{"id":642,"date":"2018-11-20T06:08:18","date_gmt":"2018-11-20T06:08:18","guid":{"rendered":"http:\/\/msp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=642"},"modified":"2019-04-30T11:55:25","modified_gmt":"2019-04-30T11:55:25","slug":"the-sturmian-theory","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp01\/chapter\/the-sturmian-theory\/","title":{"rendered":"The Sturmian theory"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
\r\n\r\n \r\n\r\n \r\n\r\nTABLE OF CONTENTS<\/strong>\r\n

1. Introduction<\/p>\r\n

2. Self-adjoint linear differential system<\/p>\r\n

3. The Sturmian theory<\/p>\r\n

3.1 The separation theorem<\/p>\r\n

3.2 Sturm\u2019s fundamental theorem<\/p>\r\n

3.3 Condition for oscillatory solution<\/p>\r\n

3.4 The first comparison theorem<\/p>\r\n

3.5 The second comparison theorem<\/p>\r\n

3.6 Sturm\u2019s oscillation theorem<\/p>\r\n

4. Application to the Sturm-Liouville system<\/p>\r\n \r\n\r\nLEARNING OBJECTIVES<\/strong>\r\n

1. In this module Sturmian theory is introduced.<\/span><\/p>\r\n

2. A self adjoint linear differential system is defined. How any system can be converted into self adjoint form is explained.<\/span><\/p>\r\n

3. The separation theorem and Sturm\u2019s fundamental theorem are stated and proved.<\/span><\/p>\r\n

4. Condition for the solution to be oscillatory is obtained.<\/span><\/p>\r\n

5. Next the first comparison theorem, the second comparison theorem and the Sturm\u2019s oscillation theorem are proved.<\/span><\/p>\r\n

6. The theory developed above is applied to the Sturm-Liouville system.<\/span><\/p>\r\n\r\n<\/div>\r\n

<\/div>\r\n \r\n
\r\n

T h e\u00a0 S t u r m i a n\u00a0 t h e o r y<\/strong><\/p>\r\n \r\n\r\n1. Introduction<\/strong><\/span>\r\n

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<\/p>\r\n\"\"\r\n

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<\/em>.<\/p>\r\n \r\n\r\n2. Self-adjoint linear differential system<\/strong><\/span>\r\n\r\nThe general form of the linear second order differential expression is\r\n\r\n\"\"\r\n\r\nThe adjoint of this expression<\/em> is defined as\r\n\r\n\"\"\r\n

A necessary and sufficient condition that the form be identical to its adjoint, that is, it is self-adjoint<\/em> is found on equating the two expressions. Thus condition for it to be self-adjoint is<\/p>\r\n\"\"\r\n\r\nso that\r\n\r\n\"\"\r\n\r\n \r\n

Even if a given expression is not self-adjoint, it can always be made so.\u00a0 This is proved by the following theorem<\/p>\r\n \r\n\r\nTheorem<\/span>\r\n\r\nGiven the equation\r\n\r\n\"\"\r\n\r\nwhere p<\/em>, q<\/em>, r<\/em> are continuous and p<\/em> is positive, it can always be put in the form\r\n\r\n\"\"\r\n\r\nProof<\/span>\r\n\r\nDivide equation (4) by p<\/em> and put it in the form\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n<\/div>\r\n

\r\n

3. The Sturmian theory<\/strong><\/span><\/p>\r\n

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<\/p>\r\n\"\"\r\n\r\nComparing with the usual form, equation (2)\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n3.1 The separation theorem<\/span>\r\n\r\nThe separation theorem<\/em> states that\r\n\r\n\"\"\r\n\r\nProof<\/span><\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n\"\"\r\n\r\n\"\"\r\n
<\/div>\r\n
\r\n\r\n \r\n\r\n3.2 Sturm\u2019s fundamental theorem<\/span>\r\n

