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Eigenvalues of singular differential operators by finite difference methods, I
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 38,244-254
(1972)
Eigenvalues of Singular Differential Operators Finite Difference Methods, I* JOHN Department
by
V. BAXLEY
of Mathematics, Wake Forest University, Winston-Salem, North Carolina Submitted by Peter D. Lax Received November
30, 1970
1. INTRODUCTION Let 7 be the formal differential operator given by 5%= - &
[p(x) U’]‘,
0 < x < 1.
We assume that m(x) > 0, p(x) > 0 and both m(x), p(x) are infinitely ferentiable functions on (0, 11. We further assume that j”:m(r) [J‘thdt]
dx = M < CD.
dif-
(14
Thus, 7 may be singular at the endpoint x = 0. Note, however, that the analysis here includes the regular case. Our concern is with approximating the eigenvalues of certain self-adjoint operators defined by 7 in the Hilbert spaceL2(m; 0, 1) = L2(m) of all measurable functions defined for 0 < x < 1 for which si if I2 m dx < CO.We face, in general, two problems. The first is the problem of defining a suitable Hilbert space operator; we must be careful that the operator has eigenvalues or else our question is not meaningful. The operators we investigate will be strictly positive with compact inverses; the well-known theory of compact operators will then guarantee that the spectrum consists only of eigenvalues * This work was supported in part by NASA University Sustaining Research Grant No. NGR-34-001-005 and by the Oak Ridge National Laboratory, Oak Ridge, Tennessee, which is operated by Union Carbide Corporation for the U. S. Atomic Energy Commission.
244 0 1972 by Academic Press, Inc.
EIGENVALUES OF DIFFERENTIAL OPERATORS
245
and that the eigenfunctions are complete in L2(m). The second problem concerns the finite difference operators to be used. Although it is not necessary, we shall for convenience restrict ourselves to the natural simple finite difference approximations. In a later paper, generalizations of the results here to higher order differential operators and a wider class of approximating operators will be presented. The techniques and results described here were motivated by earlier work on Toeplitz matrices both by the author [I] and S. V. Parter [5, 61.
2. THE APPROPRIATE DIFFERENTIAL OPERATORS We begin with definitions and explanation of notation. The inner product and norm in L2(m) will be denoted by (., *) and [I . 11, respectively. If m(x) E p(x) E 1, our operator is just ru = - u”. Two classical boundary value problems for this operator consist of imposing either the fixed-end conditions u(0) = ~(1) = 0 or the conditions u’(0) = u(1) - 0. We shall consider below two boundary value problems for the general operator in (1 .I), which in the case m(x) = p(x) = 1 reduce to the two simple problems just mentioned. By C”(0, 1) is meant the collection of all infinitely differentiable functions defined on (0, 1); ?I E C,“(O, 1) if and only if u E Cm(O,1) and u has compact support in (0, 1); and u E Cr”(0, 1) if and only if zi E Com(O,1) and u vanishes in some neighborhood of x = 1. We define the two operators L and T in L2(m) by
Lu = rl.4,
for every u E D(L) = Com(O,l),
(2.1)
Tu=ru,
for every u E D(T) = Clm(O,1).
(24
We remark that (1.2) implies that C,“(O, 1) C L2(m). It is easy to verify that T is symmetric and semibounded (below by 0); that is, (Tu, u) > 0, for all u E D(T). Since L is a restriction of T, L is also symmetric and semibounded. It follows from Friedrichs’ theory (see [4; or 3, pp. 1240-12421) that L and T have particular self-adjoint extensions F and G, respectively, which preserve the lower bound. We call F and G the Friedrichs’ extensions of L and T, respectively. LEMMA 2.1.
Ifu E Clm(O, 1) and 0 < x1 < x2 < 1, then
1u(x2) - U(Xl)l” <(774, u)s21 “a b(t)]-1 dt.
246
RAXLEY
Proof.
Write u(x2) -- $x1) == s:l’ u’(t) dt I
and use Schwarz’s inequality. Let L* (resp., T*) be the Hilbert space adjoint of L (resp., 7’). Then D(L) C D(F) C D(L*) and D(T) C D(G) C D(T*). We shall characterize D(F) and D(G) using boundary conditions. THEOREM 2.2.
If u E D(F), then
(a)
u(l) = 0 and
(i)
lir$ u(x) = 0,
(ii)
1u(x)l” = 0 (1: [p(t)]-l dt)
;f
i,‘[p(t)]-l as
dt < co, x + O+,
(b) I 44 - 4d12 < (Fu,4 j” lNY 21
(4
(% 4 < Jvk
dt
if
jl [p(t)]-l
dt = 0~).
forO
4.
Proof. We proceed as in the proof of [2, Theorem 21. If u E D(F), there exists ulcE D(L) such that 11ulc - u II+ 0 and (Lu, , uk) --f (Fu, u) as k * 00. From Lemma 2.1,
I%(X2) - %(X1)12 <(LUk 94s
1:MW 4
k = 1, 2,...,
(2.3)
for O,
EIGENVALUES
OF DIFFERENTIAL
OPERATORS
247
for 0 < x1 < xa < 1, which is (b) and letting x, = x > 0, xs = 1 in (b) gives (ii). Letting x1 = x, xa = 1 in (b) gives
I 44l" < (Fu,4 j12[p(t)]-ldt and (c) follows. Using the same techniques, we obtain: 2.3. If u E D(G), then
THEOREM
(a)
u(1) = 0.
(b)
1u(xa) - u(xJ]” < (Gu, ZJ)& [p(t)]-l dt, for 0 < xl < x2 < 1.
(4
(u, u) < WGu, 4.
THEOREM 2.4. All eigenvalues of F (req. G) are strictly positive and hence F-l (resp. G-1) exists. Further F-l and G-l are compact operators.
Proof. Consider the case of F. It follows from Theorem 2.2(c) that all the eigenvalues of F are strictly positive. To show F-l is compact, suppose ]jFu, I] < K < co for k = 1,2 ,... . From Theorem 2.2(c), we get for u E D(F),
Ilull
k = 1, 2,... .
Applying Theorem 2.2(b), we get
I +(x2>
- U412 < &u-c2j” kW-l 4
h = 1, 2,...,
El
forO
248
BAXLEY
THEOREM 2.5. (a) u E D(F) if and only if u E D(L*) and u satis$es the boundary conditions of Theorem 2.2(a). (b) u E D(G) if and only if u E D(T*) and u satis$es the boundary condition u(l) = 0 of Theorem 2.3(a).
Proof. Suppose u satisfies the conditions of Theorem 2.2(a). Since 0 is not in the spectrum of F, the range of F is all of L2(m) and there exists v E D(F) such thatFv = L*u. We complete the proof of (a) by showing that u = v. Let w = u - V. Then by Theorem 2.2(a) v satisfies the boundary conditions there; u satisfies these conditions by hypothesis; thus w satisfies these conditions. Since L*w = 0, then rw = 0; hence
w(x) = Cl 1’ [p(t)]-l z
dt + C, .
But the boundary conditions imply C, = C’s = 0; so w = 0 and u = v. The same techniques prove (b); compare [ 1; Lemma 4.21. It follows from Theorem 2.5 that the eigenvalues of both F and G are simple. For, if ur and us both satisfy Fu = Au (or Gu = Au), then %(l) = u,(l) = 0 and the Wronskian of u1 and us vanishes. Thus u1 and us are linearly dependent and the solution space of Fu = Au (or Gu = Au) is at most one-dimensional. We now investigate the relationship between F and G. Since T is strictly positive and symmetric, the deficiency indices of T are both equal to the dimension of the null space N(T*) = {u E D(T*) : T*u = 0} of T*. Since each u E D( T*) satisfies p(x) u’(x) + 0 as x + 0+ (this boundary condition is thus “built-in” for functions in D(G) and results from the choice of D(T) as C1”(O, l)), it follows that N( T*) has dimension 1. Thus, as is expected from the general theory in [3], G is obtained by the imposition of the one boundary condition u(1) = 0 on D(T*). Making a similar analysis of L, we see that N(L*) is spanned by the functions Ul(X) = 1
and
us(x)= 1’r [p(W’ dt,
if us E La(m). Thus the deficiency indices of L are 2 if us ALL and 1 if us 4 LZ(m). Therefore, if us $ L2(m), F will be obtained by the imposition of the one boundary condition u(l) = 0 on D(L*). Since D(T*) C D(L*), then D(G) C D(F), but F and G are both self-adjoint; so F = G. However, this result can be significantly strengthened, as Theorem 2.7 below shows. The proof of the following lemma is an immediate consequence of the description of the Friedrichs extension in [3; Corollary X11.5.31.
EIGENVALUES OF DIFFERENTIAL
249
OPERATORS
LEMMA 2.6. Let A be any semibounded, symmetric operator, let A’ be the Friedrichs extension of A, and let B be a restriction of A’ to a linear manifold D(B) where D(A) C D(B) C D(A’). Then the Friedrichs extension of B is A’. THEOREM
2.7. If Ji [p(t)]-l
dt = co, then F = G.
Proof. By Theorem 2,5(a), D(T) = CIm(O, 1) CD(F). Thus T is the restriction of F to D(T), and applying Lemma 2.6, we get F = G. As an example, consider the case m(x) = xoL,p(x) = x”+l (a > - 1 is constant), which arose in [I]. Then si [p(t)]-l dt = co if 01> 0 and st [p(t)]-l dt < co if - 1 < 01< 0. However, the deficiency indices of L are (2, 2) if - 1 < 01< 1 and (1, 1) if 01> 1. Thus, for 0 < 01< 1, the boundary condition at x = 0 of Theorem 2.2(a) is necessary to define F, but it is equivalent to the condition p(x) U’(X) + 0 as x -+ Of.
3. SOME FINITE For j = 0, linear points
DIFFERENCE
OPERATORS
each positive integer n, let dx = l/(n + 2) and let xi = jdx, * 1, + 2 )...) be the lattice points on the real line. For any piecewise function htn)(x), - co < x < co, determined by its values at the xj we define the difference operators 8, and 6- by
&-
[h(“‘(xj+l)
- h(“)(x,)],
j = 1, 2,...
