Open Access

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

Boundary Value Problems20132013:49

DOI: 10.1186/1687-2770-2013-49

Received: 12 December 2012

Accepted: 21 February 2013

Published: 8 March 2013

Abstract

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.

Keywords

Coulomb potential nodal point reconstruction formula

1 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 Sturm-Liouville 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 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 [1421]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 = e 2 r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq1_HTML.gif, where r is the radius of the nucleus, e is electronic charge. According to this, we use the time-dependent Schrödinger equation
i ħ Ψ t = ħ 2 2 m 2 Ψ x 2 + U ( x , y , z ) Ψ , R 3 | Ψ | 2 d x d y d z = 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equa_HTML.gif
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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equb_HTML.gif
it will convert to energy equation dependent on the situation as follows:
ħ 2 2 m 2 Ψ ˜ + U ˜ Ψ ˜ = E Ψ ˜ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equc_HTML.gif
Therefore, energy equation in the field with the Coulomb potential becomes
ħ 2 2 m 2 Ψ ˜ + ( E + e 2 r ) Ψ ˜ = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equd_HTML.gif
If this hydrogen atom is substituted to other potential area, then the energy equation becomes
ħ 2 2 m 2 Ψ ˜ + ( E + e 2 r + q ( x , y , z ) ) Ψ ˜ = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Eque_HTML.gif
If we make the necessary transformation, then we can get a Sturm-Liouville equation with Coulomb potential
y + [ A x + q ( x ) ] y = λ y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equf_HTML.gif

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

We consider the singular Sturm-Liouville problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equ2_HTML.gif
(1.2)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equ3_HTML.gif
(1.3)
in which the function q ( x ) L 1 [ 0 , π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq2_HTML.gif, A, H are finite numbers and y ( x ) x C [ 0 , π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq3_HTML.gif. Next, we denote by φ ( x , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq4_HTML.gif the solution of (1.1) satisfying the initial condition
φ ( 0 , s ) = 0 , φ ( 0 , s ) = s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equ4_HTML.gif
(1.4)

Let λ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq5_HTML.gif be the n th eigenvalue and 0 < x 1 n < x 2 n < < x i n < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq6_HTML.gif, i = 1 , 2 , , n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq7_HTML.gif be nodal points of the n th eigenfunction. Also, let I i n = [ x i n , x i + 1 n ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq8_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq9_HTML.gif be the associated nodal length. We also define the function j n ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq10_HTML.gif by j n ( x ) = max { i : x i n < x } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq11_HTML.gif.

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 2.1 The solution of problem (1.1)-(1.3) has the following form:
φ ( x , s ) = sin s x + 0 x sin s ( x t ) s { A t + q ( t ) } φ ( t , s ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equ5_HTML.gif
(2.1)

where φ ( t , s ) t C [ 0 , π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq12_HTML.gif.

Proof Because φ ( x , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq4_HTML.gif satisfies equation (1.1), we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equg_HTML.gif
By integrating the first term twice on the right-hand side by parts and taking the conditions into account (1.2), we find that
φ ( x , s ) = sin s x + 0 x sin s ( x t ) s { A t + q ( t ) } φ ( t , s ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equh_HTML.gif

where φ ( t , s ) t C [ 0 , π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq12_HTML.gif. □

