*3.1. Stability for q* ∈ *F*

The first result in this section is the lack of any stability property when *q* ∈ *F*. In particular, we prove that inequality (4) fails for any continuous function *<sup>V</sup>*(*t*) with *<sup>V</sup>*(0) = 0.

**Theorem 2.** *Given <sup>q</sup>*<sup>0</sup> <sup>∈</sup> *F, there exists a sequence* {*qk*}*k*≥<sup>1</sup> <sup>⊂</sup> *F, such that <sup>q</sup>*<sup>0</sup> <sup>−</sup> *qk <sup>L</sup>*<sup>∞</sup> <sup>=</sup> *<sup>γ</sup>* <sup>&</sup>gt; <sup>0</sup> *for all k* ≥ 1*, while:*

$$\left\|\Lambda\_{\mathfrak{q}\_{\mathbb{O}}} - \Lambda\_{\mathfrak{q}\_{\mathbb{K}}}\right\|\_{\mathcal{L}(H^{1/2}\_{\mathfrak{q}}; H^{-1/2}\_{\mathfrak{q}})} \to 0, \quad \text{as } k \to \infty. \tag{21}$$

This result contradicts any possible stability result of the DtN map at *q*<sup>0</sup> ∈ *F*. Roughly speaking, the idea is that the eigenvalues of Λ, given in Theorem 1 above, depend continuously on *b*, unlike the *L*<sup>∞</sup> norm of the potentials. A detailed proof of the Theorem 2 is given in the Appendix A below.

#### *3.2. Partial Stability*

We now give two partial stability results when we fix *b* and *γ*, respectively.

**Theorem 3.** *Given b* <sup>∈</sup> (0, 1) *and q*1, *<sup>q</sup>*<sup>2</sup> <sup>∈</sup> *Fb, we have:*

$$\|\|q\_1 - q\_2\|\|\_{L^{q\_1}} \le \frac{15.0756}{b^4} \|\Lambda\_{q\_1} - \Lambda\_{q\_2}\|\|\_{\mathcal{L}(H^{1/2}\_{\theta}; H^{-1/2}\_{\theta})}.\tag{22}$$

*On the contrary, given <sup>γ</sup>* <sup>∈</sup> (0, 1] *and q*1, *<sup>q</sup>*<sup>2</sup> <sup>∈</sup> *<sup>G</sup>γ, we have:*

$$|b\_1 - b\_2| \le \frac{3.7489}{\gamma b^3} ||\Lambda\_{\mathfrak{q}\_1} - \Lambda\_{\mathfrak{q}\_2}||\_{\mathcal{L}(H\_\theta^{1/2}; H\_\theta^{-1/2})'} \tag{2.3}$$

*where b* <sup>=</sup> min{*b*1, *<sup>b</sup>*2}*.*

The proof of this theorem is in the Appendix A below.

Inequality (22) provides a Lipschitz stability result for Λ when *b* is fixed. This result is not new, since this situation enters in the framework in [6], as *q* takes constant values in fixed regions. The contribution here is in the dependence of the Lipschitz constant on the parameter *b*, which is associated with the size of the region, where *q* takes a different constant value. An estimate (22) shows also that the lack of Lipschitz stability is related to variations in the position of the discontinuity, which is the main idea in the negative result given in Theorem 2.

A numerical quantification of this Lipschitz stability for *b* fixed is easily obtained. We fix *<sup>b</sup>* <sup>=</sup> *<sup>b</sup>*<sup>0</sup> and consider:

$$F\_{\hbar,b\_0} = \{ q \in L^\infty(\Omega) \, : \, q(r) = \gamma \chi\_{(0,b)}(r), \, b = b\_0, \, \gamma = hj, \, j = 1, \dots, 1/h - 1 \}.$$

and for *q*<sup>0</sup> ∈ *Fh*,*b*<sup>0</sup> :

