Electroic Joural of Differetial Equatios, Vol. 7 (7, No. 4, pp. 7. ISSN: 7-669. URL: http://ejde.math.txstate.edu or http://ejde.math.ut.edu AN INVERSE STURM-LIOUVILLE PROBLEM WITH A GENERALIZED SYMMETRIC POTENTIAL ESİN İNAN ESKİTAŞÇIOĞLU, MEHMET AÇİL Commuicated by Mokhtar Kirae Abstract. We cosider the ormal form of Sturm-Liouville differetial equatios with separable boudary coditios. For this problem we kow that the potetial fuctio is determied uiquely by two spectra ad that if the potetial is symmetric, the it is determied uiquely by just oe spectrum. I this paper firstly we geeralize symmetric potetial ad the ivestigate chage of eeded data to determie potetial fuctio uiquely.. Itroductio The iverse Sturm-Liouville problem was firstly studied by Ambartsumya i 99. He cosidered the boudary value problem y (x + (λ + q(xy(x =, y ( = y ( = ad showed that if eigevalues of the problem are λ = π, the the potetial fuctio q ]. This work aturally led to questio whether the potetial is determied uiquely by oe spectrum or ot. This problem is called the iverse Sturm- Liouville problem. Borg showed that it is ot true geerally ad that two spectra are eeded to determie the potetial uiquely by chagig oe of boudary coditios. He also showed that if the potetial q is symmetric about midpoit of the iterval, π] for his problem, the it is determied uiquely by oe spectrum i 949 3]. Leviso shorteed the proof of Borg by usig complex aalysis techiques]. From the o, various versios of Borg s work have bee cosidered. Some of them ca be see i 6, 8, 9]. The study we coducted here is motivated by,, 3]. Firstly we eed to give some theorems ad defiitios. The structure of BVP we cosidered is as follows: Ly] = µy, (. y ( hy( =, (. y (a + Hy(a =, (.3 Mathematics Subject Classificatio. 3A5, 34B4, 34L6. Key words ad phrases. Boudary value ad iverse problems; Sturm-Liouville theory; umerical approximatio of eigevalues. c 7 Texas State Uiversity. Submitted December, 6. Published February 7, 7.
E. İ. ESKİTAŞÇIOĞLU, M. AÇİL EJDE-7/4 where Lu] = u + qu ad h, H, a are real costats. The operator L is a selfadjoit operator defied o L, a] provided that q satisfies suitable regularity coditios. Uder these coditios L has a discrete spectrum cosistig of the simple ad real eigevalues {µ i } i= ]. Let us cosider (. with the iitial coditios y( =, y ( = h. (.4 Theorem.. Suppose that φ(x, λ is the solutio of (. satisfyig the iitial coditios (.4 ad that q(x has m locally itegrable derivatives. The there exists fuctio K(x, t havig m + locally itegrable derivatives with respect to each of variables such that φ(x, λ = cos λx + x K(x, x = h + K(x, t cos λtdt, (.5 x q(tdt. (.6 The above theorem ad other properties of fuctio K(x, t ca be foud i ]. Lemma.. Let (a, b be a fiite iterval ad f(x L(a, b. The b lim λ a f(x cos λxdx =. The above lemma ad its proof ca be foud i 7]. Lemma.3. If a real sequece {µ } is the spectrum of BVP (.-(.3 with fuctio q(x where q L (, a, the it satisfies the asymptotic formula µ = π a + aa + o( (.7 where µ µ m for m ad a = (K(a, a + H(/(aπ. Proof. By usig the boudary coditios directly we derive µ. From Theorem., we have solutio (.5. By cosiderig (.3 we obtai µ si µa + + H cos µa + a a K(x, t x cos µtdt + K(a, a cos µa K(a, t cos µtdt ] =. (.8 The umbers {µ } are the roots of equatio (.8. Sice λ as the first approximatio for µ s is obtaied as follows: si µ a + O ( µ =. Therefore Now let us put The ( aa + γ a + O ( µ = π a + O(. µ = π a + aa + γ. ] 3 ( K(a, a + H] ( π +a a +γ a a π + O ( ]
EJDE-7/4 AN INVERSE STURM-LIOUVILLE PROBLEM 3 a µ HK(a, t + x K(x, t x=a ] cos µ tdt = from (.8. Sice HK(a, t + x K(x, t x=a ] L (, a, by usig Lemma. it follows that as a µ HK(a, t + x K(x, t x=a ] cos µ tdt Therefore a = K(a,a+H aπ ad γ as. This completes the proof. Remark.4. There is o ã differet from a such that asymptotic formula (.7 is admitted by the problem (.-(.3 with ã istead of a. Let us cosider a fuctio f :, ] R. If for x, ] ad =,,..., k, ( f(x = f x, the we will call the fuctio f as kth-order symmetric fuctio. Example.5. Defie the fuctio x + x +. < x <.5, x x +.76.5 < x <.5, f(x = x.4.5 < x <.75, x 3x +..75 < x <, The f is a secod-order symmetric fuctio. It ca be see i Figure reffig. Figure. A secod-order symmetric fuctio.
