturm’s oscillation theorem

We now look at the differential system with two point boundary conditions:

The roots of this characteristic equation are the characteristic numbers or eigenvalues. These are the values of the parameter for which the two point boundary problem has a nontrivial solution. The solutions are called the characteristic functions or eigen functions.

Theorem-1

We now further assume that

Theorem-2

The real eigenvalues of the system described by equations (23) and (24) may be arranged in increasing order of magnitude and denoted by

If the corresponding eigenfunctions are

then y_m  will have exactly m zeros in the open interval (a, b).

4.  Application to the Sturm-Liouville system

We now apply the Sturm’s oscillation theorem to the eigenvalue problem that we usually encounter. The parameter λ that we have introduced usually appears in the equation in the form

If the corresponding set of eigenfunctions are

It is now of the form of equation (23) with

Thus the conditions of theorem-1 above are not satisfied. However, since the conditions are sufficient but not necessary, it does not follow that the theorem is false. Since g changes sign in (a, b), a subinterval (a’, b’) can be found in which

and if the corresponding eigenfunctions are

SUMMARY

• In this module we introduce the Sturmian theory that deals with the nature of solutions of second order linear differential equations.
• We define the adjoint of a differential system and condition for the system being self adjoint, and explain how any system can be converted into self adjoint form.
• We first state and prove the Sturm’s separation theorem and fundamental theorem.
• Next we obtain the condition for the solution of the equation to be oscillatory.
• After that we prove the first and the second comparison theorems which deal with the increase in the number of zeros as the functions p and q are changed.
• We then take up the Sturm’s oscillation theorem which is the main result of this module.
• Finally we apply the theory developed above to the Sturm-Liouville eigenvalue problem.