(S+h(n)) (xi) =
, otherwise
a &
[h(“)(xJ
-
h(“‘(xj-J],
(3.1)
I
j = 1, 2 ,...
(S-hfn)) (xj) =
(3.2) otherwise
Now let Pn be the (n + 1)-d’rmensional space consisting of all piecewise linear functions h(“)(x) determined by their values at the lattice points x, and for which h(x) = 0 for x < 0 and for x > xn+a = 1. We use as an inner product in Pn, [h(n), gfn)] = c hcn)(xj) g(“)(xj) m(xj) Ax, i=l
where m is the function in (1.1).
250
BAXLEY
We now define a, : Pn + 9J and 6% : Pn
(& (u,h(“)) (XJ =
S+p8P)
(Xj),
% by j=
I,2 ,*a.,n+l
otherwise
10,
j-l,2
,
(3.4)
.
(3.5)
I
,***, n+l
otherwise If y is a function
in D(L) or D(T), we define rptn) E Pm by
‘&4 PY-4 = lo, LEMMA 3.1. (a) qJE D(T), then i$fp
j = 1, 2,..., n + 1 otherwise
If cpE D(L), then u&n) +L~I uniformly on [0, 11. (b) If --f Tp, unz~ormly on [0, 11.
Proof. Since the proof is a standard argument using Taylor’s theorem, we omit the details. However, we should observe that it is, in general, false that u,#lz) -+ Ts, uniformly on [0, l] for q~E D(T). Indeed, suppose CJIE D(T) and v(x) = 1 for x in some neighborhood of 0. Then, for n sufficiently large,
which,
in general, tends to infinity
as LJx -+ 0 rather than to p’(O) = 0.
4. APPROXIMATION OF THE EIGENVALUES It is easily verified that a, and 8% are self-adjoint, strictly positive operators in 9% , and thus all eigenvalues of a, and c$, are strictly positive. Let
and
be the eigenvalues, arranged in nondecreasing order and counting multiplicities, of a, and & , respectively. Further, let A,(F) and A,(G) be the eigenvalues of F and G, respectively, arranged in nondecreasing order.
EIGENVALUES
LEMMA
OF DIFFERENTIAL
251
OPERATORS
4.1.
Proof. In both cases the procedure is that used in [1 ; Lemma 5.11. For completeness, we exhibit the proof of (a). Let Y be fixed, Y > 1. Let & be the r-th normalized eigenfunction of F with eigenvalue A, . Let 0 < E < 1. We choose yr E D(L), jj q,. I( = 1 so that II 9%- 9%II < E9
r = 1, 2,..., v
(4.1)
and 16%~>~4 - PA >$s)l < E>
r, s = 1, Z...,
v.
(4.2)
Since ~4, A ,...A are mutually orthogonal, pi , vs ,..., vV are linearly independent for sufficiently small E > 0. It follows from Lemma 3.1 that
~~hvP, P?l = w% >%I,
r, s = 1, 2 )...) v.
(4.3)
Clearly,
&pP, &‘I = ($39RJ,
r, s = 1, 2,...) v.
(4.4)
Now suppose for some 6 > 0 that
liy+zup A,(u,) 3 A, + 28.
(4.5)
For n sufficiently large, the functions VT) are linearly independent. Thus, by the Courant-Fischer minimax theorem, there exists a subsequence 2, of the integers 2 and constants a,,, , a,,, ,..., a,,, such that
[“d?-@)~ P’“‘1 > fl + * [#n),pq ” ’
for n E 2, .
(4.6)
Without loss of generality, assume that each 1aisn 1 < 1. Then extract a subsequence 2, of 2, so that {a,,,}, n E 2,) converges for each i = 1,2,..., v to some ai with 1ai 1 < 1. Then {cp’“)}, 71E 2, , converges uniformly on [0, l]
252
BAXLEY
to v = C,“=, ai~i . Using (4.1)-(4.4), one sees that the quotient in (4.6) is dominated in the limit by
$ I ai I2- 2~~6 which contradicts (4.6) if E is sufficiently small. In Lemma 4.2(a) below, if si [p(t)]-l dt < co but p(t)]-l t -+ Of, we need some hypothesis which will imply that
is not bounded as
hinf [p(xj)]-1 Ax=1;[p(t)]-1 dt. 3=1 One such hypothesis (as is easy to verify) which is generally satisfied, is that [p(t)]-l is monotone near x = 0. Without further reminder, we assume henceforth that in this case some such hypothesis is made. LEMMA 4.2. (a) Suppose ucn) E Sa, and lim supn+Ju,u(fi), u(“)] -=cco. Then each subsequence of (ucn)) contains a further subsequence which converges uniformly on compact subsets of (0, l] to a limit function u which is continuous on (0, l] and satisjes the boundary conditions of Theorem 2.2(a). (b) Suppose u(“) E 9fn and lim SUP~+~[c?,@), utn)] < co. Then each subsequence of {I((“)} contains a further subsequence which converges uniformly on compact subsets of (0, l] to a limit function u which is continuous on (0, l] and satisfies the boundary condition u(1) = 0 of Theorem 2.3(a). Proof. The proof of(b) is identical with that of [I, Lemma 5.21. We now prove (a). Let 0 < x, < x, < 1. Then for u(“) E gn,