$$\mathcal{C}\_{2}(h, q\_{0}, b\_{0}) = \max\_{q \in \mathbb{F}\_{\mathbf{h}, \mathbf{b}\_{0}} \atop q \neq q\_{0}} \frac{||q\_{0} - q||\_{L^{\infty}}}{||\Lambda\_{q\_{0}} - \Lambda\_{q}||\_{\mathcal{L}(H^{1/2}\_{\emptyset}; H^{-1/2}\_{\emptyset})}},\tag{24}$$

then, *<sup>C</sup>*2(*h*, *<sup>q</sup>*0, *<sup>b</sup>*0) remains bounded as *<sup>h</sup>* <sup>→</sup> 0 for all *<sup>q</sup>*<sup>0</sup> <sup>∈</sup> *Fh*. In Figure 1, we show the behavior of *<sup>C</sup>*2(*h*, *<sup>q</sup>*0, *<sup>b</sup>*0) when *<sup>h</sup>* = <sup>10</sup>−<sup>4</sup> for different values of *<sup>b</sup>*0. To illustrate the behavior with respect to *b*<sup>0</sup> → 0, we plot on the left-hand side of Figure 1 the graphs of the functions:

$$C\_{2,\min}(b\_0) = \min\_{q \in F\_{b,b\_0}} C\_2(10^{-4}, q, b\_0), \text{ and } C\_{2,\max}(b\_0) = \max\_{q \in F\_{b\_0}} C\_2(10^{-4}, q, b\_0). \tag{25}$$

We see that both constants become larger for small values of *b*. We also see that both graphs are close in this logarithmic scale. However, the range of the interval [*C*2,min(*b*), *<sup>C</sup>*2,max(*b*)] is not small, as shown on the right-hand side of Figure 1.

**Figure 1.** Numerical estimate of the stability constant *<sup>C</sup>*<sup>2</sup> in (24) for *<sup>h</sup>* <sup>=</sup> <sup>10</sup>−4. To illustrate the behavior on *b*, we plotted the maximum and minimum value when *q* ∈ *Fh*,*<sup>b</sup>* with respect to *b* in logarithmic scale (**left**), and its range in normal scale (**right**).

Concerning inequality (23) in Theorem 3, it provides a stability result for Λ with respect to the position of the discontinuity. In particular, this provides Lipschitz stability if we consider a norm for the potentials that is sensitive to the position of the discontinuity. This is not the case for the *<sup>L</sup>*<sup>∞</sup> norm, but it is true for the *<sup>L</sup>p*-norm for some 1 <sup>≤</sup> *<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup>. For example, when *<sup>p</sup>* = 1:

$$\|\|q\_1 - q\_2\|\|\_{L^1} = \gamma \pi |b\_1^2 - b\_2^2| \le 2\pi \gamma |b\_1 - b\_2| \le \frac{7.4978\pi}{b^3} \|\Lambda\_{\mathfrak{q}\_{\parallel}} - \Lambda\_{\mathfrak{q}}\|\_{\mathcal{L}(H\_{\mathfrak{q}}^{1/2}; H\_{\mathfrak{q}}^{-1/2})}.$$

We can also check this numerically:

$$\mathbb{C}\_{\mathfrak{L}}(h,\gamma\_0,b) = \max\_{q \in \mathcal{C}\_{h,\gamma\_0}} \frac{||q\_0 - q||\_{L^1}}{||\Lambda\_{q\_0} - \Lambda\_{\mathfrak{q}}||\_{\mathcal{L}(H^{1/2}\_{\mathfrak{\mathfrak{q}}}; H^{-1/2}\_{\mathfrak{\mathfrak{q}}})},\tag{26}$$

is bounded as *h* → 0 and *b* ≥ *b*<sup>0</sup> > 0, where:

$$G\_{h,\gamma\_0} = \{ q \in L^\infty(\Omega) \; : \; q(r) = \gamma \chi\_{(0,b)}(r), \; \gamma = \gamma\_0, \; b = hj, \; j = 1, \dots, 1/h - 1 \}.$$

In Figure 2, we show the values when *<sup>h</sup>* = <sup>10</sup>−4. We can observe that the constant blows up as *b* → 0.

