Increasing stability for determining the potential in the Schr\"odinger equation with attenuation from the Dirichlet-to-Neumann map

We derive some bounds which can be viewed as an evidence of increasing stability in the problem of recovering the potential coefficient in the Schr\"odinger equation from the Dirichlet-to-Neumann map in the presence of attenuation, when energy level/frequency is growing. These bounds hold under certain a-priori regularity constraints on the unknown coefficient. Proofs use complex and bounded complex geometrical optics solutions.


Introduction
We consider the problem of recovery of the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map. The use of complex exponential solutions plays essential role in the development of this problem. The idea is dated back to Calderon and Faddeev who introduced complex exponential solutions to demonstrate uniqueness in the closely related linearized inverse conductivity problem and in the inverse potential scattering problem for the Schrödinger equation. A breakthrough was made by Sylvester and Uhlmann in [20] where they constructed almost complex exponential solutions, and proved global uniqueness of c (potential in the Schrödinger equation) in the three-dimensional case. A logarithmic stability estimate for c from the Dirichlet-to-Neumann map was obtained by Alessandrini [1] and the optimality of log-type stability (at zero energy) was demonstrated by Mandache [16]. The logarithmic stability is quite discouraging for applications, since small errors in the data of the inverse problem result in large errors in numerical reconstruction of physical properties of the medium. In particular, it severely restricts resolution in the electrical impedance tomography.
For the problem of recovering the potential in the Schrödinger equation without attenuation from the Dirichlet-to-Neumann map, the first author in [12] derived some stability estimates in different ranges of frequency, which demonstrate the increasing stability phenomena as the frequency/energy k is growing. In [12], both complexand real-valued geometrical optics solutions were used in the proof. The proof was simplified in [14] where only complex-valued geometrical optics solutions were used. Similar results were obtained by Isaev and Novikov [9] by less explicit and more complicated methods of scattering theory.
Continuing the research in [9], [12], [14] we show in this work that the stability is increasing when k is growing in the presence of constant attenuation. For this problem, we could follow the arguments in [17] to derive k-dependent stability estimates using only complex-valued geometrical optics solutions. However, in doing so, the constant associated with the Hölder part will grow exponentially in k, similar to the result in [17]. To obtain polynomially growing constants as in [12] and [14], we use both complex-and real-valued geometrical optics solutions in our proof. In addition to geometrical optics solutions, we also need explicit sharp bounds on fundamental solutions of elliptic operators with parameter k. Let ε be an operator norm of the difference of two Dirichlet-to-Neumann maps corresponding to different potentials. We give conditional estimates for difference of potentials by a function of ε which goes to zero as ε goes to zero. This function is the sum of the terms containing powers of ε and of − log ε, moreover the terms containing log ε tend to zero (as powers of k) when k → ∞.
In section 2 we state main results. In section 3 by using sharp bounds on regular fundamental solutions of some elliptic linear partial differential operators with complex coefficients containing large parameter k we construct almost complex exponential solutions to the Schrödinger equation with attenuation and give bounds on these solutions. In section 4 we similarly construct bounded almost complex exponential solutions in the "low frequency zone" which is growing with increasing k. In section 5 we use these almost exponential solutions and the Fourier transform to prove our main results. In concluding section 6 we outline challenges and possible future research.

Main results
Let Ω be a (bounded) domain in R 3 with Lipschitz boundary. We consider the Schrödinger equation with the Dirichlet boundary data Assume that the attenuation coefficient b is a non negative constant and the potential c ∈ L ∞ (Ω). Suppose that there exists a unique solution to (2.1), (2.2). Thus we can define the Dirichlet-to-Neumann map By using potential theory one can show that Λ c − Λ 0 is a continuous linear operator from L 2 (∂Ω) into L 2 (∂Ω). We denote its operator norm by ||Λ c − Λ 0 ||. We assume that volΩ ≤ 1 and that c is zero near ∂Ω. Throughout we denote C 0 generic constants whose values may change from line to line. These constants do not depend on c, k, or Ω. They are only determined by our proofs. Generic constant C(Ω, M) might in addition depend on Ω, M. We will use the norms · p (Ω) in the Lebesgue spaces L p (Ω) and · (s) (Ω) in the Sobolev spaces H s (Ω).
Then there is constant C(Ω, M) such that In the bound (2.6) the logarithmic component goes to zero as k grows. Thus this bound can be viewed as an evidence of increasing stability in recoverying c for larger frequencies/energies k. In any event the second term, namely, Cεk 2 , contributes only to (the best possible) Lipschitz stability, while logarithmic terms are decaying for larger k. While constants C 0 , C(Ω, M) are hard to evaluate for general Ω, it is very likely that when Ω is a ball or a cube one can obtain relatively simple explicit bounds on these constants.
The factor k in the second term of the bound (2.6) most likely is necessary. Indeed, one needs bounds on time derivatives in the closely related inverse problems for the wave equation. For high frequencies/energies k, with additional assumptions on E and k, one can derive a similar estimate.
Then there is constant C(Ω, M) such that .
We observe that for real-valued c and b = 0 the Dirichlet problem (2.1), (2.2) might have eigenvalues k when its solution fails to exist and be unique, so that the Dirichlet-to Neumann map is not well defined. Then one can consider instead the Neumann-to-Dirichlet map, or replace these maps by the Cauchy set with naturally defined norm.
We introduce where e 1 , e 2 , e 3 is an orthonormal basis in R 3 and √ z is the principal branch of the square root function. Then (3.
In view of (3.16) we have the bound (3.5) and due to (3.17) we have the bound (3.6). The proof is complete.

