Remarks on a New Inverse Nodal Problem

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1 Joural of Mathematical Aalysis ad Applicatios 48, 4555 doi:6maa6878, available olie at o Remars o a New Iverse Nodal Problem Ya-Hsiou Cheg, C K Law, ad Jhishe Tsay Departmet of Applied Mathematics, Natioal Su Yat-se Uiersity, Kaohsiug, Taiwa 84, Republic of Chia egyh@mathsysuedutw, law@mathsysuedutw, tsay@mathsysuedutw Submitted by Fritz Gesztesy Received October 5, 999 I a recet paper, X F Yag proved a uiqueess theorem o iverse odal problems that lis to iverse spectral theory, o oe had, ad reduces the redudacy of the classical iverse odal problems, o the other had I this ote we improve Yag s theorem by weaeig its coditios ad simplifyig its proof We also discuss variats of Yag s theorem Academic Press INTRODUCTION Cosider the SturmLiouville problem subect to boudary coditios y qž x y y, yž cos y Ž si, y cos y si, where q L Ž, ad,, Let be the th eigevalue of the operator, ad let x x be the odal poits of the th Ž Ž eigefuctio Also let s ', ad let l x x be the associated odal legth The iverse odal problem, first studied by McLaughli, is the problem of fidig the potetial q ad boudary coditios, usig oly the set of odal poits x 4 i It seems to be a good iverse problem, for ot oly is Ž q,, uiquely determied by the odal set up to a costat, 5,, 4, it ca also be recostructed from the odal set 5, 8, 4 Furthermore, 45-47X $35 Copyright by Academic Press All rights of reproductio i ay form reserved

2 46 CHENG, LAW, AND TSAY although the iverse odal problem is overdetermied, it turs out to be well-posed after a partitio 9 The developmet of the iverse spectral theory, its close relative, is quite differet The set of eigevalues, i geeral, is ot sufficiet to determie the operator, represeted by Ž q,, Oe eeds more data to determie Ž q,,, for example, two sets of eigevalues, oe set of eigevalues plus a symmetric potetial fuctio 6, ad oe set of eige- values plus owledge of q o Ž, 7 The iterested reader may read for a survey Recetly Gesztesy ad Simo also produced a umber of ew results 3, 4, Usig oe of Gesztesy ad Simo s results, Yag proved a iterestig theorem 5 o iverse odal problems which, o oe had, maes use of iverse spectral theory ad, o the other had, reduces the redudacy of odal data I short, he showed that the set of all odal poits i the subiterval Ž, b Ž b is sufficiet to obtai uiqueess The purpose of this ote is to improve Yag s theorem 5, Theorem, simplify its proof, ad also discuss various implicatios of the theorem Fix b ad, associated with the SturmLiouville problem defied by Ž q,,, let S 4 be a strictly icreasig sequece of positive itegers Suppose TŽ S Ž, : S,,,4 Let A TŽ S be such that for ay S, there is some such that Ž, A Note that there might be oly oe choice of Defie BŽ A to be a subodal set o Ž, b if ½ 5 BŽ A x Ž, b : Ž, A Clearly B A depeds o the problem q,,, the iterval, b, as well as S ad A DEFINITION We say that Ž a BŽ A is twi o Ž, b if for ay, there is a pair of adacet Ž Ž odal poits x ad x cotaied i BŽ A Ž b BŽ A is dese o Ž, b if clžbž A, b Ž c BŽ A is S-dese o Ž, b if there is a subsequece of S Žalso deoted by 4 such that for ay x Ž, b there exists i 4 such that Ž Ž xi B A such that lim xi x Obviously, if BŽ A is S-dese o Ž, b, the it is dese

3 NEW INVERSE NODAL PROBLEM 47 Throughout this paper, is reserved for the first idex of the twi odal poits for the th eigefuctio, i the case of twi subodal sets We shall also let 4 ad x 4 stad for the spectrum ad odal set for the SturmLiouville problem defied by Ž q,, THEOREM Tae b Assume that BŽ A is twi ad dese o Ž, b about Ž q,, If for ay, there is Ž, such that Ž Ž x x, Ž for all Z such that x BŽ A, the except fiitely may s, Also, Here C is a real costat,,, C, ad q q C ae o Ž, b Note that i 5, Lemma which is a ey lemma for Yag s theorem, BŽ A was required to be S-dese istead of dese We also ote that b ca be arbitrarily small Our proof will be simpler ad more direct It will be give i Sectio I Sectio 3, we shall discuss the applicatios of this theorem PROOF OF MAIN THEOREM We first quote some lemmas for the asymptotic estimates of s, the odal poits x 4, ad odal legths l 4 correspodig to Ž q,, The followig lemma is reproduced from 8, Lemma ; 4, Lemma 8 Note that there are some typo errors i 8 LEMMA Suppose that q,, L Ž,, Let if, ad if Ž a If or,, the as ž s scot scot H Ž cosž t qž t dt O '

4 48 CHENG, LAW, AND TSAY b If or, the as ž s scot scot H Ž cosž Ž t qž t dt O Here scot if ; cot otherwise LEMMA Suppose q,, L,, The as, with, Ž a o ž, if Ž I or l o, if Ž II or b O, if Ž Ia ž ž ž O, if Ž Ib x O, if Ž IIa O, if Ž IIb The lemma above ca be derived from 8, Lemma Part Ž a of Theorem 3 is due to 4, Theorem ; part Ž b is due to 8

5 NEW INVERSE NODAL PROBLEM 49 THEOREM 3 The recostructio formulas of Ž q,, from the odal set are gie by: Ž Ž a i Either, or with i tedig to, ½ lim Ž i x i, if lim i x Ž Ž i, if cot 3 ii Either, or with i tedig to, ½ lim Ž x i, if lim i x Ž Ž i, if cot Ž b i If, case I cf, the for ae x,, with x coerget to x, ½ 5 l qž x lim l H q scot scot 4 Ž ii If Ž, case Ž II, the for ae x Ž,, with x co- erget to x, l ž H q x lim ½ 5 l q scot scot 5 Here scot if ; cot otherwise LEMMA 4 Let b Suppose BŽ A is twi o Ž, b, ad for all, there is Ž, such that The except for fiitely may s, Ž Ž Ž Ž x x ad x x Ž, Ž,

6 5 CHENG, LAW, AND TSAY Moreoer, the boudary coditios Ž, ad Ž, belog to the same subcase i Proof By the asymptotic expressio of l i, i both cases, Similarly, l Ž x Ž x Ž, ž O, o Ž l o ž Thus give, whe is sufficietly large, ad Hece Ž l, Ž l This is valid for ay So we have lim 6 Next, we show that Ž, ad Ž, belog to the same case i Suppose ot; say, Ž, belogs to case Ž I while Ž, belogs to case Ž II The as, by, o o

7 NEW INVERSE NODAL PROBLEM 5 Thus by 6, o This is impossible, sice ad, N Hece Ž, ad Ž, have to belog to the same case, say, case Ž I So which implies, o o, ož Therefore if is sufficietly large, the Ž Now comparig the asymptotic expressio for x ad x i, we coclude that Ž, ad Ž, belog to the same subcase i ad for sufficietly large Proof of Theorem First we apply Lemma 4 ad the recostructio formula 3 for cot to see directly that The we see that, from the asymptotic expressios of s Ž ' Lemma, we have scot scot q o if Ž, case Ž I Ž H scot scot q o if Ž, case Ž II Now that, we have, whe is large eough, where H C ož, H C scot scot Ž q q Let ad be the ormalized eigefuctios for the SturmLiou- ville problems defied by Ž q,, ad Ž q,,, respectively, such that Ž si, cos

8 5 CHENG, LAW, AND TSAY The, by the Lagrage idetity, whe is large, x x x ž Ž x qž x qž x Ž x Ž x 7 Fix x Ž, b Sice BŽ A is dese i Ž, b, the either there is a Ž sequece of odal poits x BŽ A coverget to x or x BŽ A i Ž Hece we may itegrate 7 from to x Ž or x for the later case i to obtai Ž x i H ž x Ž H i t t t t dt, qž t qž t C o Ž t Ž t dt 8 It is ow that, from 3, Lemma 7 ad s s OŽ, cosž s x cosž s x si O,, ž Ž x Ž x siž s x siž s x O,, 3 Ž Ž Ž Ž 3 cos s x si O,, cos s x O, So, lettig i 8, we have x H Ž q q C Therefore q q C ae o Ž, b Fially, whe is large, cosider the SturmLiouville problems o the Ž Ž Ž subiterval betwee the twi odal poit x, x y qy y ad y Ž q C y Ž C y such that y x Ž y x Ž ž ž

9 NEW INVERSE NODAL PROBLEM 53 By the uiqueess of the first eigevalue of this problem, C Remar The latter part of the above proof is due to Yag It would be iterestig to ow if there is a alterative method other tha the Lagrage idetity Defie BŽ A to be T-dese o Ž, b if BŽ A is twi ad for all Ž Ž x, b, there is a subsequece of twi odal poits xi ad x i coverget to x I this case, l Ž i is well defied; we ca use the recostructio formula 4 or 6 to show directly the uiqueess of the potetial fuctio o Ž, b The proof of the theorem below follows the same lie as the above proof, ad will be omitted COROLLARY 5 Tae b Suppose BŽ A is twi ad dese o Ž,b If for all, Ž Ž x x, Ž for all Z such that x BŽ A, the, q q C ad C, for all Here C scot scot H Ž q q 3 APPLICATIONS I this sectio, we ivestigate several applicatios of Theorem Hereafter i this paper, we cosider two SturmLiouville problems de- fied by q,, ad q,, i L,, such that Hq Hq We shall explore sufficiet coditios to obtai q q ae o Ž, The first case is a aalogue of a theorem of Hochstadt ad Lieberma which states that if for all N ad q q ae o Ž,, the q q ae o Ž, THEOREM 3 Tae b Assume, q q ae o Ž b, Suppose BŽ A Ž formed from ay S is twi ad dese o Ž, b If for ay sufficietly large, there is some Ž, such that Ž Ž x x, Ž for all Z, such that x BŽ A, the Ž q, Ž q, i L Ž,,

10 54 CHENG, LAW, AND TSAY Proof By Theorem, ad q q C, where C Žscot scot H q q Now, ad Hq Hq Therefore q q ae The secod applicatio is a improvemet of Yag s theorem THEOREM 3 Let b ad b Assume BŽ A is twi ad dese o Ž, b ad for sufficietly large, 3 : 4 Ž If for ay sufficietly large N, there is Ž, such that Ž Ž x x, for all Z such that Ž, A, the Ž q,, q,, i L Ž,, Yag s theorem requires BŽ A to be S-dese istead of dese Our proof is the same as that of 5, Theorem buildig o Theorem ad usig a iverse spectral theorem by Gesztesy ad Simo 3, Theorem 3, which says that owledge of q o, for some Ž,,, ad a set S N such that S : 4 Ž N : 4, for all sufficietly large R, uiquely determie ad q o all of Ž, A third applicatio maes use of Borg s theorem, which says that if q ad q i L Ž, correspod to the same set of eigevalues for two differet boudary coditios, the q q ae THEOREM 33 Tae ay b Ž, ad S N 4 Assume that the subodal sets BŽ A ad BŽ A thus formed Žcorrespodig to Ž q,, ad Ž q,,, respectiely are both twi ad dese o Ž, b If for ay idex Ž,, x B Ž A Ž Ž Ž implies x x ad y B Ž A implies Ž Ž Ž y y Žwhe x ad y are odal poits of q correspodig to Ž, ad Ž,, the q q ae i Ž, Proof Let Ž ad Ž be the eigevalues for q Ž q corre- spodig to the boudary coditios Ž, ad Ž,, respectively Sice Hq Hq, ad the boudary coditios are the same, we have, from Corollary 5, ad, for all N Thus by Borg s theorem, q q ae o Ž,

11 NEW INVERSE NODAL PROBLEM 55 We ote that i above theorem, the iterval Ž, b ca be arbitrarily small, ulie Theorem 3 Similarly, we also have a iverse odal theorem applyig aother iverse spectral theorem of Gesztesy ad Simo, which says that owledge of q o Ž 34, plus oe full set of eigevalues for some boudary coditios ad half the set of eigevalues for some other boudary coditios determie q uiquely It would be iterestig to obtai recostructio formulas for Ž q,, for the various iverse odal problems discussed above ACKNOWLEDGMENTS We tha Professor X F Yag for sedig his paper 5 to us We also tha Hua-Huai Cher for helpful discussios CKL ad JT are partially supported by the Natioal Sciece Coucil, Taiwa, Republic of Chia REFERENCES G Borg, Eie Umehrug der SturmLiouvillesche Eigewertaufgabe, Acta Math 78 Ž 946, 96 P J Browe ad B D Sleema, Iverse odal problems for SturmLiouville equatios with eigeparameter depedet boudary coditios, Ierse Problems Ž 996, F Gesztesy ad B Simo, Iverse spectral aalysis with partial iformatio o the potetial, II The case of discrete spectrum, Tras Amer Math Soc 35 Ž, F Gesztesy ad B Simo, O the determiatio of a potetial by three spectra, Amer Math Soc Trasl Ser 89 Ž 999, O H Hald ad J R McLaughli, Solutios of iverse odal problems, Ierse Problems 5 Ž 989, H Hochstadt, The iverse SturmLiouville problem, Comm Pure Appl Math 6 Ž 973, H Hochstadt ad B Lieberma, A iverse SturmLiouville problem with mixed give data, SIAM J Appl Math 34, No 4 Ž 978, C K Law, C L She, ad C F Yag, The iverse odal problem o the smoothess of the potetial fuctio, Ierse Problems 5 Ž 999, C K Law ad Jhishe Tsay, O the well-posedess of the iverse odal problem, submitted J R McLaughli, Aalytic methods for recoverig coefficiets i differetial equatios from spectral data, SIAM Re 8 Ž 986, 537 J R McLaughli, Iverse spectral theory usig odal poits as dataa uiqueess result, J Differetial Equatios 73 Ž 988, R del Rio, F Gesztesy, ad B Simo, Iverse spectral aalysis with partial iformatio o the potetial, III Updatig boudary coditios, Iterat Math Res Notices 5 Ž 997, E C Titchmarsh, Eigefuctio Expasios, part I, d ed, Oxford Uiv Press, Lodo, 96 4 X F Yag, A solutio of the iverse odal problem, Ierse Problems 3 Ž 997, 33 5 X F Yag, A ew iverse odal problem, J Differetial Equatios, to appear