**Figure 2.** *<sup>C</sup>*2(*h*, *<sup>q</sup>*) for *<sup>b</sup>* <sup>&</sup>gt; *<sup>b</sup>*<sup>0</sup> when *<sup>h</sup>* <sup>=</sup> <sup>10</sup>−4.

#### **4. Range of the DtN Map**

In this section, we are interested in the range of Λ when *q* ∈ *F*, i.e., the set of sequences {*cn*}*n*≥<sup>0</sup> of the form (15) and (16) for all possible *<sup>b</sup>*, *<sup>γ</sup>* <sup>∈</sup> [0, 1] <sup>×</sup> [0, 1].

As *F* is a bi-parametric family of potentials, it is natural to check if we can characterize the family {*cn*}*n*≥<sup>0</sup> with only the first two coefficients *c*<sup>0</sup> and *c*1. In this section, we give numerical evidence of the following facts:


$$\begin{array}{rcl}\Lambda^h: F\_h & \to & \mathbb{R}^2\\q & \to & (c\_{0'}c\_1)\_{\prime} \end{array} \tag{27}$$

is injective. This means, in particular, that the DtN map can be characterized by the coefficients *c*<sup>0</sup> and *c*1, when restricted to functions in *Fh*. We also illustrate the set of possible coefficients *c*0, *c*1.

3. The lack of stability for Λ is associated with a higher density of points in the range of Λ*h*. This occurs when either *b* or *γ* is close to zero.

#### *4.1. Sensitivity of cn*

To analyze the relevance and sensitivity of the coefficients *cn* <sup>=</sup> *cn*(*b*, *<sup>γ</sup>*) to identify the parameters (*b*, *<sup>γ</sup>*), we computed their range when (*b*, *<sup>γ</sup>*) <sup>∈</sup> [0, 1] <sup>×</sup> [0, 1], and the norm of their gradients. As we can see in Table 1, the range decreases for large *n*. This means that, for larger values of *n*, the variability of *cn* is smaller and they are likely to be less relevant to identify *q*.

However, even if the range of *cn* becomes smaller for large *n*, they could be more sensitive to small perturbations in (*b*, *<sup>γ</sup>*) and this would make them useful to distinguish different potentials. However, this is not the case. In Figure 3, we show that for the given values of *<sup>γ</sup>* = 0.1, 0.34, 0.67, 0.99 and *<sup>b</sup>* <sup>∈</sup> (0, 1], the gradients of the first two coefficients, with respect to (*b*, *<sup>γ</sup>*), are larger than the others. Therefore, we conclude that the two first coefficients, *c*<sup>0</sup> and *c*1, are the most sensitive and, therefore relevant to identify the potential *q*.

We also see in Figure 3 that these gradients are very small for *b* << 1. This means, in particular, that identifying potentials with small *b* from the DtN map should be more difficult.

**Table 1.** Range of the coefficients, i.e., for each *cn*, the range is defined as max*q*∈*Fh cn* − min*q*∈*Fh cn*.

**Figure 3.** Norm of the gradient of the coefficients *cn*(*γ*, *<sup>b</sup>*) in terms of *<sup>b</sup>* <sup>∈</sup> (0, 1) for different values of *γ*. We can see that the gradients of higher coefficients *n* ≥ 2 are smaller than those of the first two. We can also observe that these gradients become small for small values of *b*.

#### *4.2. Range of the DtN in Terms of c*0, *c*<sup>1</sup>

Now, we focus on the range of the DtN in terms of the relevant coefficients (*c*0, *<sup>c</sup>*1), i.e., the range of the map <sup>Λ</sup>*<sup>h</sup>* in (27): *<sup>R</sup>*(Λ*h*). In Figure 4, we show this range.