uyx,) - u(n)(x,) = c (Shd~)) (Xi)Ax. j=r+l
Using Schwarz’s inequality followed by summation by parts, we get
1u(“)(x,)- u(“)(x,)j2 < [u&n),uq
(4.7) j=r+l
If 6 MW -=cco (recall the remarks made just before the statement of the Lemma), then the functions {u@)} are uniformly bounded and equicontinuous on the closed interval [0, 11. By Ascoli’s theorem, each subsequence
EIGENVALUES
OF DIFFERENTIAL
253
OPERATORS
of {un} contains a further subsequence which converges uniformly on the entire closed interval [0, l] to a continuous limit function u on [0, 11. Since U(~)(O) = @j(l) = 0, then u(0) = u(l) = 0, the required conditions. If ji [p(t)]-l dt = co, then from (4.7) we conclude only that the functions {u(“)} are uniformly bounded and equicontinuous on compact subsets of (0, 11. Then Ascoli’s theorem, together with a diagonalization argument, guarantees that each subsequence of (u(“)} contains a further subsequence that converges uniformly on compact subsets of (0, l] to a limit function u continuous on (0, 11. Since u en)(l) = 0, then u(l) = 0. Taking limits in (4.7) (through the last subsequence) yields 1U(%> - u(xd” < (l$~p[u&(“),
u(@]
1; [p(t)]-1 dt)
for 0 < X, < x2 f 1, and letting x1 = X, x2 = 1, there results 1u(X)]” < (lim sup[u,u n+oo
(‘I,@)I)(j; [p(V dt) ,
the required condition at x = 0. THEOREM 4.3. (a) For each fixed v, liq,, fixed v, limn+m &(&J = n,(G).
h(u,J = cl,(F). (b) For each
Proof. The proof in both cases is essentially the same as in [l, Lemma 5.31. We only sketch the proof of (a). Let v be fixed,
UJp
= X&r,) Al”’
and
[I@), lp]
= a,,, ,
r, s = 1, 2 )...) v.
It suffices to show that each subsequence of {X,(u,J} contains a further subsequence which converges to A”(F). By Lemmas 4.1 and 4.2, we can pass to a subsequence such that
exists and /z,(x) = lim h?“(x), ?z~+cc
r = 1, 2,...,
v
exists uniformly on compact subsets of (0, 11, and /Z,.(X)satisfies the boundary conditions of Theorem 2.2(a). Let 9 E D(L); then @) E Pn and h,(U,f) [ip,
q?‘)] = py,
u,yw]
(4.8)
254
BAXLEY
for Y = 1, 2,..., V. Using Lemma 3.1 to let n’ + co in (4.8) yields
YTPT>v) = (4 >LF)
(4.9)
for r = 1, 2,..., v and 9) E D(L). Then h, E D(L*) and L*h, = yrh, , I = 1, 2,..., v. By Theorem 2.5(a), h, E D(F), and Fh, = yrh,. , Y = 1, 2 ,..., v. It is easy to see that (h, , h,) = a,,, for Y, s = 1, 2 ,..., v, so that h, is a bona fide eigenfunction of F with eigenvalue yr . We now show that y,. = A,(F), Y = 1,2,..., V. Since y1 < A,(F) by Lemma 4.1, clearly y1 = A,(F). W e continue by induction. Suppose that yV = A,(F), for Y = 1, 2,..., K < v, but yk+l # A,+,(F). Then by Lemma 4.1, Ykfl = 4(F) < d,+,(F). But then hk,, and h, are orthogonal eigenfunctions of F corresponding to the eigenvalue A,(F), contradicting the fact that AZ(F) is a simple eigenvalue of F. We make one concluding remark. If si [p(t)]-l dt = CO,then according to Theorem 2.7, F = G. In this case, both of our difference schemes a, and 6% approximate the same eigenvalues. However, if li [p(t)]-l dt < CO,we would generally expect the eigenvalues of F and G to be different. In particular, this is the case when m(x) = p(x) = 1. Then, using a, , we approximate the positive zeros of sin X; and using 6%, we approximate the positive zeros of cos x. REFERENCES 1. J. V. BAXLEY, Extreme eigenvalues of Toeplitz matrices associated with Laguerre polynomials, Arch. Rut. Mech. Anal. 30 (1968), 308-320. 2. J. V. BAXLEY, The Friedrichs extension of certain singular differential operators,
Duke Math. J. 35 (1968), 455-462. 3. N. DUNFORD AND J. T. SCHWARTZ, “Linear Operators, “Part II, Wiley (Interscience), New York, 1963. halbbeschrtikter Operatoren, I, Math. Ann. 4. K. 0. FRIEDRICHS, Specktraltheorie 109 (1934), 465-487. 5. S. V. PARTER, On the extreme eigenvalues of truncated Toeplitz matrices, Bull. Amer. Math. Sot. 67 (1961), 191-196. 6. S. V. PARTER, On the eigenvalues of certain generalizations of Toeplitz matrices, Arch. Rat. Mech. Anal. 11 (1962), 244-257.