Sturm\u2019s fundamental theorem states that if the solutions of equation (4)<\/p>\r\n\"\"\r\n

oscillate on the interval [a<\/em>, b<\/em>], they will oscillate more rapidly<\/em> if p<\/em> and q<\/em> are diminished.<\/p>\r\n \r\n\r\nProof<\/span>\r\n

We will first prove it for the case when only q<\/em> is diminished, p<\/em> remaining unchanged. Let u<\/em> be a solution of the equation<\/p>\r\n\"\"\r\n\r\nand v<\/em> a solution of\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n

This is a special case of what is usually called Green\u2019s formula<\/em> or theorem<\/em>.<\/p>\r\n\"\"\r\n\r\nWe now take up the general case which is also sometimes called the Sturm-Picone theorem<\/em>. In this case the two equations are\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nThis is known as <\/span>Picone formula<\/em>.<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n<\/div>\r\n

\r\n

3.3 Condition for oscillatory solution<\/span><\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\nFirst consider the comparison equation\r\n\r\n\"\"\r\n

The solutions of equation (8) do not oscillate in [a<\/em>, b<\/em>] more rapidly than those of (13). But the solutions of equation (13) are known.<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\nNow consider the other comparison equation<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

3.4 The first comparison theorem<\/span><\/p>\r\nConsider the systems\r\n\r\n\"\"\r\n\r\nand\r\n\r\n\"\"\r\n\r\nWe are given\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nProof<\/span><\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n

\r\n\r\nThe integral on the right hand side is obviously positive. However, the quantity on the left hand side is\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n3.5 The second comparison theorem<\/span>\r\n\r\nLet c<\/em> be any interior point of the interval [a<\/em>, b<\/em>] which is not a zero of either u<\/em> or v<\/em>. The second comparison<\/em> theorem <\/em>states that if c <\/em>is such that u <\/em>and v <\/em>have the same number of zeros in the interval a<x <c<\/em> , then\r\n\r\n\"\"\r\n\r\nProof<\/span>\r\n\r\n\"\"\r\n\r\nThe required result follows immediately from this inequality.\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

3.6 Sturm\u2019s oscillation theorem<\/span><\/p>\r\nWe now look at the differential system with two point boundary conditions:\r\n\r\n\"\"\r\n

<\/div>\r\n
\r\n\r\n\"\"\r\n\r\n\"\"\r\n

The roots of this characteristic equation<\/em> are the characteristic numbers<\/em> or eigenvalues<\/em>. 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 <\/em>or eigen functions.<\/em><\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\nTheorem-1<\/span>\r\n\r\n\"\"\r\n\r\nWe now further assume that\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\nTheorem-2<\/span><\/span>\r\n\r\nThe real eigenvalues of the system described by equations (23) and (24) may be arranged in increasing order of magnitude and denoted by<\/span>\r\n\r\n\"\"\r\n\r\nIf the corresponding eigenfunctions are<\/span>\r\n\r\n\"\"\r\n\r\nthen ym<\/sub><\/em>\u00a0 will have exactly <\/span>m<\/em> zeros in the open interval (<\/span>a<\/em>, <\/span>b<\/em>).<\/span>\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n

4.\u00a0 Application to the Sturm-Liouville system<\/strong><\/span><\/p>\r\n

We now apply the Sturm\u2019s oscillation theorem to the eigenvalue problem that we usually encounter. The parameter \u03bb<\/em> that we have introduced usually appears in the equation in the form<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nIf the corresponding set of eigenfunctions are\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nIt is now of the form of equation (23) with<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n

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 <\/span>g<\/em> changes sign in (<\/span>a<\/em>, <\/span>b<\/em>), a subinterval (<\/span>a\u2019<\/em>, <\/span>b\u2019<\/em>) can be found in which<\/span><\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nand if the corresponding eigenfunctions are\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\nSUMMARY<\/strong>\r\n\r\n<\/div>\r\n