**Figure 4.** Range of the discrete Dirichlet-to-Neumann (DtN) map in (27) (*<sup>h</sup>* = <sup>10</sup>−2).

Coordinate lines for fixed *<sup>γ</sup>* and *<sup>b</sup>* are given in Figure 5. We can observe that *<sup>R</sup>*(Λ*h*) is a convex set between the curves:

$$\begin{aligned} r\_{low} &: \{ (c\_0(\gamma, 1), c\_1(\gamma, 1)), \text{ with } \gamma \in [0, 1] \}, \\ r\_{up} &: \{ (c\_0(1, b), c\_1(1, b)), \text{ with } b \in [0, 1] \}. \end{aligned}$$

Note also that in the *c*0, *c*<sup>1</sup> plane, the length of the coordinate lines associated with *b* constant are segments that become smaller as *b* → 0. Analogously, the length of those associated with constant *γ* become smaller as *γ* → 0. Thus, the region where either *b* or *γ* are small produces a higher density of points in the range of Λ*h*. This corresponds to the upper left part of its range (see Figure 4). On the contrary, this Figure provides numerical evidence of the injectivity of <sup>Λ</sup>*<sup>h</sup>* as well. In fact, any point inside *<sup>R</sup>*(Λ*h*) is the intersection of two coordinate lines associated with some unique *b*<sup>0</sup> and *γ*0.

**Figure 5.** Coordinate lines of the map <sup>Λ</sup>*<sup>h</sup>* defined in (27) (*<sup>h</sup>* = <sup>10</sup>−2). The upper figure contains the coordinate lines associated with *b* constant, while the lower one corresponds to *γ* constant.

The higher density of points in the upper left hand-side of the range of Λ*<sup>h</sup>* should correspond to potentials *<sup>q</sup>* with a large stability constant *<sup>C</sup>*2(*h*, *<sup>q</sup>*), defined as:

$$C\_2(h, q) = \max\_{q \in F} \frac{||q\_0 - q||\_{L^1}}{||\Lambda\_{q\_0} - \Lambda\_q||\_{\mathcal{L}(H^{1/2}\_{\emptyset}; H^{-1/2}\_{\emptyset})}}.$$

In Figure 6, we show the level sets of *<sup>C</sup>*2(*h*, *<sup>q</sup>*) for *<sup>h</sup>* = <sup>10</sup>−<sup>4</sup> and different *<sup>q</sup>* <sup>∈</sup> *Fh*. The region with a larger constant corresponds to small values of *b* (upper right figure) and larger values of *c*<sup>1</sup> (upper left and lower figures). On the contrary, the region with a lower stability constant is for *<sup>b</sup>* close to *<sup>b</sup>* = 1, which corresponds to the lower part of the range of Λ*<sup>h</sup>* when *c*<sup>0</sup> is small.

**Figure 6.** Level sets of the *<sup>C</sup>*2(*b*, *<sup>γ</sup>*) for *<sup>q</sup>* <sup>∈</sup> *Fh* and *<sup>h</sup>* <sup>=</sup> <sup>10</sup>−<sup>4</sup> in terms of (*b*, *<sup>γ</sup>*) (**upper left**) and in terms of (*c*0, *<sup>c</sup>*1) (**upper righ**t), and a close up of the upper left region in this last figure is in the lower figure. Regions separated by level sets are indicated: Region I corresponds to the potentials with a stability constant larger that 107, region II corresponds to those with a stability constant lower that 10<sup>7</sup> but larger than 106, and so on.

