Chapter 8 Sturm-Liouville Theory

CHAPTER 8

Sturm-Liouville Theory

8.1. The Differential Equation In a celebrated group of papers Sturm and Liouville treated boundary problems for a second-order ordinary differential equation, which we shall write (Y’/Y)’

+ (hp + q)y

a

= 0,

< x < b,

’

= d/dx,

(8.1.1)

where p, q and r are suitably smooth functions of x and h is a scalar parameter; in addition, the functions p , r are commonly required to be positive, except possibly at the end-points x = a, b. T h e formal analogy between (8.1.1) and the three-term recurrence relation (4.1. l), which we may bring out better by writing the latter in the form

is reflected in a rather complete correspondence between the results for the two cases. As mentioned in Chapter 0, the analogy is also borne out by a common physical model, the vibrating string, either continuously loaded or else weightless and bearing discrete particles. T h e analogy is substantiated by various mathematical formalisms which unify the two cases. As such a formalism we shall treat in this chapter the first-order system 24’ = YO,

Y’

=

-(hp

+ q ) u.

(8.1.2-3)

In the first place, if we have a solution of (8.1.1) then a pair of solutions of (8.1.2-3) is given by y = u, y ’ / r = v ; the reverse deduction may also be made if r > 0. Secondly, we may also consider (8.1.2-3) in the case that r vanishes over a subinterval of (a, b). Suppose in particular that (a, b) breaks up into a sequence of intervals (a, b,), (b, , a,) (a1, bl), ( b , , as), ..., in which alternately r = 0, or p = q = 0. T h e 202

8.1.

203

THE DIFFERENTIAL EQUATION

relations (8.1.2-3) yield on integration recurrence relations of the form U(b0) =

244,

f@,)

-

4 . )

=-42)

.(al) - u(b,) = o(b,)

T

dx,

(A J p dx b0

.(al)

+ I””4 dx) ,

= s(b,),

bo

and so on. With the identifications

it may be shown that

which is essentially a three-term recurrence formula of the above type. The transition from (8.1.1) to (8.1.2-3) has also a bearing on the topic of quasidifferential equations, in which we minimize the differentiability requirements on the coefficients. Assuming thatp, q E C(a, b), T E C’(a,b), and that T > 0, we may write (8.1.1) as

a “solution” being sought in the class C”(a, b). This is an unnecessarily restrictive procedure ; (8.1.1) retains sense without assuming T to be differentiable, or y to be twice differentiable, if we merely ask that y’/r be continuously differentiable, without attempting to differentiate it as a product. Such an interpretation is included in (8.1.2-3), with the assumptions that p, q, Y E C(a, b), solutions being sought in C’(a,b). We obtain a fairly wide framework if we consider the system (8.1.2-3) with the hypotheses that p, q, T be piecewise continuous, having at most a denumerable number of discontinuities which are simple jumps. This will include both the case of the three-term recurrence formula, and the case of (8.1.1) when the coefficients are continuous. However, the more general case in which the coefficients are integrable in the sense of Lebesgue provides an adequate foundation. I n this chapter it will be assumed that the following hold: (i) p, q, Y E L(a, b), where (a, b) is a finite real interval; (ii) for a x b we have p>o, r>o;

< <

(8.1.4)

204

8.

(iii) for any x, a

< x < b,

l z p ( t ) dt a

STURM-LIOUVILLE THEORY

> 0,

f p ( t ) dt 2

> 0,

~ ( tdt)

> 0;

(8.1.5)

a

(iv) if for some x, ,x2 we have (8.1.6)

then (8.1.7)

In the requirement (i), subject top, 4, Y E L(a, b), the additional restriction that ( a , b) be finite is a matter of convenience. In (iii) the requirements ensure that the system (8.1.2-3) does actually involve the parameter h at its end-points. Slightly greater generality still is to be obtained by replacing (8.1.2-3) by the Stieltjes integral equations

(8.1.8-9)

Basic requirements would be that p, , 4 , , and rl should be of bounded variation, p, and I, being nondecreasing; to simplify matters we could assume them all continuous. The system (8.1.2-3) is a particular case, where p, , q1 , and I, are integrals of p, q, and Y. Conversely, if p, , q1 , and I, satisfy the above requirements, and are in addition absolutely continuous, we may take p, q, and r to be their derivatives and reason back to (8.1.2-3). This procedure is excluded in the event that p, , q1 , or rl contains a singular, not absolutely continuous, component. In what follows we consider the system in differential form (8.1.2-3), subject to the assumptions (i)-(iv) above. A solution will be a pair of absolutely continuous functions u, z, satisfying (8.1.2-3) almost everywhere. The reader who wishes may specialize this situation to that in which p, q, and Y are piecewise continuous, (8.1.2-3) holding everywhere except at discontinuities of p, q, and I, or, more specially still, to that in which p, q, and r are continuous, and (8.1.2-3) hold everywhere.

8.2.

205

EXISTENCE, UNIQUENESS, AND BOUNDS

8.2. Existence, Uniqueness, and Bounds for Solutions We recall the standard fact that(8.1.2-3) has a unique solution forwhich

u(a), v ( a ) have prescribed values. A solution such that u(a) = .(a) = 0, or such that u(x) = V(X) = 0 for any x in [a, b], must necessarily be the trivial solution given by U ( X ) = w(x) = 0.

We need the observation that for a nontrivial solution there holds the inequality u(x) l a dx > 0. (8.2.1) In other words, the equality U(X)

(8.2.2)

lads = 0

must imply that u = D = 0. If p ( x ) is positive and continuous, it is immediate that u = 0, whence if ~ ( x )is also positive we have w = 0, from the first of (8.1.2-3). T o derive the same result under the assumptions of (i)-(iv) of Section 8.1, we may argue in the first place that w must be constant. For it follows from (8.1.3) that ~(x) .(a) = --h

I Z p ( t )u ( t ) dt

-

a

a

q(t) u ( t ) dt,

(8.2.3)

and both the integrals on the right must vanish. In the case of the first such integral we have

which vanishes by (8.2.2). Suppose, if possible, that the last integral in (8.2.3) does not vanish for some x. We then have dt

and hence, for some

K

> 0,

> 0,

1: I

q(t) u(t> I dt

:

Hence there must exist arbitrarily small intervals (xl , x2) such that

8.

206

STURM-LIOUVILLE THEORY

and so, since u ( t ) is continuous and so bounded, such that (8.2.4)

< <

while 1 u ( t ) 1 > K for some t , x1 t x 2 . By taking the interval (xl, x2) small enough, we may ensure that 1 u ( t ) [ > K in (xl , x2), again since u ( t ) is continuous in [a, b]. From (8.2.4) it follows, by (iv) of Section 8.1, that

and since 1 u(t) 1

> &K

in this interval,

in contradiction to (8.2.2). Hence the right of (8.2.3) vanishes and W(X) is constant. To complete the proof, suppose first that this constant is zero. Supposing if possible that u 0, we have from (8.1.2) that u is constant, and so a non-zero constant. In this case (8.2.2) is impossible in view of (8.1.5). Supposing again that w is a non-zero constant, it follows from (8.1.2) that u is monotonic, and does not vanish at both x = a, x = b, in view of the last of (8.1.5). Hence, by the continuity of u, there is an E > 0 such that u(t) has a positive lower bound in at least one of the intervals (a, a E), ( b - E, b), which again conflicts with (8.2.2), in view of (8.1.5). Hence both u, ZI must vanish identically, as was to be proved. For fixed initial values u(a), w(a), let us now consider the dependence of the solution of (8.1.2-3) upon A. It is a standard result that the dependence of the solution on A is analytic for all complex A. Writing the solution u(x, A), w(x, A), we have that these are entire functions of A. Certain conclusions can be drawn from the fact that they are entire functions of order at most More precisely, there hold bounds of the form

+

+

4.

u(x, A), w(x, A) = O{exp (const.

41h I)}.

(8.2.5)

To prove this we use the fact, a consequence of (8.1.2-3), that

8.3.

Since, if I A

207

THE BOUNDARY PROBLEM

1 # 0, 2 I WJI

< {I

A

+I

I I la

1”/1/1A

I 9

we deduce that

I (dldx) log {I A I I la

+I

12}

< d1A I ( r + PI + I Q 1w1A I,

and (8.2.5) follows on integrating with respect to x and taking exponentials.

8.3. The Boundary Problem Preliminary conclusions can now be drawn concerning the eigenvalue problem in which we fix real numbers a, /Iand ask for nontrivial solutions of (8.1.2-3) sych that sin a

= 0,

(8.3.1)

u(b) cosp - w(b) sinp

= 0.

(8.3.2)

u(a) cos a - .(a)

To treat this problem we choose a solution of (8.1.2-3) such that u(a) = sin a,

.(a) = cos a,

(8.3.3)

so that (8.3.1) is satisfied. Writing u(x, A), o(x, A) for this solution, we find that the eigenvalues are then the roots of u(b, A) cos /3 - a(b, A) sin p = 0.

(8.3.4)

We first verify that they are all real. Supposing that h is a complex eigenvalue, we have that will also be an eigenvalue. For since the coefficients p , q and I in (8.1.2-3) are all real-valued, u(x, A) = u(x, A), and similarly for o, so that if (8.3.2) holds for A, it will also hold for A. At this point we need the identity, of Lagrange type,

x

the analog of (4.2.1), which it in fact includes.

208

8.

STURM-LIOUVILLE THEORY

For the proof we note that, by (8.1.2-3),

( W x ) { u ( x , A) v(x, PI

- 4 x 3 P) 4% A)

A ) v(x, P) - 4 x 9

= y+,

- y q x , P) v ( x , 4 =

(A

-

+

P) 4% PI p(.>

4 (PP + 9) 4 x 9 PI

u(x, PI

+,4.

(AP

+ 9) 4 x 9 4

The result (8.3.5) now follows on integration, using the fact that the left of (8.3.5) vanishes when x = a, in view of (8.3.3), which holds for all A. Taking in particular p = 1, we have ~ ( xA) , ~ ( x , - u(x,

A)

A) o(x, A)

=

(A

-

A) r p ( t ) I u(t, A) l2 dt. a

(8.3.6)

Putting x = b, the left-hand side vanishes by the assumed boundary condition (8.3.2), and so ( A - A)

s” p ( t ) I a

u(t, A)

12

dt = 0.

