Abstract
In this paper, we are concerned with an inverse problem for the SturmLiouville 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 nth 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 SturmLiouville problem.
MSC: 34L05, 45C05.
Keywords:
Coulomb potential; nodal point; reconstruction formula1 Introduction
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 [1]. 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 SturmLiouville operator from a spectral function or from two spectra is given in [47].
We note that the details of the inverse problem for singular equations are given in the monographs [811] and references therein.
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 SturmLiouville 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 [1421]etc.
In this paper, we deal with an inverse nodal problem for the SturmLiouville 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].
Before giving the main results, we mention some physical properties of the SturmLiouville
operator with Coulomb potential. Learning about the motion of electrons moving under
the Coulomb potential is of significance in quantum theory. Solving these types of
problems allows us to find energy levels not only for a hydrogen atom but also for
single valence electron atoms such as sodium. For hydrogen atom, the Coulomb potential
is given by
where Ψ is the wave function, ħ is Planck’s constant and m is the mass of electron. In this equation, if the Fourier transform is applied
it will convert to energy equation dependent on the situation as follows:
Therefore, energy equation in the field with the Coulomb potential becomes
If this hydrogen atom is substituted to other potential area, then the energy equation becomes
If we make the necessary transformation, then we can get a SturmLiouville equation with Coulomb potential
where λ is a parameter which corresponds to the energy [22].
We consider the singular SturmLiouville problem
in which the function
Let
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.
Lemma 2.1The solution of problem (1.1)(1.3) has the following form:
where
Proof Because
By integrating the first term twice on the righthand side by parts and taking the conditions into account (1.2), we find that
where
Lemma 2.2The eigenvalues of problem (1.1)1.3) are the roots of (1.3). This spectral characteristic satisfies the following asymptotic expression[23]:
where
Lemma 2.3Assume that
Proof By using some iterations and trigonometric calculations in (2.1), we obtain
If
Now, we take
for some integer i, then
Therefore
The nodal length is
This completes the proof of Lemma 2.3. □
Lemma 2.4Suppose
Proof Since
This proves Lemma 2.4. □
Theorem 2.1The potential function
for almost every
Proof When we consider (2.4) in the form
so that
By Lemma 2.4
for almost every
It remains to show that for almost every
tends to zero as
By Lemma 2.3,
and so it tends to zero as
On the other hand,
Because
Lemma 2.5We take a sequence
Proof By (2.4) and observation that the integral
and for
Lemma 2.6Suppose that
Proof Firstly, let us show that if q is continuous on
If x is close enough to a, the difference can be arbitrarily small. Then, for all
In the above process, we assume that
From the above process and Lemma 2.5, when k is large enough, the first two terms are arbitrarily small. Hence, as
□
Theorem 2.2
Proof When we consider the value of
It suffices to show that as
By using (2.4) we have
Hence, we only need to prove that for
and
From Lemma 2.6, the first limit holds and the second limit also holds. On the other hand, the sequence of functions
converges to 0 for almost every
and
Then, we may apply the Lebesque dominated convergence theorem to show that (2.5) is valid. The proof of Theorem 2.2 is completed. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
MS wrote the first draft and ESP corrected and improved the final version. Both authors read and approved the final draft.
Acknowledgements
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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