{"id":724,"date":"2018-11-20T10:33:30","date_gmt":"2018-11-20T10:33:30","guid":{"rendered":"http:\/\/msp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=724"},"modified":"2019-04-30T11:56:31","modified_gmt":"2019-04-30T11:56:31","slug":"the-sturm-liouville-eigenvalue-problem","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp01\/chapter\/the-sturm-liouville-eigenvalue-problem\/","title":{"rendered":"The Sturm-Liouville Eigenvalue problem"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
\r\n\r\nTABLE OF CONTENTS<\/strong>\r\n

1. The boundary value problem<\/p>\r\n

2. The Sturm-Liouville eigenvalue problem<\/p>\r\n

3. Application of the Sturmian theory<\/p>\r\n

3.1 Reality of eigenvalues<\/p>\r\n

3.2 Orthogonality of eigenfunctions<\/p>\r\n

3.2.1 Normalization<\/p>\r\n

3.3 Nature of eigenvalues<\/p>\r\n

4. Expansion in terms of eigenfunctions<\/p>\r\n

5. An illustrative Example<\/p>\r\n \r\n\r\nLEARNING OBJECTIVES<\/strong>\r\n

1. Two point boundary conditions are introduced and conditions for existence of a unique solution to inhomogeneous equations or inhomogeneous boundary conditions determined.<\/span><\/p>\r\n

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.<\/span><\/p>\r\n

3. It is proved that eigenvalues with regular boundary conditions are simple.<\/span><\/p>\r\n

4. Expansion of a function in terms of eigenfunctions of Sturm-Liouville eigenvalue problem is obtained.<\/span><\/p>\r\n

5. A detailed example to illustrate all the aspects of eigenvalues and eigenfunctions discussed above is provided.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

T h e\u00a0 S t u r m - L i o u v i l l e\u00a0 e i g e n v a l u e\u00a0 p r o b l e m<\/strong><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n1. The boundary value problem<\/strong><\/span>\r\n

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.<\/p>\r\nWe wish to find a solution of the equation\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n

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.<\/p>\r\n \r\n\r\nTheorem-1<\/span>\r\n\r\nThe boundary value problem for the equation\r\n\r\n\"\"\r\n

with boundary conditions (2) has a unique solution if, and only if, the solution with the homogeneous boundary conditions<\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\nhas only the trivial solution.\r\n\r\n \r\n\r\nProof<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nThen\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nHence following the same procedure as above, we have<\/span>\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
\r\n\r\nBut the coefficient matrix is non-singular [equation (7)], so we have\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
\r\n

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.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 QED<\/p>\r\n \r\n\r\nTheorem-2<\/span>\r\n

The next theorem is about the inhomogeneous equation (1). It states that the boundary value problem (1) together with (2):<\/p>\r\n\"\"\r\n

has a unique solution if the homogeneous system, equations (3) and (4), has only the trivial solution.<\/p>\r\n\r\n<\/div>\r\n

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

\r\n\r\nOr\r\n\r\n\"\"\r\n

Since the homogeneous problem has only the trivial solution, the matrix on the left can be inverted, so that<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\nExample<\/span>\r\n\r\nConsider the equation\r\n\r\n\"\"\r\n\r\nThis has the general solution\r\n\r\n\"\"\r\n\r\nThe boundary value problem\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nmust have a unique solution.\u00a0 The unique solution is\r\n\r\n\"\"\r\n\r\nOn the other hand, for the boundary value problem\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n2.\u00a0 The Sturm-Liouville eigenvalue problem<\/span><\/strong><\/span>\r\n\r\nWe now wish to consider eigenvalue problem of the form<\/span>\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n

Here \u03bb<\/em> is a parameter which is quite often the energy or frequency. We consider boundary conditions of the regular type<\/p>\r\n\"\"\r\n\r\nas well as of the periodic type\r\n\r\n\"\"\r\n\r\n\"\"\r\n

