Inverse Nodal Problems for Differential Equation on the Half-line

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1 Australia Joural of Basic ad Applied Scieces, 3(4): , 2009 ISSN Iverse Nodal Problems for Differetial Equatio o the Half-lie A. Dabbaghia, A. Nematy ad Sh. Akbarpoor 1 Islamic Azad Uiversity, Neka Brach, Neka, Ira Departmet of Mathematics, Uiversity of Mazadara, Babolsar, Ira 3 Islamic Azad Uiversity, Joubar Brach, Joubar, Ira 2 Abstract: By usig of the Jost soluthio of the Sturm-Liouville equatio o the half-lie we obtai odal poits ad odal legths. Furthermore, by usig odal poits we have show that the potetial fuctio ca be established uiquely. Key words: Nodal poits; Iverse problem; Asymptotic form; Eigevalue; Sturm-Liouville. Mathematics Subject Classificatio: 34B24, 34L20. INTRODUCTION Let us cosider the Sturm-Liouville problem 2 q(x) L(0, ) is a complex-valued fuctio ad h is a complex umber. Let ë = ñ, ñ = ó + iô ad let for defiiteess ô := Imñ 0. Iverse odal problems cosist i recoverig operators from give odes(zeros) of their eigefuctios. From the physical poit of view this correspods to fidig, e.g., the desity of a strig or a beam from the zero-amplitude positios of their eigevibratios. Mclaughli seems to be the first to cosider this sort of iverse problem (Mclaughli,J.R., 1988). Later o, some remarkable results were obtaied. For example, X.F. Yag got the uiqueess for geeral boudary coditios usig the same method as Mclaughli (Yag, X.F., 1997); C.K. Law ad Chig-Fu Yag (Koyubaka, H., 2006) have recostructed the potetial fuctio ad its derivatives from odal data. The iverse odal problems were studied for secod-order differetial equatios with a liear depedece o the spectral parameter (Koyubaka, H., 2006, Shieh, C.T., V.A. YurkoIverse 2008). I (Freilig, G., V.A. Yurko, 2001) authors cosidered iverse problem of (1)-(2) they proved a expasio theorem ad solved the iverse problem of the Sturm-Liouville equatio from its Weyl fuctio. I this work, we ivestigate iverse odal problem for the boudary value problem (1)-(2)(i.e., we will cosider iverse problems of recoverig h ad q(x) from the give odal characteristics). 2 Iverse Nodal Problems: The study of iverse odal o a fiite iterval cosidered by O.H. Hald (1998) ad J. R. Mciaughli (1988). I this sectio, we cosider the iverse odal problems o the half-lie. Put (1) (2) Theorem 1. Equatio (1) has a uique solutio y=e(x,ñ), ñ Ù, x 0, satisfyig the itegral equatio (3) Correspodig Author: A. Dabbaghia, Islamic Azad Uiversity, Neka Brach, Neka, Ira a.dabbaghia@umz.ac.ir 4498

2 The fuctio e(x, ñ) has the followig properties: (i) For x, ad each fixed ä > 0, Aust. J. Basic & Appl. Sci., 3(4): , 2009 (4) uiformly i Ù ä. (1)havig this property.. Moreover, e(x, ñ) is the uique solutio of (ii) uiformly for x 0. (v) (iii) For each fixed x 0, ad v = 0, 1, the fuctios e (x, ñ) are aalytic for Im ñ > 0, ad are cotiuous for ñ Ù. (iv) For real ñ 0, the fuctios e(x, ñ) ad e(x, ñ) form a fudametal system of solutios for (1), ad is the Wroskia. The fuctio e(x,ñ) is called the Jost solutio for (1). proof. (Freilig, G., V.A. Yurko, 2001). By virtue of Theorem 1, the fuctio Ä (ñ) is aalytic for Imñ > 0, ad cotiuous for ñ Ù. It follows from (5) that for (5) (6) (7) (8) Substitutig (8) ito the relatio Ä(ñ) = 0, we get (9) I the followig Theorem we obtai asymptotic expressios for the poits (j=1,2,...,- 1, =1,2,...) at which y, the eigefuctio correspodig to the eigevalue ë of the problem (1)-(2), vaishes. Theorem 2. We cosider the boudary value problem (1)-(2). The, the odal poits of the problem (1)-(2) are 4499

3 (10) ad the odal legth is (11) proof. Let us suppose that (12) By (12) the Jost solutio has the represetatio ad kerel k(x, t) is expressed i terms of q([5,chapter3,sectio1]) k(x, t) is cotiuously differetiable with respect to their argumets ad (13) (14) (15) (16) The solutio e(x, ñ) also satisfies (17) see[7,appedix II]. Hece, we use the classical estimate (18) M is a costat. Thus e(x, ñ) will vaish i the itervals whose ed poits are solutios to. This equatio ca also be writte as (19) After some straightforward computatios, we get 4500

4 the, expadig arccos(m), we obtai that (20) The odal legth is (21) this completes the proof. Now, we will give a uiqueess theorem. Theorem 3. Suppose that q is itegrable. The h ad are uiquely determied by ay dese set of odal poits. proof. Assume that we have two problems of the type (1)-(2) with. Let the odal poits satisfyig from a dese set i [0, ). We take solutios of (1)-(2) as ö for (h,q) ad. It follows from (1) that. (22) Let. To show that h =, we itegrate both sides of (22) from 0 to ad usig the boudary coditio (2) we obtai (23) We ote that are uiformly bouded i ad the are uiformly bouded i ad x [0, ). We ow select a subsequece of odes from the dese set. If the subsequece teds to zero, the the rights side of (23) is equal to zero. Hece we get h =. 4501

5 We take a sequece accumulatig at a arbitrary x [0, ) ad usig the above techique, sice h = From the asymptotic forms of we have We take a sequece accumulatig at a arbitrary x [0, ). Hece ad this holds for all, we ca therefore coclude that is uiquely determied by a dese set of odes. This completes the proof. corollary 4. For the problem (1)-(2) the potetial q is uiquely determied by a dese set of odes ad the costat proof. Suppose that. Sice h =, it follows that. Hece, we ca coclude from Theorem 3 that q = almost every o [0, ). REFERENCES Freilig, G., V.A. Yurko, Iverse Sturm-Liouville problems ad their applicatios, NOVA sciece publishers, New York. Hald, O.H., J.R. Mclaughli, Solutios of iverse odal problems, Iverse Problem, 14: Koyubaka, H., A New iverse problem for the diffusio operator, Applied Mathematics Letter, 19: Law, C.K., Chig-Fu Yag,Recostructig the potetial fuctio ad its derivatives usig odal data, Iverse Problem,14: Marcheko,V.A., Sturm-Liouville operators ad applicatios, Birkhauser Verlag, Basel. Mclaughli,J.R., Iverse spectral theory usig odal poits as data-a uiqueess result, J. Differetial Equatios, 73: Naimark, M.A., Liear Differetial operators, New York. Shieh, C.T., V.A. YurkoIverse odal ad iverse spectral problems for discotiuous boudary value problems, J. Math. Aal. Appl., 347: Yag, X.F., A solutio of the iverse odal problem, Iverse Problem,13: