Inverse Problem for a Curved Quantum Guide

Abstract

We consider the Dirichlet Laplacian operator −Δ on a curved quantum guide in ℝ^𝑛(𝑛=2,3) 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 𝑛=2

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 [1–7].

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 Ω⊂ℝ² which is not self-intersecting, with Dirichlet’s boundary conditions, denoted by −Δ^Ω_𝐷. We proceed as in [1]. We denote by Γ=(Γ₁,Γ₂) the function 𝐶³-smooth (see [7, Remark 5]) which characterizes the reference curve and by 𝑁=(𝑁₁,𝑁₂) the outgoing normal to the boundary of Ω. We denote by 𝑑 the fixed width of Ω and by Ω₀∶=ℝ×]−𝑑/2,𝑑/2[. Each point (𝑥,𝑦) of Ω is described by the curvilinear coordinates (𝑠,𝑢) as follows: 𝑓∶Ω₀⟶Ωwith𝑥,𝑦=𝑓(𝑠,𝑢)=Γ(𝑠)+𝑢𝑁(𝑠). (1.1) We assume Γ^′₁(𝑠)²+Γ^′₂(𝑠)²=1 and we recall that the signed curvature 𝛾 of Γ is defined by 𝛾(𝑠)=−Γ₁^′′(𝑠)Γ^′₂(𝑠)+Γ₂^′′(𝑠)Γ^′₁(𝑠), (1.2) named so because |𝛾(𝑠)| represents the curvature of the reference curve at 𝑠. We recall that a guide is called simply bent if 𝛾 does not change sign in ℝ. We assume throughout this paper the following.

Assumption 1.1. One has the following.(i)𝑓 is injective.(ii)𝛾∈𝐶²(ℝ)∩𝐿^∞(ℝ),𝛾≢0, (i.e., Ω is nontrivially curved). (iii)𝑑/2<1/‖𝛾‖_∞, where ‖𝛾‖_∞∶=sup_𝑠∈ℝ|𝛾(𝑠)|=‖𝛾‖_𝐿∞(ℝ).(iv)𝛾(𝑠)→0 as |𝑠|→+∞ (i.e., Ω is asymptotically straight).

Note that by the inverse function theorem, the map 𝑓 (defined by (1.1)) is a local diffeomorphism provided 1−𝑢𝛾(𝑠)≠0, for all 𝑢,𝑠, which is guaranteed by Assumption 1.1 and since 𝑓 is assumed to be injective, the map 𝑓 is a global diffeomorphism. Note also that 1−𝑢𝛾(𝑠)>0 for all 𝑢 and 𝑠. More precisely, 0<1−(𝑑/2)‖𝛾‖_∞≤1−𝑢𝛾(𝑠)≤1+(𝑑/2)‖𝛾‖_∞ for all 𝑢,𝑠. The curvilinear coordinates (𝑠,𝑢) are locally orthogonal, so by virtue of the Frenet-Serret formulae, the metric in Ω is expressed with respect to them through a diagonal metric tensor, (e.g., [4]) 𝑔_𝑖𝑗=1−𝑢𝛾(𝑠)²001. (1.3) The transition to the curvilinear coordinates represents an isometric map of 𝐿²(Ω) to 𝐿²(Ω₀,𝑔^1/2𝑑𝑠𝑑𝑢) where 𝑔(𝑠,𝑢)^1/2∶=1−𝑢𝛾(𝑠) (1.4) is the Jacobian 𝜕(𝑥,𝑦)/𝜕(𝑠,𝑢). So we can replace the Laplacian operator −Δ^Ω_𝐷 acting on 𝐿²(Ω) by the Laplace-Beltrami operator 𝐻_𝑔 acting on 𝐿²(Ω₀,𝑔^1/2d𝑠d𝑢) relative to the given metric tensor (𝑔_𝑖𝑗) (see (1.3) and (1.4)) where 𝐻_𝑔∶=−𝑔^−1/2𝜕_𝑠𝑔^−1/2𝜕_𝑠−𝑔^−1/2𝜕_𝑢𝑔^1/2𝜕_𝑢. (1.5) We rewrite 𝐻_𝑔 (defined by (1.5)) into a Schrödinger-type operator acting on 𝐿²(Ω₀,𝑑𝑠𝑑𝑢). Indeed, using the unitary transformation 𝑈_𝑔∶𝐿²Ω₀,𝑔^1/2𝑑𝑠𝑑𝑢⟶𝐿²Ω₀,𝑑𝑠𝑑𝑢𝜓⟼𝑔^1/4𝜓, (1.6) setting 𝐻_𝛾∶=𝑈_𝑔𝐻_𝑔𝑈_𝑔⁻¹, (1.7) we get 𝐻_𝛾=−𝜕_𝑠𝑐_𝛾(𝑠,𝑢)𝜕_𝑠−𝜕²_𝑢+𝑉_𝛾(𝑠,𝑢), (1.8) with 𝑐_𝛾()=1𝑠,𝑢1−𝑢𝛾(𝑠)²,𝑉_𝛾()𝛾𝑠,𝑢=−²(𝑠)41−𝑢𝛾(𝑠)²−𝑢𝛾^′′(𝑠)21−𝑢𝛾(𝑠)³−5𝑢²𝛾^′2(𝑠)41−𝑢𝛾(𝑠)⁴. (1.9)(1.10)

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

Assumption 1.2. 𝛾∈𝐶²(ℝ) and 𝛾^(𝑘)∈𝐿^∞(ℝ) for each 𝑘=0,1,2 where 𝛾^(𝑘) denotes the 𝑘th derivative of 𝛾.