4 E. İ. ESKİTAŞÇIOĞLU, M. AÇİL EJDE-7/4 I this part we cosider the BVP. Mai results y (x + (λ q(xy(x =, (. y ( hy( =, y ( + Hy( =. (. Theorem.. Let be q L, ], q is a secod-order symmetric fuctio ad h = H. The for (.-(. the potetial q is determied uiquely by its half spectrum except fiite umbers of its eigevalues. Proof. To prove this theorem let us cosider the BVP y (x + q(xy(x = λy(x, (.3 y ( h y( =, y (/ + H y(/ =. (.4 From the theory about symmetric potetial we kow that for all x, ] whe q is symmetric about midpoit, oe spectrum is sufficiet to determie potetial q uiquely. Furthermore sice q(x = q( x for all x, /], the same situatio holds for (-(. We ited to show that the spectrum of (-( represets half spectrum of the problem (.-(. except for a fiite umbers of its eigevalues. From Lemma. we see that spectrum of (-( has the asymptotic form ( λ k = kπ + ã k + o k ad that the spectrum of (.-(. has also asymptotic form λk = π + a ( + o. Also we have ã = K(/, / + H/ = ] K(/, / + H π/ π = π = π ( h + / ( H + h + = π H + K(, ] = a ] q(xdx + H ] q(xdx where K ad K are kerels of solutios which represet (.-(. ad (-( respectively. Havig the above results we obtai λ ( λ = o for large. This completes the proof. Corollary.. Let be q L, ], q is a kth-order symmetric fuctio ad h = H. The for (.-(. the potetial q is determied uiquely by its spectrum k except for a fiite umbers of eigevalues of spectrum. The above corollary ca be show easily by usig iductio with the above process. This result leads to a questio: What is the situatio whe q is a ifiiteorder symmetric fuctio? Our related theorem is as follows.
EJDE-7/4 AN INVERSE STURM-LIOUVILLE PROBLEM 5 Theorem.3. Let q C, ], H = h ad q be a ifiite-order symmetric fuctio. The for problem (.-(. the potetial fuctio q is determied uiquely by just its oe eigevalue. Proof. Sice q C, ] from Weierstrass approximatio theorem (see ] there exists polyomial P with degree m o a, b] which represets potetial q. The for k it follows that P (x = P ( x. By rewritig this for k =,,..., ad k the addig those for P (x = c + c x + + c m x m we obtai P (x = { c + c ( x + ( x + + ( ] +... + c m ( x m + + ( m]} = c + m c k ( x k + + ( x k]. k= By takig limits for the geeral term of the sum as it follows that ( x k + + ( x k lim = lim { ( + k + + ( k x ( + k + + ( k + + ( k x k = ( k x k by usig Stolz-Cesáro theorem for each term i parethesis. We ca assume that m = k without loss of geerality. Thus we see that P (x = c +c x + +c k x k. Besides we have P ( = P ( for x =. So for =, 3,..., k + we obtai 4... k c 4 8... 4k c 4..... =.. k 4k... k By usig the LU decompositio 5] for coefficiet matrix, say A, it follows that k 4k 6k... k 3 4k+ 6k+... det(a = 45 6k+6........... M k k = ( k ( 4 k... ( (k k(k+(k+ 3 where M = ( ( 4... ( (k. So c k c = c 4 = = c k =. Cosequetly q(x = c. Fially by usig (.-(. with the fact h = H we see that potetial q ca be determied by just its oe eigevalue. }
