I n this section we will show t h a t if the constant h in the boundary conditions is given then the lowest eigenvalue is superfluous. More precisely we will prove t h a t to each per- turbation of a finite number of the eigenvalues in the reduced spectrum there corresponds a potential. This is in some respects more general, in others less general t h a n Borg's result, see [3, p. 71]. For example, in our case there is no restriction on the size of the perturba- tion of the eigenvalues and the boundary conditions can be perturbed as well. On the other hand, Borg permits a small perturbation of all the eigenvalues. Another difference lies in the class of potentials considered. Borg admits all square-integrable functions as
288 O:H: H~a~D
compa~son: potentials;: while we restrict= Our attention tO potentials, which are: essentiall:~
bounded: This restriction is imposed b y our method, of proof, and the result':can presumably be: e x t e n d e 4 t o all intograble comparison: potentials.
L E ~ 2. Consider the eigenvalue problem (1.2) where ~ is es8entiaUy bounded and satis/y ~(x)= ~ ( ~ - x)/or almost all x. Let ~o, ~1,: .... be: the eigenvalues o/(1.2). Let the constant h and 21 <23 <... be given and assume that ,~ = "~/or all j >~ N. Then there exists an essentially bounded/unction q(,x),, which satisfies q(x) w q ( ~ - x ) almoat everywhere, such<that ]~ are the eigenvalues o] (1,1).
Proo/. Le~ ~0 < 2 r From Lemma 1 follows that there exist a constant h and a function q such t h a t ~j are the eigenvalues of (1 ::l). Since ~ is essentially bounded we conclude from Theorem 2 that IIqllo~ is fifiite. To prove Lemma 2 it is sufficient to show that if h* is given and 2~ = ~j for all' j >~ 1 then there exists a ~0 ~ ~1 Such' t h a t
o)
(7.1)Assume namely t h a t there exists a 3. 0 in ( - 0% ]1) such t h a t (7.1) is satisfied. According to Lemma 1 there exist a constant h and a function q such that ~0, 2~1 ... are the eigenvalues of (1:1). We can then use eq. (4.5) and conclude by comparing eq. (1.7! and (7.1! t h a t h = h*.
T h a t ~0 is uniquely determined from h and ~1, ~3 .... follows from Corollary 2.
Let h =h* be given. To show t h a t eq.. (7.1) does have a solution we will show that the graph in Fig. 3 is quahtatively correct. Sinc e ~ is symmetric we see that u ( z ) ~ - 1 and o5-->0 as 207~2r Thus the right-hand side of (7.1) converges t o - co when ~0~]~. We will now show that the right-hand side of (7.!) tends toward + oo as ~,--> - oo. Since 05 = - ~ f i - fi' we can rewrite eq. (7.1) as
" ' " 1 ' "
Consider the auxiliary equations ~0" + (l~01 -+.
J[, ll
= 0 with initial conditions ~0 -~ 1 and~0: = / / a t x =0. If 20 is less than - ( 3 ] ~ I +0.1) 3 -Jl~lloo then it !ollows from Sturm's compari- son theorem t h a t
1 ~ 2
~ < - ~<- . ( 7 . 3 )
W e will~ n o w show :that ,1~2' is less t h a n (4]~:}](~26 I':-~ [I qH o~) when :-~:2o~iS sufficiently large: ]~y combining~ this Tesutt with (,7.2): and (7.3) we see that the right-hand side o f (7A,)'
THE INVERSE STURM--LIOUVILLE PROBLEM WITH SYMMETRIC POTENTIALS 289 tends to + o~ as 2o-+ - ~ , and the proof will be complete. Let q be a bounded function and let q0 be the solution of q0"= ( q - 2 ) ~ with initial conditions ~0 = 1 and q0' = h at x = 0.
We will' assume t h a t
Let ~v =qr T~ estimate q a n d to we use t h e method of Successive approximations and define
(Pn+l = 1~+
f[~,dt
If ~o a n d ~P0 are conHnuous funcgions theh~n and ~bn coriVerge uniformly to ~'and y~/Specifi:
cMly we' choose
% = i + h x + ~-x ~
~po=h+ ~x
O~where g = : [ 2 [ = [ [ q ~ >~0. T h u S : ~ l - ~ o . I t follows f r o m ( 7 . 4 ) t h a t ~0 > 0..,Sjn,oe [~[ -~q>
we can estimate : ~ by
, x x[ 3h~
Thus Yh >~Oo and we find b y induction thab "~n ~>~o and ~Pn ~>~o and consequently ~ >~0 and v2>~o. B y evaluating ~o.at x=7~ and using (7 4) we see finally t h a t 1/~(~.) is less t h a n 4/(7ca). This completes the proof.
The proof of Lemma 2 provides a numerical method for solviiig the inverse S t u r m - Liouville problem with prescribed boundary conditions. A more practical ,technique is to find the ;root 2~'~f t h e nonlinear equati0n i n ' S t e F 4 ~ see Secti6rt:4. Here we nOte that the calculations in Step 2~ need only be perforzned once for each,j > 0, and t h a t the modifica- tions in Step 3 ~ a r e trivial:
The resultw presented in this section ar e incomplete. : w e have not ~rbved a global well-posedness result like T h e o r e m 2 nor a local existence result 5ke Theorem 3, In the case h=)~ Borg has obtained a local existence and well-posedness reSu{t; see :[3, "~). 71]. The graph i n Fig. 3 indicates that if h - h is very Iarge then the lowest eigenvaIue 20 is quite sensitive to small changes ofvh.
290 O.H. HALD 8. F u r t h e r results
The theory for the inverse Sturm-Liouville problem with fixed boundary conditions is in m a n y ways simpler t h a n the theory for mixed boundary conditions. The reason is be of interest to discover the weakest possible assumptions on the eigenvalues under which Theorem 1 is valid. The representation theorem leads naturally to Hochstadt's algorithm, see [9]. Since the boundary conditions are fixed there is no need to modify the original the National Science Foundation under Grant No. MCS75-10363 A01.
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Received December 20, 1976
Received in revised ]orm M a y 5, 1978
1 8 t - 782902 Acta mathematica 141. Imprim6 le 8 D~cembre 1978