Remarks 1. Since Ω is nontrivially curved and asymptotically straight, the operator −Δ^Ω_𝐷 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 𝑑 is sufficiently small and the curvature 𝛾 is rapidly decaying at infinity; see [3] under the assumption that the curvature 𝛾 has a compact support).
Furthermore, note that such operator 𝐻_𝛾 admits bound states and that the minimum eigenvalue 𝜆₁ is simple and associated with a positive eigenfunction 𝜙₁ (see [9, Section 8.17]). Then, note that by [10, Theorem 7.1] any eigenfunction of 𝐻_𝛾 is continuous and by [11, Remark 25 page 182] any eigenfunction of 𝐻_𝛾 belongs to 𝐻²(Ω₀).
Finally, note also that (𝜆,𝜙) is an eigenpair (i.e., an eigenfunction associated with its eigenvalue) of the operator 𝐻_𝛾 acting on 𝐿²(Ω₀,𝑑𝑠𝑑𝑢) means that (𝜆,𝑈_𝑔⁻¹𝜙) is an eigenpair of −Δ^Ω_𝐷 acting on 𝐿²(Ω). So the data of one eigenfunction of the operator 𝐻_𝛾 is equivalent to the data of one eigenfunction of −Δ^Ω_𝐷.
We first prove that the data of one eigenpair determines uniquely the curvature.

Theorem 1.3. Let Ω be the curved guide in ℝ² defined as above. Let 𝛾 be the signed curvature defined by (1.2) and satisfying Assumptions 1.1 and 1.2. Let 𝐻_𝛾 be the operator defined by (1.8) and (𝜆,𝜙) be an eigenpair of 𝐻_𝛾.
Then 𝛾²(𝑠)()=−4Δ𝜙𝑠,0𝜙(𝑠,0)−4𝜆, (1.11) for all 𝑠 when 𝜙(𝑠,0)≠0.

Note that the condition 𝜙(𝑠,0)≠0 in Theorem 1.3 is satisfied for the positive eigenfunction 𝜙₁ and for all 𝑠∈ℝ. Then, we prove later in the paper under the following assumption.

Assumption 1.4. 𝛾∈𝐶⁵(ℝ) and 𝛾^(𝑘)∈𝐿^∞(ℝ) for each 𝑘=0,…,5, that one weak solution 𝜙 of the problem 𝐻_𝛾𝜙=𝑓inΩ₀𝜙=0on𝜕Ω₀, (1.12) (where 𝑓 is a known given function) is in fact a classical solution and the data of 𝜙 determines uniquely the curvature 𝛾.

Theorem 1.5. Let Ω be the curved guide in ℝ² defined as above. Let 𝛾 be the signed curvature defined by (1.2) and satisfying Assumptions 1.1 and 1.4. Let 𝐻_𝛾 be the operator defined by (1.8). Let 𝑓∈𝐻³(Ω₀)∩𝐶(Ω₀) and let 𝜙∈𝐻¹₀(Ω₀) be a weak solution of (1.12).
Then we have 𝛾²(𝑠)=−4(Δ𝜙(𝑠,0)/𝜙(𝑠,0))−4(𝑓(𝑠,0)/𝜙(𝑠,0)) for all 𝑠 when 𝜙(𝑠,0)≠0.

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

Theorem 1.6. Let Ω be the curved guide in ℝ² defined as above. Let 𝛾 be the signed curvature defined by (1.2) and satisfying Assumptions 1.1 and 1.2. We assume also that 𝛾 is a nonnegative function. Let 𝐻_𝛾 be the operator defined by (1.8). Let 𝑓∈𝐿²(Ω₀) be a non null function and let 𝜙 be a weak solution in 𝐻¹₀(Ω₀) of (1.12). Assume that there exists a positive constant 𝑀 such that |𝑓(𝑠,𝑢)|≤𝑀|𝜙(𝑠,𝑢)| a.e. in Ω₀. Then (𝑓,𝜙) determines uniquely the curvature 𝛾.

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

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 ℝ³.

2. Proofs of Theorems 1.3, 1.5, and 1.6

2.1. Proof of Theorem 1.3

Recall that 𝜙 is an eigenfunction of 𝐻_𝛾, belonging to 𝐻²(Ω₀). Since 𝜙 is continuous and 𝐻_𝛾𝜙=𝜆𝜙, then 𝐻_𝛾𝜙 is continuous too. Thus, noticing that 𝑐_𝛾(𝑠,0)=1, we deduce the continuity of the function (𝑠,0)↦Δ𝜙(𝑠,0) and from (1.8) to (1.10), we get ()−𝛾−Δ𝜙𝑠,0²(𝑠)4𝜙(𝑠,0)=𝜆𝜙(𝑠,0), (2.1) and equivalently, 𝛾²(𝑠)()=−4Δ𝜙𝑠,0𝜙()()𝑠,0−4𝜆if𝜙𝑠,0≠0. (2.2)