6 E. İ. ESKİTAŞÇIOĞLU, M. AÇİL EJDE-7/4 Coclusio. I this study, we cosidered the variatio of eeded data to be the spectrum of the operator to determie uiqueess of the potetial fuctio that has property q(x = q( x by chagig atural umbers. I the recet times the studies which iclude recostructio of the potetial have bee coducted. Some of them ca be see i 4, 5, 4]. For a applicatio of this study, Theorem., Corollary., ad Theorem.3 ca be cosidered. We thik that oe ca obtai a approximatio to the potetial by usig the sets {λ }, {λ, λ }, {λ, λ, λ 3, λ 4 },.... It is clear that obtaiig a good approximatio is difficult. However it ca be cosidered to be a iitial potetial i umerical algorithms like the algorithm described i 4]. Refereces ] Ambartsumya, V. A.; Abereie Frage der Eigewerttheorie, Zeitschrift fúr Physik, 53 (99, pp. 69-695. ] Bidig, P. A.; Browe, P. J.; Watso, B. A.; Iverse spectral problems for left-defiite Sturm- Liouville equatios with idefiite weight, Joural of Mathematical Aalysis ad applicatios, 7 (, pp. 383-48. 3] Borg, G.; Eie Umkehrug der Sturm-Liouvillesche Eigewertaufgabe. Bestimmug der Differetialgleichug durch die Eigewerte, Acta Mathematica, 78 (946, pp. -96. 4] Efremova, L. S.; Freilig, G.; Numerical solutio of iverse spectral problems for SturmLiouville operators with discotiuous potetials, Cet. Eur. J. Math. ( 3, pp. 44-5. 5] Freilig, G.; A Numerical Algorithm for Solvig Iverse Problems for Sigular SturmLiouville Operators, Advaces i Dyamical Systems ad Applicatios. ISSN 973-53 Volume Number (7, pp. 955. 6] Gelfad, I. M.; Levita, B. M.; O the determiatio of a differetial equatio from its spectral fuctio, Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya, 95, 5 (4, pp. 39-36; 955. traslated i America Mathematical Society Traslate,, 53. 7] Gradshtey, I. S.; Ryzhik, I. M.; Tables of Itegrals, Series, ad Products, 6th ed. Sa Diego, CA: Academic Press,, pp.. 8] Hoschtadt, H.; The iverse Sturm-Liouville Problem, Commuicatios Pure ad Applied Mathematics, 6 (973, pp. 75-79. 9] Hoschtadt, H.; Lieberma, B.; A Iverse Sturm-Liouville problem with mixed give data, SIAM Joural o AppliedMathematics, 978, 34(4, pp. 676-68. ] Jeffreys, H.; Jeffreys, B. S.; Methods of Mathematical Physics 3rd ed., 988, pp. 446-448, Cambridge Uiversity Press, Cambridge, Eglad. ] Leviso, N.; The Iverse Sturm-Liouville Problem, Mat. Tideskr. B., 5 (949, pp. 5-3. ] Levita, B. M.; Gasymov, M. G.; The determiatio of a differetial equatio by two its spectra, Uspekhi Matematicheskikh Nauk, 964, 9 (, pp. 3-63. 3] McLaughli, J. R.; Aalytical methods for recoverig coefficiets i differetial equatios from spectral data, SIAM Review, 8 (986, pp. 53-7. 4] Röhrl, N.; A Least Squares Fuctioal for Solvig Iverse Sturm-Liouville Problems, Dyamical Iverse Problems: Theory ad Applicatio; Volume 59 of the series CISM iteratioal ceter for mechaical scieces,, pp. 63-83. 5] Vadeberghe, L.; Lecture Notes o Applied Numerical Computig, The EE3 lecture otes ad course reader for the - Fall Quarter. Esi İa Eskitaşçioğlu Departmet of Mathematics, Faculty of Scieces, Yüzücü Yıl Uiversity, 658, Va, Turkey E-mail address: iaciar@myet.com Mehmet Açil Departmet of Mathematics, Faculty of Scieces, Yüzücü Yıl Uiversity, 658, Va, Turkey E-mail address: mehmetacil76@gmail.com