Abstract
In this paper, we consider the Schrödinger equation in the unit ball in . We study the inverse problem of identifying the potential from the Dirichlet to Neumann map which associates to all possible functions on the boundary and the measurements of the normal derivative of the solution of Schrödinger equation on . Using spherical harmonics tools, we determine an explicit expression for the potential on the edge of the domain from an explicit formula for the Dirichlet to Neumann map in a unit ball in dimension 3. We theoretically and numerically present an example.
1. Introduction
The electrical impedance tomography is an imaging method which consists in determining the electrical proprerty of a medium by making voltage and current measurements on its boundary. In the mathematical literature, this is also known as “Calderón’s problem” from Calderón’s pioneer contribution [1, 2]. Calderón was motivated by oil prospection.
This pioneer contribution motivated many developments in inverse problems, particularly in the construction of “complex geometrical optics” solutions of partial differential equations to solve several inverse problems or solutions of the Faddeev type [3]. The problem that Calderón considered was whether one can determine the electrical conductivity of a medium by making voltage and current measurements at the boundary of the medium [2]. EIT is also used in medical imaging, including early diagnosis of breast cancer [4] and monitoring pulmonary function [5]. It is also used to detect leaks in buried pipes [6]. It is applied in the location of oil and mineral deposits in the Earth’s interior, creation of astrophysical images from telescope data, finding cracks and interfaces within materials, shape optimization, model identification in growth processes, and modeling in the life sciences among others.
Studying an inverse problem requires good knowledge of the theory for the corresponding direct problem.
The direct problem in EIT is to find the potential differences measured at the electrodes by knowing these following informations: the shape of the medium, the conductivity distribution of the medium, the position of the electrodes, and the intensity of currents injected at the electrodes. Therefore, solving the direct problem involves modeling the behavior of the tomography when taking EIT measurements. If the current is injected into the medium, then the tomography takes the EIT measurements, and the voltage is measured.
To describe the mathematical problem, we consider a conductor filling a bounded isotropic domain with a smooth boundary (or Lipschitz boundaries), and the electrical conductivity at each point of is represented by a positive function satisfying and [7].
Under the assumption of no source or sink of current in , when a voltage potential is applied on the boundary , the induced potential in is ; that is the weak solution of
Here, we can define the Dirichlet to Neumann map (DN map) formally asand Calderón’s problem would recover from .
The study of the problem (1), for and regular , is reduced to the corresponding problem for the Schrödinger equation (3) with a so called potential of conductivity type [7]. That is why we are motivated to study the problem ofThe inverse problem for the Schrödinger equation is analogous to Calderón’s problem. The inverse problem for the Schrödinger equation is to determine the potential function from the measurements of for all possible functions on the boundary of .
That is, the knowledge of the map , also called Dirichlet-to-Neumann map for the Schrödinger equation, which to any associates the normal derivative . This is why we are motived to determine an explicit formula of the potential knowing of an explicit formula of the Dirichlet-to-Neumann map , for all given .
In this paper, our aim is to determine the potential on the edge of the unit ball from the Dirichlet to Neumann map , for any given .
As contribution, we give an explicit expression for the potential radial near the edge of the unit ball in dimension 3, from an explicit formula for the Dirichlet to Neumann map , corresponding to the Schrödinger operator .
The paper is organized as follows: In Section 2, we study the existence of a solution of the inverse problem for the Schrödinger equation and its uniqueness and the stability of the inverse potential problem. In Section 3, we establish a result giving the explicit expression of the potential on the boundary of the unit ball in dimension 3. In Section 4, we present an example of potential both theoretically and numerically. In Section 5, we present conclusions and perspectives.
2. Inverse Potential Problem
Let be the unit ball in defined by and assumed that . 0 is not a Dirichlet eigenvalue of in .
We focus to determine from the knowledge of the function , and the explicit formula of the Dirichlet to Neumann map for the Schrödinger equation is defined formally aswhere is the outer unit normal vector on , and the unique solution ofThese choices guarantee the existence of a solution of (5) by the Fourier method, and 0 is not an eigenvalue that ensures the uniqueness of the solution.
The map depends linearly on . encodes the measurements of for all possible functions on the boundary of .
The inverse potential problem we pose is to obtain information on the potential from to .
Let us introduce some results that will be useful in the study.
2.1. Preliminaries
In the following, the solution of (5) is written in polar coordinates . A scaled potential is defined by a piecewise constant function, which has points of discontinuity.
For all denotes the Bessel function of the first type or the Bessel function of the second type , and denotes the modified Bessel function of the first type or the modified Bessel function of the second type (see equations (22) and (23)).
is the spherical harmonics defined on .