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 𝜔⊂ℝ^𝑛, if 𝜙∈𝐻¹₀(𝜔) satisfies _{𝜔𝑖,𝑗}𝑎_𝑖𝑗𝜕𝜙𝜕𝑥_𝑖𝜕𝜓𝜕𝑥_𝑗=_𝜔𝑓𝜓∀𝜓∈𝐻¹₀(𝜔) (2.3) then if 𝜔 is of class 𝐶²𝑓∈𝐿²(𝜔),𝑎_𝑖𝑗∈𝐶¹(𝜔),𝐷^𝛼𝑎_𝑖𝑗∈𝐿^∞(𝜔)∀𝑖,𝑗,∀𝛼,|𝛼|≤1imply𝜙∈𝐻²(𝜔) (2.4) and for 𝑚≥1, if 𝜔 is of class 𝐶^𝑚+2𝑓∈𝐻^𝑚(𝜔),𝑎_𝑖𝑗∈𝐶^𝑚+1(𝜔),𝐷^𝛼𝑎_𝑖𝑗∈𝐿^∞(𝜔)∀𝑖,𝑗,∀𝛼,|𝛼|≤𝑚+1imply𝜙∈𝐻^𝑚+2(𝜔). (2.5)

Now we can prove Theorem 1.5.

We have 𝐻_𝛾𝜙=𝑓, so _Ω0𝑐_𝛾𝜕_𝑠𝜙𝜕_𝑠𝜓+𝜕_𝑢𝜙𝜕_𝑢𝜓=_Ω0𝑓−𝑉_𝛾𝜙𝜓,∀𝜓∈𝐻¹₀Ω₀ (2.6) with 𝑐_𝛾 defined by (1.9) and 𝑉_𝛾 defined by (1.10).

Using Assumption 1.4, since 𝛾^(𝑘)∈𝐿^∞(Ω₀) for 𝑘=0,1,2 then 𝑉_𝛾∈𝐿^∞(Ω₀) and 𝑓−𝑉_𝛾𝜙∈𝐿²(Ω₀). From the hypotheses 𝛾∈𝐶¹(ℝ) and 𝛾′∈𝐿^∞(ℝ), we get that 𝑐_𝛾∈𝐶¹(Ω₀),𝐷^𝛼𝑐_𝛾∈𝐿^∞(Ω₀) for any 𝛼,|𝛼|≤1, and so, using Lemma 2.1 for (2.6), we obtain that 𝜙∈𝐻²(Ω₀).

By the same way, we get that 𝑓−𝑉_𝛾𝜙∈𝐻¹(Ω₀),𝑐_𝛾∈𝐶²(Ω₀) and 𝐷^𝛼𝑐_𝛾∈𝐿^∞(Ω₀) for any 𝛼,|𝛼|≤2 (from 𝛾∈𝐶³(ℝ),𝛾^(𝑘)∈𝐿^∞(ℝ) for any 𝑘=0,…,3). Using Lemma 2.1, we obtain that 𝜙∈𝐻³(Ω₀).

We apply again Lemma 2.1 to get that 𝜙∈𝐻⁴(Ω₀) (since 𝑓−𝑉_𝛾𝜙∈𝐻²(Ω₀),𝑐_𝛾∈𝐶³(Ω₀),𝐷^𝛼𝑐_𝛾∈𝐿^∞(Ω₀) for all 𝛼,|𝛼|≤3, from the hypotheses 𝛾∈𝐶⁴(ℝ) and 𝛾^(𝑘)∈𝐿^∞(ℝ) for 𝑘=0,…,4).

Finally, using Assumption 1.4 and Lemma 2.1, we obtain that 𝜙∈𝐻⁵(Ω₀).

Due to the regularity of Ω₀, we have 𝜙∈𝐻⁵(ℝ²) and Δ𝜙∈𝐻³(ℝ²). Since ∇(Δ𝜙)∈(𝐻²(ℝ²))² and 𝐻²(ℝ²)⊂𝐿^∞(ℝ²), we can deduce that Δ𝜙 is continuous (see [11, Remark 8 page 154]).

Therefore, we can conclude by using the continuity of the function (𝑠,0)↦−𝜕_𝑠𝑐_𝛾(𝑠,0)𝜕_𝑠𝜙(𝑠,0)−𝜕²_𝑢𝜙(𝑠,0)=𝑓(𝑠,0)−𝑉_𝛾(𝑠,0)𝜙(𝑠,0). (2.7) Therefore, we get −Δ𝜙(𝑠,0)−(𝛾²(𝑠)/4)𝜙(𝑠,0)=𝑓(𝑠,0) and equivalently, 𝛾²(𝑠)()=−4Δ𝜙𝑠,0𝜙()𝑓()𝑠,0−4𝑠,0𝜙()()𝑠,0if𝜙𝑠,0≠0. (2.8)

2.3. Proof of Theorem 1.6

We prove here that (𝑓,𝜙) determines uniquely 𝛾 when 𝛾 is a nonnegative function.

For that, assume that Ω₁ and Ω₂ are two quantum guides in ℝ² with same width 𝑑. We denote by 𝛾₁ and 𝛾₂ the curvatures, respectively, associated with Ω₁ and Ω₂, and we suppose that each 𝛾_𝑖 satisfies Assumption 1.2 and is a nonnegative function. Assume that 𝐻_𝛾1𝜙=𝑓=𝐻_𝛾2𝜙.

Then 𝜙 satisfies −𝜕_𝑠𝑐_𝛾1(𝑠,𝑢)−𝑐_𝛾2(𝑠,𝑢)𝜕_𝑠𝜙(𝑠,𝑢)+𝑉_𝛾1(𝑠,𝑢)−𝑉_𝛾2(𝑠,𝑢)𝜙(𝑠,𝑢)=0. (2.9) Assume that 𝛾₁≢𝛾₂.

