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International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 651390, 12 pages
http://dx.doi.org/10.1155/2012/651390
Research Article

Inverse Problem for a Curved Quantum Guide

1UT1 Ceremath, Université de Toulouse, 21 Allées de Brienne, 31042 Toulouse Cedex 9, France
2Institut de Mathématiques de Toulouse, Université Paul Sabatier, UMR 5219, 31062 Toulouse Cedex 9, France
3LATP, Université d'Aix-Marseille, UMR 7353, 39 rue Joliot Curie, 13453 Marseille Cedex 13, France

Received 28 March 2012; Accepted 22 June 2012

Academic Editor: Wolfgang Castell

Copyright © 2012 Laure Cardoulis and Michel Cristofol. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the Dirichlet Laplacian operator on a curved quantum guide in with an asymptotically straight reference curve. We give uniqueness results for the inverse problem associated to the reconstruction of the curvature by using either observations of spectral data or a boot-strapping method.

1. Introduction and Main Results in Dimension

The spectral properties of curved quantum guides have been studied intensively for several years, because of their applications in quantum mechanics electron motion. We can cite among several papers [17].

However, inverse problems associated with curved quantum guides have not been studied to our knowledge, except in [8]. Our aim is to establish uniqueness results for the inverse problem of the reconstruction of the curvature of the quantum guide: the data of one eigenpair determines uniquely the curvature up to its sign and similar results are obtained by considering the knowledge of a solution of Poisson's equation in the guide.

We consider the Laplacian operator on a nontrivially curved quantum guide which is not self-intersecting, with Dirichlet’s boundary conditions, denoted by . We proceed as in [1]. We denote by the function -smooth (see [7, Remark 5]) which characterizes the reference curve and by the outgoing normal to the boundary of . We denote by the fixed width of and by . Each point of is described by the curvilinear coordinates as follows: We assume and we recall that the signed curvature of is defined by named so because represents the curvature of the reference curve at . We recall that a guide is called simply bent if does not change sign in . We assume throughout this paper the following.

Assumption 1.1. One has the following.(i) is injective.(ii) (i.e., is nontrivially curved). (iii) where .(iv) as (i.e., is asymptotically straight).

Note that by the inverse function theorem, the map (defined by (1.1)) is a local diffeomorphism provided , for all , which is guaranteed by Assumption 1.1 and since is assumed to be injective, the map is a global diffeomorphism. Note also that for all and . More precisely, for all . The curvilinear coordinates are locally orthogonal, so by virtue of the Frenet-Serret formulae, the metric in is expressed with respect to them through a diagonal metric tensor, (e.g., [4]) The transition to the curvilinear coordinates represents an isometric map of to where is the Jacobian . So we can replace the Laplacian operator acting on by the Laplace-Beltrami operator acting on relative to the given metric tensor (see (1.3) and (1.4)) where We rewrite (defined by (1.5)) into a Schrödinger-type operator acting on . Indeed, using the unitary transformation setting we get with

We will assume throughout all this paper that the following assumption is satisfied.

Assumption 1.2. and for each where denotes the derivative of .

Remarks 1. Since is nontrivially curved and asymptotically straight, the operator has at least one eigenvalue of finite multiplicity below its essential spectrum (see [4, 7]; see also [1] under the additional assumptions that the width is sufficiently small and the curvature is rapidly decaying at infinity; see [3] under the assumption that the curvature has a compact support).
Furthermore, note that such operator admits bound states and that the minimum eigenvalue is simple and associated with a positive eigenfunction (see [9, Section 8.17]). Then, note that by [10, Theorem 7.1] any eigenfunction of is continuous and by [11, Remark 25 page 182] any eigenfunction of belongs to .
Finally, note also that is an eigenpair (i.e., an eigenfunction associated with its eigenvalue) of the operator acting on means that is an eigenpair of acting on . So the data of one eigenfunction of the operator is equivalent to the data of one eigenfunction of .
We first prove that the data of one eigenpair determines uniquely the curvature.

Theorem 1.3. Let be the curved guide in defined as above. Let be the signed curvature defined by (1.2) and satisfying Assumptions 1.1 and 1.2. Let be the operator defined by (1.8) and be an eigenpair of .
Then for all when .

Note that the condition in Theorem 1.3 is satisfied for the positive eigenfunction and for all . Then, we prove later in the paper under the following assumption.

Assumption 1.4. and for each , that one weak solution of the problem (where is a known given function) is in fact a classical solution and the data of determines uniquely the curvature .

Theorem 1.5. Let be the curved guide in defined as above. Let be the signed curvature defined by (1.2) and satisfying Assumptions 1.1 and 1.4. Let be the operator defined by (1.8). Let and let be a weak solution of (1.12).
Then we have for all when .

In the case of a simply bent guide (i.e., when does not change sign in ), we can restrain the hypotheses upon the regularity of . We obtain the following result.