(1972)
Eigenvalues of Singular Differential Operators Finite Difference Methods, I* JOHN Department
by
V. BAXLEY
of Mathematics, Wake Forest University, Winston-Salem, North Carolina Submitted by Peter D. Lax Received November
30, 1970
1. INTRODUCTION Let 7 be the formal differential operator given by 5%= - &
[p(x) U’]‘,
0 < x < 1.
We assume that m(x) > 0, p(x) > 0 and both m(x), p(x) are infinitely ferentiable functions on (0, 11. We further assume that j”:m(r) [J‘thdt]
dx = M < CD.
dif-
(14
Thus, 7 may be singular at the endpoint x = 0. Note, however, that the analysis here includes the regular case. Our concern is with approximating the eigenvalues of certain self-adjoint operators defined by 7 in the Hilbert spaceL2(m; 0, 1) = L2(m) of all measurable functions defined for 0 < x < 1 for which si if I2 m dx < CO.We face, in general, two problems. The first is the problem of defining a suitable Hilbert space operator; we must be careful that the operator has eigenvalues or else our question is not meaningful. The operators we investigate will be strictly positive with compact inverses; the well-known theory of compact operators will then guarantee that the spectrum consists only of eigenvalues * This work was supported in part by NASA University Sustaining Research Grant No. NGR-34-001-005 and by the Oak Ridge National Laboratory, Oak Ridge, Tennessee, which is operated by Union Carbide Corporation for the U. S. Atomic Energy Commission.
244 0 1972 by Academic Press, Inc.
EIGENVALUES OF DIFFERENTIAL OPERATORS
245
and that the eigenfunctions are complete in L2(m). The second problem concerns the finite difference operators to be used. Although it is not necessary, we shall for convenience restrict ourselves to the natural simple finite difference approximations. In a later paper, generalizations of the results here to higher order differential operators and a wider class of approximating operators will be presented. The techniques and results described here were motivated by earlier work on Toeplitz matrices both by the author [I] and S. V. Parter [5, 61.
2. THE APPROPRIATE DIFFERENTIAL OPERATORS We begin with definitions and explanation of notation. The inner product and norm in L2(m) will be denoted by (., *) and [I . 11, respectively. If m(x) E p(x) E 1, our operator is just ru = - u”. Two classical boundary value problems for this operator consist of imposing either the fixed-end conditions u(0) = ~(1) = 0 or the conditions u’(0) = u(1) - 0. We shall consider below two boundary value problems for the general operator in (1 .I), which in the case m(x) = p(x) = 1 reduce to the two simple problems just mentioned. By C”(0, 1) is meant the collection of all infinitely differentiable functions defined on (0, 1); ?I E C,“(O, 1) if and only if u E Cm(O,1) and u has compact support in (0, 1); and u E Cr”(0, 1) if and only if zi E Com(O,1) and u vanishes in some neighborhood of x = 1. We define the two operators L and T in L2(m) by
Lu = rl.4,
for every u E D(L) = Com(O,l),
(2.1)
Tu=ru,
for every u E D(T) = Clm(O,1).
(24
We remark that (1.2) implies that C,“(O, 1) C L2(m). It is easy to verify that T is symmetric and semibounded (below by 0); that is, (Tu, u) > 0, for all u E D(T). Since L is a restriction of T, L is also symmetric and semibounded. It follows from Friedrichs’ theory (see [4; or 3, pp. 1240-12421) that L and T have particular self-adjoint extensions F and G, respectively, which preserve the lower bound. We call F and G the Friedrichs’ extensions of L and T, respectively. LEMMA 2.1.
Ifu E Clm(O, 1) and 0 < x1 < x2 < 1, then
1u(x2) - U(Xl)l” <(774, u)s21 “a b(t)]-1 dt.
246
RAXLEY
Proof.
Write u(x2) -- $x1) == s:l’ u’(t) dt I
and use Schwarz’s inequality. Let L* (resp., T*) be the Hilbert space adjoint of L (resp., 7’). Then D(L) C D(F) C D(L*) and D(T) C D(G) C D(T*). We shall characterize D(F) and D(G) using boundary conditions. THEOREM 2.2.
If u E D(F), then
(a)
u(l) = 0 and
(i)
lir$ u(x) = 0,
(ii)
1u(x)l” = 0 (1: [p(t)]-l dt)
;f
i,‘[p(t)]-l as
dt < co, x + O+,
(b) I 44 - 4d12 < (Fu,4 j” lNY 21
(4
(% 4 < Jvk
dt
if
jl [p(t)]-l
dt = 0~).
forO
4.
Proof. We proceed as in the proof of [2, Theorem 21. If u E D(F), there exists ulcE D(L) such that 11ulc - u II+ 0 and (Lu, , uk) --f (Fu, u) as k * 00. From Lemma 2.1,
I%(X2) - %(X1)12 <(LUk 94s
1:MW 4
k = 1, 2,...,
(2.3)
for O,
EIGENVALUES
OF DIFFERENTIAL
OPERATORS
247
for 0 < x1 < xa < 1, which is (b) and letting x, = x > 0, xs = 1 in (b) gives (ii). Letting x1 = x, xa = 1 in (b) gives
I 44l" < (Fu,4 j12[p(t)]-ldt and (c) follows. Using the same techniques, we obtain: 2.3. If u E D(G), then
THEOREM
(a)
u(1) = 0.