Step 1. First, we consider the case where (for example) 𝛾₁(𝑠)<𝛾₂(𝑠) for all 𝑠∈ℝ.
Let 𝜖>0,𝜔_𝜖∶=ℝ×𝐼_𝜖 with 𝐼_𝜖=]−𝜖,0[. Multiplying (2.9) by 𝜙 and integrating over 𝜔_𝜖, we get _𝜔𝜖𝑐_𝛾1−𝑐_𝛾2𝜕_𝑠𝜙²−_𝜕𝜔𝜖𝑐_𝛾1−𝑐_𝛾2𝜕_𝑠𝜙𝜙𝜈_𝑠+_𝜔𝜖𝑉_𝛾1−𝑉_𝛾2𝜙²=0. (2.10) Since 𝜖≪1, 𝑉_𝛾𝑖(𝑠,𝑢)≃−𝛾²_𝑖(𝑠)/4 for 𝑖=1,2, and so 𝑉_𝛾1(𝑠,𝑢)−𝑉_𝛾2(𝑠,𝑢)>0 in 𝜔_𝜖.
Moreover, since 𝑐_𝛾1(𝑠,𝑢)−𝑐_𝛾2()=𝑢𝛾𝑠,𝑢¹(𝑠)−𝛾₂(𝑠)2−𝑢𝛾₁(𝑠)+𝛾₂(𝑠)1−𝑢𝛾₁(𝑠)²1−𝑢𝛾₂(𝑠)², (2.11) we have 𝑐_𝛾1(𝑠,𝑢)>𝑐_𝛾2(𝑠,𝑢) in 𝜔_𝜖.
Since _𝜕𝜔𝜖𝑐_𝛾1−𝑐_𝛾2𝜕_𝑠𝜙𝜙𝜈_𝑠=0 (2.12) Thus, from (2.10)−(2.12), we get _𝜔𝜖𝑐_𝛾1−𝑐_𝛾2𝜕_𝑠𝜙²+_𝜔𝜖𝑉_𝛾1−𝑉_𝛾2𝜙²=0, (2.13) with 𝑐_𝛾1−𝑐_𝛾2>0 in 𝜔_𝜖 and 𝑉_𝛾1−𝑉_𝛾2>0 in 𝜔_𝜖. We can deduce that 𝜙=0 in 𝜔_𝜖.
Using a unique continuation theorem (see [12, Theorem XIII.63 page 240]), from 𝐻_𝛾𝜙=𝑓, noting that −Δ(𝑈_𝑔⁻¹𝜙)=𝑈_𝑔⁻¹𝑓=𝑔^−1/4𝑓, (recall that 𝑈_𝑔 is defined by (1.6)) and so by |𝑓|≤𝑀|𝜙| we have |Δ(𝑈_𝑔⁻¹𝜙)|≤𝑀|𝑔^−1/4𝜙| with 𝑔>0 a.e., and we can deduce that 𝜙=0 in Ω₀. So we get a contradiction (since 𝐻_𝛾𝜙=𝑓 and 𝑓 is assumed to be a non null function).

Step 2. From Step 1, we obtain that there exists at least one point 𝑠₀∈ℝ such that 𝛾₁(𝑠₀)=𝛾₂(𝑠₀). Since 𝛾₁≢𝛾₂, we can choose 𝑎∈ℝ and 𝑏∈ℝ∪{+∞} such that (for example) 𝛾₁(𝑎)=𝛾₂(𝑎),𝛾₁(𝑠)<𝛾₂(𝑠) for all 𝑠∈]𝑎,𝑏[ and 𝛾₁(𝑏)=𝛾₂(𝑏)if𝑏∈ℝ.
We proceed as in Step 1, considering, in this case, 𝜔_𝜖∶=]𝑎,𝑏[×𝐼_𝜖. We study again (2.10) and as in Step 1, we have _𝜕𝜔𝜖𝑐_𝛾1−𝑐_𝛾2𝜕_𝑠𝜙𝜙𝜈_𝑠=0. (2.14) Indeed from (2.11) and 𝛾₁(𝑎)=𝛾₂(𝑎) we have 𝑐_𝛾1(𝑎,𝑢)=𝑐_𝛾2(𝑎,𝑢) and so ⁰_−𝜖𝑐_𝛾1(𝑎,𝑢)−𝑐_𝛾2(𝑎,𝑢)𝜕_𝑠𝜙(𝑎,𝑢)𝜙(𝑎,𝑢)𝑑𝑢=0. (2.15) By the same way, if 𝑏∈ℝ, we also have 𝑐_𝛾1(𝑏,𝑢)=𝑐_𝛾2(𝑏,𝑢). Thus, (2.10) becomes (2.13) with 𝑐_𝛾1−𝑐_𝛾2>0 in 𝜔_𝜖 and 𝑉_𝛾1−𝑉_𝛾2>0in𝜔_𝜖. So 𝜙=0 in 𝜔_𝜖 and as in Step 1, by a unique continuation theorem, we obtain that 𝜙=0 in Ω₀. Therefore, we get a contradiction.
Note that the previous theorem is true if we replace the hypothesis “𝛾 is nonnegative” by the hypothesis “𝛾 is nonpositive.” Indeed, in this last case, we just have to take 𝐼_𝜖=]0,𝜖[ and the proof rests valid.