Theorem 1.6. Let be the curved guide in defined as above. Let be the signed curvature defined by (1.2) and satisfying Assumptions 1.1 and 1.2. We assume also that is a nonnegative function. Let be the operator defined by (1.8). Let be a non null function and let be a weak solution in of (1.12). Assume that there exists a positive constant such that a.e. in . Then determines uniquely the curvature .

Note that the above result is still valid for a nonpositive function .

This paper is organized as follows. In Section 2, we prove Theorems 1.3, 1.5, and 1.6. In Sections 3 and 4, we extend our results to the case of a curved quantum guide defined in .

2. Proofs of Theorems 1.3, 1.5, and 1.6

2.1. Proof of Theorem 1.3

Recall that is an eigenfunction of , belonging to . Since is continuous and , then is continuous too. Thus, noticing that , we deduce the continuity of the function and from (1.8) to (1.10), we get and equivalently,

2.2. Proof of Theorem 1.5

First, we recall from [11, Remark 25 page 182] the following lemma.

Lemma 2.1. For a second-order elliptic operator defined in a domain , if satisfies then if is of class and for , if is of class

Now we can prove Theorem 1.5.

We have , so with defined by (1.9) and defined by (1.10).

Using Assumption 1.4, since for then and . From the hypotheses and , we get that for any , and so, using Lemma 2.1 for (2.6), we obtain that .

By the same way, we get that and for any (from for any ). Using Lemma 2.1, we obtain that .

We apply again Lemma 2.1 to get that (since for all , from the hypotheses and for ).

Finally, using Assumption 1.4 and Lemma 2.1, we obtain that .

Due to the regularity of , we have and . Since and , we can deduce that is continuous (see [11, Remark 8 page 154]).

Therefore, we can conclude by using the continuity of the function Therefore, we get and equivalently,

2.3. Proof of Theorem 1.6

We prove here that determines uniquely when is a nonnegative function.

For that, assume that and are two quantum guides in with same width . We denote by and the curvatures, respectively, associated with and , and we suppose that each satisfies Assumption 1.2 and is a nonnegative function. Assume that .

Then satisfies Assume that .

Step 1. First, we consider the case where (for example) for all .
Let with . Multiplying (2.9) by and integrating over , we get Since , for , and so in .
Moreover, since we have in .
Since Thus, from (2.10)−(2.12), we get with in and in . We can deduce that in .
Using a unique continuation theorem (see [12, Theorem XIII.63 page 240]), from , noting that , (recall that is defined by (1.6)) and so by we have with a.e., and we can deduce that in . So we get a contradiction (since and is assumed to be a non null function).

Step 2. From Step 1, we obtain that there exists at least one point such that . Since , we can choose and such that (for example) for all and .
We proceed as in Step 1, considering, in this case, . We study again (2.10) and as in Step 1, we have Indeed from (2.11) and we have and so By the same way, if , we also have . Thus, (2.10) becomes (2.13) with in and . So in and as in Step 1, by a unique continuation theorem, we obtain that in . Therefore, we get a contradiction.
Note that the previous theorem is true if we replace the hypothesis “ is nonnegative” by the hypothesis “ is nonpositive.” Indeed, in this last case, we just have to take and the proof rests valid.

3. Uniqueness Result for a -Quantum Guide

Now, we apply the same ideas for a tube in . We proceed here as in [7]. Let , be a curve in . We assume that is a -smooth curve satisfying the following hypotheses

Assumption 3.1. possesses a positively oriented Frenet frame with the following properties (i), (ii)for all ,(iii)for all , for all lies in the span of .

Recall that a sufficient condition to ensure the existence of the Frenet frame of Assumption 3.1 is to require that for all the vectors are linearly independent.

Then we define the moving frame along by following [7]. This moving frame better reflects the geometry of the curve and it is still called the Tang frame because it is a generalization of the Tang frame known from the theory of three-dimensional waveguides.

Given a bounded open connected neighborhood of , let denote the straight tube . We define the curved tube of cross-section about by with and being a real-valued differentiable function such that the torsion of . This differential equation is a consequence of the definition of the moving Tang frame (see [7, Remark 3]).

Note that is a rotation matrix in chosen in such a way that are orthogonal “coordinates” in . Let be the first curvature function of . Recall that since is a nonnegative function. We assume throughout all this section that the following hypothesis holds:

Assumption 3.2. One has the following.

does not overlap.

Assumption 3.2 assures that the map (defined by (3.1)) is a diffeomorphism (see [7]) in order to identify with the Riemannian manifold where is the metric tensor induced by , that is, , ( denoting the Jacobian matrix of ). Recall that (see [7]) with Note that Assumption 3.2 implies that for all and . Moreover, setting we can replace the Dirichlet Laplacian operator acting on by the Laplace-Beltrami operator acting on relative to the metric tensor . We can rewrite into a Schrödinger-type operator acting on . Indeed, using the unitary transformation setting we get where denotes the derivative relative to and denotes the derivative relative to and with We assume also throughout all this section that the following hypotheses hold:

Assumption 3.3. One has the following.
(i)(ii).