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]:
s n = λ n = n + 1 2 + A 2 π ln ( n + 1 2 ) ( n + 1 2 ) + c 0 ( n + 1 2 ) + O ( ln n n 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equ6_HTML.gif
(2.2)
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equi_HTML.gif
Lemma 2.3 Assume that q L 1 ( 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq13_HTML.gif. Then, as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq14_HTML.gif,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equ7_HTML.gif
(2.3)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equ8_HTML.gif
(2.4)
Proof By using some iterations and trigonometric calculations in (2.1), we obtain
φ ( x , s ) = sin s x + sin s x 2 s 0 x sin 2 s t { A t + q ( t ) } d t cos s x 2 s 0 x ( 1 cos 2 s t ) { A t + q ( t ) } d t + o ( 1 s 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equj_HTML.gif
If φ ( x , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq4_HTML.gif is equal to zero and cos λ x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq15_HTML.gif is not close to zero, then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equk_HTML.gif
Now, we take s = s n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq16_HTML.gif and x = x i n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq17_HTML.gif. Because Taylor’s expansion for the arctangent function is given by
arctan x = π i k = 0 ( 1 ) k + 1 x 2 k + 1 2 k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equl_HTML.gif
for some integer i, then
s n x i n = π i + 1 2 s n 0 x i n ( 1 cos 2 s t ) { A t + q ( t ) } d t + o ( 1 s n 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equm_HTML.gif
Therefore
x i n = π i s n + 1 2 s n 2 0 x i n ( 1 cos 2 s t ) { A t + q ( t ) } d t + o ( 1 s n 3 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equn_HTML.gif
The nodal length is
l i n = x i + 1 n x i n , l i n = π s n + 1 2 s n 2 x i n x i + 1 n ( 1 cos 2 s t ) { A t + q ( t ) } d t + o ( 1 s n 3 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equo_HTML.gif

This completes the proof of Lemma 2.3. □

Lemma 2.4 Suppose f L 1 ( 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq18_HTML.gif. Then, for almost every x ( 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq19_HTML.gif with j = j n ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq20_HTML.gif,
lim n s n π x j n x j + 1 n f ( t ) d t = f ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equp_HTML.gif
Proof Since f L 1 ( 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq18_HTML.gif, d d x a x f ( t ) d t = f ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq21_HTML.gif almost everywhere. Thus, given any ζ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq22_HTML.gif, when n is sufficiently large and for almost every x ( 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq23_HTML.gif,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equq_HTML.gif

This proves Lemma 2.4. □

Theorem 2.1 The potential function q ( x ) L 1 ( 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq24_HTML.gif satisfies
q ( x ) = lim n [ 2 s n 2 ( s n l j n π 1 ) s n A ln ( x j + 1 n x j n ) + s n A π x j n x j + 1 n cos 2 s n t t d t ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equr_HTML.gif

for almost every x ( 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq23_HTML.gif with j = j n ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq20_HTML.gif. We note that the asymptotic expression for s n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq25_HTML.gif in Theorem 2.1 implies that q ( x ) = lim n F n ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq26_HTML.gif.

Proof When we consider (2.4) in the form
l j n = π s n + 1 2 s n 2 x j n x j + 1 n ( 1 cos 2 s t ) { A t + q ( t ) } d t + o ( 1 s n 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equs_HTML.gif
so that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equt_HTML.gif
By Lemma 2.4
lim n s n π x j n x j + 1 n q ( t ) d t = q ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equu_HTML.gif

for almost every x ( 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq23_HTML.gif.

It remains to show that for almost every x ( 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq23_HTML.gif,
T n ( x ) : = s n π x j n x j + 1 n cos 2 s n t q ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equv_HTML.gif
tends to zero as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq14_HTML.gif. Take a sequence of continuous functions q k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq27_HTML.gif which converges to q in L 1 ( 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq28_HTML.gif. Then q k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq27_HTML.gif has a subsequence converging to q almost everywhere in ( 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq29_HTML.gif. We call this subsequence q k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq27_HTML.gif. Take any x such that q k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq30_HTML.gif converges to q ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq31_HTML.gif. Then for a given ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq32_HTML.gif, we can fix a large k such that | q k ( x ) q ( x ) | < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq33_HTML.gif. Hence
T n ( x ) = s n π x j n x j + 1 n cos 2 s n t [ q ( t ) q k ( t ) ] d t + s n π x j n x j + 1 n cos 2 s n t [ q k ( t ) q k ( x ) ] d t + s n π x j n x j + 1 n cos 2 s n t q k ( x ) d t = A n + B n + C n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equw_HTML.gif
By Lemma 2.3,
C n = q k ( x ) 2 π [ sin ( 2 s n x j + 1 n ) sin ( 2 s n x j n ) ] = q k ( x ) O ( 1 n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equx_HTML.gif
and so it tends to zero as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq14_HTML.gif. By Lemma 2.4, the first term A n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq34_HTML.gif satisfies, when n is sufficiently large,
| A n | = | s n π x j n x j + 1 n cos 2 s n t [ q ( t ) q k ( t ) ] d t | s n π x j n x j + 1 n | q ( t ) q k ( t ) | d t < | q ( x ) q k ( x ) | + ε < 2 ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equy_HTML.gif
On the other hand,
| B n | = | s n π x j n x j + 1 n cos 2 s n t [ q k ( t ) q k ( x ) ] d t | s n π x j n x j + 1 n | q k ( t ) q k ( x ) | d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equz_HTML.gif