3. Uniqueness Result for a ℝ³-Quantum Guide

Now, we apply the same ideas for a tube Ω in ℝ³. We proceed here as in [7]. Let 𝑠↦Γ(𝑠),Γ=(Γ₁,Γ₂,Γ₃), be a curve in ℝ³. We assume that Γ∶ℝ→ℝ³ is a 𝐶⁴-smooth curve satisfying the following hypotheses

Assumption 3.1. Γ possesses a positively oriented Frenet frame {𝑒₁,𝑒₂,𝑒₃} with the following properties (i)𝑒₁=Γ′, (ii)for all 𝑖∈{1,2,3},𝑒_𝑖∈𝐶¹(ℝ,ℝ³),(iii)for all 𝑖∈{1,2}, for all 𝑠∈ℝ,𝑒^′_𝑖(𝑠) lies in the span of 𝑒₁(𝑠),…,𝑒_𝑖+1(𝑠).

Recall that a sufficient condition to ensure the existence of the Frenet frame of Assumption 3.1 is to require that for all 𝑠∈ℝ the vectors Γ^′(𝑠),Γ^′′(𝑠) are linearly independent.

Then we define the moving frame {𝑒₁̃,𝑒₂̃,𝑒₃̃} along Γ 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 𝐶⁵ bounded open connected neighborhood 𝜔 of (0,0)∈ℝ², let Ω₀ denote the straight tube ℝ×𝜔. We define the curved tube Ω of cross-section 𝜔 about Γ by Ω∶=𝑓(ℝ×𝜔)=𝑓Ω₀,𝑓𝑠,𝑢₂,𝑢₃∶=Γ(𝑠)+³_𝑖=2𝑢_𝑖³_𝑗=2𝑅_𝑖𝑗(𝑠)𝑒_𝑗(𝑠)=Γ(𝑠)+³_𝑖=2𝑢_𝑖𝑒_𝑖̃(𝑠), (3.1) with 𝑢=(𝑢₂,𝑢₃)∈𝜔 and 𝑅(𝑠)∶=𝑅_𝑖𝑗(𝑠)_{𝑖,𝑗∈{2,3}}=(𝜃(𝑠(𝜃(𝑠(𝜃(𝑠(𝜃(𝑠cos))−sin))sin))cos)), (3.2)𝜃 being a real-valued differentiable function such that 𝜃^′(𝑠)=𝜏(𝑠) the torsion of Γ. This differential equation is a consequence of the definition of the moving Tang frame (see [7, Remark 3]).

Note that 𝑅 is a rotation matrix in ℝ² chosen in such a way that (𝑠,𝑢₂,𝑢₃) are orthogonal “coordinates” in Ω. Let 𝑘 be the first curvature function of Ω. Recall that since Ω⊂ℝ³,𝑘 is a nonnegative function. We assume throughout all this section that the following hypothesis holds:

Assumption 3.2. One has the following.
𝑘∈𝐶²(ℝ)∩𝐿^∞(ℝ),𝑎∶=sup_𝑢∈𝜔‖𝑢‖_ℝ2<1/‖𝑘‖_∞,𝑘(𝑠)→0as|𝑠|→+∞
Ω does not overlap.

Assumption 3.2 assures that the map 𝑓 (defined by (3.1)) is a diffeomorphism (see [7]) in order to identify Ω with the Riemannian manifold (Ω₀,(𝑔_𝑖𝑗)) where (𝑔_𝑖𝑗) is the metric tensor induced by 𝑓, that is, (𝑔_𝑖𝑗)∶=^𝑡𝐽(𝑓)⋅𝐽(𝑓), (𝐽(𝑓) denoting the Jacobian matrix of 𝑓). Recall that (𝑔_𝑖𝑗)=diag(ℎ²,1,1) (see [7]) with ℎ𝑠,𝑢₂,𝑢₃∶=1−𝑘(𝑠)cos(𝜃(𝑠))𝑢₂+sin(𝜃(𝑠))𝑢₃. (3.3) Note that Assumption 3.2 implies that 0<1−𝑎‖𝑘‖_∞≤1−ℎ(𝑠,𝑢₂,𝑢₃)≤1+𝑎‖𝑘‖_∞ for all 𝑠∈ℝ and 𝑢=(𝑢₂,𝑢₃)∈𝜔. Moreover, setting 𝑔∶=ℎ², (3.4) we can replace the Dirichlet Laplacian operator −Δ^Ω_𝐷 acting on 𝐿²(Ω) by the Laplace-Beltrami operator 𝐾_𝑔 acting on 𝐿²(Ω₀,ℎd𝑠d𝑢) relative to the metric tensor (𝑔_𝑖𝑗). We can rewrite 𝐾_𝑔 into a Schrödinger-type operator acting on 𝐿²(Ω₀,𝑑𝑠𝑑𝑢). Indeed, using the unitary transformation 𝑊_𝑔∶𝐿²Ω₀,ℎ𝑑𝑠𝑑𝑢⟶𝐿²Ω₀,𝑑𝑠𝑑𝑢𝜓↦𝑔^1/4𝜓, (3.5) setting 𝐻_𝑘∶=𝑊_𝑔𝐾_𝑔𝑊_𝑔⁻¹, (3.6) we get 𝐻_𝑘=−𝜕_𝑠ℎ⁻²𝜕_𝑠−𝜕²_𝑢2−𝜕²_𝑢3+𝑉_𝑘, (3.7) where 𝜕_𝑠 denotes the derivative relative to 𝑠 and 𝜕_𝑢𝑖 denotes the derivative relative to 𝑢_𝑖 and with 𝑉_𝑘𝑘∶=−²4ℎ²+𝜕²_𝑠ℎ2ℎ³−5𝜕_𝑠ℎ²4ℎ⁴. (3.8) We assume also throughout all this section that the following hypotheses hold:

Assumption 3.3. One has the following.
(i)𝑘^′∈𝐿^∞(ℝ),𝑘^′′∈𝐿^∞(ℝ)(ii)𝜃∈𝐶²(ℝ),𝜃^′=𝜏∈𝐿^∞(ℝ),𝜃^′′∈𝐿^∞(ℝ).