(b)
1u(xa) - u(xJ]” < (Gu, ZJ)& [p(t)]-l dt, for 0 < xl < x2 < 1.
(4
(u, u) < WGu, 4.
THEOREM 2.4. All eigenvalues of F (req. G) are strictly positive and hence F-l (resp. G-1) exists. Further F-l and G-l are compact operators.
Proof. Consider the case of F. It follows from Theorem 2.2(c) that all the eigenvalues of F are strictly positive. To show F-l is compact, suppose ]jFu, I] < K < co for k = 1,2 ,... . From Theorem 2.2(c), we get for u E D(F),
Ilull
k = 1, 2,... .
Applying Theorem 2.2(b), we get
I +(x2>
- U412 < &u-c2j” kW-l 4
h = 1, 2,...,
El
forO
248
BAXLEY
THEOREM 2.5. (a) u E D(F) if and only if u E D(L*) and u satis$es the boundary conditions of Theorem 2.2(a). (b) u E D(G) if and only if u E D(T*) and u satis$es the boundary condition u(l) = 0 of Theorem 2.3(a).
Proof. Suppose u satisfies the conditions of Theorem 2.2(a). Since 0 is not in the spectrum of F, the range of F is all of L2(m) and there exists v E D(F) such thatFv = L*u. We complete the proof of (a) by showing that u = v. Let w = u - V. Then by Theorem 2.2(a) v satisfies the boundary conditions there; u satisfies these conditions by hypothesis; thus w satisfies these conditions. Since L*w = 0, then rw = 0; hence
w(x) = Cl 1’ [p(t)]-l z
dt + C, .
But the boundary conditions imply C, = C’s = 0; so w = 0 and u = v. The same techniques prove (b); compare [ 1; Lemma 4.21. It follows from Theorem 2.5 that the eigenvalues of both F and G are simple. For, if ur and us both satisfy Fu = Au (or Gu = Au), then %(l) = u,(l) = 0 and the Wronskian of u1 and us vanishes. Thus u1 and us are linearly dependent and the solution space of Fu = Au (or Gu = Au) is at most one-dimensional. We now investigate the relationship between F and G. Since T is strictly positive and symmetric, the deficiency indices of T are both equal to the dimension of the null space N(T*) = {u E D(T*) : T*u = 0} of T*. Since each u E D( T*) satisfies p(x) u’(x) + 0 as x + 0+ (this boundary condition is thus “built-in” for functions in D(G) and results from the choice of D(T) as C1”(O, l)), it follows that N( T*) has dimension 1. Thus, as is expected from the general theory in [3], G is obtained by the imposition of the one boundary condition u(1) = 0 on D(T*). Making a similar analysis of L, we see that N(L*) is spanned by the functions Ul(X) = 1
and
us(x)= 1’r [p(W’ dt,
if us E La(m). Thus the deficiency indices of L are 2 if us ALL and 1 if us 4 LZ(m). Therefore, if us $ L2(m), F will be obtained by the imposition of the one boundary condition u(l) = 0 on D(L*). Since D(T*) C D(L*), then D(G) C D(F), but F and G are both self-adjoint; so F = G. However, this result can be significantly strengthened, as Theorem 2.7 below shows. The proof of the following lemma is an immediate consequence of the description of the Friedrichs extension in [3; Corollary X11.5.31.
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LEMMA 2.6. Let A be any semibounded, symmetric operator, let A’ be the Friedrichs extension of A, and let B be a restriction of A’ to a linear manifold D(B) where D(A) C D(B) C D(A’). Then the Friedrichs extension of B is A’. THEOREM
2.7. If Ji [p(t)]-l
dt = co, then F = G.
Proof. By Theorem 2,5(a), D(T) = CIm(O, 1) CD(F). Thus T is the restriction of F to D(T), and applying Lemma 2.6, we get F = G. As an example, consider the case m(x) = xoL,p(x) = x”+l (a > - 1 is constant), which arose in [I]. Then si [p(t)]-l dt = co if 01> 0 and st [p(t)]-l dt < co if - 1 < 01< 0. However, the deficiency indices of L are (2, 2) if - 1 < 01< 1 and (1, 1) if 01> 1. Thus, for 0 < 01< 1, the boundary condition at x = 0 of Theorem 2.2(a) is necessary to define F, but it is equivalent to the condition p(x) U’(X) + 0 as x -+ Of.
3. SOME FINITE For j = 0, linear points
DIFFERENCE
OPERATORS
each positive integer n, let dx = l/(n + 2) and let xi = jdx, * 1, + 2 )...) be the lattice points on the real line. For any piecewise function htn)(x), - co < x < co, determined by its values at the xj we define the difference operators 8, and 6- by
&-
[h(“‘(xj+l)
- h(“)(x,)],
j = 1, 2,...
