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.
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.
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