Because q k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq27_HTML.gif is continuous, this term is arbitrarily every x ( 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq35_HTML.gif. Hence we conclude that lim n T n ( x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq36_HTML.gif. This proves Theorem 2.1. □

Lemma 2.5 We take a sequence f k C [ 0 , π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq37_HTML.gif converges to f L 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq38_HTML.gif, then, for any large enough n, with j = j n ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq20_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq39_HTML.gif
s n π x j n x j + 1 n [ f k ( t ) f ( t ) ] d t 1 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equaa_HTML.gif
Proof By (2.4) and observation that the integral x j n x j + 1 n [ f k ( t ) f ( t ) ] d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq40_HTML.gif is constant on any nodal interval I j n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq41_HTML.gif, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equab_HTML.gif

and for k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq39_HTML.gif this term converges to zero. □

Lemma 2.6 Suppose that q L 1 ( 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq42_HTML.gif, then as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq43_HTML.gif with j = j n ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq20_HTML.gif,
s n π x j n x j + 1 n q ( t ) d t q ( x ) 1 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equac_HTML.gif
Proof Firstly, let us show that if q is continuous on [ 0 , π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq44_HTML.gif, the result is satisfied. Let M = max x [ 0 , π ] | q ( x ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq45_HTML.gif. By using the intermediate value theorem, there exists ξ ( a , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq46_HTML.gif such that
| 1 x a a x q ( t ) d t q ( x ) | = | q ( ξ ) q ( x ) | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equad_HTML.gif
If x is close enough to a, the difference can be arbitrarily small. Then, for all ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq32_HTML.gif, when n is large enough, with j = j n ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq20_HTML.gif we get
| s n π x j n x j + 1 n q ( t ) d t q ( x ) | | s n ( x x j n ) π [ 1 x x j n x j n x q ( t ) d t q ( x ) ] | + | s n ( x j + 1 n x ) π [ 1 x j + 1 n x x x j + 1 n q ( t ) d t q ( x ) ] | + | q ( x ) | | ( s n l j n π 1 ) | 2 s n l j n ε π + M ε ( M + 2 + 2 ε ) ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equae_HTML.gif
In the above process, we assume that x x j n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq47_HTML.gif. The estimate also holds if x = x j n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq48_HTML.gif. Hence if q C [ 0 , π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq49_HTML.gif, s n π x j n x j + 1 n q ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq50_HTML.gif converges to q ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq31_HTML.gif uniformly on ( 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq51_HTML.gif. Thus s n π x j n x j + 1 n q ( t ) d t q ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq52_HTML.gif can be arbitrarily small. Because C [ 0 , π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq53_HTML.gif is dense in L 1 ( 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq54_HTML.gif, for any q L 1 ( 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq13_HTML.gif, there exists a sequence q k C [ 0 , π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq55_HTML.gif convergent to q in L 1 ( 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq56_HTML.gif. Hence, fix n sufficiently large,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equaf_HTML.gif
From the above process and Lemma 2.5, when k is large enough, the first two terms are arbitrarily small. Hence, as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq57_HTML.gif,
s n π x j n x j + 1 n [ f k ( t ) f ( t ) ] d t 1 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equag_HTML.gif

 □

Theorem 2.2 F n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq58_HTML.gif converges to q in L 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq59_HTML.gif.