(S+h(n)) (xi) =
, otherwise
a &
[h(“)(xJ
-
h(“‘(xj-J],
(3.1)
I
j = 1, 2 ,...
(S-hfn)) (xj) =
(3.2) otherwise
Now let Pn be the (n + 1)-d’rmensional space consisting of all piecewise linear functions h(“)(x) determined by their values at the lattice points x, and for which h(x) = 0 for x < 0 and for x > xn+a = 1. We use as an inner product in Pn, [h(n), gfn)] = c hcn)(xj) g(“)(xj) m(xj) Ax, i=l
where m is the function in (1.1).
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We now define a, : Pn + 9J and 6% : Pn
(& (u,h(“)) (XJ =
S+p8P)
(Xj),
% by j=
I,2 ,*a.,n+l
otherwise
10,
j-l,2
,
(3.4)
.
(3.5)
I
,***, n+l
otherwise If y is a function
in D(L) or D(T), we define rptn) E Pm by
‘&4 PY-4 = lo, LEMMA 3.1. (a) qJE D(T), then i$fp
j = 1, 2,..., n + 1 otherwise
If cpE D(L), then u&n) +L~I uniformly on [0, 11. (b) If --f Tp, unz~ormly on [0, 11.
Proof. Since the proof is a standard argument using Taylor’s theorem, we omit the details. However, we should observe that it is, in general, false that u,#lz) -+ Ts, uniformly on [0, l] for q~E D(T). Indeed, suppose CJIE D(T) and v(x) = 1 for x in some neighborhood of 0. Then, for n sufficiently large,
which,
in general, tends to infinity
as LJx -+ 0 rather than to p’(O) = 0.
4. APPROXIMATION OF THE EIGENVALUES It is easily verified that a, and 8% are self-adjoint, strictly positive operators in 9% , and thus all eigenvalues of a, and c$, are strictly positive. Let
and
be the eigenvalues, arranged in nondecreasing order and counting multiplicities, of a, and & , respectively. Further, let A,(F) and A,(G) be the eigenvalues of F and G, respectively, arranged in nondecreasing order.
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LEMMA
OF DIFFERENTIAL
251
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4.1.
Proof. In both cases the procedure is that used in [1 ; Lemma 5.11. For completeness, we exhibit the proof of (a). Let Y be fixed, Y > 1. Let & be the r-th normalized eigenfunction of F with eigenvalue A, . Let 0 < E < 1. We choose yr E D(L), jj q,. I( = 1 so that II 9%- 9%II < E9
r = 1, 2,..., v
(4.1)
and 16%~>~4 - PA >$s)l < E>
r, s = 1, Z...,
v.
(4.2)
Since ~4, A ,...A are mutually orthogonal, pi , vs ,..., vV are linearly independent for sufficiently small E > 0. It follows from Lemma 3.1 that
~~hvP, P?l = w% >%I,
r, s = 1, 2 )...) v.
(4.3)
Clearly,
&pP, &‘I = ($39RJ,
r, s = 1, 2,...) v.
(4.4)
Now suppose for some 6 > 0 that
liy+zup A,(u,) 3 A, + 28.
(4.5)
For n sufficiently large, the functions VT) are linearly independent. Thus, by the Courant-Fischer minimax theorem, there exists a subsequence 2, of the integers 2 and constants a,,, , a,,, ,..., a,,, such that
[“d?-@)~ P’“‘1 > fl + * [#n),pq ” ’
for n E 2, .
(4.6)
Without loss of generality, assume that each 1aisn 1 < 1. Then extract a subsequence 2, of 2, so that {a,,,}, n E 2,) converges for each i = 1,2,..., v to some ai with 1ai 1 < 1. Then {cp’“)}, 71E 2, , converges uniformly on [0, l]
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to v = C,“=, ai~i . Using (4.1)-(4.4), one sees that the quotient in (4.6) is dominated in the limit by
$ I ai I2- 2~~6 which contradicts (4.6) if E is sufficiently small. In Lemma 4.2(a) below, if si [p(t)]-l dt < co but p(t)]-l t -+ Of, we need some hypothesis which will imply that
is not bounded as
hinf [p(xj)]-1 Ax=1;[p(t)]-1 dt. 3=1 One such hypothesis (as is easy to verify) which is generally satisfied, is that [p(t)]-l is monotone near x = 0. Without further reminder, we assume henceforth that in this case some such hypothesis is made. LEMMA 4.2. (a) Suppose ucn) E Sa, and lim supn+Ju,u(fi), u(“)] -=cco. Then each subsequence of (ucn)) contains a further subsequence which converges uniformly on compact subsets of (0, l] to a limit function u which is continuous on (0, l] and satisjes the boundary conditions of Theorem 2.2(a). (b) Suppose u(“) E 9fn and lim SUP~+~[c?,@), utn)] < co. Then each subsequence of {I((“)} contains a further subsequence which converges uniformly on compact subsets of (0, l] to a limit function u which is continuous on (0, l] and satisfies the boundary condition u(1) = 0 of Theorem 2.3(a). Proof. The proof of(b) is identical with that of [I, Lemma 5.21. We now prove (a). Let 0 < x, < x, < 1. Then for u(“) E gn,
