Optimality of increasing stability for an inverse boundary value problem

In this work we study the optimality of increasing stability of the inverse boundary value problem (IBVP) for Schr\"{o}dinger equation. The rigorous justification of increasing stability for the IBVP for Schr\"{o}dinger equation were established by Isakov \cite{Isa11} and by Isakov, Nagayasu, Uhlmann, Wang of the paper \cite{INUW14}. In \cite{Isa11}, \cite{INUW14}, the authors showed that the stability of this IBVP increases as the frequency increases in the sense that the stability estimate changes from a logarithmic type to a H\"{o}lder type. In this work, we prove that the instability changes from an exponential type to a H\"older type when the frequency increases. This result verifies that results in \cite{Isa11}, \cite{INUW14} are optimal.


Introduction
In this paper, we study the instability phenomenon of the inverse boundary value problem (IBVP) for the Schrödinger equation with a frequency. This IBVP is notoriously ill-posed. Ignoring the effect of the frequency, a logarithmic type estimate were first derived in [Ale88] and this logarithmic estimate was shown be optimal in the form of exponential instability in [Man01]. Such exponential instability was also established for different inverse problems in [DR03,RS18,ZZ19]. However, by taking account of the frequency, it was observed numerically [CHP03] or [KW20] that the stability of some inverse problems will improve as the frequency increases. The increasing stability phenomena were rigorously proved in other situations [DI07, DI10, HI04, ILX20, INUW14, Isa07, Isa11, KU19, LLU19, NUW13], not only for inverse problems, but also for the unique continuation property.
In this work, we will study the counterpart of the increasing stability by investigating how the exponential instability is affected by the frequency. Here we consider the Schrödinger equation with a potential and a frequency (1.1) ∆ + q(x) + κ 2 u = 0 in Ω ⊂ R d with d ≥ 2. The Cauchy data corresponding to the Schrödinger equation (1.1) is defined by u is a solution to (1.1) .
Let dist (C q 1 , C q 2 ) be the Hausdorff distance between two Cauchy data with respect to q 1 , q 2 . Under some appropriate assumptions, it was shown in [INUW14] that (1.2) q H −s (R d ) ≤ C κ + log 1 dist (C q 1 , C q 2 ) −2s−d whereq is the zero extension of q 1 − q 2 . The estimate (1.2) clearly indicates that the logarithmic part decreases as κ increases and the estimate changes from a logarithmic type to a Hölder type. Isakov in [Isa11], as well as Isaev and Novikov in [IN12], proved a similar estimate in terms of the Dirichlet-to-Neumann map.
We now briefly describe the problem considered in this work. For simplicity, we consider (1.1) in Ω = B 1 and ∂Ω = S d−1 = ∂B 1 . If κ 2 / ∈ spec (−(∆ + q)), where spec(−(∆ + q)) denotes the Dirichlet spectra of −(∆ + q) in Ω, then the Dirichlet-to-Neumann map of the Schrödinger equation (1.1), Λ q : u| S d−1 → ∂ r u| S d−1 is well-defined. It is well-known that Λ q ∈ L(H 1 2 (S d−1 ), H − 1 2 (S d−1 )), where L(X , Y) denotes the space of bounded linear operators from X to Y. Clearly, given any s ≥ 1 2 , we have Λ q ∈ L(H s (S d−1 ), H −s (S d−1 )). To simplify the notations, we simply denote We now state the main results of this paper. where R is related to the size of the potentials. Then there exist potentials q 1 , q 2 ∈ C α (B 1 ) such that where C R is a positive constant depends only on the parameter R.