It is interesting to analyze the set of potentials with the same coefficient *c*<sup>0</sup> or *c*1. We provide, in Figure 7, the coordinate lines of the inverse map (Λ*h*)<sup>−</sup>1. When increasing the value of either *c*<sup>0</sup> (light lines) or *c*<sup>1</sup> (dark lines), we obtain lines closer to the left part of the (*b*, *<sup>γ</sup>*) region. We can see that the angle between coordinate lines becomes very small for small *b*. In this region, close points could be the intersection of the coordinate lines associated with not so close parameters (*b*, *<sup>γ</sup>*). This agrees with the region where the stability constant is larger.

**Figure 7.** Coordinate lines of the map (Λ*h*)−<sup>1</sup> defined in (27).

#### **5. Conclusions**

We considered the relationship between the potential in the Schrödinger equation and the associated DtN map in one of the simplest situations, i.e., for a subset of radial one-step potentials in two-dimension. In particular, we focused on two difficult problems: The stability of the map Λ (defined in (3)) and its range. In this case, the map Λ is easily characterized in terms of the Bessel functions and this allows us to give some analytical and numerical results for these problems. We proved the lack of any possible stability

result by adapting the argument in [7] [Alessandrini, 1988] for the conductivity problem. We also obtained some partial Lipschitz stability when the position of the discontinuity is fixed in the potential, as well as numerical evidence of the stability with respect to the *L*<sup>1</sup> norm. Finally, we characterized numerically the range of Λ in terms of the first two eigenvalues of the DtN map and provided some insight into the regions where the stability of Λ is worse. As a future line of work, it could be interesting to consider the problem in a more complicated stage, for instance, one can study not only one-step radial potentials *q* in the problem, but could add more steps into the definition of the potentials.

**Author Contributions:** Conceptualization, S.L. and S.M.; Formal analysis, S.L. and S.M.; Investigation, S.L. and S.M.; Methodology, S.L. and S.M.; Visualization, S.L.; Writing—original draft, S.L. and S.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received of the project PDI2019-110712GB-100 .

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The first author was partially supported by project MTM2017-85934-C3-3-P from the MICINN (Spain). The second author was partially supported by project PDI2019-110712GB-100 of the Ministerio de Ciencia e Innovación, Spain. We want to thank to J.A. Barceló and C. Castro their contribution to the research.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

To prove Theorems 2 and 3, we need the following technical results regarding the the Bessel functions.

**Lemma A1.** *Let <sup>J</sup>μ*(*r*) *be the Bessel functions of the first kind of order <sup>μ</sup>* <sup>&</sup>gt; <sup>−</sup><sup>1</sup> <sup>2</sup> *. It is well known (see [10]) that:*

$$J\_{\mu}(r) = \frac{r^{\mu}}{2^{\mu}\Gamma(\mu+1)} + S\_{\mu}(r),$$

*where:*

$$S\_{\mu}(r) = \frac{r^{\mu}}{2^{\mu}\Gamma\left(\mu + \frac{1}{2}\right)\Gamma\left(\frac{1}{2}\right)} \int\_{-1}^{1} (\cos rt - 1) \left(1 - t^{2}\right)^{\mu - \frac{1}{2}} dt.$$

*For n* <sup>=</sup> 0, 1, 2, ··· *and r* <sup>∈</sup> (0, 1), *the following holds:*

$$-\frac{r^{n+2}}{2^{n+1}\Gamma\left(n+\frac{3}{2}\right)\sqrt{\pi t}}\int\_0^1 \left(1-t^2\right)^{n+\frac{1}{2}}dt \le S\_n(r) \tag{A1}$$

$$\leq -\frac{r^{n+2}\cos r}{2^{n+1}\Gamma\left(n+\frac{3}{2}\right)\sqrt{\pi t}} \int\_0^1 \left(1 - t^2\right)^{n+\frac{1}{2}} dt,$$

$$0 < \frac{r^n}{2^{n+1}n!} \leq f\_n(r) \leq \frac{r^n}{2^n n!},\tag{A2}$$

*and:*

$$0 < \frac{r^n}{2^{n+2}n!} \le f\_{n+1}'(r) \le \frac{r^n}{2^{n+1}n!}.\tag{A3}$$