Remarks 2. Note that as for the 2-dimensional case, such operator (defined by (3.3)−(3.8)) admits bound states and that the minimum eigenvalue is simple and associated with a positive eigenfunction (see [7, 9]). Still note that is an eigenpair of the operator acting on means that is an eigenpair of acting on (with defined by (3.5)). Finally, note that by [10, Theorem 7.1] any eigenfunction of is continuous and by [11, Remark 25 page 182] any eigenfunction of belongs to .
As for the 2-dimensional case, first we prove that the data of one eigenpair determines uniquely the curvature.

Theorem 3.4. Let be the curved guide in defined as above. Let be the first curvature function of . Assume that Assumptions 3.1 to 3.3 are satisfied. Let be the operator defined by (3.3)−(3.8) and be an eigenpair of .
Then for all when .

Then, One has the following.

Assumption 3.5. One has the following.(i) for all ,(ii) for all ,where (resp., ) denotes the ith derivative of (resp. of ), we obtain the following result.

Theorem 3.6. Let be the curved guide in defined as above. Let be the first curvature function of . Assume that Assumptions 3.1 to 3.5 are satisfied. Let be the operator defined by (3.3)−(3.8). Let and let be a weak solution of in .
Then is a classical solution and for all when .

Remarks 3. Recall that in , is a nonnegative function and that the condition imposed on () in Theorems 3.4 and 3.6 is satisfied by the positive eigenfunction .
As for the two-dimensional case, we can restrain the hypotheses upon the regularity of the functions and .
For a guide with a known torsion, we obtain the following result.

Theorem 3.7. Let be the curved guide in defined as above. Let be the first curvature function of and let be the second curvature function (i.e., the torsion) of . Denote by a primitive of and suppose that for all . Assume that Assumptions 3.1 to 3.3 are satisfied. Let be the operator defined by (3.3)−(3.8). Let be a non null function and let be a weak solution of in . Assume that there exists a positive constant such that a.e. in .
Then the data determines uniquely the first curvature function if the torsion is given.

4. Proofs of Theorems 3.4, 3.6, and 3.7

4.1. Proof of Theorem 3.4

Recall that is an eigenfunction of . Since is continuous, and then is continuous. Therefore, for , we get: and equivalently, if .

4.2. Proof of Theorem 3.6

We follow the proof of Theorem 1.5. We have with . So with defined by (3.3) and defined by (3.8).

From Assumptions 3.2 and 3.3, since are bounded, we deduce that . Therefore, . Moreover, we have also and for any . Thus, using Lemma 2.1 for (4.1), we obtain that .

By the same way, we get that and for any (since and all of their derivatives are bounded). Using Lemma 2.1, we obtain that .

We apply again Lemma 2.1 to get that (since for all , from the hypotheses and for ).

Finally, using Assumption 3.5 and Lemma 2.1, we obtain that . Due to the regularity of (see [11, Note page 169]), we have and . Since and , we can deduce that is continuous (see [11, Remark  8 page 154]).

Thus, we conclude as in Theorem 1.5 and for , we get and equivalently, if .

4.3. Proof of Theorem 3.7

We prove here that determines uniquely .

Assume that and are two guides in . We denote by and the first curvatures functions associated with and and we denote by a primitive of the common torsion of and . We suppose that and satisfy Assumptions 3.2 and 3.3 and that for all . Assume that .

Then satisfies where (associated with ) is defined by (3.3), is defined by (3.8), (associated with ) is defined by (3.3), and is defined by (3.8).

Assume that .

Step 1. First, we consider the case where (for example) for all . Recall that each is a nonnegative function.
Let and denote by with small enough to have (recall that ).
Multiplying (4.2) by and integrating over , we get
Since for , and so in .
Moreover, note that with .
Since and for all , we have . Therefore, by (4.4), we deduce that in .
Thus, .
Note also that Therefore, from (4.3) and (4.5) we get with in and in .
From (4.6) we can deduce that in . Using a unique continuation theorem (see [12, Theorem XIII.63 page 240]), from , noting that , by a.e. in , we can deduce that in . So we get a contradiction since is assumed to be a non null function.

Step 2. From Step 1, we obtain that there exists at least one point such that . Since , we can choose and such that (for example) for all and if . We proceed as in Step 1, considering in this case . From , we get that . Therefore, we obtain . So (4.3) becomes (4.6) with in and in . So in and as in Step 1, by a unique continuation theorem, we obtain that in . Therefore, we get a contradiction.

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