Remark 1.2. There are two reasons in using H d+4 2 norm in the Dirichlet-to-Neumann map. One is to bound the · d+4 2 →− d+4 2 norm of the Dirichlet-to-Neumann map by the X s of its corresponding discretization (see (3.21)). The other is to bound the cardinality of the δ-net Y (see Remark 3.7). Remark 1.3. In the proof, we actually construct q ℓ (ℓ = 1, 2) with q ℓ =q ℓ + i for some real-valued functionsq ℓ , where i = √ −1. Therefore, κ 2 / ∈ ∪ ℓ=1,2 spec (−(∆ + q ℓ )) and so Λ q ℓ are well-defined. We want to point out that the discrepancy parameter θ is independent of κ and the estimate on the right hand side of (1.6) is expressed explicitly in κ. Furthermore, C R in (1.6) only depends on the size of the potentials and is independent of κ. The assumption of ℑ(q ℓ ) = 1 is to achieve this purpose with the help of explicit elliptic estimates. On the other hand, in the usual stability estimate of the inverse boundary value problem for the Schrödinger equation, potentials q ℓ , ℓ = 1, 2, are a priori given such that we can assume that κ 2 is not a Dirichlet eigenvalue of −(∆+ q ℓ ). In which case, we can consider the Dirichlet-to-Neumann maps [Isa11]. Instead of Dirichlet-to-Neumann maps, we can also use the Cauchy data in the stability estimate [INUW14]. However, in the instability estimate, potentials q ℓ are determined posteriorly. Therefore, either assuming that κ 2 is not a Dirichlet eigenvalue of −(∆ + q ℓ ) or considering the Cauchy data are not feasible.
By the standard elliptic estimates, one can show that for any q 1 , q 2 ∈ L ∞ (B 1 ) provided Λ q 1 and Λ q 2 are well-defined, where the constant C(q 1 , q 2 , κ) depends on q 1 , q 2 , and κ. Estimate (1.7) can be viewed as an well-posedness estimate of the mapping q → Λ q .
However, it provides no information of the ill-posedness of Λ q → q. Theorem 1.1 is valid for all frequencies κ. The estimate (1.6) apparently demonstrates that its right hand side is dominated by the Hölder component θ 1 2α when κ is large. To further elucidate our result, it is interesting to compare Theorem 1.1 and a similar instability estimate proved by Isaev in [Isae13]. In our work, we first give an arbitrary wave number κ 2 and a perturbation θ (independent of κ), and then construct suitable potentials q 1 , q 2 satisfying (1.5) and (1.6). In [Isae13], Isaev constructed some wave number κ 2 and some potentials q ∈ C d+1 (B 1 ) such that . In other words, (1.8) validates the optimality of the increasing stability estimate obtained in [IN12] (similar to (1.2)) for potentials near zero. We want to point out that the potentials q 1 , q 2 constructed in Theorem 1.1 are not necessarily small. Unlike the κ-independent discrepancy parameter θ in (1.5), the lower bound of q L ∞ (B 1 ) = q − 0 L ∞ (B 1 ) depends on κ in view of (1.8). However, Theorem 1.1 is not optimal in the lower frequency. It is expected that the IVBP is exponentially unstable if κ is small. To verify this, we prove a similar estimate in the lower frequency.
Estimate (1.6) shows that the instability changes from an exponential type to a Hölder type when κ 2 increases, conversely, (1.9) shows that the instability is an exponential type when κ 2 is small as in [Man01]. In particular, for any perturbation θ, we have the following dichotomy: exponential instability when κ 2 < θ * i.e. when κ 2 is small , Hölder instability when κ 2 > θ * i.e. when κ 2 is large , .
Such transition of instability was also proved for an inverse problem in the stationary radiative transport equation in [ZZ19]. Our study is also inspired by their result. We also want to comment that the stability estimate of determining a potential in the wave equation by the knowledge of the hyperbolic Dirichlet-to-Neumann map was shown to be a Hölder type, see [Sun90]. In other words, in the high frequency, the inverse boundary value for (1.1) is as stable as the inverse problem for the wave equation. We can actually combine Theorem 1.1 and Theorem 1.4 into a single theorem.
Theorem 1.6. Let α > 0 and R > 0 be any given constants. There exists a positive constant θ 0 = θ 0 (α, R) such that the following statement holds: For each κ > 0 and 0 < θ < θ 0 , there exist potentials q 1 , q 2 ∈ C α (B 1 ) such that (1.5) holds and for some constants C R depending only on R and absolute constant c 0 .
This paper is organized as follows. We list some preliminary materials in Section 2. Section 3 is a collection of some useful estimates needed in the proofs of main theorems. We then prove Theorem 1.1 and Theorem 1.4 in Section 4. Like other works in the instability of the inverse problem, our proof is based on Kolmogorov's entropy theorem [KT61].