As proved in Section 8.2, the integral can only vanish in the case of the trivial solution, which is excluded by (8.3.1). Hence A must be real. We have the essentials of the proof of the following

Theorem 8.3.1. Subject to the assumptions (i)-(iv) of Section 8.1, the boundary problem (8.1.2-3), (8.3.1-2) has at most a denumerable set of eigenvalues A,, A, , ..., all of which are real, and which are such that (8.3.7)

for every c

> 0. T h e eigenfunctions u(x, A,.)

are orthogonal according to

/ : p ( x ) ~ ( x4) , ~ ( xA,) , dx = 0,

y

# s.

(8.3.8)

The eigenvalues are the zeros of the entire function on the left of (8.3.4). We have just shown that this function does not vanish when A is complex. Hence it does not vanish identically, and hence its zeros form a denumerable set, at most, with no finite limit. They also satisfy (8.3.7), since this function is of order at most %.T h e orthogonality (8.3.8) follows from (8.3.5) on taking x = b, h = A,, p = A,, and using (8.3.2).

8.4.

209

OSCILLATORY PROPERTIES

When we come to the expansion theorem it will be convenient to use normalized versions of the eigenfunctions. If we define (8.3.9) (8.3.10)

we may replace (8.3.8) by the orthonormal relations (8.3.11 )

8.4. Oscillatory Properties Since the boundary problem (8.3.1-2) prescribes the values of the ratio u : v at x = a, b, an important role is played by the discussion of the functions u/v, v/u, and 8 = tan-l(u/v) in respect of their dependence on x and A. T h e functions are also connected in an obvious manner with the zeros of u. T h e dependence of u/v on x is characterized by a Riccati-type differential equation. We assume h real. Theorem 8.4.1. For a nontrivial solution of (8.1.2-3), the functions u/v, v/u satisfy, when finite, the differential equations (u/v)’ = r

+ (XP + 4 )

(w/u)’ = -Y(v/u)a

-

(8.4.1)

(U/v)2,

(Xp

+ 4).

(8.4.2)

In particular, as x increases, u/v cannot tend to zero from above; as x decreases, u/v cannot tend to zero from below. On differentiating u/v, v/u and using (8.1.2-3) we get at once (8.4.1-2). We have only to verify the last statements. Suppose if possible that u/v --t 0 as x increases, say, as x --t x 2 , u/v being positive in a leftneighborhood of x 2 , Then v/u 4 + w as x + x2 - 0. Suppose that v/u is finite for x1 x < x2 . Noting that, by (8.4.2), (vlu)’ -(A$ q), since Y 2 0, and integrating over ( x l , x), where x1 < x < x 2 , we deduce that

<

<

+

210

8.

STURM-LIOUVILLE THEORY

Making x --+ x, , the left tends to fm, since vju + while the right remains finite, since p, q E L(a, b), giving a contradiction. T h e proof of the final statement in the theorem is analogous. The dependence of u/v, vju for fixed x on varying real A is monotonic. , A) = v (a ) fixed. We assume u(a, A) = ~ ( a )v(a, +a),

Theorem 8.4.2. If v ( x , A) # 0, (8.4.3)

while if u(x, A) # 0,

I n particular, for a nontrivial solution, u(b, A)/v(b, A) and v(b, A)/u(b,A) are respectively strictly increasing and strictly decreasing functions of A when finite. We use the result [cf. (4.2.3)]

a

v(x, A) - u(x, A)

ah

a

- u(x, A) ah w(x, A)

=

I zU p ( t ) {u(t,

dt.

(8.4.5)

This follows from (8.3.5) on dividing by (A - p), making p + A, and using 1’Hbpital’s rule. From this (8.4.3-4) follow immediately. As regards the last statement in the theorem, it was shown in Section 8.2 that the integral on the right of (8.4.3-4) is not zero, if the solution u, w is nontrivial, and taking here A to be real and x = b. T o avoid complications with the infinities of u/v, v / u , we introduce the angular variable 8 = tan-l(u/v), or more precisely, 8 = arg {w

+ iu}.

(8.4.6)

We assume in addition that u, v have fixed initial values for x = a, and all A, given by (8.3.3). Since u, v are functions of x, A, so also is 8, and we define initially

qa,A)

= a,

(8.4.7)

in view of (8.3.3). For other x and A, 8(x, A) is given by (8.4.6) except for an arbitrary multiple of 27r, since u and w cannot vanish simultaneously. This multiple of 27r is to be fixed so that 8(x, A) satisfies (8.4.7) and is continuous in x and A. Since the (x, A)-region, namely, a x b, -m < A < 00, is simply-connected, this defines e(x, A) uniquely. T h e following properties of 8(x, A) are contained in previous results.

< <

8.4. OSCILLATORY PROPERTIES

21 1

Theorem 8.4.3. (i) 8(x, A) satisfies the differential equation, with :espect to x, 8’ = r cosz 8 (hp + q) sin2 8. (8.4.8)

+

(ii) As x increases, 8 cannot tend to a multiple of x from above; as x decreases, 8 cannot tend to a multiple of x from below. (iii) As A increases, for fixed x, 8 is nondecreasing; in particular, 8(b, A) is a strictly increasing function of A.

As regards the differential equation (8.4.8) we have from (8.4.6) that 8’= u’v - v’u Y2

+

v2

which gives (8.4.8), since tan 8 = u / v . The statement (ii) follows from the last part of Theorem 8.4.1 and likewise (iii) from the last part of Theorem 8.4.2. From (8.4.6) it is evident that the zeros of ~ ( xA) , are the same as the occasions on which 8(x, A) is a multiple of x . Considering particularly the zeros of u(x, A) for fixed A as x increases from a to b, we see that zeros of u will occur as 8 increases through, or increases to, a multiple of T ; by (ii) of the theorem, it is not possible for 8 to decrease to a multiple of x as x increases. Supposing that 0 a < x , as x increases from a to b, 8 may reach in succession a finite number of the values x , 277, ... . Since it cannot decrease to a multiple of x , it reaches multiples of x in ascending order. It reaches 8 = 0 only insofar as it starts there, and cannot reach negative values at all. It may exceptionally happen that B(x, A) is a multiple of x , nx, say, for some x in an interval in which r = 0. In this case by (8.4.8), d(x, A) = nx throughout this interval; likewise u E 0 throughout this interval. The term zero may occasionally bear the interpretation of an interval of zeros. With this qualification we have

<

Theorem 8.4.4. As A increases, the zeros of u(x, A) move to the left, except for a zero at x = a in the event that a = 0, or for an interval of zeros containing x = a, in the event that r E 0 in a right-neighborhood of a. Suppose first that 0 < a < x , and that for some x’, A’, we have B(x‘, A’) = nx for some positive integral n. By Theorem 8.4.3 (iii) we then have 8(x’, A”) >, nx for all A” > A’. In fact, we must have

212

8.

STURM-LIOUVILLE THEORY

B(x’, A”) > nrr. T o see this we refer back to (8.4.3); since 0 < a < n, we have u(a, A) # 0, and so, by the first of (8.1.5), the integral on the right of (8.4.3) is positive for x = Hence u(x’, h)/v(x‘, A) is a strictly increasing function of A, when finite; from (8.4.4) we see similarly that its reciprocal is strictly decreasing when finite. Hence B(x’, h) is strictly increasing as a function of A, so that B(x’, A”) > nr if A” > A‘. Since B(u, A) = a, where a < n, it follows that there is a root of the equation B(x, A“) = nn such that u < x < as was to be proved. ‘ I t remains to deal with the case that a = 0, or that u(a, A‘) = 0. It will again be sufficient to show that XI.

XI,

> 0, where as before x’ is such that B(x’, A’) = nrr contrary, that = 0.

> 0.

Suppose, on the

Then, as shown in Section 8.2, v ( x ) must be constant in u < x < a non-zero constant since u(a, A‘) = 0 and so v ( a , A’) # 0. However, when 0 = i r r , we have v = 0, and so 0 cannot reach the value i r r , in (a, x’); it therefore cannot reach a value nn > 0 in (a, x’], contrary to hypothesis. This completes the proof of the theorem. We now consider the boundary problem (8.3.1-2), taking it that

XI,

0

<

0.

< n,

0


<

71,

(8.4.9)

and prove the “oscillation theorem,” according to which the eigenvalues may be uniquely associated with the numbers of zeros of the eigenfunctions.

Theorem 8.4.5. The eigenvalues A, of the problem (8.1.2-3), (8.3.1-2) form a sequence A, < A, < ..., possibly finite, such that (8.4.10)

The eigenfunctions u ( x , A,) have, with a suitable interpretation, just n zeros in (a, b). The interpretation in question relates to possible intervals of zeros in the event that r = 0 throughout an interval. Two zeros xl, x2 of u(x, A) such that Jz2 r ( t ) dt = 0 are not to be reckoned as distinct. If 21 either x, = a or x2 = 6 they are not to be counted at all; we confine attention to zeros in the interior of (a, b).

8.4. oscI LLATORY

213

PRO PERTIES

Since the case of finite orthogonal polynomials, whose zeros are eigenvalues of a certain boundary problem, is included in the assumptions of Section 8.1, we cannot assert the existence of an infinity of eigenvalues in Theorem 8.4.5; in degenerate cases there may be none at all. Supplementary conditions, ensuring the existence of an infinity of eigenvalues, will be given in Theorem 8.4.6. T h e result of Theorem 8.4.5 is that if there are any eigenvalues, they can be arranged in increasing order, the corresponding eigenfunctions having 0, 1, ... zeros in the manner just described. Since the function B(b, A) is continuous and strictly increasing in A, the equation (8.4.10) will have exactly one real root A, for a certain sequence of n values. We have to show that the lowest member of this sequence, if nonempty, is n = 0. We prove this by showing that B(b, A) -+ 0 as A --t - w . Noting that B(a, A) = a 2 0, and that as x increases B(x, A) cannot decrease to 0, or decrease from 0, by Theorem 8.4.3 (ii), we have that B(x, A) 2 0 for all real A and a x b. Since also B(x, A) is nondecreasing as a function of A, we have that there exists the limit B(x, - w ) = lim B(x, A) as A --+ and furthermore that B(x, - 0 0 ) 0. We have to prove that B(b, = 0. We have in particular that B(b, A) is bounded for A 0, lying between B(b, 0) and 0. Integrating (8.4.8) over (a, 6 ) we have that

< <

>

--a),

<

--a))

8(b, A) - OL

=

j:

{I

cos2 8

+ hp sin28 + q sin28) dx

is uniformly bounded for A < 0. Since q, Y E L(a, b), it follows that 1 A j J:p sin2 B dx is uniformly bounded for X < 0. We draw the following conclusion, to be used several times in the proof of Theorem 8.4.5. If the interval (xl, x2) is such that f X 2 p ( tdt)

> 0,

(8.4.1 1)

XI

then there is an x 3 , x1

< x3 < x 2 , such that I sin 8 I

< const. I h

I-1l2,

(8.4.12)

where the constant may depend on xl, x2 but not on A, though x3 may vary with A. I n other words, if it is known that sin B is bounded from zero in (xl , x2), for X < 0, independently of A, then we must have j : : p ( t ) dt

= 0.

(8.4.13)

2 14

8.

STURM-LIOUVILLE

THEORY

A second property needed is that O(x, -m) tinuously. Taking h < 0 in (8.4.8), we have 8'

<

1

Integrating over ( x 4 , x5), where x4

can only increase con-

+ I q I.

< x 5 , we have, if h < 0, (8.4.14)

Making h - t

-00,

we deduce that

We proceed to the proof that B(b, -00) = 0. We first observe that e(xt, -m) < Q T for some xt with a xt < b. T o see this we take an a' such that e(x, 0) < T - r] for a x a' and some r] > 0,

<

< <

which is possible since O(a, 0) = a < n, note that r ' p ( t ) dt > 0 by a (8.1.5), and apply the conclusion (8.4.11-12). We see that for large h < 0, [a, a'] contains an x for which sin 0 is arbitrarily small. Since 0 O(x, A) O(x, 0) < T - r ] , this means that 0 is arbitrarily small, for large h < 0 and some x in [a, a']. Hence for large h < 0 there is at any rate an xt for which 0(xt, A) < +T, and so e(xt, -m) < i n , as was to be proved. In the next step we prove that e(x, -a) < Q n for xt < x < b. Let 36 denote the upper bound of xtt b with the property that O(x, -m) < Q T for xt x xtt. Suppose first that x6 < b. We assert that O(xa, -m) = Q n . For if O(xs , -m) < Q n , it would follow from (8.4.15) that O(x, -m) < Q n for x in some right-neighborhood of x,; for this purpose we apply (8.4.15) with x, , x in place of x 4 , x 5 , where x > x,, and is suitably close to x, . Similarly, if O(x,,, -m) > Q T , it would follow that O(x, --a) > Q T for x in some left-neighborhood of x6 , as we see by applying (8.4.15) with x, x,, in place of x 4 , xs and taking x < x6 and suitably close to it. Both of these situations conflict with the definition of x, , and we conclude that O(xs, -00) = Q n . In the event that x, = b, we have O(x,,, -00) &T, since otherwise e(x, -m) > n in a left-neighborhood of x,, . We now show that it is in fact impossible that O(x,, -m) = Q n . Supposing the latter to hold, we choose an x7 < x, such that

<

<

< <

<

<

9

(8.4.16) 27

8.4. OSCILLATORY

215

PROPERTIES

and It then follows from (8.4.15) that O(x,, --) - O(x7, --) < so B(x7, --) > By the same argument, we have in fact 8(x, --) > 47. for x7 x x, , and indeed O(x, A) > for the same x and all real A, since O(x, A) is nondecreasing in A. Since 8(x7, -a) < Q T , by the definition of x 7 , we have O(x7, A) < * T for large negative A, say, A < A‘. By (8.4.14), with x, , x in place of xp , x 5 , it then follows that O(x, A) < $ T for A < A’ and x7 x x, . Hence, for such x and A, we have $ 7 ~< 8(x, A) < $T, and so sin2 8 > 8 . By the argument of (8.4.11-13) we deduce that Jx8p(t)dt = 0, and z7 so also Jxe I q(t) I dt = 0, so that p, q vanish almost everywhere in +7 (x7 , x6). We may therefore replace (8.4.8) in this interval by 0’ = r cos2 8, or (tan 0)’ = Y , whence

ST.

4~,

< <

< <

tan B(x, A)

- tan B(x,,

A) =

lX

~ ( tdt)

,

27

<

for x7 x < . For A < x’, tan 8(x7, A) will be finite, since O(x7 , A). < Q T , and so tan O(x, A) remains finite as x + x, from below. Hence O(x, , A) < Q T for A < A’, giving a contradiction. We deduce that 8(x, .< Q T for xt x b, so that in particular 8(b, --) < i n . Suppose if possible that O(b, = 7’ > 0. Applying (8.4.14-15) as previously, we choose x8 < b such that

< <

--a))

--a))

< <

and deduce that for x, x b and large negative A, say, A < A”, there holds i 7‘ < O(x, A) < & T i q’,so that sin2 8 > sin2 (Q7’)> 0. By the argument of (8.4.11-13), this implies that J b p ( t ) dt = 0, which X8 conflicts with (8.1.5). Hence O(b, = 0, and the proof of Theorem 8.4.5 is complete. Finally, we note conditions which exclude the event that zc(b, A), w(b, A) are polynomials in A and ensure the existence of an infinity of eigenvalues.

+

--a))

Theorem 8.4.6. In addition to the assumptions (i)-(iv) of Section 8.1, let there be an infinite sequence to< <: ... of points of (a, b) such that /::‘p(t)

dt

> 0,

K

= 0, 1,

... ,

(8.4.17)

8.

216

STURM-LIOUVILLE THEORY

and ,:::::~(t) dt

> 0,

k

= 0,

1, ... .

(8.4.18)

Then the problem (8.1.2-3), (8.3.1-2) has an infinity of eigenvalues. For the proof it will be sufficient to show that 8(b, A) becomes arbitrarily large as A -+ f-, or again that as x increases from a to b, 8(x, A) increases through a number of multiples of T which increases indefinitely with A. It will be convenient to prove this instead for a modified phase u / v , where A > 0, the variable 8, = 8,(x, A) defined by tan 8, = arbitrary additive multiple of T being fixed by 18 - I < 8.r. Since tan 8, = tan 8, the two variables 8, 8, will equal a multiple of T , together, and will increase and decrease together. It will thus be sufficient to show that, as x increases from a to b, 8,(x, A) increases through a number of multiples of r which tends to infinity with A. For this purpose we set up the differential equation satisfied by 8,. We have

. 0;

sec2 0,

= W(U/W)’ =X~/~(U’W - UW’)/W~

+ W ( h p + q) (u/w)Z = h112r + A-V((hp + q) tan2 0, . =N 2 T

Hence, for A

> 0, 0;

= All2 T

It follows that, for A

cos28,

+ X112psin20, + h-V

q sina 0,

.

(8.4.19)

> 0, e; 2

-A-V

I I

(8.4.20)

and so, for A 2 1, (8.4.21)

Hence Bl(b, A) - 8,(a, A) is bounded from below, uniformly for A 2 1 ; let us suppose if possible that it is bounded from above, uniformly for A 2 1, say, by

O l ( 4 4 - 0,(a, 4

< c1 -

(8.4.22)

I n showing that this is impossible, we show first that this hypothesis would imply that Bl(x, A) is of bounded variation over (a, b), uniformly for A 3 1. Since the left of (8.4.22) may be written All2

f a

(T

cos2el

+ p sin20,) dx + k 1 1 2 fq sin2el dx a

8.5.

217

AN INTERPOLATORY PROPERTY

we deduce that, for h 2 1, All2

s” a

(I

cos20,

+ p sin20,) dx < c1 +

b

a

I q I dx

= c2 ,

(8.4.23)

say. Hence (8.4.24)

say, so that 8, is of bounded variation uniformly for A 2 1. We now compare (8.4.17) with the fact that, by (8.4.23),

J::yl

p sin2

dx

<

A-112

c2

.

Writing rlzk for the left of (8.4.17), we deduce that sin2

A)

d A-l12 c2/vZk

(8.4.25)

for at any rate one x E [ t Z k52k+,]. , With a similar notation for the left of (8.4.18), we have in the same way cos2

A)

<

~ - 1 1 2Cz/v2r+l

Making A large, we may ensure that

8, is arbitrarily close to a multiple of

.

(8.4.26)

[ t o tl] , contains an x such that and that [el, t2]contains an x

T,

such that 8, is arbitrarily close to an odd multiple of Q T , and so on alternately. Hence by taking h large, the variation of B,(x, A) over (a, 6) can be made as large as we please, and we have a contradiction. Hence B,(b, A) - B,(a, A) can be made arbitrarily large, and B,(x, A) increases through an arbitrarily large number of multiples of T as x goes from a to b, which completes the proof of Theorem 8.4.6.

8.5. An Interpolatory Property In this and the next sections we consider the eigenfunction expansion, the expansion of a function from some general class in a series of the u(x, An), n = 0, 1, ... , in extension of the Fourier sine or cosine series. Of the many possible proofs of this expansion, we select that due to Prufer, which proceeds entirely in the real domain and makes no use of the theory of integral equations, or its equivalents. It rests on an interpolatory property of the eigenfunctions, a special case of a group

218

8.

STURM-LIOUVILLE THEORY

of properties which have interest independently of the eigenfunction expansion. Defining u,(x) by (8.3.9-lo), the property in question is Theorem 8.5.1. Let the boundary problem (8.1.2-3), (8.3.1-2) admit the eigenvalues A,, A, , ..., A, , ... , for some m > 0. Then an expression of the form

4-4 = 2 anun(x), m-1

(8.5.1)

0

where the a, are real and not all zero, cannot vanish at all the zeros of urn(.). We assume that the eigenvalues are arranged in increasing order, and that additional conditions, such as those given by Theorem 8.4.6 have ensured the existence of at least m 1 eigenvalues; it is not, however, necessary at the moment that there should be an infinity of eigenvalues. A more general result, the Cebygev property, due in this case to Sturm, asserts that w(x) as given by (8.5.1) cannot have as many as m zeros; in our present case certain conventions must be set up as to when zeros are regarded as distinct. So far as the expansion theorem is concerned, however, the more restricted result will suffice, that the zeros of w(x) cannot include all the zeros of u,(x) in a < x < b. The proof depends on the following lemma, also needed for the proof of the eigenfunction expansion.

