Ch 11.4: Singular Sturm-Liouville Problems

1. Ch 11.4: Singular Sturm-Liouville Problems • In the preceding sections of this chapter we considered the Sturm-Liouville problems • Until now, we have always assumed the problem is regular. • That is, that p is differentiable, q and r are continuous and that p(x) > 0 and r(x) > 0 on the closed interval 0 x 1. • There are equations of physical interest in which some of these conditions are not satisfied.

2. Example 1: Bessel’s Equation • Consider Bessel’s equation of order  so that p(x) = x, q(x) = 2/x, and r(x) = x. • Thus p(0) = 0, r(0) = 0, q(x) is unbounded and hence discontinuous as x 0. • However, the conditions imposed on regular Sturm-Liouville problems are met elsewhere in the interval. • That is, that p is differentiable, q and r are continuous and that p(x) > 0 and r(x) > 0 on 0 x 1.

3. Example 2: Legendre’s Equation • Consider Legendre’s equation of order  where • Here the required conditions on p, q, and r are satisfied on the interval [0, 1], except at x =1 where p(1) = 0. • The required conditions are that p is differentiable, q and r are continuous and that p(x) > 0 and r(x) > 0 on 0 x 1.

4. Singular Sturm-Liouville Problems • We use the term singular Sturm-Liouville problem to refer to a certain class of boundary value problems for the equation in which p is differentiable, q and r are continuous and that p(x) > 0 and r(x) > 0 on the open interval 0 < x < 1, but at least one of these functions fails to satisfy them at one or both of the boundary points. • We also prescribe suitable boundary conditions of a kind to be described in more detail later in this section. • Singular problems also occur if the interval is unbounded, for example, 0 x < . We will not consider this case.

5. Example 3: Singular Problem (1 of 6) • Consider the equation or • This equation arises in describing free vibrations of a circular elastic membrane, and is discussed further in Section 11.5. • If we introduce the new independent variable t defined by then

6. Example 3: Bessel’s Equation (2 of 6) • Thus • Substituting these expressions into our differential equation we obtain • This equation then reduces to Bessel’s equation of order zero (see Section 5.8):

7. Example 3: General Solution (3 of 6) • The general solution to Bessel’s equation of order zero is • Hence the general solution of our original equation is where

8. Graphs of Bessel Functions, Order Zero (4 of 6) • Recall from Section 5.8 that J0 andY0 are Bessel functions of the first and second kinds, respectively, of order zero. • The graphs of J0 andY0 are given below. • Note that J0(0) = 1 and Y0(x)  - as x 0. • Also, J0has infinitely many solutions of J0(x) = 0.

9. Example 3: Trivial Solution (5 of 6) • Hence the general solution of our original equation is • Suppose we seek a solution to the boundary value problem • Since J0(0) = 1 and Y0(x)  - as x 0, the condition y(0) = 0 can be satisfied only by choosing c1 = c2 = 0. • Thus the boundary value problem has only the trivial solution. • This illustrates the general situation, that it is necessary to consider a modified type of boundary condition at a singular boundary point.

10. Example 3: Modified Boundary Condition (6 of 6) • Suppose we require that y and y'remain bounded as x 0. • Recalling that the general solution of our original equation is we see that the boundary value condition at x = 0 is satisfied by taking c2 = 0. • The second boundary condition, y(1) = 0, then yields • It can be shown that there is an infinite set of discrete positive roots, yielding eigenvalues 1 < 2 < … < n < … of the given problem, with eigenfunctions (up to a multiplicative constant)

11. Lagrange’s Identity for Singular Sturm-Liouville Problems (1 of 4) • Because of their importance in applications, it is worthwhile to investigate singular Sturm-Liouville problems further. • There are two main questions that are of concern: • Precisely what type of boundary conditions can be allowed in a singular Sturm-Liouville problem? • To what extent do the eigenvalues and eigenfunctions of a singular problem share the properties of the eigenvalues and eigenfunctions of regular Sturm-Liouville problems? In particular, are the eigenvalues real, are the eigenfunctions orthogonal, and can a given function be expanded as a series of eigenfunctions? • Both of these questions can be answered by a study of Lagrange’s identity

12. Improper Integral (2 of 4) • We therefore investigate the conditions for which Lagrange’s identity holds for singular problems, where the integral in may now be an improper integral. • Suppose x = 0 is a singular boundary point and x = 1 is not. • The boundary condition y(1) = 0 is imposed at x = 1, but we leave the boundary condition at x = 0 unspecified for now. • Choose  > 0 and consider the integral • Assume u and v have at least two continuous derivatives on the interval x 1.

