Inverse problems for the fourth order Schr\"odinger equation on a finite domain

In this paper we establish a global Carleman estimate for the fourth order Schr\"odinger equation posed on a $1-d$ finite domain. The Carleman estimate is used to prove the Lipschitz stability for an inverse problem consisting in retrieving a stationary potential in the Schr\"odinger equation from boundary measurements.


Introduction
The fourth order Schrödinger equation arises in many scientific fields such as quantum mechanics, nonlinear optics and plasma physics, and has been intensively studied with fruitful references. The well-posedness and existence of the solutions has been shown (for instance, see [11,18,19]) by means of the energy method and harmonic analysis. In this paper, we are interested in the inverse problem for the fourth order Schrödingier equation posed on a finite interval.
However, for the higher order equations, due to the increased complexity, there are few papers investigating the stability of the inverse problems via Carleman estimates. In [22], Zhang solves the exact controllability of semilinear plate equations via a Carleman estimate of the second order Schrödinger operator. Zhou ([24]) considers the observability results of the fourth order parabolic equation and Fu ([10]) derives the sharp observability inequality for the plate equation. In both papers, they show the Carleman estimates for the corresponding fourth order operators for 1 − d cases, respectively.
To our knowledge, the result of determination of a time-independent potential for the fourth order Schrödinger equation from the boundary measurements on the endpoint is new. Furthermore, our work in this paper is the first one dealing with the Carleman estimate of the fourth order Schrödinger equation.
To begin with, we introduce a suitable weight function: Let λ ≫ 1 be a sufficiently large positive constant depending on Ω. For t ∈ (0, T ) and following [9], we introduce the functions with a positive constant µ. Denote by We also introduce the set The first main result is the following global Carleman estimate for system (1.1). Theorem 1.1 There exist three constants µ 0 > 1, C 0 > 0 and C > 0 such that for all µ ≥ µ 0 and for all λ ≥ C 0 (T + T 2 ), holds true for all u ∈ Z, where the constants µ 0 , C 0 and C only depend on x 0 . Remark 1.1 Note that for simplicity, we give the exact form of the function ψ(x) in (1.2). In fact, the statement holds true for any function satisfying It is worthy to mention that, by taking x 0 > 1, one could switch the observation data in (1.4) to the left end-point x = 0.
Remark 1.2 [23] shows an observability inequality which estimates initial data by the measurement of ∆u for a Schrödinger equation without the potential q on Γ 0 = {x ∈ ∂Ω; (x − x 0 ) · ν(x) ≥ 0} using a multiplier identity and Holmgren's uniqueness theorem. Observability inequalities are technically related to our inverse problem (see [20]). However, the approach in [23] can not be applied to our problem, even though there are less observability data are considered. In what follows, we shall denote by u p the solution of the system (1.1) associated with the potential p.
The rest of the paper is organized as follows. In Section 2, we state a weighted point wise inequality for the fourth order Schrödinger operator. In Section 3, we establish a global Carleman estimate for a fourth order Schrödinger equation with a potential. The proof of Theorem 1.2 is given in Section 4. Finally we list several comments and some open problems for the future work.

A weighted point-wise estimate for the fourth order operator
In this section, we shall establish a weighted identity for 1-d Schrödinger operator, which will pay an important role in the proof of the Carleman estimate (1.4).

3)
where Ψ is a real value function in C 2 (lR). Moreover, we have Proof. We may assume that u is sufficiently smooth. Since v = θu and notice the definitions of a i , i = 0, 1, 2, 3 in (2.2), it is esay to get We divide P u into I 1 and I 2 as in (2.3). Multiplying θP u by its conjugate we have where I ij denotes the sum of the i-th term of I 1 times the j-th term ofĪ 2 in I 1Ī2 and its conjugate part inĪ 1 I 2 .
The computations will be treated in the following two parts. Part I: We compute I 1j , j = 1, 2, 3, 4. We first have On the other hand, it is easy to verify that (2.8) Moreover, (2.9) By replacing a 1 in (2.8) by a 3,xx , substituting it into the last term of (2.9), we have Taking the exact form of a 0 , a 1 , a 3 in (2.2) into account, one can verifty thatã 0 is exactly the one in (2.2). Furthermore, Meanwhile, for the first term of I 14 , recalling that (2.12) with Part II: We compute the rest of I ij , with some extra terms coming from (2.13). Set C 24 = a 3 Ψ − a 3ã0 , which is the same notation as in (2.2). We have the following identity: (2.14) Consequently, it holds (2.15) Now we compute I 3j , j = 1, 2, 3, 4. It holds For the term I 41 , it holds:

(2.19)
I 42 is considered with an extra term from I 14 as follows: Note that it is not hard to verify that C 41 has the form as in (2.2). Finally, the last two terms I 43 and I 44 equal to (2.23) By (2.7)-(2.23), combining all " ∂ ∂t -terms", all " ∂ ∂x -terms" and (2.6) we arrive at the desired inequality (2.1).