+

Lemma 8.5.2. Let the real-valued absolutely continuous functions g(x), h(x) satisfy g' = rh, (8.5.2) g(a) cos o - h(a)sin

CL

= 0,

g(b) cos

- h(b) sin /3 = 0,

(8.5.3)

and let g vanish at all the zeros of u, in (a, 6). Then (8.5.4)

Completing the notation (8.3.10) for the normalized eigenfunctions, so that we write D,(x) for ~ ( xA,), u:, = TD,

,

v:,

=

-(Amp

+ q) u, .

(8.5.5-6)

8.5.

219

AN INTERPOLATORY PROPERTY

We obtain the required result in a formal way if we integrate (8.5.7) over (a, b), noting that the first term on the right is non-negative, and that integrating the term on the left gives zero, since (glum) (hum - gwm)

(8 S.8)

0

-+

as x -+ a and as x -+ b. T o justify this in more detail we consider the E, ? - z), where 4 , are ~ integral over an interval of the form ( f consecutive zeros of u,, so that u,(f) = urn(?)= 0, u , ~ ( x )# 0 for .$ < x < 9. Since the first term on the right of (8.5.7) is non-negative, we have

+

for small E > 0. We wish to make z --+ 0, and assert that (8.5.8) is also true when x -+ f 0, x -+ 7 - 0, so that the left of (8.5.9) yields zero as E + 0. By hypothesis, we have u, -+ 0, g -+ 0 as x -+ 4 0, and so in order to prove (8.5.8) for x -+ 5 0 it will be sufficient to show that g/u,, is bounded. Since urn([)= 0, g ( e ) = 0 we have

+

+

+

um(f

+

I,

e+r

6)

=

r ( t ) w,(t) dt,

g(f

+

c) =

fyt

Since v,(.$) cannot vanish with urn((),and since v, we have for small e inequalities of the form

so that glum is bounded, and (8.5.8) holds as x --f f

r ( t ) h(t) dt.

, h are continuous,

+ 0. In an entirely

220

8.

STURM-LIOUVILLE THEORY

similar way, the result may be proved for x --+ r ] - 0. Hence making E ---t 0 in (8.5.9) we have

T o complete the proof we observe that this is also true when 6 = a and 7 is the smallest zero of u,,(x) which is greater than a. If the boundary condition at x = a is that u,(a) = 0, that is, if sin a = 0, this has already been proved. If sin a # 0, then glum is finite at x = a, while hum - gv, = 0 at x = a by (8.3.1) and (8.5.3). Hence (8.5.8) is true for x -+ a 0, so that (8.5.10) is available. Similarly, it is available if 7 = b and 6 is the nearest zero of u, to the left of b. We now note that the interval (a, b) comprises a finite number of intervals of the above forms, that is to say, intervals bounded by consecutive zeros of u, or by a zero of u,, and an end-point of (a, b). I n exceptional cases, there may in addition be intervals throughout which u,, vanishes; in terms of the phase variable 8 = B(x, A,) defined in Section 8.4, there will be m 1 intervals in which 8 goes from a to n, from x to 277, and finally from mx to mx 8, and possibly others in which 6 remains a mu1tip)e of n. Intervals of this latter form, in which u, = 0 and so in whichg 3 0, clearly do not contribute to the integrals in (8.5.4). Hence on summing the results (8.5.10) we have

+

+

+

(8.5.1 1)

which is equivalent to (8.5.4), completing the proof of the lemma. Passing to the proof of Theorem 8.5.1, we suppose if possible that w ( x ) as given by (8.5.1) vanishes at all the zeros of u,(x). We apply the result of the lemma, with w in place of g, and w1 in place of h where

We have then w‘ = rwl in view of (8.5.5), while the boundary conditions (8.5.3) hold since they are satisfied by u, , vn . Evaluating for this case the right of (8.5.4), we have

8.5.

AN INTERPOLATORY PROPERTY

22 1

by (8.5.6). Hence the right of (8.5.4) gives

by the orthonormality (8.3.11). In a similar way the left of (8.5.4) becomes b

m-1

a

0

h m j pw2dx = h m z a : .

Hence from (8.5.4) we have

or

Since the A, are in increasing order, this implies that all the a, vanish. This proves Theorem 8.5.1. T h e following interpolatory property follows at once.

Theorem 8.5.3. Let b, , ..., 6 , be any constants, and let xl, ..., x, be zeros of u,(x) which are distinct from each other and from the endpoints a, b, no two such points lying in an interval in which U, = 0. Then there is a unique set of constants a, , ..., am-l such that

2 anun(x,)

m-1 0

=

b, ,

s =

I,

...,m.

(8.5.12)

For if there were not always such a unique set, there would be a set of a , , not all zero, such that

2 anun(x,)

m-1

= 0,

s =

1, ...,m.

(8.5.1 3)

0

Denoting this expression as before by w ( x ) , we should have that w(x) vanished at all the zeros of u,(x). If x = a, or x = b, or both, were zeros of u,, according to the boundary conditions, then these points would also be zeros of w . Any further zeros of u, would not be essentially distinct from 'these, but would lie together with one of the x, , or one

222

8.

STURM-LIOUVILLE THEORY

of a or b, in an interval in which u, = 0 and so in which Y = 0 almost everywhere; however, in such an interval all the un would be constant, and so also w, which accordingly would vanish throughout such an interval. Hence w would vanish at all the zeros of u, contrary to Theorem 8.5.1. The criterion for the zeros x, and the end-points a , b to be distinct in the above sense may be put explicitly as Y l ( 4

< +1)

< ..-< T l ( X r n ) < Y l ( h

(8.5.14)

where as previously rl(x) = J" ~ ( tdt. )

8.6. The Eigenfunction Expansion

The interpolation theorem just proved may be stated in the form that, given any function ~ ( x )a, x b, and any m, we can find a linear combination of uo(x),..., U , - ~ ( X ) which coincides with it at the zeros of u,(x) in a < x < b ; strictly speaking, the zeros should be distinct from each other and from the end-points, and there must of course be at least m 1 eigenvalues. This is already a form of expansion theorem. Furthermore, making m + m and assuming that there are an infinity of eigenvalues, we obtain approximations which are correct at a larger and larger number of points in ( a , b). It was shown by Prufer that there exists a rigorous argument leading from the interpolatory property to the eigenfunction expansion. If the expansion

< <

+

W

dx)=

Glun(x)

(8.6.1)

0

holds, with say absolute and uniform convergence, the coefficients may be found by multiplying by p ( x ) un(x) and integrating over (a, b). By the orthonormal property (8.3.11) this yields (8.6.2)

The first step is to establish the validity of the expansion in meansquare, with respect to the measure p ( x ) dx, in the sense that (8.6.3)

8.6. as m

+ 03.

223

THE EIGENFUNCTION EXPANSION

As follows from (8.6.2), (8.3.11), this may also be written (8.6.4)

as m + m, or, what is the same thing, (8.6.5)

the Parseval equality. Having established the expansion in the meansquare sense (8.6.3), improvements of two kinds may be undertaken. It may be possible to show, often by less delicate arguments, that the expansion actually converges in the uniform sense; its uniform limit must then also be cp(x), at least when p ( x ) is positive and continuous. In another direction, it may be possible to show that the class of v(x) originally considered are dense in the mean-square sense, in some larger space, and so to extend the validity of the mean-square result (8.6.3). The central result is

< < b,

Theorem 8.6.1. Let functions cp, t,b, x be defined in a x absolutely continuous with derivatives satisfying

v and t,b being

v’ = 4,

#’

+ 9v = Px,

where p1I2x is of integrable square over (a,b). Let also &heboundary conditions v(a)cos OL - #(a) sin OL p ( b ) cos 9, - #(b) sin ,9

(8.6.6-7)

v, $

satisfy

= 0,

(8.6.8)

= 0.

(8.6.9)

Then the expansion (8.6.1) is true in the mean-square sense (8.6.3-5). We shall confine attention to the event that there are actually an infinity of eigenvalues. Sufficient conditions for this were noted in Theorem 8.4.6; it is, for example, sufficient that there be an interval in (u, b) in which p , r are both continuous and positive. The contrary situation was considered in Chapter 4. For the proof we take zeros xl, ..., x, of u,,,(x) in (a, b), distinct from each other and from the end-points a, b in the sense (8.5.14); this means that in the event of a whole interval of zeros, in which r = 0. we take only one zero from this interval. We form the interpolatory ) these points, choosing the an so that approximation to ~ ( x at m-1

~ a , U , ( x , )= P(%), n=O

s = 1, * * * I m,

(8.6.10)

224

8.

STURM-LIOUVILLE THEORY

which is possible by Theorem 8.5.3, and define the difference

2

m-1

g(x) = d x ) -

0

W n ( 4 .

(8.6.11)

The required result (8.6.4) is then obtained by applying Lemma 8.5.2 to the function g(x). T o complete the formalism of Lemma 8.5.2, we note that (8.5.2) holds where h(x) is given by (8.6.12)

by (8.5.5), (8.6.6); the h(x) so defined is absolutely continuous, and together with g(x) satisfies the boundary conditions (8.5.3), in view of (8.6.8-9)and the boundary conditions(8.3.1-2) of the eigenvalue problem. Next we note that g(x) vanishes with u,(x). This is obvious in the case of isolated zeros x, of unL(x),or again in the case of zeros at x = u, b of uJx), prescribed by the boundary conditions. For the case when u,,(x) has an interval of zeros, in which necessarily I = 0 or r1 is constant, and containing one representative zero x, , we note that g(x) will also be constant throughout this interval, by (8.5.2), and so will vanish throughout. It remains to substitute for g, h in (8.5.4) according to (8.6.11-12) and to evaluate the result. In the following calculations, sums will, unless otherwise indicated, be from 0 to m - 1, integrals and variations from x = a to x = b. On the left of (8.5.4) we get

Turning to the right of (8.5.4), we note first that

8.6.

THE EIGENFUNCTION EXPANSION

225

Hence the right of (8.5.4) gives

In order to evaluate this we have to calculate integrals of the form Ju,px dx. Using (8.5.5-6), (8.6.6-7) and integration by parts we have

where in setting the integrated term equal to zero we have relied on the boundary conditions. Hence (8.6.14) may be written

= -I

p ~ dx x - 2hncE

+ 2 A , , ( c n - an)z.