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 \u03bb<\/em>. In that case \u03bb<\/em> is called an eigenvalue<\/em> and the corresponding solution an eigenfunction<\/em> associated with the eigenvalue \u03bb<\/em>. Our aim now is to find the eigenvalues and corresponding eigenfunctions and study their general properties.<\/p>\r\n \r\n\r\nExample<\/span>\r\n\r\nConsider the boundary value problem\r\n\r\n\"\"\r\n\r\nThe characteristic equation is\r\n\r\n\"\"\r\n\r\nThere are three cases to be considered:<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n3.\u00a0 Application of the Sturmian theory<\/span><\/strong><\/span>\r\n

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<\/span><\/p>\r\n

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

We will now prove certain theorems regarding the eigenvalues and eigenfunctions of the Sturm-Liouville system.\u00a0<\/span>But first we prove the following theorem for a self-adjoint system.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\nTheorem-1<\/span>\r\n\r\n \r\n\r\nIf\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\nProof<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

3.1 Reality of eigenvalues<\/span><\/p>\r\n \r\n\r\nTheorem-2<\/span>\r\n\r\nAll eigenvalues of the Sturm-Liouville problem are real.\r\n\r\n \r\n\r\nProof<\/span>\r\n\r\nLet\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nEquate real and imaginary parts of the above equation and we get<\/span>\r\n\r\n\"\"\r\n

\r\n

Now multiply the first equation by v<\/em>, the second by u<\/em> and subtract the resulting first equation from the second:<\/p>\r\n\"\"\r\n\r\nSince the solution satisfies the boundary conditions<\/span>\r\n\r\n\"\"\r\n\r\nFurther since the boundary conditions are linear this implies<\/span>\r\n\r\n\"\"\r\n\r\nSo <\/span>u<\/em> and <\/span>v<\/em> satisfy the conditions of theorem-1; as a result the first integral in equation (17) vanishes and<\/span>\r\n\r\n\"\"\r\n\r\n<\/div>\r\n\"\"\r\n

\r\n\r\n \r\n

3.2 Orthogonality of eigenfunctions<\/span><\/p>\r\n \r\n\r\nTheorem-3<\/span>\r\n\r\n\"\"\r\n\r\n \r\n\r\nProof<\/span>\r\n\r\n\"\"\r\n\r\nso the above equation becomes<\/span>\r\n\r\n\"\"\r\n\r\nSince the two eigenvalues are distinct, it follows that<\/span>\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n

3.2.1 Normalization<\/span><\/span><\/p>\r\n

We know that if <\/span>y<\/em> is a solution of a linear homogeneous differential equation then so is any constant multiple of <\/span>y<\/em>. The same of course is true for eigenfunctions as well. We can always use this arbitrariness to \u201cnormalize\u201d the eigenfunction in any suitable manner. Usually the eigenfunction is normalized so that<\/span><\/p>\r\n\"\"\r\n

If a given eigenfunction <\/span>y<\/em>(<\/span>x<\/em>) is not normalized, let <\/span>\u03c6<\/em>(<\/span>x<\/em>) = <\/span>cy<\/em>(<\/span>x<\/em>).\u00a0 Then <\/span>\u03c6<\/em>(<\/span>x<\/em>) is properly normalized provided<\/span><\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n3.3 Nature of eigenvalues<\/span>\r\n\r\n \r\n\r\nTheorem-4<\/span>\r\n

The eigenvalues of the regular Sturm-Liouville eigenvalue problem are simple. An eigenvalue is said to be simple <\/em>if there is only one linearly independent eigenfunction corresponding to it. In other words, if u <\/em>and v <\/em>are two eigenfunctions with eigenvalue \u03bb<\/em>, then = for some constant c<\/em>.<\/p>\r\n \r\n\r\nProof<\/span>\r\n

Let u<\/em> and v<\/em> be eigenfunctions of regular Sturm-Liouville eigenvalue problem with eigenvalue \u03bb<\/em>.\u00a0 Then<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

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

Theorem-5<\/span><\/p>\r\n\"\"\r\n\r\n \r\n\r\n4. Expansion in terms of eigenfunctions<\/strong><\/span>\r\n

