Reconstruction of potential function for Sturm-Liouville operator with Coulomb potential
© Panakhov and Sat; licensee Springer. 2013
Received: 12 December 2012
Accepted: 21 February 2013
Published: 8 March 2013
In this paper, we are concerned with an inverse problem for the Sturm-Liouville operator with Coulomb potential using a new kind of spectral data that is known as nodal points. We give a reconstruction of q as a limit of a sequence of functions whose n th term is dependent only on eigenvalue and its associated nodal data. It is mentioned that this method is based on the works of Law and Yang, but we have applied the method to the singular Sturm-Liouville problem.
KeywordsCoulomb potential nodal point reconstruction formula
Inverse problems of spectral analysis imply the reconstruction of a linear operator from some or other of its spectral characteristics. Such characteristics are spectra (for different boundary conditions), normalizing constants, spectral functions, scattering data, etc. An early important result in this direction, which gave vital impetus for further development of inverse problem theory, was obtained in . At present, inverse problems are studied for certain special classes of ordinary differential operators. Inverse problems from two spectra are the most simple in their formulation and well studied in [2, 3]. An effective method of constructing a regular and singular Sturm-Liouville operator from a spectral function or from two spectra is given in [4–7].
In some recent interesting works [12, 13], Hald and McLaughlin and Browne and Sleeman have taken a new approach to inverse spectral theory for the Sturm-Liouville problem. The novelty of these works lies in the use of nodal points as the given spectral data. In recent years, inverse nodal problems have been studied by several authors [14–21]etc.
In this paper, we deal with an inverse nodal problem for the Sturm-Liouville operator with Coulomb potential. We have reconstructed the potential function q from the nodal points of eigenfunctions, provided q is smooth enough. The method is based on a series of works by Law and Yang [14, 17].
where λ is a parameter which corresponds to the energy .
Let be the n th eigenvalue and , be nodal points of the n th eigenfunction. Also, let be the i th nodal domain of the n th eigenfunction and let be the associated nodal length. We also define the function by .
2 Main results
In this section, we try to obtain some asymptotic results and a reconstruction formula for the potential q, which has been obtained as a solution of an inverse nodal problem.
where . □
This completes the proof of Lemma 2.3. □
This proves Lemma 2.4. □
for almost every with . We note that the asymptotic expression for in Theorem 2.1 implies that .
for almost every .
Because is continuous, this term is arbitrarily every . Hence we conclude that . This proves Theorem 2.1. □
and for this term converges to zero. □
Theorem 2.2 converges to q in .
Then, we may apply the Lebesque dominated convergence theorem to show that (2.5) is valid. The proof of Theorem 2.2 is completed. □
The authors would like to thank the editor and referees for their valuable comments and remarks which led to a great improvement of the article.
- Ambartsumyan VA: Über eine frage der eigenwerttheorie. Z. Phys. 1929, 53: 690-695. 10.1007/BF01330827View ArticleGoogle Scholar
- Levitan BM: On the determination of the Sturm-Liouville operator from one and two spectra. Math. USSR, Izv. 1978, 12: 179-193. 10.1070/IM1978v012n01ABEH001844View ArticleGoogle Scholar
- Isaacson EL, Trubowitz E: The inverse Sturm-Liouville problem. I. Commun. Pure Appl. Math. 1983, 36: 767-783. 10.1002/cpa.3160360604MathSciNetView ArticleGoogle Scholar
- Gelfand IM, Levitan BM: On the determination of a differential equation by its spectral function. Izv. Akad. Nauk SSSR, Ser. Mat. 1951, 15: 309-360. Ams, 253-304 (1955)MathSciNetGoogle Scholar
- Hochstadt H: The inverse Sturm-Liouville problem. Commun. Pure Appl. Math. 1973, 26: 715-729. 10.1002/cpa.3160260514MathSciNetView ArticleGoogle Scholar
- Pöschel J, Trubowitz E: Inverse Spectral Theory. Academic Press, Boston; 1987.Google Scholar
- Rundell W, Sack EP: Reconstruction of a radially symmetric potential from two spectral sequences. J. Math. Anal. Appl. 2001, 264: 354-381. 10.1006/jmaa.2001.7664MathSciNetView ArticleGoogle Scholar
- Carlson R: Borg-Levinson theorem for Bessel operator. Pac. J. Math. 1997, 177: 1-26. 10.2140/pjm.1997.177.1View ArticleGoogle Scholar
- Chadan K, Colton D, Paivarinta L, Rundell W: An Introduction to Inverse Scattering and Inverse Spectral Problems. SIAM, Philadelphia; 1997.View ArticleGoogle Scholar
- Panakhov ES, Sat M: On the determination of the singular Sturm-Liouville operator from two spectra. Comput. Model. Eng. Sci. 2012, 84: 1-11.MathSciNetGoogle Scholar
- Hald OH: Discontinuous inverse eigenvalue problem. Commun. Pure Appl. Math. 1984, 37: 539-577. 10.1002/cpa.3160370502MathSciNetView ArticleGoogle Scholar
- Browne PJ, Sleeman BD: Inverse nodal problems for Sturm-Liouville equation with eigenparameter dependent boundary conditions. Inverse Probl. 1996, 12: 377-381. 10.1088/0266-5611/12/4/002MathSciNetView ArticleGoogle Scholar
- Hald OH, McLaughlin JR: Solution of inverse nodal problems. Inverse Probl. 1989, 5: 307-347. 10.1088/0266-5611/5/3/008MathSciNetView ArticleGoogle Scholar
- Chen YT, Cheng YH, Law CK, Tsa J: Convergence of reconstruction formula for the potential function. Proc. Am. Math. Soc. 2002, 130: 2319-2324. 10.1090/S0002-9939-02-06297-4View ArticleGoogle Scholar
- Yang FX: A solution of the inverse nodal problem. Inverse Probl. 1997, 13: 203-213. 10.1088/0266-5611/13/1/016View ArticleGoogle Scholar
- McLaughlin JR: Inverse spectral theory using nodal points as a data - a uniqueness result. J. Differ. Equ. 1988, 73: 354-362. 10.1016/0022-0396(88)90111-8View ArticleGoogle Scholar
- Law CK, Shen CL, Yang CF: The inverse nodal problem on the smoothness of the potential function. Inverse Probl. 1999, 15: 253-263. 10.1088/0266-5611/15/1/024MathSciNetView ArticleGoogle Scholar
- Yurko VA, Freiling G: Inverse nodal problems for differential operators on graphs with a cycle. Tamkang J. Math. 2010, 41: 15-24.MathSciNetGoogle Scholar
- Yang CF: Inverse nodal problems for the Sturm-Liouville operator with eigenparameter dependent boundary conditions. Oper. Matrices 2012, 6(1):63-77.MathSciNetView ArticleGoogle Scholar
- Koyunbakan H, Panakhov ES: A uniqueness theorem for inverse nodal problem. Inverse Probl. Sci. Eng. 2007, 15: 517-524. 10.1080/00423110500523143MathSciNetView ArticleGoogle Scholar
- Koyunbakan H: Reconstruction of potential function for diffusion operator. Numer. Funct. Anal. Optim. 2009, 30: 111-120. 10.1080/01630560802279256MathSciNetView ArticleGoogle Scholar
- Blohincev DI: Foundations of Quantum Mechanics. GITTL, Moscow; 1949.Google Scholar
- Amirov RK, Çakmak Y, Gulyaz S: Boundary value problem for second order differential equations with Coulomb singularity on a finite interval. Indian J. Pure Appl. Math. 2006, 37: 125-140.MathSciNetGoogle Scholar