Let us introduce this theorem for the explicit formula of the DN map in the unit ball in given by Fagueye in [8].
Theorem 1. Let be the unit ball in and the scaled potentialwhere , with and such that the Dirichlet problem for is well posed.
Then, there is an explicit formula for the Dirichlet-to-Neumann map defined as follows:where , with depending on and .
Proof. Theorem 1: for details, see [8].
Lemma 1. In polar coordinates with and , if , then equation (5) admits a unique solution of the formwhere satisfies the problem of
Proof Lemma 1. We have assumed 0 is not an eigenvalue of , then the problem (5) admits a unique solution with and in the unit ball .
If , then we can write .
Using the separation of variables , the completeness property of the spherical harmonics implies that any well-behaved function defined in , that is which is single valued, continuous, and finite, can be written as for some choice of coefficients .
Choosing , then we have . Taking in , we obtain , and then, in .
From , we obtain . And satisfying (5) gives uswhere , and .
Since , we haveWe know is an orthonormal basis, then verifies (9).
Proposition 1. Let be a bounded Lipschitz domain in complex value potential, and . We assume that 0 is not an eigenvalue for the operator . Then, there exists a unique weak solution to the Dirichlet problem ofMoreover,where depends on , and .
(For details of Proposition 1, see [9]).
Lemma 2. Let be a bounded open set with a smooth boundary, where , and and be two functions in such that the Dirichlet problems for and in are well posed. Then, for any , one haswhere is the solution of in with boundary values .
(For details of lemma, see [7]).
Theorem 2. Let be a boundary. denotes the outer unit normal vector on , and is one of its components.(1)(Green formula) If and , then(2)(Integration by parts) If and , then(3)(Divergence theorem) If is a vector field, then
Lemma 3. Let be included in . Let be a continuous function in . If for all function with compact support in , we haveThen, is zero in .
(For proof of Lemma 3, see [10]).Let us suppose that there is such that . We assume that (otherwise, we take −). By continuity, there exists a small open neighborhood of such that for all . Thus, for any positive and nonzero test function , whose support is included in , we have which is a contradiction with the hypothesis on . Then, for all . In order to reconstruct the potential from and , let us give results concerning its existence.
2.2. Existence of Solution of the Inverse Problem
In this section, the inverse potential problem of the Schrödinger equation (5) we pose is to determine the potential function from the knowledge of for all given .
Let be the subset of such that
If and the potential a piecewise constant radial, then is defined by (7) in Theorem 1, and the subset .
Our aim is to determine from the Cauchy data .
The question what we will want to answer is as follows:
Is there nonzero and solution of problem 4 such that ?
According to [11, 12], there is nonzero and solution of problem 4 such that .
Now, we are interested first in the uniqueness result and the stability of the reconstruction of .
2.3. Uniqueness and Stability for the Inverse Potential Problem
For and the potential function , the uniqueness result for the inverse potential problem, i.e., uniquely determines , was proved by Salo in [7], Sylvester and Uhlmann in [13], Uhlmann in [2], and Haberman and Tataru in [14].
Uhlmann in [2] gives the following uniqueness result in dimension
Theorem 3. Let be a bounded Lipschitz domain in . We ssume that , then .
Salo in [7] also gives the result as follows:
Theorem 4. Let be the unit ball in , and let and be two functions in such that the Dirichlet problems for and in are well posed. If , then in .
After ensuring the uniqueness, we will be interested in the stability of the reconstruction of ; that is, we will want to answer the question as follows:
If are two potentials such that is close to , does this imply that is close to ?
In [15], Beretta et al. studied the Lipschitz stability of an inverse boundary value problem for a Schrödinger type equation by assuming that the potential is a piecewise constant complex function in dimension .
In [16], Beretta and Francini proved the Lipschitz stability for the electrical impedance tomography problem for a complex piecewise conductivity satisfying and for some , in a bounded domain with a smooth boundary.
In [17] Uhlmann has given a depth dependent logarithmic type stability estimate of an inverse boundary value problem for a Schrödinger type equation.
The type of depth dependent stability estimate has been proved in [18] for the case of some electrical inclusions.
In [8], Fagueye confirmed the Lipschitz stability in the following theorem for the reconstruction of the potential near to the border by giving an estimate constant.
Theorem 5. Let be the unit ball in , and the scaled potential verifieswhere , with and , and such that the Dirichlet problem for is well posed. We assume that , and there is a positive constant such thatThen, there is a constant such that
Estimate (23) solves the stability problem near to the border.Taking when in 1, this following theorem ensures the Lipschitz stability in the ball [7].