Remarks 2. Note that as for the 2-dimensional case, such operator 𝐻_𝑘 (defined by (3.3)−(3.8)) admits bound states and that the minimum eigenvalue 𝜆₁ is simple and associated with a positive eigenfunction 𝜙₁ (see [7, 9]). Still note that (𝜆,𝜙) is an eigenpair of the operator 𝐻_𝑘 acting on 𝐿²(Ω₀,𝑑𝑠𝑑𝑢) means that (𝜆,𝑊_𝑔⁻¹𝜙) is an eigenpair of −Δ^Ω_𝐷 acting on 𝐿²(Ω) (with 𝑊_𝑔 defined by (3.5)). Finally, note that by [10, Theorem 7.1] any eigenfunction of 𝐻_𝑘 is continuous and by [11, Remark 25 page 182] any eigenfunction of 𝐻_𝑘 belongs to 𝐻²(Ω₀).
As for the 2-dimensional case, first we prove that the data of one eigenpair determines uniquely the curvature.

Theorem 3.4. Let Ω be the curved guide in ℝ³ defined as above. Let 𝑘 be the first curvature function of Ω. Assume that Assumptions 3.1 to 3.3 are satisfied. Let 𝐻_𝑘 be the operator defined by (3.3)−(3.8) and (𝜆,𝜙) be an eigenpair of 𝐻_𝑘.
Then 𝑘²(𝑠)=−4(Δ𝜙(𝑠,0,0)/𝜙(𝑠,0,0))−4𝜆 for all 𝑠 when 𝜙(𝑠,0,0)≠0.

Then, One has the following.

Assumption 3.5. One has the following.(i)𝑘∈𝐶⁵(ℝ),𝑘^(𝑖)∈𝐿^∞(ℝ) for all 𝑖=0,…,5,(ii)𝜃∈𝐶⁵(ℝ),𝜃^(𝑖)∈𝐿^∞(ℝ) for all 𝑖=1,…,5,where 𝑘^(𝑖) (resp., 𝜃^(𝑖)) denotes the ith derivative of 𝑘 (resp. of 𝜃), we obtain the following result.

Theorem 3.6. Let Ω be the curved guide in ℝ³ defined as above. Let 𝑘 be the first curvature function of Ω. Assume that Assumptions 3.1 to 3.5 are satisfied. Let 𝐻_𝑘 be the operator defined by (3.3)−(3.8). Let 𝑓∈𝐻³(Ω₀)∩𝐶(Ω₀) and let 𝜙∈𝐻¹₀(Ω₀) be a weak solution of 𝐻_𝑘𝜙=𝑓 in Ω₀.
Then 𝜙 is a classical solution and 𝑘²(𝑠)=−4(Δ𝜙(𝑠,0,0)/𝜙(𝑠,0,0))−4(𝑓(𝑠,0,0)/𝜙(𝑠,0,0)) for all 𝑠 when 𝜙(𝑠,0,0)≠0.

Remarks 3. Recall that in ℝ³, 𝑘 is a nonnegative function and that the condition imposed on 𝜙 (𝜙(𝑠,0,0)≠0) in Theorems 3.4 and 3.6 is satisfied by the positive eigenfunction 𝜙₁.
As for the two-dimensional case, we can restrain the hypotheses upon the regularity of the functions 𝑘 and 𝜃.
For a guide with a known torsion, we obtain the following result.

Theorem 3.7. Let Ω be the curved guide in ℝ³ defined as above. Let 𝑘 be the first curvature function of Ω and let 𝜏 be the second curvature function (i.e., the torsion) of Ω. Denote by 𝜃 a primitive of 𝜏 and suppose that 0≤𝜃(𝑠)≤𝜋/2 for all 𝑠∈ℝ. Assume that Assumptions 3.1 to 3.3 are satisfied. Let 𝐻_𝑘 be the operator defined by (3.3)−(3.8). Let 𝑓∈𝐿²(Ω₀) be a non null function and let 𝜙∈𝐻¹₀(Ω₀) be a weak solution of 𝐻_𝑘𝜙=𝑓 in Ω₀. Assume that there exists a positive constant 𝑀 such that |𝑓(𝑠,𝑢)|≤𝑀|𝜙(𝑠,𝑢)| a.e. in Ω₀.
Then the data (𝑓,𝜙) determines uniquely the first curvature function 𝑘 if the torsion 𝜏 is given.