(8.6.16)

The result (8.5.4) now states that the expression (8.6.13) does not exceed the expression (8.6.16). On slight rearrangement this gives

Since the A, are in increasing order, the last sum is non-negative and may be omitted, as also the first sum on the right; this yields the main result (8.6.17)

226

8.

STURM-LIOUVILLE THEORY

The desired conclusion, that the expression in the braces {} on the left tends to zero as m 400, now follows, provided that there is an infinity of eigenvalues. This proves (8.6.4), and its equivalents (8.6.3) and (8.6.5). For a later purpose we note that (8.6.18)

This comes from applying the Bessel inequality to -x, whose Fourier coefficients are h,c, , by (8.6.15). This proves (8.6.4), and its equivalents (8.6.3) and (8.6.5). We show later that the expansion is uniformly and absolutely convergent under the same assumptions; the proof of this will depend on the Green’s function, considered in Sections 8.8-9.

8.7. Second-Order Equation with Discantinuities By way of illustration we formulate the oscillation and expansion theorems for the special case of a second-order differential equation d2YldP

+ PP(5) + d03r

(8.7.1)

= 0,

to hold in (0, 1) except at a finite number of points where discontinuities in y’ are prescribed, the change in y’ being proportional to y . Let 8, be such that 0 = to< 5, < ... < 5, < tm+,= 1, let p , q be continuous in each interval [t,, 5,+,], and let p be positive. Let y satisfy tnfl),be continuous at each t,, the (8.7.1) in each interval discontinuity in y’ at 5, being specified by

(en,

+ 0)-

~ ’ ( t n

0) = -(AP(n)

~ ~ ’ (t n

+ q‘n’)y(En),

1

< n d m,

(8.7.2)

the pen), q(,) being constants, the pcn1 > 0. If for simplicity we take as boundary conditions

the oscillation theorem will assert that there is an infinity of eigenvalues, all real, and forming an increasing sequence with no finite limit, corresponding eigenfunctions having 0, 1, 2, ... zeros in 0 < < 1. We may

8.7.

SECOND-ORDER EQUATION WITH 'DISCONTINUITIES

227

derive this from Theorems 8.4.5-6 by considering the first-order system for u(x), v ( x ) given by

ul=v,

11'

= v,

0'

=

-[Ap(x)

11'

= 0,

0'

=

-(Ap

v'=

(nl

+ q(x)] + q'")) u,

11,

-[Ap(x-l)+q(~-l)]u,

< x < 61 , El < x < tl + 1, 0

&+1
and so on, terminating with

(8.7.4) (8.7.5) (8.7.6)

the boundary conditions being

It is readily seen that this system has the same eigenvalues as (8.7.1-3), and that zeros of y correspond to zeros of u, with the qualification n - 1, n), and that u is constant in intervals of the form (& so may exceptionally have an interval of zeros. Let now the eigenfunctions of (8.7.1-3) corresponding to eigenvalues A, be written y,(x). They will be orthogonal, and we may suppose them normalized by multiplication by constant factors, according to

+

en +

for j , K = 0, 1, ...; this is the same as (8.3.11) in our present case. The formalities of the expansion will be that for a function z(5)from some general class we define the Fourier coefficient

and consider the validity of the expansion

Specializing the result of Theorem 8.6.1 to the continuous domain, let us assume that z(6) is continuous in [0, 13, and that it is continuously twice differentiable in each of the intervals [t,, the derivatives

228

8.

STURM-LIOUVILLE THEORY

at the end-points of these intervals being one-sided derivatives; it is not assumed that the left and right derivatives at tl,..., tmcoincide. Assume finally that the boundary conditions (8.7.3) hold, that is, that z(0) = z(1) = 0. Then (8.7.1 1) is true at any rate in the mean-square sense. In the present case (8.6.3) becomes the following. Write (8.7.12)

Then, as m

--+

00,

(8.7.13)

In particular, if pen) > 0, we see that z,(gn) --+ 0 as m --+ 00, so that (8.7.11) is true in the ordinary sense for ( = g,, , mean-square convergence implying ordinary convergence. We may, of course, extend the boundary conditions to y(1) cos /? - y'(1) sin /? = 0.

y(0) cos a - y'(0) sin a = 0,

(8.7.14-15)

More generally we may superimpose a discontinuity of the form (8.7.2) upon the boundary data. Taking (8.7.14) in the form y(0) cot (Y = y'(0) and applying (8.7.2) with n = 0, the effective boundary condition at E=Ois y'(O+) = y(0) {cot a - Ap'O' - p), (8.7.16) where actually the q ( O ) is redundant. Applying similar considerations to the other end of the interval, we are led to the boundary problem formed by (8.7.1-2) together with ~ ' ( 0 )= ~ ( 0(cot )

- Ap'O)),

~ ' ( 1 )= ~ ( 1(Cot ) /?

+ Ap(m+l)),

(8.7.17-18)

where the constants p(O),$cm+l) are non-negative. T h e main effect of these boundary conditions will be that the sums in (8.7.9-lo), (8.7.13) must now be taken over 0, m 1. I n the case q(() = 0, qcn) = 0, we may interpret the equation considered here as that of a string of density $( (), loaded additionally with particles of masses pen). As already explained in Section 0.8, the case in which the string has a particle at each end, a weightless portion of string being fixed to each end so that the system can vibrate, leads to a boundary problem with the eigenvalue parameter appearing in the boundary conditions.

+

8.8.

THE GREEN’S FUNCTION

229

8.8. The Green’s Function

Returning to the general theory of the boundary problem (8.1.2-3), (8.3.1-2), we give the analog, indeed extension, of the results of Section 6.4. We start with the inhomogeneous problem, in which we suppose given a function x ( x ) E L (u ,b), and ask for absolutely continuous functions 97, $ satisfying the differential equations $‘

9J’ = y$v

+ (AP + 4)9J = x,

(8.8.1-2)

and the boundary conditions (8.6.8-9). Provided that A is not an eigenvalue, we show that (8.8.1-2) has a unique solution which is expressed by

so far as ‘p is concerned; this corresponds to (6.4.2). T h e kernel G(x, t , A) is the Green’s function which provides two of the main proofs of the eigenfunction expansion. Here we use it to establish the uniform and absolute convergence of this expansion, under the conditions of Section 8.6. I n addition, it has sign-definite properties which may be used to prove separation theorems, as was done in Section 6.3. Our first task is to establish the existence of Green’s function and to find it explicitly. This we do by solving (8.8.1-2), together with the boundary conditions, by the method of the variation of parameters. In addition to the solutions u ( x ) = u(x, A), and w(x) = w(x, A) of (8.1.2-3) fixed by the initial conditions (8.3.3), we define a second pair of solutions by means of the terminal conditions ut(b) = sin 8,

d ( b ) = cos 8,

(8.8.4)

corresponding to the solution w,(A) of the recurrence relation, defined in (6.1.12-13). Provided that A is not an eigenvalue, ut and w t will not be merely a constant multiple of u and v. Their functional determinant or Wronskian will be written w = w(A) = u(x) d ( x ) - ut(x) .(X) =

u(b) cos 8 - o(b)sin 8.

(8.8.5) (8.8.6)

It follows from the differential equations (8.1.2-3) that the Wronskian appearing in (8.8.5) is independent of x for a x b; putting x = b, we get the left of (8.3.4), whose vanishing determines the eigenvalues. With these notations we can specify the Green’s function.

< <

230

8.

STURM-LIOUVILLE THEORY

Theorem 8.8.1. If x EL(u,b) and X is not an eigenvalue, the unique solution of the inhomogeneous system (8.8.1-2) subject to the boundary conditions (8.6.8-9) has q~ given by (8.8.3), where (8.8.7)

The last formulas correspond, in the discrete case, to (6.4.5-6). We have omitted for brevity the dependence on X in u, ut. Following the method of the variation of parameters, we seek solutions of (8.8.1-2) in the form ql

where s

= 3-24

= s(x), s t = st(x)

+ stut,

*

=m

+ stof,

(8.8.8-9)

are to be found. The boundary conditions

(8.6.8-9) will be satisfied if we choose s(b)

= 0,

St(U)

= 0,

(8.8.10-11)

in view. of the initial values of tl, v and the final values of at, vt. It remains to ensure that (8.8.1-2) hold. We must first have (su

+ stut)’ = r(sw + StWt),

and since u‘ = rv, ut’ = r v t , we must have s‘u

+ st’ut = 0.

(8.8.12)

For (8.8.2) we must have

and since

‘u’

=

-(Xp

+ q) u, v t ’ = -(A9 + q) u t , this implies that slw

+ St’Wt = x.

(8.8.13)

Using (8.8.5) we deduce that St’ = ux/w.

s( = -utx/w,

From (8.8.10-11) it now follows that

5

b

S(X) =

2

u t ( t ) ~ ( tw-I ) dt,

st(x> =

5’

a

u ( t ) ~ ( tw-l )

dt. (8.8.14-15)

Substituting in (8.8.8) we get (8.8.3), with G given by (8.8.7).

8.8.

23 1

THE GREEN'S FUNCTION

T o verify the solution we set up the corresponding expression for

#(x). Substituting for s, st in (8.8.9) we obtain

the full expression for

being

).(.I I

b

~(x= ) w-l

2

u t ( t ) ~ ( tdt)

+ ut(x) /' u ( t ) ~ ( tdt) 1 . a

(8.8.17)

It is a routine matter to verify that these satisfy (8.8.1-2) and the boundary conditions (8.6.8-9). The uniqueness of the solution of (8.8.1-2), (8.6.8-9) follows from the fact that if there were two solutions, their difference would satisfy (8.1.2-3) and (8.3.1-2), so that either this difference would vanish or h would be an eigenvalue. The following formal properties of the Green's function are more or less immediate. Theorem 8.8.2. If h is not an eigenvalue, (i) the Green's function is symmetric, that is, G(x, t , 4 = GO, x, 4,

(8.8.18)

(ii) it is continuous in x, t jointly, and absolutely continuous in x, for fixed t, or in t for fixed x; (iii) its partial derivatives have discontinuities when x = t, according to

a

I A) I

- G(x, t , A) ax

a

- G(x, t , at

?-.t+O

t==z+o

a

- - G(x, t , A) ax -

a at G(x, t , A)

I

=~(t),

a

< t < b,

(8.8.19)

= Y(x),

a

< x < 6;

(8.8.20)

2-t-0

I

t-2-0

the pair G, H satisfy in either x or t the differential equations (8.1.2-3) when x f t, and the -boundary conditions (8.3.1-2); (v) for any eigenfunction u,,(x) we have (8.8.22)

232

8. STURM-LIOUVILLE THEORY

(vi) if p is also not an eigenvalue, there holds the resolvent equation G(x, t , A) - G(x, t , P ) = (P - 4

I

b a

G(x,

5, P ) G(5,t , A) p ( 5 ) d5.