Theorem 6. Kenig–Sjöstrand–Uhlmann assume that is convex and is any open subset of . If for all , and if , then in .
In the next section, we will look at the reconstruction of the potential from the Cauchy data .
3. Reconstruction of the Potential
First, we show that the Dirichlet to Neumann map depends continuously on the potential .
Theorem 7. The assumptions are in Theorem 5.
Then, there is a constant such that
Inequality (24) gives the continuous dependence of the Dirichlet to Neumann map with respect to the potential .
Proof of Theorem 7. using the results of Lemma 2 with , the unit ball, and , we haveUsing Cauchy–Schwarz theorem, we haveUsing the inequality (9) in the Proposition 1 we have alsowhere depends on and .
Taking , we obtainWe know that for all ,Then, the relation (28) implies thatLet us introduce this result by giving the explicit formula of the radial potential for our inverse problem.
Theorem 8. Let the unit ball be defined by ; its surface and with and .
We assume that , are not all zero.
If 0 is not a Dirichlet eigenvalue of in , then there exist a unique nonzero solution of the problemand a unique nonzero radial function , which is uniquely determined at the edge from the data bywhere is a constant for any .
And in the bull,
Proof of Theorem 8. Let us consider the problem (5). The existence and uniqueness of its solution are ensured by the assumptions.
According to Lemma 1, we haveAccording to [11, 12], there is nonzero and solution of problem (5) such that .
We know that with and is the unique solution of 4 with .
Then, using the weak reformulation of the equation , we haveAccording to the Divergence theorem, Green’s formula, and the orthonormal basis , this implies thatorWe know that the gradient of the multipole expansion of the scalarcan be expressed in terms of spherical vector harmonics as follows:where , and are unit vectors such that and a function depending on .
Then, using Bessel Parseval’s identity, we can writewhich is equivalent toWe note that the first term of left-hand side of this equality does not depend on . By differentiating both members of the equality with respect to , we obtainBy integrating over , since only depends on , we havewhereThen, since for all in the solution of (5), we haveAnd, since in , we havewhere is a constant for any .
Using the Cauchy data and Bessel Parseval’s identity, since , we havewhere is a constant for any
This result shows that can be recovered from the Cauchy data near the edge .
To recover it in the unit ball, we use always the Laplacian method in spherical coordinates as follows:whereSinceand , thenis equivalent toAs the solution must be finite at the origin, then we have the following condition:Since for all in the solution of (5), we can writeFrom (32) and (33), it is easy to verify that .
In the below section, we will determine both theoretically and numerically from a Cauchy data example .
4. Numerical Simulations
In this section, we will numerically determine the radial potential with ,… in order to simplify the calculations. Since Theorem 3, the map which associates a Dirichlet-to-Neumann map to all potentials is injective.
Let us consider as the solution to Laplace’s equation in spherical coordinates and introduce the spherical harmonics, :
The phase factor , introduced originally by Condon and Shortly, is convenient for applications in quantum mechanics. Let us introducewhere the ∗ means the complex conjugation.
Let us list the spherical harmonics for and non-negative values of as
Let us assume that the Dirichlet-to-Neumann map can be characterized by the first eigenvalues defined in theorem (7).
Then, for all we define such that
Let us introduce the first vector spherical harmonics introduced in equation (39) and defined bywith depending on is a combination of unit vectors and . We have for
According the explicit expression of the radial potential near the boundary of the unit ball defined in (32), Theorem 1, and the infectivity of the map , we have by calculations
Some values of the potential on the edge of following the values of are presented in Table 1.
They are also represented in Figure 1.
The same approach allows to determine both theoretically and numerically the radial potential near the edge for or .
5. Conclusion
We can conclude that the potential can be recovered explicitly from the Cauchy data near the edge of the unit ball in dimension 3 both theoretically and numerically. This paper can potentially provide accurate physical knowledge of the nature and properties of the potential , near the edge of the unit ball, that is subject to a particle with a wave function and that is a solution of the Schrödinger equation satisfying the boundary conditions. For perspectives, we will determine an explicit formula of the potential in any domain with a Lipschitzian border in dimension 3 and in dimension 2. We will be also interested in the case where the domain edge is not Lipschitzian [19].Another project is to see how to apply these results in the resolution of problems related to greenhouse gases and air pollution.
Appendix
We consider the spherical Bessel functions asthat satisfies the equation
The modified spherical Bessel function isthat satisfies the equation if , then
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The author declares that there are no conflicts of interest.