4. Proofs of Theorems 3.4, 3.6, and 3.7

4.1. Proof of Theorem 3.4

Recall that 𝜙 is an eigenfunction of 𝐻_𝑘. Since 𝜙 is continuous, 𝐻_𝑘𝜙=𝜆𝜙 and 𝜙∈𝐻²(Ω₀) then 𝐻_𝑘𝜙 is continuous. Therefore, for 𝑢=(𝑢₂,𝑢₃)=(0,0), we get: −Δ𝜙(𝑠,0,0)−(𝑘²(𝑠)/4)𝜙(𝑠,0,0)=𝜆𝜙(𝑠,0,0) and equivalently, 𝑘²(𝑠)=−4(Δ𝜙(𝑠,0,0)/𝜙(𝑠,0,0))−4𝜆 if 𝜙(𝑠,0,0)≠0.

4.2. Proof of Theorem 3.6

We follow the proof of Theorem 1.5. We have 𝐻_𝑘𝜙=𝑓 with 𝜙∈𝐻¹₀(Ω₀). So _Ω0ℎ⁻²𝜕_𝑠𝜙𝜕_𝑠𝜓+𝜕_𝑢2𝜙𝜕_𝑢2𝜓+𝜕_𝑢3𝜙𝜕_𝑢3𝜓=_Ω0𝑓−𝑉_𝑘𝜙𝜓,∀𝜓∈𝐻¹₀Ω₀, (4.1) with ℎ defined by (3.3) and 𝑉_𝑘 defined by (3.8).

From Assumptions 3.2 and 3.3, since 𝑘,𝑘^′,𝑘^′′,𝜃^′,𝜃^′′ are bounded, we deduce that 𝑉_𝑘∈𝐿^∞(Ω₀). Therefore, 𝑓−𝑉_𝑘𝜙∈𝐿²(Ω₀). Moreover, we have also ℎ⁻²∈𝐶¹(Ω₀) and 𝐷^𝛼(ℎ⁻²)∈𝐿^∞(Ω₀) for any 𝛼,|𝛼|≤1. Thus, using Lemma 2.1 for (4.1), we obtain that 𝜙∈𝐻²(Ω₀).

By the same way, we get that 𝑓−𝑉_𝑘𝜙∈𝐻¹(Ω₀),ℎ⁻²∈𝐶²(Ω₀) and 𝐷^𝛼(ℎ⁻²)∈𝐿^∞(Ω₀) for any 𝛼,|𝛼|≤2 (since 𝑘∈𝐶³(ℝ),𝜃∈𝐶³(ℝ) and all of their derivatives are bounded). Using Lemma 2.1, we obtain that 𝜙∈𝐻³(Ω₀).

We apply again Lemma 2.1 to get that 𝜙∈𝐻⁴(Ω₀) (since 𝑓−𝑉_𝛾𝜙∈𝐻²(Ω₀),𝑐_𝛾∈𝐶³(Ω₀),𝐷^𝛼𝑐_𝛾∈𝐿^∞(Ω₀) for all 𝛼,|𝛼|≤3, from the hypotheses 𝛾∈𝐶⁴(ℝ) and 𝛾^(𝑘)∈𝐿^∞(ℝ) for 𝑘=0,…,4).

Finally, using Assumption 3.5 and Lemma 2.1, we obtain that 𝜙∈𝐻⁵(Ω₀). Due to the regularity of Ω₀ (see [11, Note page 169]), we have 𝜙∈𝐻⁵(ℝ³) and Δ𝜙∈𝐻³(ℝ³). Since ∇(Δ𝜙)∈(𝐻²(ℝ³))³ and 𝐻²(ℝ³)⊂𝐿^∞(ℝ³), we can deduce that Δ𝜙 is continuous (see [11, Remark  8 page 154]).

Thus, we conclude as in Theorem 1.5 and for 𝑢=(𝑢₂,𝑢₃)=(0,0), we get −Δ𝜙(𝑠,0,0)−(𝑘²(𝑠)/4)𝜙(𝑠,0,0)=𝑓(𝑠,0,0) and equivalently, 𝑘²(𝑠)=−4(Δ𝜙(𝑠,0,0)/𝜙(𝑠,0,0))−4(𝑓(𝑠,0,0)/𝜙(𝑠,0,0)) if 𝜙(𝑠,0,0)≠0.

4.3. Proof of Theorem 3.7

We prove here that (𝑓,𝜙,𝜃) determines uniquely 𝑘.

Assume that Ω₁ and Ω₂ are two guides in ℝ³. We denote by 𝑘₁ and 𝑘₂ the first curvatures functions associated with Ω₁ and Ω₂ and we denote by 𝜃 a primitive of 𝜏 the common torsion of Ω₁ and Ω₂. We suppose that 𝑘₁,𝑘₂ and 𝜃 satisfy Assumptions 3.2 and 3.3 and that 0≤𝜃(𝑠)≤𝜋/2 for all 𝑠∈ℝ. Assume that 𝐻_𝑘1𝜙=𝑓=𝐻_𝑘2𝜙.

Then 𝜙 satisfies −𝜕_𝑠ℎ₁⁻²𝑠,𝑢₂,𝑢₃−ℎ₂⁻²𝑠,𝑢₂,𝑢₃𝜕_𝑠𝜙𝑠,𝑢₂,𝑢₃+𝑉_𝑘1𝑠,𝑢₂,𝑢₃−𝑉_𝑘2𝑠,𝑢₂,𝑢₃𝜙𝑠,𝑢₂,𝑢₃=0, (4.2) where ℎ₁ (associated with 𝑘₁) is defined by (3.3), 𝑉_𝑘1 is defined by (3.8), ℎ₂ (associated with 𝑘₂) is defined by (3.3), and 𝑉_𝑘2 is defined by (3.8).

Assume that 𝑘₁≢𝑘₂.