Consider a function f<\/em>(x<\/em>) defined on the interval [a<\/em>, b<\/em>]. Let the function be \u201creasonably well behaved\u201d. Most functions we have to deal with satisfy the criterion of \u201creasonableness\u201d. 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.<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n5. An illustrative Example<\/strong><\/span>\r\n

Let us look at the familiar and the simplest equation, the harmonic oscillator, under various types of boundary conditions:<\/p>\r\n\"\"\r\n\r\nThe general solution of this equation is\r\n\r\n\"\"\r\n\r\n<\/div>\r\nCase-1<\/span><\/span>\r\n

The boundary conditions are<\/span><\/p>\r\n\"\"\r\n

For\u00a0\u03bb<\/em>\u00a0\u2264 0, the boundary conditions demand <\/span>a<\/em> = <\/span>b<\/em> = 0, so we have only the null solution.<\/span><\/p>\r\n

For\u00a0<\/span>\u03bb<\/em> > 0, the boundary conditions demand <\/span>b<\/em> = 0 and for a nontrivial solution<\/span><\/p>\r\n\"\"\r\n

Thus the only eigenvalues and corresponding eigenfunctions are<\/span><\/p>\r\n\"\"\r\n

As expected, all eigenvalues are real and simple and eigenfunctions orthogonal.<\/span><\/p>\r\n\r\n

\r\n\r\nCase-2<\/span>\r\n\r\nThe boundary conditions are\r\n\r\n\"\"\r\n

For\u00a0\u03bb<\/em>\u00a0\u2264 0, the boundary conditions demand a<\/em> = b<\/em> = 0, so we have only the null solution.<\/p>\r\n

For\u00a0\u03bb<\/em> > 0, the boundary conditions demand b<\/em> = 0 and for a nontrivial solution<\/p>\r\n\"\"\r\n

Thus this time the eigenvalues and corresponding eigenfunctions are<\/p>\r\n\"\"\r\n

For the same equation the eigenvalues and eigenvectors change with the boundary conditions.<\/p>\r\n \r\n\r\nCase-3<\/span>\r\n\r\nThis time the boundary conditions are\r\n\r\n\"\"\r\n

For \u03bb<\/em> = 0, the boundary conditions are satisfied with y<\/em> = a<\/em>. Thus the constant function y<\/em> = a<\/em> is an eigenfunction with eigenvalue \u03bb<\/em> = 0. As before, in this case also for \u03bb<\/em>< 0, boundary conditions allow only the null solution.<\/p>\r\n

For\u00a0\u03bb<\/em> > 0, the boundary conditions demand a<\/em> = 0 and for a nontrivial solution<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\nCase-4<\/span>\r\n

Now we take the boundary conditions to be slightly more complicated:<\/p>\r\n\"\"\r\n\r\n\"\"\r\n

Hence <\/span>\u03bb<\/em> = 0 is not an eigenvalue.<\/span><\/p>\r\n\"\"\r\n

Boundary conditions demand the following:<\/span><\/p>\r\n\"\"\r\n

For a nontrivial solution we obtain the condition<\/span><\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nOn applying the boundary conditions we get<\/span>\r\n\r\n\"\"\r\n\r\nFor a nontrivial solution we obtain the condition<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

Case-5<\/span><\/p>\r\n

All the four boundary conditions that we have considered so far were regular boundary conditions. We now look at the periodic boundary condition:<\/span><\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n

On solving these two equations for <\/span>a<\/em> and <\/span>b<\/em> we have<\/span><\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nOr\r\n\r\n\"\"\r\n

For a nontrivial solution the determinant of the coefficient matrix must be zero.\u00a0 This leads to the condition<\/span><\/p>\r\n\"\"\r\n\r\nHence the eigenvalues are\r\n\r\n\"\"\r\n

However both the coefficients, <\/span>a<\/em> and <\/span>b<\/em>, are indeterminate. Hence for each eigenvalue there are two linearly independent eigenfunctions:<\/span><\/p>\r\n\"\"\r\n\r\n\"\"\r\n

<\/div>\r\n \r\n\r\nSUMMARY<\/strong>\r\n