*More explicit estimates for S*0(*r*) *and S*2(*r*) *are given by:*

$$-\frac{r^2}{4} \le \mathcal{S}\_0(r) \le -\frac{r^2 \cos r}{4} \le 0,\tag{A4}$$

$$-\frac{r^4}{15\pi} 0.4909 \le S\_2(r) \le -\frac{r^4 \cos r}{15\pi} 0.4909. \tag{A5}$$

**Proof.** To prove (A1), we use:

$$\frac{r^2t^2}{2}\cos r \le 1 - \cos(rt) \le \frac{r^2t^2}{2}, \quad r, t \in (0, 1), \tag{A6}$$

and:

$$\int\_0^1 t^2 \left(1 - t^2\right)^{n - \frac{1}{2}} dt = \frac{1}{2\left(n + \frac{1}{2}\right)} \int\_0^1 \left(1 - t^2\right)^{n + \frac{1}{2}} dt.$$

From (A1) and the well-known identities:

$$\begin{aligned} \Gamma\left(\frac{1}{2}\right) &= \sqrt{\pi}, \\ \Gamma(r+1) &= r\Gamma(r), \quad r > 0, \\ 2l'\_{n+1}(r) &= l\_n(r) - l\_{n+2}(r), \quad r > 0, \end{aligned}$$

(see [11]), we get (A2), (A3), (A4), and (A5).

The following lemma is used in the proof of Theorem 3.

**Lemma A2.** *For* <sup>0</sup> <sup>&</sup>lt; *<sup>r</sup>* <sup>≤</sup> *<sup>s</sup>* <sup>&</sup>lt; <sup>1</sup> *and n* <sup>=</sup> 0, 2*, we have:*

$$\int\_0^1 (1 - \cos(rt)) \left(1 - t^2\right)^{n - \frac{1}{2}} dt \le \frac{\pi r^2}{28n + 8}r$$

*and:*

$$\int\_0^1 (\cos(rt) - \cos(st)) \left(1 - t^2\right)^{n - \frac{1}{2}} dt \le \frac{\pi (s^2 - r^2)}{28n + 8}.$$

**Proof.** The previous estimates are a consequence of (A6) and the inequality:

$$2\cos r - \cos s = 2\sin\frac{s+r}{2}\sin\frac{s-r}{2} \le \frac{s^2 - r^2}{2}.$$

**Proof of Theorem 2.** We take *<sup>γ</sup>* <sup>=</sup> 1 without loss of generality. For *<sup>b</sup>*<sup>0</sup> <sup>∈</sup> (0, 1), we consider the fixed potential:

$$q\_0(r, \theta) = \begin{cases} 1, & 0 < r < b\_{0\prime} \\ 0, & b\_0 \le r < 1 \end{cases}$$

and a positive integer *<sup>k</sup>*(*b*0) satisfying *<sup>b</sup>*<sup>0</sup> <sup>+</sup> <sup>1</sup> *<sup>k</sup>*(*b*0) <sup>&</sup>lt; 1. We define the potentials:

$$q\_k(r, \theta) = \begin{cases} 1, & 0 < r < b\_{k\_{\prime}} \\ 0, & b\_k \le r < 1 \end{cases} \quad k = 1, 2, \cdots, \tag{A7}$$

with *bk* <sup>=</sup> *<sup>b</sup>*<sup>0</sup> <sup>+</sup> <sup>1</sup> *<sup>k</sup>*(*b*0)+*<sup>k</sup>* .