Preliminaries
2.1. Existence of θ-discrete set for some neighborhood. Fixing r 0 ∈ (0, 1). Given any α > 0, θ > 0, and β > 0, we consider the following set where B r 0 denotes the ball of radius r 0 centered at the origin. The following proposition can be found in [ZZ19, Lemma 5.2] (or in [KT61] in a more abstract form), see also [Man01, Lemma 2] for a direct proof for the special case when r 0 = 1 2 . Proposition 2.1. There exists a constant µ > 0 such that the following statement holds for all β > 0 and for all θ ∈ (0, µβ): Moreover, the cardinality of Z, denoted by |Z|, is bounded below by 2.2. Matrix representation via spherical harmonics. As in [Man01, ZZ19], we will use the set of d-dimensional spherical harmonics: which is a complete orthogonal set in is also equivalent to the following norm: see e.g. [Man01].

Given any bounded linear operator
, we define a mjnk := AY mj , Y nk and consider the Banach space: (see [Man01,ZZ19]). The following proposition can be found in [ZZ19, Lemma 5.3], which is crucial in our work.
Proposition 2.2 suggests that one can interpret a mjnk := AY mj , Y nk as the matrix representation of the bounded linear operator A :

Some useful estimates
In this section, we would like to derive some useful estimates that are needed in the proofs of main theorems.
Proof. Multiplying (3.1) by v and integrating over B 1 , we have Taking the imaginary part of (3.4), we have The following elliptic estimate is useful in our proof.
Assume that (3.2) holds. Then there exists a positive constant C d such that Here and after, C d (and C ′ d , C ′′ d ) are general constants depending only on dimension d. Proof. Letφ ∈ H 2 (B 1 ) be an extension of φ satisfying the inequality which implies (3.6).
. For each κ 2 > 0, we consider the mapping Let u mj ∈ H 1 (B 1 ) be the unique solution to Similarly, letů mj ∈ H 1 (B 1 ) be the unique solution to We also define v mj ∈ H 1 (B 1 ) the unique solution to and letv mj ∈ H 1 (B 1 ) be the unique solution to By (2.2) and (3.6), we can derive WriteỸ mj (x) := |x| m Y mj (x/|x|), which is harmonic in B 1 . We can prove Proof. We begin to prove

Using integration by parts, we have
In the computation of (3.12), we used the fact u mj =ů mj on S d−1 andỸ nk is harmonic in B 1 . Since and supp (q) ⊂ B r 0 , (3.12) implies which is exactly (3.11).
Next, we want to establish , similar to the derivation of (3.12), we have and hence (3.13).
3.3. Construction of a net. In view of (3.21) in Remark 3.5, we want to construct a δnet Y of ((Γ q ref mjnk (B ∞,r 0 +,R )), • Xs ), which is not too-large. Precisely, we want to obtain the following proposition.

Proofs of the main results
In this section, we would like to present detailed proofs of Theorem 1.1 and Theorem 1.4. Before proving our results, we first give some observations. Taking q ref = i (hence λ = 1) and s = d+4 2 (i.e. τ = 1), see also Remark 1.3, we see that (3.22) becomes Note that for all κ 2 > 0. Therefore, given any θ satisfies there exists a unique δ ∈ (0, Φ) such that and (4.1) is reduced to In view of these observations, we are ready to prove Theorem 1.1, the case of higher frequency.
that is, Combining these two cases yields which gives (1.6). Since Z is θ-discrete subset with respect to the norm • L ∞ (B 1 ) , (1.5) follows immediately, which is our desired result. Now, we want to prove Theorem 1.4 in which the frequency is small. First of all, we improve Proposition 3.2 without the restriction (3.2).
Proposition 4.1. Let κ 2 > 0, q ∈ L ∞ (B 1 ), φ ∈ H 3 2 (B 1 ), and u ∈ H 1 (B 1 ) be the solution to (3.5). If where κ 1 is the first Dirichlet eigenvalue of −∆ on B 1 (which depends only on dimension d), then for some constant C d depending only on dimension d.
Remark 4.3. Since the potential q is real-valued, u mj = v mj andů mj =v mj .
Let M (l) mjnk (l = 1, 2) be defined in (3.14), then we again obtain (3.15), that is, (2) mjnk (0)|. Following exactly the same arguments in the proof of Lemma 3.4, we have Consequently, we can replace the function Φ in (3.16) and (3.17) by the constant 1. The observation significantly improves our estimates later.
It follows from Proposition 2.2 that Again, we estimateδ by θ.
Putting these two cases together, we obtaiñ which gives (1.9). By the fact that Z is θ-discrete subset with respect to the norm • L ∞ (B 1 ) , we then obtain (1.5). The proof is now completed.