Reconstruction of potential function for Sturm-Liouville operator with Coulomb potential

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.

MSC:34L05, 45C05.

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 Sturm-Liouville operator from a spectral function or from two spectra is given in [4–7].

We note that the details of the inverse problem for singular equations are given in the monographs [8–11] 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 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].

Before giving the main results, we mention some physical properties of the Sturm-Liouville 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 U=\frac{-e^2}{r}, where r is the radius of the nucleus, e is electronic charge. According to this, we use the time-dependent Schrödinger equation

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 Sturm-Liouville equation with Coulomb potential

where λ is a parameter which corresponds to the energy [22].

We consider the singular Sturm-Liouville problem

(1.1)

(1.2)

(1.3)

in which the function q(x)\in L^1[0,\pi ], A, H are finite numbers and \frac{y(x)}{x}\in C[0,\pi ]. Next, we denote by \phi (x,s) the solution of (1.1) satisfying the initial condition

Let {\lambda }_n be the n th eigenvalue and , i=1,2,\dots ,n-1 be nodal points of the n th eigenfunction. Also, let I_i^n=[x_i^n,x_{i+1}^n] be the i th nodal domain of the n th eigenfunction and let l_i^n=|l_i^n|=x_{i+1}^n-x_i^n be the associated nodal length. We also define the function j_n(x) by .

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 2.1 The solution of problem (1.1)-(1.3) has the following form:

where \frac{\phi (t,s)}{t}\in C[0,\pi ].

Proof Because \phi (x,s) satisfies equation (1.1), we get

By integrating the first term twice on the right-hand side by parts and taking the conditions into account (1.2), we find that

where \frac{\phi (t,s)}{t}\in C[0,\pi ]. □

Lemma 2.2 The eigenvalues of problem (1.1)-1.3) are the roots of (1.3). This spectral characteristic satisfies the following asymptotic expression [23]:

where

Lemma 2.3 Assume that q\in L^1(0,\pi ). Then, as n\to \mathrm{\infty },

(2.3)

(2.4)

Proof By using some iterations and trigonometric calculations in (2.1), we obtain

If \phi (x,s) is equal to zero and cos\lambda x is not close to zero, then

Now, we take s=s_n and x=x_i^n. Because Taylor’s expansion for the arctangent function is given by

for some integer i, then

Therefore

The nodal length is

This completes the proof of Lemma 2.3. □

Lemma 2.4 Suppose f\in L^1(0,\pi ). Then, for almost every x\in (0,\pi ) with j=j_n(x),

Proof Since f\in L^1(0,\pi ), \frac{d}{dx}{\int }_a^{x}f(t)dt=f(x) almost everywhere. Thus, given any \zeta >0, when n is sufficiently large and for almost every x\in (0,\pi ),

This proves Lemma 2.4. □

Theorem 2.1 The potential function q(x)\in L^1(0,\pi ) satisfies

for almost every x\in (0,\pi ) with j=j_n(x). We note that the asymptotic expression for s_n in Theorem 2.1 implies that q(x)={lim}_{n\to \mathrm{\infty }}{F}_n(x).

Proof When we consider (2.4) in the form

so that

By Lemma 2.4

for almost every x\in (0,\pi ).

It remains to show that for almost every x\in (0,\pi ),

tends to zero as n\to \mathrm{\infty }. Take a sequence of continuous functions q_k which converges to q in L^1(0,\pi ). Then q_k has a subsequence converging to q almost everywhere in (0,\pi ). We call this subsequence q_k. Take any x such that q_k(x) converges to q(x). Then for a given \epsilon >0, we can fix a large k such that . Hence

By Lemma 2.3,

and so it tends to zero as n\to \mathrm{\infty }. By Lemma 2.4, the first term A_n satisfies, when n is sufficiently large,

On the other hand,

Because q_k is continuous, this term is arbitrarily every x\in (0,\pi ). Hence we conclude that {lim}_{n\to \mathrm{\infty }}{T}_n(x)=0. This proves Theorem 2.1. □

Lemma 2.5 We take a sequence f_k\in C[0,\pi ] converges to f\in L^1, then, for any large enough n, with j=j_n(x) as k\to \mathrm{\infty }

Proof By (2.4) and observation that the integral {\int }_{x_j^n}^{x_{j+1}^n}[f_k(t)-f(t)]dt is constant on any nodal interval I_j^n, we obtain

and for k\to \mathrm{\infty } this term converges to zero. □

Lemma 2.6 Suppose that q\in L^1(0,\pi ), then as n\to \mathrm{\infty } with j=j_n(x),

Proof Firstly, let us show that if q is continuous on [0,\pi ], the result is satisfied. Let M={max}_{x\in [0,\pi ]}|q(x)|. By using the intermediate value theorem, there exists \xi \in (a,x) such that

If x is close enough to a, the difference can be arbitrarily small. Then, for all \epsilon >0, when n is large enough, with j=j_n(x) we get

In the above process, we assume that x\ne x_j^n. The estimate also holds if x=x_j^n. Hence if q\in C[0,\pi ], \frac{s_n}{\pi }{\int }_{x_j^n}^{x_{j+1}^n}q(t)dt converges to q(x) uniformly on (0,\pi ). Thus \parallel \frac{s_n}{\pi }{\int }_{x_j^n}^{x_{j+1}^n}q(t)dt-q(x)\parallel can be arbitrarily small. Because C[0,\pi ] is dense in L^1(0,\pi ), for any q\in L^1(0,\pi ), there exists a sequence q_k\in C[0,\pi ] convergent to q in L^1(0,\pi ). Hence, fix n sufficiently large,

From the above process and Lemma 2.5, when k is large enough, the first two terms are arbitrarily small. Hence, as k\to \mathrm{\infty },

 □

Theorem 2.2 F_n converges to q in L^1.

Proof When we consider the value of F_n, we obtain that

It suffices to show that as n\to \mathrm{\infty }

By using (2.4) we have

Hence, we only need to prove that for n\to \mathrm{\infty }

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 x\in (0,\pi ). Furthermore,

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. □

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.

Affiliations

1. Department of Mathematics, Firat University, Elazig, 23119, Turkey

   • Etibar S Panakhov

2. Department of Mathematics, Erzincan University, Erzincan, 24100, Turkey

   • Murat Sat

Corresponding author

Correspondence to Murat Sat.

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.

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