Step 1. First, we consider the case where (for example) 𝑘₁(𝑠)<𝑘₂(𝑠) for all 𝑠∈ℝ. Recall that each 𝑘_𝑖 is a nonnegative function.
Let 𝜖>0 and denote by 𝐽_𝜖∶=]−𝜖,0[×]−𝜖,0[,𝑂_𝜖∶=ℝ×𝐽_𝜖 with 𝜖 small enough to have 𝐽_𝜖⊂𝜔 (recall that Ω₀=ℝ×𝜔).
Multiplying (4.2) by 𝜙 and integrating over 𝑂_𝜖, we get _𝑂𝜖ℎ₁⁻²−ℎ₂⁻²𝜕_𝑠𝜙²+_𝜕𝑂𝜖ℎ₁⁻²−ℎ₂⁻²𝜕_𝑠𝜙𝜙𝜈_𝑠+_𝑂𝜖𝑉_𝑘1−𝑉_𝑘2𝜙²=0. (4.3)
Since 𝜖≪1,𝑉_𝑘𝑖≃−𝑘²_𝑖(𝑠)/4 for 𝑖=1,2, and so 𝑉_𝑘1(𝑠,𝑢₂,𝑢₃)−𝑉_𝑘2(𝑠,𝑢₂,𝑢₃)>0 in 𝑂_𝜖.
Moreover, note that ℎ₁⁻²𝑠,𝑢₂,𝑢₃−ℎ₂⁻²𝑠,𝑢₂,𝑢₃=𝛼𝑠,𝑢₂,𝑢₃𝑘₁(𝑠)−𝑘₂(𝑠)ℎ₁𝑠,𝑢₂,𝑢₃+ℎ₂𝑠,𝑢₂,𝑢₃ℎ²₁𝑠,𝑢₂,𝑢₃ℎ²₂𝑠,𝑢₂,𝑢₃, (4.4) with 𝛼(𝑠,𝑢₂,𝑢₃)∶=cos(𝜃(𝑠))𝑢₂+sin(𝜃(𝑠))𝑢₃.
Since (𝑢₂,𝑢₃)∈𝐽_𝜖 and 0≤𝜃(𝑠)≤𝜋/2 for all 𝑠∈ℝ, we have 𝛼(𝑠,𝑢₂,𝑢₃)<0. Therefore, by (4.4), we deduce that ℎ₁⁻²−ℎ₂⁻²>0 in 𝑂_𝜖.
Thus, ∫_𝑂𝜖(ℎ₁⁻²−ℎ₂⁻²)(𝜕_𝑠𝜙)²+∫_𝑂𝜖(𝑉_𝑘1−𝑉_𝑘2)𝜙²≥0.
Note also that _𝜕𝑂𝜖ℎ₁⁻²−ℎ₂⁻²𝜕_𝑠𝜙𝜙𝜈_𝑠=0. (4.5) Therefore, from (4.3) and (4.5) we get _𝑂𝜖ℎ₁⁻²−ℎ₂⁻²𝜕_𝑠𝜙²+_𝑂𝜖𝑉_𝑘1−𝑉_𝑘2𝜙²=0, (4.6) with ℎ₁⁻²−ℎ₂⁻²>0 in 𝑂_𝜖 and 𝑉_𝑘1−𝑉_𝑘2>0 in 𝑂_𝜖.
From (4.6) we can deduce that 𝜙=0 in 𝑂_𝜖. Using a unique continuation theorem (see [12, Theorem XIII.63 page 240]), from 𝐻_𝑘1𝜙=𝑓, noting that −Δ(𝑊_𝑔⁻¹𝜙)=𝑊_𝑔⁻¹𝑓=𝑔^−1/4𝑓, by |𝑓|≤𝑀|𝜙| a.e. in Ω₀, we can deduce that 𝜙=0 in Ω₀. So we get a contradiction since 𝑓 is assumed to be a non null function.

Step 2. From Step 1, we obtain that there exists at least one point 𝑠₀∈ℝ such that 𝑘₁(𝑠₀)=𝑘₂(𝑠₀). Since 𝑘₁≢𝑘₂, we can choose 𝑎∈ℝ and 𝑏∈ℝ∪{+∞} such that (for example) 𝑘₁(𝑎)=𝑘₂(𝑎),𝑘₁(𝑠)<𝑘₂(𝑠) for all 𝑠∈]𝑎,𝑏[ and 𝑘₁(𝑏)=𝑘₂(𝑏) if 𝑏∈ℝ. We proceed as in Step 1, considering in this case 𝑂_𝜖∶=]𝑎,𝑏[×𝐽_𝜖. From 𝑘₁(𝑎)=𝑘₂(𝑎), we get that ℎ₁⁻²(𝑎,𝑢₂,𝑢₃)=ℎ₂⁻²(𝑎,𝑢₂,𝑢₃). Therefore, we obtain ∫_𝜕𝑂𝜖(ℎ₁⁻²−ℎ₂⁻²)(𝜕_𝑠𝜙)𝜙𝜈_𝑠=0. So (4.3) becomes (4.6) with ℎ₁⁻²−ℎ₂⁻²>0 in 𝑂_𝜖 and 𝑉_𝑘1−𝑉_𝑘2>0 in 𝑂_𝜖. So 𝜙=0 in 𝑂_𝜖 and as in Step 1, by a unique continuation theorem, we obtain that 𝜙=0 in Ω₀. Therefore, we get a contradiction.