(8.8.23)

Of these (i)-(iv) need only a straightforward verification. For (v), we rewrite (8.5.5-6) as u:, = rvn

,

4

+ (Ap + 4)

= (A - h,)pu,

% I

-

Comparing this with (8.8.1-3) we deduce that

I

b

u n ( ~ )=

a

G(x, t , A) (A - h ) p ( t )un(t)4

which is the required result. For (vi) we consider (8.8.1-2) for fixed x and varying A, writing the solutions of (8.8.1-2) and (8.6.8-9) F ~+ a,. We have then

d = .+a

9

+;

+ (+ + 4)pa =

XI

and the latter may be rewritten as

tfii + (PP + q) Fa = (P - X)Pcpa + X. Hence

may be expressed both in the form (8.8.3) and also in the form

j: G(x, 4 PI {(P - 4m v,n(t)+ X ( t ) ) dt =

s”

G(x, 1, P ) X ( t ) dt

+ (P - 4 J

b

G(x, 5, P ) P ( 5 ) d5

s”

(8.8.24) G(5, t, 4 X(t) dt,

on substituting for ~ ~ on ( tthe ) basis of (8.8.3). Since (8.8.3) and (8.8.24) are the same for all continuous functions x, and since G is continuous, there must hold the identity (8.8.23).

8.9. Convergence of the Eigenfunction Expansion We now use the Green’s function to establish the absolute convergence of the eigenfunction expansion under the conditions of Section 8.6; we shall also consider the uniformity of the convergence, in regard to x and in regard to varying boundary conditions. At the center of these investigations is the Fourier expansion of the Green’s function G(x, t, A)

8.9.

233

CONVERGENCE OF EIGENFUNCTION EXPANSION

in a series of the u,(t), taking x fixed. Taking it that h is not an eigenvalue, the Fourier coefficients are given by (8.8.22) as n = 0, 1, ...,

un(x>

h -An'

(8.9.1)

and so we have the formal Fourier expansion, the bilinear formula (8.9.2)

Without considering, in the first place, whether this formula is true, in either the pointwise or the uniform or the mean-square sense, we base ourselves on the Bessel inequality (8.9.3)

J a

I n particular, taking h = i, (8.9.4)

and so (8.9.5)

for some c independent of x ; for it follows from (8.8.7) that G(x, t, i) is bounded, uniformly in x and t, so that the right of (8.9.4) is bounded, uniformly in x. We now strengthen the result of Section 8.6. We have Theorem 8.9.1. Let q(x) satisfy the assumptions of Theorem 8.6.1. Then the eigenfunction expansion (8.6.1) is true, the series on the right being absolutely and uniformly convergent for a x b. We first prove that the series in (8.6.1) is absolutely and uniformly convergent, and then consider whether its sum is equal to ~ ( x ) .For any integers m,m' with 0 < m < m' we have

< <

(8.9.8)

8.

234

STURM-LIOUVILLE THEORY

by (8.9.5). In view of (8.6.18-19), the last sum tends to zero as m + 00, m'+ 00 independently of x. This proves the absolute and uniform convergence. It follows that the sum

2 03

Vl(4 =

(8.9.9)

GLun(x)

is continuous; we have to show that it is the same as p(x). Making m + in (8.6.3), we have that

(8.9.10) In the special case in which p ( x ) is positive and continuous in [a, b ] , it follows at once that V(X) = V&) (8.9.11) for all x in [a, b]. More generally, suppose first that x E (a, b ) is a point of increase of p l ( x ) = J"p(t) dt, that is to say, that

r+f

p(t) dt

x-

for arbitrarily small E > 0 so that for x - 6

E

s"" x-f

> 0. Since V,

I V(t> - P d t ) I 2


E,

>0

(8.9.12)

f

p1 are continuous, we may choose

*

IV(4

-Vl(4

I 9

and so that also (8.9.12) holds. This gives

P(t) I P(t) - V l W

l2 dt 2 I +{V(X)

-Vl(4)

l2 p x-f + ' P ( t ) dt,

whence, if p(x) # A x ) ,

rf 2- f

P(t) I VP)

- Vdt)

lP

dt

> 0,

which is impossible by (8.9.10). Thus (8.9.11) holds for all x such that (8.9.12) holds for arbitrarily small E > 0. In view of (8.1.5), an entirely similar argument shows that (8.9.12) holds at x = a and at x = b. Suppose now that (xl, x2) is an interval in which pl(x) is constant, and that it is not contained in any larger such interval. We have therefore (8.9.13)

8.9. that

Q

CONVERGENCE OF EIGENFUNCTION EXPANSION

235

< x1 < x p < b, by (8.1.5), and furthermore that = Vl@l),

&l)

94x2)

= Vl(XZ),

(8.9.14)

since (8.9.12) cannot hold when x = xl, x = x 2 . By (8.1.6-7) we have = q = 0 almost everywhere in (xl,xz), and so the w, are constant in (xl, xz), by (8.5.6) and likewise I/J by (8.6.7). By (8.6.6) we have then

p

d x ) = dXl)

+

$(XI)

42

and from (8.5.5) u,(x) = un(xl)

+ w,(xl)

r ( t ) dt,

x1

Q x d x2 ,

(8.9.15)

21

s2

r ( t ) dt,

21

x1

< x < x2.

(8.9.16)

If J2*r ( t ) dt = 0,that is to say, if r(t) = 0 almost everywhere in (xl , xz), 21 then q~ is constant in (xl, x2), and likewise the u, and so also vl. Hence it follows from (8.9.14) that p)(x) = tpl(x) in (xl, x2). If again Jx2 r ( t ) dt > 0,we have +1

the last series necessarily converging, since ql(x2) is finite. Comparing this with (8.9.15) with x = x2 and using (8.9.14) we deduce that

However, by the argument just used.

<

for x1 x < x 2 , using (8.9.14). Comparing this with (8.9.15) we have (8.9.11) for x1 < x < x2 , and so, together with the previous results, it holds generally. Hence the eigenfunction is valid in the sense of pointwise convergence, completing the proof of the theorem.

8.

236

STURM-LIOUVILLE THEORY

We proceed to a partial justification of the bilinear formula (8.9.2), and to the uniformity of the convergence in regard to the boundary conditions. We commence with the expansion of the iterated Green’s function appearing on the right of (8.8.23).

Theorem 8.9.2. Let A, p be not eigenvalues. Then

the series on the right being uniformly and absolutely convergent in a
<

< <

where x satisfies the assumptions of Theorem 8.6.1, then ‘p can be expanded in a uniformly and absolutely convergent series (8.6.l), since the assumptions (8.6.6-9) concerning ‘p(x) are equivalent to a representation in the form (8.9.18), this being the basic property of the Green’s function. In particular, it is sufficient that x(() be continuous, and we may therefore apply the result with ~ ( 5 = ) G((,t, A); we also replace X by p in (8.9.18). It follows that (8.9.19)

with absolute and uniform convergence for any fixed 1. The coefficients ck are given by

= (p -A n y (A

- A,)-l

U&),

by two applications of (8.8.22), using the symmetry property (8.8.18). Hence we have (8.9.17) with pointwise convergence, and with absolute and uniform convergence for fixed t and varying x, a x b. In order to obtain the convergence uniformly in x and t we need the following theorem of Dini, which we cite as

< <

8.9.

237

CONVERGENCE OF EIGENFUNCTION EXPANSION

Lemma 8.9.3. Let the series Z: g,(x) of non-negative functions g,(x), continuous on some closed compact set S, converge and have a sum s(x) which is also continuous on S. Then the series converges uniformly on S. To indicate the proof, suppose that the result is untrue, so that there is an E > 0 and a sequence n1 < n2 < ... of positive integers and an associated sequence x k E S such that zckgrL(xk)> e. Here, by the compactness of S , we may take it that the sequence x k has a limit xo E S. Writing s,(x) = go(.) ... g,.-l(x), choose n‘ such that

+ +

1 4x0)’ - %@o) I < 6/39 and a 8

> 0 such that

I ~ ( x k) Snr(Xk) 1

< I ~(xb)-

sn*(xk)

I

< I S b k ) - 4x0) I + 1 4x0) - %@o) I + I %@o)

-S

n h J

I<

€9

so that Zzkg,(xk) < c. This gives a contradiction, proving Dini’s theorem. While we have in mind first the case in which S consists of an interval on the real line, we use later the case in which it is a plane point set. If in (8.9.17) we take t = x, and p = A, not being an eigenvalue, we have

Here the terms on the right are non-negative, while the sum on the left is continuous in x; the latter may be seen more clearly by transforming the left of (8.9.20) by use of (8.8.23), when it becomes, if A is complex,

(A

- X)-l {G(x, t , A)

-

Hence by Dini’s theorem the series on the right of (8.9.20-21) is uniformly convergent for a x b. The statement that the series in (8.9.17) is uniformly convergent in x and t jointly now follows by means of the Cauchy inequality.

< <

238

8.

STURM-LIOUVILLE THEORY

The result (8.9.21) may be put as

Theorem 8.9.4. If A is complex, the bilinear formula (8.9.2) holds when we take imaginary parts of both sides. An inessential modification of the above arguments gives Theorem 8.9.5.

The series

<

converges uniformly, for fixed complex A, for all a x < b and real a, j3 appearing in the boundary conditions. For the left of (8.9.21) is easily seen to be continuous in x, a, and j3, being periodic in a and j3. Finally we have as a consequence

Theorem 8.9.6. Let ~ ( x )satisfy the assumptions of Theorem 8.6.1 for given a and all B, so that ~ ( b = ) 0, $(b) = 0. Then the eigenfunction expansion (8.6.1) is convergent uniformly in x and j3. Taking A = i in (8.9.21), the left is continuous in x and j3, and so the conditions of Lemma 8.9.3 are satisfied, the set S now being a x b, 0 j3 27r. Hence the series in (8.9.5) converges uniformly in x and B. We now employ the argument of (8.9.6-7) in the sense that the first factor in (8.9.7) is bounded, by (8.6.18), while the second tends to zero as m, m14 m, uniformly in x and j3, by the uniformity of the convergence of the series (8.9.5).