Proof When we consider the value of F n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq58_HTML.gif, we obtain that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equah_HTML.gif
It suffices to show that as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq14_HTML.gif
s n ( 2 s n 2 l j n π 2 s n A ln ( x j + 1 n x j n ) + A π x j n x j + 1 n cos 2 s n t t d t ) q 1 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equai_HTML.gif
By using (2.4) we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equaj_HTML.gif
Hence, we only need to prove that for n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq14_HTML.gif
s n π x j n x j + 1 n q ( t ) d t q 1 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equak_HTML.gif
and
s n π x j n x j + 1 n cos 2 s n t { A t + q ( t ) } d t 1 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equ9_HTML.gif
(2.5)
From Lemma 2.6, the first limit holds and the second limit also holds. On the other hand, the sequence of functions
c n ( x ) = s n π x j n x j + 1 n cos 2 s n t { A t + q ( t ) } d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equal_HTML.gif
converges to 0 for almost every x ( 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_IEq23_HTML.gif. Furthermore,
| c n ( x ) | s n π x j n x i + 1 n | A t + q ( t ) | d t = g n ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equam_HTML.gif
and
0 π g n ( x ) d x = i = 0 n 1 s n l j n π x j n x j + 1 n | A t + q ( t ) | d t = [ 1 + O ( ln n n ) ] q 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-49/MediaObjects/13661_2012_Article_304_Equan_HTML.gif

Then, we may apply the Lebesque dominated convergence theorem to show that (2.5) is valid. The proof of Theorem 2.2 is completed. □

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, Firat University
(2)
Department of Mathematics, Erzincan University

References

  1. Ambartsumyan VA: Über eine frage der eigenwerttheorie. Z. Phys. 1929, 53: 690-695. 10.1007/BF01330827View Article
  2. 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 Article
  3. Isaacson EL, Trubowitz E: The inverse Sturm-Liouville problem. I. Commun. Pure Appl. Math. 1983, 36: 767-783. 10.1002/cpa.3160360604MathSciNetView Article
  4. 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)MathSciNet
  5. Hochstadt H: The inverse Sturm-Liouville problem. Commun. Pure Appl. Math. 1973, 26: 715-729. 10.1002/cpa.3160260514MathSciNetView Article
  6. Pöschel J, Trubowitz E: Inverse Spectral Theory. Academic Press, Boston; 1987.
  7. 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 Article
  8. Carlson R: Borg-Levinson theorem for Bessel operator. Pac. J. Math. 1997, 177: 1-26. 10.2140/pjm.1997.177.1View Article
  9. Chadan K, Colton D, Paivarinta L, Rundell W: An Introduction to Inverse Scattering and Inverse Spectral Problems. SIAM, Philadelphia; 1997.View Article
  10. Panakhov ES, Sat M: On the determination of the singular Sturm-Liouville operator from two spectra. Comput. Model. Eng. Sci. 2012, 84: 1-11.MathSciNet
  11. Hald OH: Discontinuous inverse eigenvalue problem. Commun. Pure Appl. Math. 1984, 37: 539-577. 10.1002/cpa.3160370502MathSciNetView Article
  12. 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 Article
  13. Hald OH, McLaughlin JR: Solution of inverse nodal problems. Inverse Probl. 1989, 5: 307-347. 10.1088/0266-5611/5/3/008MathSciNetView Article
  14. 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 Article
  15. Yang FX: A solution of the inverse nodal problem. Inverse Probl. 1997, 13: 203-213. 10.1088/0266-5611/13/1/016View Article
  16. 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 Article
  17. 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 Article
  18. Yurko VA, Freiling G: Inverse nodal problems for differential operators on graphs with a cycle. Tamkang J. Math. 2010, 41: 15-24.MathSciNet
  19. Yang CF: Inverse nodal problems for the Sturm-Liouville operator with eigenparameter dependent boundary conditions. Oper. Matrices 2012, 6(1):63-77.MathSciNetView Article
  20. Koyunbakan H, Panakhov ES: A uniqueness theorem for inverse nodal problem. Inverse Probl. Sci. Eng. 2007, 15: 517-524. 10.1080/00423110500523143MathSciNetView Article
  21. Koyunbakan H: Reconstruction of potential function for diffusion operator. Numer. Funct. Anal. Optim. 2009, 30: 111-120. 10.1080/01630560802279256MathSciNetView Article
  22. Blohincev DI: Foundations of Quantum Mechanics. GITTL, Moscow; 1949.
  23. 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.MathSciNet

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© Panakhov and Sat; licensee Springer. 2013

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