A Uniqueness the Theorem for Singular Sturm-Liouville Problem

Transcription

1 C.Ü. Fen-Edebiyat Fakültesi Fen Bilimleri Dergisi 3Cilt Sayı A Uniqueness the Theorem for Singular Sturm-Liouville Problem Etibar S. PANAKOV and ikmet KOYUNBAKAN Firat University Department of Mathematics 39 Elazığ / TURKEY epenahov@hotmail.com hkb63@yahoo.com Received:9.3. Accepted: 6.3. Abstract. In this paper we show that If q is prescribed on the ] suffices to determine q on the interval then the one spectrum.the potential function q in a Sturm Liouville problem is uniquely determined with one spectra by using the ochstadt and Lieberman s method []. Key Words: Sturm-Liouville problem Spectrum Singüler Sturm-Liouville Problemi için Teklik Teoremi Özet: Bu makalede gösterdi ki q ] aralığında tanımlanmış ise aralığı üzerinde q fonksiyonunu belirlemek için bir spektrum yeterlidir. Sturm-Liouville probleminde q potansiyel fonksiyonu ochstadt ve Lieberman metodu kullanılarak bir spektruma göre tek olarak belirlenir. Anahtar Kelimeler: Sturm-Liouville problem Spectrum Introduction. In this paper we shall be concerned with an inverse Sturm-Liouville operator. We consider the operator / y q y y

2 with the boundary conditions y lim / Γ y cos β y sin β. 3 The operator L is Self-Adjoint on the conditions has a discret spectrum { So we obtain a new spectrum { n}. n } L [. If condition 3 is replaced by ] and with 3 boundary y cosγ y sinγ. In this paper we will consider a variation of the above inverse problem in that we will not require any information about a second spectrum but rather suppose q is known almost everywhere on. This information together with the spectrum { n } of the problem 3 will be shown to determine q uniquely on ]. Theorem : We get the operator with the boundary conditions and 3. Let be the spectrum of L with and 3. Consider a second operator y q where q is summable on the interval ] and on the interval / y y { } n q q 6. Suppose that the spectrum of L with the 3 is also { n }. Then q q almost everywhere on ]. Proof : Before proving the theorem we will first mention some results which will be need later. We take the following problems and / y q y y 7 y lim / Γ 5 8

3 y q y lim / Γ / y y 9 As known [6] the Bessel s functions of the first kind of order is following asymptotic relations: J cos O J sin O. It addition It can be shown [5] that there eist a kernel t continuous on [ ] [ ] such that every solution of 7 and 8 can be epressed in the form Where the kernel y t t J t J tdt t d t q d. is solution of following problem / t t t / q t t t 3 Analogous results to 3 hold for y in terms of a kernel t which has similar properties of the. Using equation 3 and Its for y we find that t t yy% J t % t J J t dt t J t dt % s J s ds. If the range of t and t is etended respect to the second argument and some straightforward computations we rewrite as 3

4 y y cos cos d 5 where t s s ds s s. 6 Now we define the function ds Ω y cos β y sin β. 7 The zeros of Ω are the eigenvalues of L or L subject to -3 and if the asymptotic results of y and y are considered the is a entire function of order of Ω. If we multiply 7 by y and 9 by y and subtract we obtain after integration Using we obtain y y y y q q yyd. 8 Now [ y y' y y ' ] q q d. 9 q q and K yyd. If the properties of y and are considered the function K is a entire function and y for since the first term of 9 is zero n In addition using 3 and for K n. < K M 3

5 where M is constant. Now Ω Ψ K Ψ is a entire function. Asymptotic form of Ω and with 3 O Ψ. So From the Liouville Theorem for all Ψ 5 or. 6 K From now on substituting 5 into d d Cos Cos. 7 This can be written as d d Cos d. Letting for real we see from Riemann -Lebesque Lemma that we must have d 9 and d d Cos 3 But from the completeness of the functions Cos we see that < < d 3 Since equation 3 is a Volterra integral equations it has only the zero solution. ence 5

6 q q almost everywhere. References. Gasymov M.G. The Definition of Sturm-Liouville Operator from Two Spectra DAN SSSR Vol.6 No: ochstadt. and Lieberman B. An Invers Sturm Liouville Problem with Mied Given Data. Siam J. Appl. Math. Vol.3 No: Levitan B.M. On the Determination of the Sturm - Liouville Operator from One and Two Spectra Math. USSR Izvestija vol Panakhov E. S. The definition of differential operator with peculiarity in zero on two spectrum. VINITI N Volk V.Y. On Inverse Formulas for a Differential Equation with a Singularity at Usp.Mat.Nauk N.S Watson G. A Treatise on the Theory of Bessel Functions Cambridge University Press Cambridge 96 6