< <

< <

8.10. Spectral Functions For investigations in which the eigenvalue problem is varied at the end x = b of the basic interval, it is convenient to have the expansion theorem in terms of eigenfunctions with fixed initial values at x = a. We therefore express the eigenfunction expansion in terms of the functions u(x, A,), where as in Section 8.3 we have u(a, A,) = sin a, w(a, A,) = cos a. In terms of the normalized eigenfunctions uJx) the expansion theorem states that (8.10.1)

8.10.

SPECTRAL FUNCTIONS

239

this series being absolutely and uniformly convergent under the conditions of Theorem 8.6.1, as was proved in Theorem 8.9.1. Since un(x) = u(x, A,) where p, is given by (8.3.9), (8.10.1) is equivalent to (8.10.2)

As on previous occasions, we may put this into Stieltjes integral form by defining the spectral function (8.10.3) .(A) =

-2

a
pi1,

h

< 0,

(8.10.4)

with the understanding that ~ ( 0 = ) 0; .(A) will of course be a nondecreasing right-continuous step function. The eigenfunction expansion (8.6.1-2) may then be put in the form of a pair of reciprocal integral transforms. Defining, as a sort of generalized Fourier coefficient,

With a view to manipulations concerning more than one spectral function, we collect here estimates relating to the convergence of the eigenfunction expansion (8.10.6). We write, for A > 0, =

/

A

-A

u(x,

dT(h) Y(h).

(8.10.7)

Assuming cp to satisfy the assumptions of Theorem 8.6.1, we have first that the expansion is valid in the mean-square sense given by (8.6.3) or (8.6.4). In the present notation these results may be written respectively as (8.10.8)

or

240

8.

STURM-LIOUVILLE THEORY

by (8.3.10) and (8.6.2). Thus, taking first (8.10.9), (8.10.10)

Since the A, are bounded from below, and A, < A, < ..., we may drop the restriction - A < A, if - A < A, , and the left of (8.10.9) may then be written (8.10.11)

which tends to zero as A -+ m, by (8.6.4). Thus (8.10.9) holds. In a similar way, if - A < A,, the left of (8.10.8) is the same as (8.10.12)

which tends to zero as A + m, by (8.6.3). This proves (8.10.8). Next we replace (8.10.9) by a bound for the left-hand side. By (8.10.9) and (8.10.10), or by the Parseval equality (8.6.5), the left of (8.10.9) is the same as

In view of (8.6.18) we have that a

p ( x ) I ~ ( x l2) dx -

(" I y(A) l2 d+) < -A

s"

p ( x ) I x(x)

la

dx.

(8.10.13)

a

We use this bound later for the limiting transition b + m. 8.1 1. Explicit Expansion Theorem

We shall now prove the analog of Theorems 4.9.1 and 7.7.1. In the latter results we proved that polynomials orthogonal on the real axis or on the unit circle were orthogonal with respect to a weight function given explicitly in terms of the polynomials themselves ; the orthogonality applied to a finite number of the polynomials, and could have been expressed as an expansion theorem on the lines of (4.4.5-7). In the present Sturm-Liouville case, the place of a sequence of polynomials of degrees 0, 1, 2, ... , is taken by the functions u(x, A), where in place

8.1 1.

24 1

EXPLICIT EXPANSION THEOREM

of the degree of the polynomial we have the continuous variable x . These are orthogonal with respect to integration over A only in a rather questionable sense, and we use here instead the formulation as an expansion theorem. T h e ordinary expansion theorem involves the determination of eigenvalues, which are in general the roots of transcendental equations. We use the term expZicit to describe the result of the present section, since it is expressible directly in terms of solutions of the differential equations.

Theorem 8.11.1. Let p)(x) satisfy the assumptions of Theorem 8.6.1 for all j3, that is to say, we have p)(b) = $(b) = 0. Then (8.1 1.1)

where y(A) is the extended Fourier coefficient defined in (8.10.5). T h e proof is similar to that of Theorem 4.9.1, and proceeds by averaging the ordinary eigenfunction expansion with respect to the angle j3 determining the boundary condition at x = b, the condition at x = a remaining fixed. We write

P(4

J p(t) b

=

U2(t,

4 dt,

(8.11.2)

where u2(t,A) denotes {u(t, A)}2, so that in the notation (8.3.9) we have

p n = p(An). T h e expansion (8.10.2) is then

(8.11.3)

Considering A, as a function of j3, namely, the root of B(b, A,) = j3 we propose to calculate dAn/dj3, that is to say, the value of

+ m,

pep, ~ y a ~ } - l when A

=

A,.

Now by (8.4.6) we have

-awl A) - {un(b, A) w(b, A) - u(b, A) W I ( b , A)}

ax

{U2(b,

A)

+ W y b , A)}-l.

By (8.4.5) this gives -ae(b9A )

ax

- p(A) ( U Z ( b , A)

+ "2(b, A)}-1,

(8.1 1.4)

242

8.

STURM-LIOUVILLE THEORY

whence (8.1 1.5)

Hence the eigenfunction expansion (8.1 1.3) may be written (8.11.6)

To complete the proof of the theorem we integrate with respect to of course, the left of (8.11.1). The series on the right of (8.11.6) is uniformly convergent, by Theorem 8.9.6, and may therefore be integrated term by term, so that we get

fl over (0, x ) . The left of (8.11.6) gives,

u(x, A) y(A) {u2(b,A)

where we have written A, = A,@?),

+ w2(b, A)}-'

dh,

(8.1 1.7)

and h,(+O) in place of h,(O) since

fl was restricted in (8.4.9) to 0 < fl < T . Since X,(fl) is monotonic increasing in fl, and since every finite real h is an eigenvalue for some fl,

the sum on the right of (8.11.7) adds up' to the integral over the real axis appearing in (8.11.1). T o be precise, as fl --t +0, A,@) ---t -00, since it was proved in Section 8.4 that d(b, A) tends to zero from above as h ---t --. Hence the first term in the series in (8.11.7) gives the integral in (8.1 1.7) over (--, h0(r)).The remaining terms on the right of (8.11.7) give the integrals over (h,-l(x), A,(T)), n = 1, 2, ... , since, as is easily seen, X,(+O) = hn-l(x). This completes the proof. The result remains in force if there are only a finite number of eigenvalues, but is then equivalent to Theorem 4.9.1. By applying the same process to the Parseval equality associated with the ordinary eigenfunction expansion we get Theorem 8.11.2. x

Under the assumptions of Theorem 8.11.1,

/ l p ( x ) I cp(x) l2 dx

=

--m

I y(A) l2 { ~ 2 ( bA),

+ ~ 2 ( bA)}-l ,

dh.

(8.1 1.8)

We use (8.10.13), of which the left-hand side may be written, with the notation (8.11.2),

8.12.

EXPANSIONS OVER A HALF-AXIS

243

or, in view of (8.11.5),

Integrating (8.10.13) with respect to

over (0,r) thus gives

< 7rA-* J ~ ( x I)X(X) la dx. 1,

a

and (8.1 1.8) follows on making

(i --f

(8.11.9)

00.

8.12. Expansions over a Half-Axis In this section we apply the limiting transition b to the eigenfunction expansion, in Parseval equality form, keeping fixed a and the boundary condition at x = a. This situation is analogous to that in which we have an expansion theorem associated with a finite sequence of recurrence relations, and consider the effect on this theorem of increasing without limit the number of stages in the set of recurrence formulas; particular cases of this process were undertaken in Sections 2.3, 5.2, and 7.3. Once more, the simplest procedure is to show that the spectral function T(A) is bounded, for fixed A, as b --+ 00, and to use the Helly-Bray theorems. The argument is adapted only to the proof of the existence of at least one spectral function, in the limiting sense, and does not touch on the question of uniqueness. We assume in this section that the assumptions (i)-(iv) of Section 8.1 hold for a sequence of intervals (a, b), where a is fixed and b = b, , b, , ... , where b, as m --t 00. We now write T ~ , ~ ( Afor ) the step function defined by (8.10.34). Our first step is to prove its boundedness. We have Theorem 8.12.1. There is a function c(h) independent of b = b , , m = 1,2, ... , such that

and of

The proof proceeds by applying the Bessel inequality to a function which is initially unity in some small interval and thereafter is zero. In the above-mentioned discrete cases a similar argument was used,

244

8.

STURM-LIOUVILLE THEORY

relying on certain Parseval equalities. In the present case we take a function rp&) = 1 ( a < x < a’) (8.12.2) =0

(x

2

a’),

where a‘ = a’(h) is to be chosen later. With respect to the orthonormal its Fourier set {un(x)}rassociated with some finite b = b, and some /I, coefficient in the sense (8.6.2) will be

taking it that b,, > a’. Although this function does not satisfy our assumptions for the expansion theorem, we can nevertheless use the Bessel inequality, which tells us that (8.12.3)

or Jrn --a

If

P ( 4 4 6 I.) dx

l2

d7dCL) Q

T’

We now show that for given h we can choose a’

f(4dx.

(8.12.4)

> a and c > 0 so that (8.12.5)

from which the result (8.12.1) will follow easily. For on taking on the left of (8.12.4) only the integral over (- I h I, I h I) and using the bound (8.12.5) it will follow that

recalling that T*,&) Q 0 when p < 0, we deduce (8.12.1). That (8.12.5) can be arranged to hold is easily seen if sin a = u(a,X)# 0. Since u(x, A) is continuous in both variables we may choose a‘ > a so that u(x, p ) 3 &sinor > O i f a x a’and I p I [ h 1. Wehavethen

< <

<

which is positive by the first of (8.1.5). Suppose next that sin a that is, that a = 0 since we take 0 a < x.

<

=

0,

8.12.