13. Homogeneous Boundary Condition at Nonsingular Point(3 of 4) • Now • As in Section 11.2, we integrating twice by parts to obtain • The boundary term at x = 1 is again eliminated if both u and v satisfy the boundary condition y(1) = 0 there. Thus • It follows that

14. Boundary Condition at Singular Point (4 of 4) • Therefore Lagrange’s identity holds iff for every pair of functions u, v in the class under consideration. • This limit equation is therefore the criterion that determines what boundary conditions are allowable at x = 0 if that point is a singular boundary point. • If x = 1 is the singular boundary point, then the criterion is

15. Self-Adjoint Boundary Value Problems • As in Section 11.2, a singular boundary value problem for is said to be self-adjoint if Lagrange’s identity holds, possibly as an improper integral, for each pair of functions u and v with the following properties: (1) They are twice continuously differentiable on (0, 1); (2) They satisfy a boundary condition of the form 1y(a) + 2y'(a) = 0 at each regular boundary point a, and at each singular boundary point they satisfy a boundary condition sufficient to ensure that if x = 0 is a singular boundary point, or if x = 1 is singular, then

16. Singular Sturm-Liouville Problem Definition • If at least one boundary point is singular, then the equation together with two boundary conditions of the type described on the previous slide are said to form a singular Sturm-Liouville problem.

17. Example 4: Self-Adjoint Problem • Consider our previous boundary value problem • Then p(x) = x, and hence if u and v are twice continuously differentiable and satisfy the boundary conditions above, then • Thus this boundary value problem is a self-adjoint.

18. Continuous Spectrum • The most striking difference between regular and singular Sturm-Liouville problems is that in a singular problem the eigenvalues may not be discrete. • That is, the problem may have nontrivial solutions for every value of , or every value of  in some interval. • In such a case the problem has a continuous spectrum. • It may happen that a singular problem has a mixture of discrete eigenvalues and also a continuous spectrum. • Finally, it is also possible that only a discrete set of eigenvalues exists. This occurred in Example 3 for the modified boundary condition at x = 0.

19. Eigenfunction Expansions • A systematic discussion of singular Sturm-Liouville problems is beyond the scope of our treatment here. • We will restrict ourselves to some examples related to physical applications, where in each example it is known that there is an infinite set of discrete eigenvalues. • If a singular Sturm-Liouville problem does only have a discrete set of eigenvalues and eigenfunctions, then Lagrange’s identity is valid and can be used to show that the eigenvalues are real and the eigenfunctions are orthogonal with respect to the weight function r. • The expansion of a given function in terms of a series of eigenfunctions then follows as in Section 11.2.

20. Example 5: Boundary Value Problem (1 of 3) • Consider our previous boundary value problem • The eigenfunctions of this problem satisfy the orthogonality relation with respect to the weight function r(x) = x.

21. Example 5: Eigenfunction Expansion (2 of 3) • Suppose that f is a given function, and assume • Multiplying this equation by and integrating term-by-term, we obtain • By orthogonality, it follows that

22. Example 5: Convergence of Series (3 of 3) • Thus for a given function f, we have • The convergence of this series is established by an extension of Theorem 11.2.4 to cover this case. • This theorem can be shown to hold for other sets of Bessel functions, which are solutions of appropriate boundary value problems, as well as for Legendre polynomials, and for solutions of a number of other singular Sturm-Liouville problems of considerable interest.

23. Integral Representations • The singular problems mentioned in this section, with a discrete set of eigenvalues, are not typical. • In general, singular boundary value problems are characterized by continuous spectra. • The corresponding set of eigenfunctions are therefore not denumerable, and series expansions of the type described by Theorem 11.2.4 do not exist; they are instead replaced by integral representations.