^{1,2}

^{3}

^{1}

^{2}

^{3}

We consider the Dirichlet Laplacian operator

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 [

However, inverse problems associated with curved quantum guides have not been studied to our knowledge, except in [

We consider the Laplacian operator on a nontrivially curved quantum guide

One has the following.

Note that by the inverse function theorem, the map

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

Since

Furthermore, note that such operator

Finally, note also that

We first prove that the data of one eigenpair determines uniquely the curvature.

Let

Then

Note that the condition

Let

Then we have

In the case of a simply bent guide (i.e., when

Let

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

This paper is organized as follows. In Section

Recall that

First, we recall from [

For a second-order elliptic operator defined in a domain

Now we can prove Theorem

We have

Using Assumption

By the same way, we get that

We apply again Lemma

Finally, using Assumption

Due to the regularity of

Therefore, we can conclude by using the continuity of the function

We prove here that

For that, assume that

Then

First, we consider the case where (for example)

Let

Moreover, since

Since

Using a unique continuation theorem (see [

From Step

We proceed as in Step

Note that the previous theorem is true if we replace the hypothesis “

Now, we apply the same ideas for a tube

for all

for all

Recall that a sufficient condition to ensure the existence of the Frenet frame of Assumption

Then we define the moving frame

Given a

Note that

One has the following.

Assumption

One has the following.

Note that as for the 2-dimensional case, such operator

As for the 2-dimensional case, first we prove that the data of one eigenpair determines uniquely the curvature.

Let

Then

Then, One has the following.

One has the following.

Let

Then

Recall that in

As for the two-dimensional case, we can restrain the hypotheses upon the regularity of the functions

For a guide with a known torsion, we obtain the following result.

Let

Then the data

Recall that

We follow the proof of Theorem

From Assumptions

By the same way, we get that

We apply again Lemma

Finally, using Assumption

Thus, we conclude as in Theorem

We prove here that

Assume that

Then

Assume that

First, we consider the case where (for example)

Let

Multiplying (

Since

Moreover, note that

Since

Thus,

Note also that

From (

From Step