245

EXPANSIONS OVER A HALF-AXIS

To begin with, suppose in addition that

s:

> a. Since

for all x

r ( t ) dt

>0

(8.12.6)

v ( x , p) is continuous we may choose a’ so that $ for a x a‘ and I p I 1 h I. Writing r,(x) for the integral in (8.12.6) we have from (8.1.2) that u(x, p ) 2 &r1(x), for a x a’ and I p I 1.h I, so that

w(x, p)

2

$.(a,

< <

p) =

< <

<

<

(8.12.7)

and it will be sufficient to show that the last integral is positive. This follows from (8.1.5). Choose in fact an a” > a such that

r’

and then we have

a

p ( x ) rl(x) dx >, ~ Y , ( w )

r’ a”

p(x) dx > 0,

which again justifies (8.12.5). Suppose finally that u(a, A) = 0 and that (8.12.6) fails for some x In this case there will be an a, > a such that j l l P ( x ) I u(x, A )

dx = 0,

> a.

(8.12.8)

since r vanishes in a neighborhood of a so that u is constant and so zero in such a neighborhood. As was shown in Section 8.2, it follows from (8.12.8) that v ( x , A) is constant in (a, ul), and so equal to unity, whence u(x, A) = rl(x). Thus the solution is independent of h for a x a , . We take a, to be the greatest number with the property (8.12.8). Then (8.12.9)

< <

for all x

> a, , and

so (8.12.10)

for all x of a.

> a,.

We now apply the previous arguments with a, in place

246

8.

STURM-LIOUVILLE THEORY

Since it follows from (8.12.8) that T l p ( x ) u(x, A) dx = 0,

J a

r’

we have to show that there is an a‘

> a,

p ( x ) u(x, p) d x

such that

>c >0

(8.12.1 1)

a1

<

for 1 p I I X 1. If u(al , p) = rl(al) > 0, the existence of a’ follows as before, using (8.12.10). Suppose again that u(al , p ) = 0. In this case, we assert,

s‘

r ( t )di

>0

(8.12.12)

a1

for any x > a,. For otherwise there would be an a, > a, such that = 0 almost everywhere in (a,, a2),so that u would be constant there, and so zero, in contradiction to the assumption that a, is the greatest number with the property (8.12.8). This brings us back to the situation in which u(a, A)’ = 0 and (8.12.6) holds, which has already been dealt with, so that (8.12.11) may be taken to hold in this case also. This completes the proof of Theorem 8.12.1. The existence of at any rate one limiting spectral function is now more or less immediate. I

) is Theorem 8.12.2. There is a nondecreasing function ~ ( h which = 0, such that the Parseval equality right-continuous, with ~(0)

) the following conditions: holds for functions ~ ( x satisfying

(i) ~ ( x )is defined and absolutely continuous for x 2 a, vanishing outside some finite interval ; (ii) there are functions #(x), ~ ( x )vanishing , outside some finite interval, = r#, Y ,I qF = p x for x 2 a, 6 being absolutely consuch that tinuous and p1j2x of integrable square over (a, .); C$

+

(iii) ~ ( acos ) a - +(a) sin a = 0.

247

8.13. NESTING CIRCLES

Since the sequence of spectral functions Tb,,fi(h), where for definiteness we keep j3 fixed, is uniformly bounded in any finite A-interval, it contains a convergent subsequence. We take T(A) as the limit of this subsequence, normalized to ensure right-continuity and that ~ ( 0 = ) 0. T o justify (8.12.13) we write (8.10.13) in the form

Here we have taken it that m is so large that 9,x are zero in (b, , m); for simplicity let us assume also that A, - A are not points of discontinuity of any of the Tb,,fi(A) or oy these excluded values forming a denumerable set. Making rn + 8 , through the subsequence which makes the spectral functions converge, we obtain

~(x),

and the required result follows on making A + a. 8.13. Nesting Circles

The following alternative proof of the boundedness of the family of spectral functions T ~ , ~ ( A for ) , increasing b, is more elaborate than that given in the last section, but provides some information on the uniqueness of the limiting spectral function. The argument is similar to that of Sections 5.4-5, and is given in outline only. The first step is to construct a function previously termed here a characteristic function, whose poles are at the eigenvalues A,&, the residues being the corresponding normalization constants pn , Such a function will be set up in the next chapter in terms of the resolvent kernel, an extension of the notion of the Green's function. Here we set it up directly in terms of the solution u(x, A), u(x, A) of (8.1.2-3) such that u(a, A) = sin a, v (a , A) = cos a, and a second solution of (8.1.2-3), which we denote by ul(x, A), q ( x , A), such that ul(u, A) = cos a, q ( u , A) = - sin a. We then define [cf. (4.5.4)] (8.13.1)

The definition may be motivated as follows. We define a third solution of (8.1.2-3) by u ~ ( xA), = Y ~ ( xA) , - fi((x, A),

w ~ ( xA), = w ~ ( xA),

-fe(x, A)

(8.13.2-3)

248

8.

STURM-LIOUVILLE THEORY

where f is to be determined so that u 2 , v2 should satisfy the boundary condition at x = b, namely, u2(b,A) cos j3 - wz(b,A) sin j3

= 0.

(8.13.4)

This leads to f as given by (8.13.1). The function (8.13.1) has the following analytic properties.

Theorem 8.13.1. For complex A, Im A and Imfb,s(A) have the opposite signs. For real A, ,fb.p(A) is real, and finite except at the A,, where its residue is p, . We make here the assumptions of Section 8.1; the A,, are the roots of (8.3.4), the p , being given by (8.3.9). It is obvious from (8.13.1) that fb,@(A) is regular except at the zeros of the denominator, which are the A,, and that it is otherwise real for real A. Its residue at X = A, is ul(b, A,) cos j3 - q ( h , A,) sin j3 ul(b, A,) cos j3 - q ( b , A,) sin j3

Using (8.3.2) this is the same as

T o evaluate the numerator we may replace b by a, giving the value 1, while the denominator is p , , by (8.4.5). Thus Im A and Imfb,s(A) certainly have the opposite sign when A has the form A, f k for small E > 0. T o complete the proof it will be sufficient to show that Imfb,a(A) does not vanish when A is complex. Supposingfb,B(A)to be real, then the u2(x, A), v2(x, A) given by (8.13.2-3) with this value off would satisfy the boundary problem given by the differential equations (8.1.2-3); they would also satisfy the boundary condition (8.13.4), and the initial condition (sin a f cos a) u,(a, A) (cos a - f sin a ) v,(a, A) = o since u,(a, A) -- cos 01 - f sin a, v,(a, A) = - sin 01 - f cos a. With f real, this is a boundary problem of the same type as that of Section 8.3, with a different a, and since u 2 , v2 do not vanish identically, A must be real. Hence fb,@(h) is complex with A, completing the proof. From a standard property of functions which map the upper and lower half-planes into each other, we have

+

+

I for any complex A.

P i 1 Im {(A

- An>-1>

Id

8.13. Taking A

=

249

NESTING CIRCLES

i we have (8.13.5)

whence, for any real A’, Tb.B(A’)

= 0{1

+ A’2),

(8.13.6)

where Tb,,9(A) is the function defined in (8.10.34). For the final link in the chain bounding the spectral function we need to show that f b $ ( A ) is bounded as b -+ 00 for fixed complex A, such as h = i; it will then follow that (8.13.6) holds uniformly in A‘ and b, and for that matter p. We assume that the conditions laid down in Section 8.1 hold for all b > a, or less restrictively that conditions (i) and (ii) hold for all b > a, and that (iii) and (iv) hold for some b > a. For this purpose we define the circle C(b,A) which is the locus of (8.13.1) for real 8, which is the same as the circle described by (8.13.7) as z describes the real axis, including 00; here we assume A complex. Denoting by D(b, A) the disk bounded by C(b, A), we have that D(b, A ) is the map under z +fb,c(h) of either the upper or the lower halfplanes. Taking for definiteness I m h > 0, we assert that D(b, A) is in fact the map of I m z < 0. For if I m A > 0, we have from (8.3.6) that I m (u(b, A)/o(b, A)} > 0, so that fb,e(A) is finite if I m A > 0, I m z < 0. If we interpret D(b, A ) as the closed disk, we have the nesting property given by

Theorem 8.13.2. For b’ > b, D(b, A) 3 D(b’, A). Writingfforfl,,(A), and solving (8.13.7) for z in terms off, we obtain z = uz(b, A)/a,(b, A), where u, and w, are given by (8.13.2-3); we no longer impose (8.13.4), which applies to the special choice z = tan /3. T h e - set I m z 0 is thus given by I m (uz(b,h)v,(b, A) - u,(b, A)w,(b, A)} \< 0. We now observe that

<

in a similar way to (8.3.6). Also, _ _ _ _ _ _

u2(a,A ) wz(a,A) - uz(a,A ) w,(a, A) = (cos a - f sin a) (- sin a - fcos a )

- (cos a -fsin

a ) (- sin a - f c o s a) = f

-f.

8.

250 Hence the set I m z

STURM-LIOUVILLE THEORY

< 0 is also characterized by

As b increases, this inequality becomes more stringent, so that the f-locus which satisfies it shrinks, or at least does not expand. This proves the theorem, and therewith the existence of at least one limiting spectral function. Still assuming that Im A > 0, we have that the circles C(b, A) “nest,” apart from intervals in which p = 0, where they will be constant. For fixed A, we thus have the distinction between limit-circle and limitpoint cases, which may be investigated by calculating the radius of C(b,A). Writing u for u(b, A), and so on, the disk D(b, A) is the f-set given by I m (u26, - C202) 0, or

<

Im{(u, -fu)

(4 -fB) - (a, - fa) (0, - f w ) )

< 0.

This may be brought to the form

the right-hand side being the squared radius. Since u,(b, A) w(b, A ) - u(b, A) w,(b, A)

= ul(u, A) w(a, A) - u(a, A ) wl(a, A) =

1

by (8.1.2-3), and the denominator is given by (8.3.6), we find that the radius of C(b, A) is [cf. (5.4.6)]

The limit-circle and limit-point cases can now be identified as those in which, respectively, /%(s)

I u(x, A) la dx < 00,

= 00.

(8.1 3.10-1 1)

We state without proof the following properties, which may be established similarly to their analogs in Chapter 5: (i) in the limit-circle case, all solutions satisfy J ” p I u is, are of integrable square;

la dx < 00, that

8.13. NESTING CIRCLES

25 1

(ii) if the limit-circle holds for one complex A, it holds for all complex A, and likewise for the limit-point case; (iii) for every complex A, there is at least one nontrivial solution of integrable square; (iv) if all solutions are of integrable square for one A, then this is the case for all A. In the proof of (i) and (iii) we use (8.13.8); for (ii) and (iv) we use the variation of parameters, rather as in Section 5.6.