(3.1)
We now give the proof of Theorem 1.1.
Proof. The proof is divided into several steps.

Boundary observations: Proof of Theorem 1.2
In this section, we show the proof of Theorem 1.2, which is a direct application of the Carleman inequality (1.4). The standard procedure can be found in [2,17]. Proof of Th. 1.2. Pick any p, q as in the statement of the theorem, and introduce the difference y := u p − u q of the corresponding solutions of (1.1).
Then for any m ≥ 0 there exists a constant C > 0 such that for any q ∈ L ∞ (Ω) with q L ∞ (Ω) ≤ m and for all f ∈ L 2 (lR; Ω), the solution of (4.1) satisfies Proof of Proposition 4.1. Let f ∈ L 2 (lR; Ω) and R ∈ H 1 (0, T ; L ∞ (Ω)) be such that R(0, x) ∈ lR a.e. in Ω, and let y be the solution of (4.1). We take the even-conjugate extensions of y and R to the interval (−T, T ); i.e., we set y(t, x) = y(−t, x) for t ∈ (−T, 0) and similarly for R. Since R(0, x) ∈ lR a.e. in Ω, we have that R ∈ H 1 (−T, T ; L ∞ (Ω)), and y satisfies the system (4.1) in (−T, T ) × Ω. In the case when R(0, x) ∈ ilR, the proof is still valid by take odd-conjugate extensions.
Changing t into t + T , we may assume that y and R are defined on (0, 2T ) × Ω, instead of (−T, T ) × Ω.
Let z(t, x) = y t (2T − t, x). Then z satisfies the following system: x ∈ Ω. (4.3) We shall apply Theorem 1.1, with 2T instead of T . Therefore, here we consider and To use the Theorem 1.1, we introduce v = θz and I 1 is taken as in (2.3). Now set Then we have The last inequality comes from the fact that ϕ is bounded from below. On the other hand, for each p ∈ L ∞ (Ω, lR), we define the operator for all t ∈ (0, T ), u xx (·, 1) ∈ L 2 (0, T ), u xxx (·, 1) ∈ L 2 (0, T ) . As a direct consequence of Theorem 1.1, we have the following slightly revised Carleman estimate: Proposition 4.2 Given m ≥ 0, there exist µ 0 ≥ 1, λ 0 ≥ 0 and C > 0 such that for each for all λ ≥ λ 0 , µ ≥ µ 0 and z ∈ Z p .
Proof. The term |I 1 | 2 can be added by directly taking (2.6) into account. Moreover, the operator P can be changed to P p since p is assumed to be uniformly bounded and the cost is a slight change of C with respect to the upper bound m.
On the other hand, since v = θz, we have hence, Using the hypothesis on R(T, x), it follows that  1. There is another formulation for stationary inverse problems known as the Dirichlet-to-Neumann map. For instance, Bukhgeim and Uhlmann ( [1]) show that the potential can be uniquely determined by the boundary data for (∆+q)u = 0. It would be interesting to find out what happens for the fourth order Schrödinger operator. However, the relationship between the two problems is not really clear.
2. In this paper we derive a boundary Carleman estimate for the fourth order Schrödinger operator. It is well known that based on (1.4), we can derive the observability inequality and, consequently, prove the controllability property of the controlled system with two boundary controls. As a direct consequence of this methodology, it is very likely to expect that the controllability property holds for the fourth order Schrödinger equation with nontrivial potential q. Such result is much more general than the existing one in [23], which is for trivial potential q, even though only one boundary control is needed. It would be interesting to know whether two controls on the boundary are necessary with the nontrivial potential q.
3. It is well known that the Carleman estimate is a useful tool to analyze inverse problems. In fact, it has been studied for second order Schrödinger operator not only in bounded domain, but also in an unbounded strip ( [5]) or on a tree ( [12]). One could expect similar results in different domains. Meanwhile, it is still a challenging problem whether one can construct Carleman inequalities for fourth order equations on higher dimensions.
4. Note that there are fruitful literatures considering the numerical approximation results for the second order Schrödinger equations. Similar to the discrete Carleman estimate constructed by parabolic equation (see [4]), it would be interesting to find out the discrete analogue of (1.4) for space semi-discretized Schrödinger equation as the first step to solve discrete problems.