✷
In the following lemma we will derive boundary estimates of almost complex exponential solutions constructed in Lemma 3.1. Moreover, from trace theorems for Sobolev spaces by (
As in the proof of Lemma 3.1 the operator F (v(; j)) in the right side of (4.10) maps the ball B(ρ) = {v : ||v|| (1) (4.12) because 1 − θ √ 2M > 1 2 due to the second condition (4.1). Repeating similar arguments in Lemma 3.1, the operator F is continuous from H 1 (Ω) into H 2 (Ω) and therefore compact from H 1 (Ω) into itself. Now this operator maps convex closed set B(ρ) ⊂ H 1 (Ω) into itself and is compact, hence by Schauder-Tikhonov Theorem it has a fixed point v(; j) ∈ B(ρ). Due to (4.12) we have the bound (4.5) and due to (3.17) we have the bound (4.6).

✷
We also estimate the boundary values of the almost real exponential solution u(; j).

Proofs of stability estimates
The following standard orthogonality result (see, for example, [1], [10]) follows by simple application of the Green's formula.
then the second bound of (3.1) is clearly satisfied, and the first bound of (3.1) holds provided k < E, as guaranteed by (2.5). Now we choose with some R > 2 to be selected later on. Then the right side of (5.6) is positive and hence our choice of τ is possible. Indeed, due to (5.7) it suffices to show that We recall the elementary inequality (A + B) λ ≤ A λ + B λ for any positive numbers A, B and λ ∈ (0, 1). Using this inequality with λ = 2 3 and A = E 2 , B = k 2 we conclude that the needed inequality follows from This inequality in turn follows from three inequalities where a = 2 −2/3 . The first and second ones hold since we assumed 2 < E and the third one is obvious. The second condition (3.1) follows from our choice of τ in (5.6). Indeed, due to (5.6) this condition becomes Due to (5.6) and Thus we obtain from (5.5) that Choosing R = max{(4C 3 ) 1 3 , 2} we will absorb the first term on the right side by the left side. The proof is complete.
Using the bounded almost exponential solutions (4.2) in the identity (5.1) and (5.3) as above we obtain that where we have utilized (4.5) and (4.13). As above, using the Parseval identity and polar coordinates we conclude that Hence we obtain from (5.10) Choosing ρ = (2C 3 ) −1 (E α + k β ), we can absorb the first term on the right side by the left side and obtain the bound (2.8). It only remains to show that with this choice the first bound of (4.1) is satisfied. To this end, in view of (2.7), we deduce that The proof is complete.

Conclusion
In this paper we showed that stability of recovery of the Schrödinger potential is increasing in presence of constant attenuation which is a feature of most applied problems. There is a belief that stability in the continuation and inverse problems always grows with frequency k. As shown in [15], in general stability of the continuation for the Helmholtz equation might deteriorate. In [8], [11] it was shown that the stability of the continuation is improving under some (convexity type) conditions. In [13] it was demonstrated that in some cases these convexity conditions can be relaxed. Now we outline possible future developments and challenges. We believe that constants C 0 in the stability estimates may be evaluated more explicitly by using periodic Faddeev type solutions [6] or more suitable regular fundamental solutions based on the work of Hörmander in 1955 [7]. We expect to obtain explicit bounds at least when Ω is the unit ball. One expects to obtain improving stability results when c is not necessarily zero near ∂Ω by using methods of [21] or of singular solutions [10] for boundary reconstruction. It would be interesting to use the new ideas in [4] to handle the two-dimensional case. It is not clear now how to get better stability for both c and constant b or for b close to a constant. Most likely, one will need to use the Fourier Integral Operators instead of the Fourier Transform.
There is a need in an additional numerical evidence of increasing stability in the important inverse problem we considered in this paper, as well as in similar inverse medium problems studied in particular in [3], [18]. In our view, even numerical results for the linearized problem (like Born approximation) would be convincing and interesting.
We hope to demonstrate the increasing stability for hard or transparent (convex) obstacles from the Dirichlet-to-Neumann map by combining the results of [11], [13] with the methods of [2]. So far increasing stability of reconstruction of obstacles was observed numerically, but there are no analytic results explaining it and suggesting better numerical methods.
It is still an open question whether logarithmic stability of recovery of near field from far field pattern [5], [10], section 6.1 is improving with growing frequency. To show it one can try to adjust the methods of [13] to handle Hankel functions.
Probably, it will be difficult to show increasing stability for the coefficient a 0 in the equation (−∆ − k 2 a 2 0 (x))u = 0. At present, there are only some preliminary results (in the plane case) [19] under some non trapping conditions on a 0 and bounds with constants exponentially growing with k [17]. We observe that the conductivity equation div(a∇u) + k 2 u = 0 can be transformed into this Helmholtz type equation with a 0 = a − 1 2 , c = a − 1 2 ∆a 1 2 , so increasing stability for the conductivity equation remains a challenge.