uyx,) - u(n)(x,) = c (Shd~)) (Xi)Ax. j=r+l
Using Schwarz’s inequality followed by summation by parts, we get
1u(“)(x,)- u(“)(x,)j2 < [u&n),uq
(4.7) j=r+l
If 6 MW -=cco (recall the remarks made just before the statement of the Lemma), then the functions {u@)} are uniformly bounded and equicontinuous on the closed interval [0, 11. By Ascoli’s theorem, each subsequence
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OF DIFFERENTIAL
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OPERATORS
of {un} contains a further subsequence which converges uniformly on the entire closed interval [0, l] to a continuous limit function u on [0, 11. Since U(~)(O) = @j(l) = 0, then u(0) = u(l) = 0, the required conditions. If ji [p(t)]-l dt = co, then from (4.7) we conclude only that the functions {u(“)} are uniformly bounded and equicontinuous on compact subsets of (0, 11. Then Ascoli’s theorem, together with a diagonalization argument, guarantees that each subsequence of (u(“)} contains a further subsequence that converges uniformly on compact subsets of (0, l] to a limit function u continuous on (0, 11. Since u en)(l) = 0, then u(l) = 0. Taking limits in (4.7) (through the last subsequence) yields 1U(%> - u(xd” < (l$~p[u&(“),
u(@]
1; [p(t)]-1 dt)
for 0 < X, < x2 f 1, and letting x1 = X, x2 = 1, there results 1u(X)]” < (lim sup[u,u n+oo
(‘I,@)I)(j; [p(V dt) ,
the required condition at x = 0. THEOREM 4.3. (a) For each fixed v, liq,, fixed v, limn+m &(&J = n,(G).
h(u,J = cl,(F). (b) For each
Proof. The proof in both cases is essentially the same as in [l, Lemma 5.31. We only sketch the proof of (a). Let v be fixed,
UJp
= X&r,) Al”’
and
[I@), lp]
= a,,, ,
r, s = 1, 2 )...) v.
It suffices to show that each subsequence of {X,(u,J} contains a further subsequence which converges to A”(F). By Lemmas 4.1 and 4.2, we can pass to a subsequence such that
exists and /z,(x) = lim h?“(x), ?z~+cc
r = 1, 2,...,
v
exists uniformly on compact subsets of (0, 11, and /Z,.(X)satisfies the boundary conditions of Theorem 2.2(a). Let 9 E D(L); then @) E Pn and h,(U,f) [ip,
q?‘)] = py,
u,yw]
(4.8)
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for Y = 1, 2,..., V. Using Lemma 3.1 to let n’ + co in (4.8) yields
YTPT>v) = (4 >LF)
(4.9)
for r = 1, 2,..., v and 9) E D(L). Then h, E D(L*) and L*h, = yrh, , I = 1, 2,..., v. By Theorem 2.5(a), h, E D(F), and Fh, = yrh,. , Y = 1, 2 ,..., v. It is easy to see that (h, , h,) = a,,, for Y, s = 1, 2 ,..., v, so that h, is a bona fide eigenfunction of F with eigenvalue yr . We now show that y,. = A,(F), Y = 1,2,..., V. Since y1 < A,(F) by Lemma 4.1, clearly y1 = A,(F). W e continue by induction. Suppose that yV = A,(F), for Y = 1, 2,..., K < v, but yk+l # A,+,(F). Then by Lemma 4.1, Ykfl = 4(F) < d,+,(F). But then hk,, and h, are orthogonal eigenfunctions of F corresponding to the eigenvalue A,(F), contradicting the fact that AZ(F) is a simple eigenvalue of F. We make one concluding remark. If si [p(t)]-l dt = CO,then according to Theorem 2.7, F = G. In this case, both of our difference schemes a, and 6% approximate the same eigenvalues. However, if li [p(t)]-l dt < CO,we would generally expect the eigenvalues of F and G to be different. In particular, this is the case when m(x) = p(x) = 1. Then, using a, , we approximate the positive zeros of sin X; and using 6%, we approximate the positive zeros of cos x. REFERENCES 1. J. V. BAXLEY, Extreme eigenvalues of Toeplitz matrices associated with Laguerre polynomials, Arch. Rut. Mech. Anal. 30 (1968), 308-320. 2. J. V. BAXLEY, The Friedrichs extension of certain singular differential operators,
Duke Math. J. 35 (1968), 455-462. 3. N. DUNFORD AND J. T. SCHWARTZ, “Linear Operators, “Part II, Wiley (Interscience), New York, 1963. halbbeschrtikter Operatoren, I, Math. Ann. 4. K. 0. FRIEDRICHS, Specktraltheorie 109 (1934), 465-487. 5. S. V. PARTER, On the extreme eigenvalues of truncated Toeplitz matrices, Bull. Amer. Math. Sot. 67 (1961), 191-196. 6. S. V. PARTER, On the eigenvalues of certain generalizations of Toeplitz matrices, Arch. Rat. Mech. Anal. 11 (1962), 244-257.