We have *<sup>q</sup>*<sup>0</sup> <sup>−</sup> *qk <sup>L</sup>*<sup>∞</sup> <sup>=</sup> 1 and to have (21), we have to prove for *<sup>g</sup>* <sup>∈</sup> *<sup>H</sup>*1/2 # that:

$$\|\left(\Lambda\_{q\_0} - \Lambda\_{q\_k}\right)g\|\_{H\_{\theta}^{-1/2}}^2 \le \mathbb{C}|b\_0 - b\_k|^2 \|g\|\_{H\_{\theta}^{1/2}}^2 \le \frac{\mathsf{C}}{k^2} \|g\|\_{H\_{\theta}^{1/2}}^2\tag{A8}$$

where *C* is a constant independent of *k* and *g*.

If *<sup>g</sup>*(*θ*) = <sup>∑</sup>*n*∈*<sup>Z</sup> gneinθ*, by (15) and (16), we have:

$$\|\left(\Lambda\_{q0}-\Lambda\_{q\_k}\right)g\|\_{H\_\bullet^{1-1/2}}^2 \le \left|\frac{b\_k f\_1(b\_k)}{b\_k f\_1(b\_k) \log b\_k + f\_0(b\_k)} - \frac{b\_0 f\_1(b\_0)}{b\_0 f\_1(b\_0) \log b\_0 + f\_0(b\_0)}\right|^2 |g\_0|^2$$

$$+ \sum\_{n=1}^\infty \left|\frac{I\_{n-1}(b\_k) - b\_k^{2n} f\_{n+1}(b\_k)}{f\_{n-1}(b\_k) + b\_k^{2n} f\_{n+1}(b\_k)} - \frac{I\_{n-1}(b\_0) - b\_0^{2n} f\_{n+1}(b\_0)}{f\_{n-1}(b\_0) + b\_0^{2n} f\_{n+1}(b\_0)}\right|^2 (1+n^2)^{1/2} \left(|g\_n|^2 + |g\_{-n}|^2\right)$$

$$= I\_0^2 |g\_0|^2 + \sum\_{n=1}^\infty I\_n^2 (1+n^2)^{1/2} \left(|g\_n|^2 + |g\_{-n}|^2\right).$$

We start by estimating *I*0.

From (A2), (A1), and (A4) *<sup>J</sup>*1(*r*) <sup>≤</sup> *<sup>r</sup>* <sup>2</sup> , when *<sup>r</sup>* <sup>∈</sup> (0, 1) and:

$$r J\_1(r) \log r + J\_0(r) \ge \frac{r^2 \log r}{2} + 1 - \frac{r^2}{4}, \quad r \in (0, 1).$$

Since *<sup>r</sup>*<sup>2</sup> log *<sup>r</sup>* <sup>2</sup> <sup>+</sup> <sup>1</sup> <sup>−</sup> *<sup>r</sup>*<sup>2</sup> <sup>4</sup> is a decreasing function in (0, 1), we have:

$$\|r\|\_1(r)\log r + f\_0(r) \ge \frac{3}{4}, \quad r \in (0, 1). \tag{A9}$$

A simple calculation and this inequality gives us:

$$I\_0 \lesssim b\_k b\_0 f\_1(b\_k) f\_1(b\_0) |\log b\_k - \log b\_0| + f\_1(b\_k) f\_0(b\_0) |b\_k - b\_0|$$

$$+ b\_0 f\_0(b\_k) |f\_1(b\_k) - f\_1(b\_0)| + b\_0 f\_1(b\_k) |f\_0(b\_k) - f\_0(b\_0)|.$$

where the symbol denotes that the left-hand side is bounded by a constant times the right-hand one. Thus, combining the mean value theorem, the identity *J* <sup>0</sup>(*r*) = <sup>−</sup>*J*1(*r*), the fact that *bk*, *<sup>b</sup>*<sup>0</sup> <sup>∈</sup> (0, 1) and (A2), we easily get:

$$I\_0 \stackrel{<}{\sim} \frac{1}{b\_0} |b\_k - b\_0|. \tag{A10}$$

,

Now, we deal with *Ik*, *<sup>k</sup>* <sup>=</sup> 1, 2, ··· . We use the mean value Theorem, *bk*, *<sup>b</sup>*<sup>0</sup> <sup>∈</sup> (0, 1), *b*2*<sup>n</sup> <sup>k</sup>* <sup>−</sup> *<sup>b</sup>*2*<sup>n</sup>* 0 - |*bk*−*b*0| *<sup>n</sup>* , (A2), and (A3) to obtain:

$$I\_n \stackrel{<}{\sim} \frac{I\_{n+1}(b\_k) f\_{n-1}(b\_0) |b\_k^{2n} - b\_0^{2n}| + b\_0^{2n} f\_{n-1}(b\_0) |f\_{n+1}(b\_k) - f\_{n+1}(b\_0)|}{f\_{n-1}(b\_k) f\_{n-1}(b\_0)}$$

$$+ \frac{b\_k^{2n} f\_{n+1}(b\_0) |f\_{n-1}(b\_k) - f\_{n-1}(b\_0)|}{f\_{n-1}(b\_k) f\_{n-1}(b\_0)} \stackrel{<}{\sim} \frac{b\_k - b\_0}{n} \le b\_k - b\_0.$$

From this estimate and (A10), we have (A8).

**Remark A1.** *Theorem 2 can be extended to the case that q*<sup>0</sup> *is null. In this case, we take in (A7) <sup>k</sup>*(*b*0) = <sup>0</sup> *and from (17):*

$$\begin{aligned} \left\| \left( \Lambda\_{q\_0} - \Lambda\_{q\_k} \right) g \right\|\_{H\_\bullet^{-1/2}}^2 &\le \left| \frac{b\_k f\_1(b\_k)}{b\_k f\_1(b\_k) \log b\_k + f\_0(b\_k)} \right|^2 \left| \mathcal{g}\_0 \right|^2 \\\ &+ \sum\_{n=1}^\infty \left| 1 - \frac{f\_{n-1}(b\_k) - b\_k^{2n} f\_{n+1}(b\_k)}{f\_{n-1}(b\_k) + b\_k^{2n} f\_{n+1}(b\_k)} \right|^2 (1 + n^2)^{1/2} \left( \left| \mathcal{g}\_n \right|^2 + \left| \mathcal{g}\_{-n} \right|^2 \right) \end{aligned}$$

*by using bk* <sup>∈</sup> (0, 1)*, (A9), and (A2):*

$$\begin{aligned} &\lesssim b\_k^4 |\g\_0|^2 + \sum\_{n=1}^\infty \frac{b\_k^{4n} f\_{n+1}^2(b\_k)}{f\_{n-1}^2(b\_k)} (1+n^2)^{1/2} \left( |\mathcal{g}\_n|^2 + |\mathcal{g}\_{-n}|^2 \right), \\ &\lesssim b\_k^4 |\mathcal{g}\_0|^2 + \sum\_{n=1}^\infty \frac{b\_k^{2n+4}}{n(n+1)} (1+n^2)^{1/2} \left( |\mathcal{g}\_n|^2 + |\mathcal{g}\_{-n}|^2 \right) \lesssim \frac{1}{k^4} \|\mathcal{g}\|\_{H^{1/2}\_\bullet}^2. \end{aligned}$$

**Proof of Theorem 3.** Let *<sup>q</sup>*1(*x*) = *<sup>γ</sup>*1*χB*(0,*b*<sup>1</sup>)(*x*), *<sup>q</sup>*2(*x*) = *<sup>γ</sup>*2*χB*(0,*b*<sup>2</sup>)(*x*) in *Fb* and *<sup>g</sup>*(*θ*) = 1 <sup>2</sup>1/4 *<sup>e</sup>iθ*.

$$\left\|\Lambda\_{q\_1} - \Lambda\_{q\_2}\right\|\_{\mathcal{L}(H\_\bullet^{1/2}; H\_\bullet^{-1/2})}^2 \ge \left\|\left(\Lambda\_{q\_1} - \Lambda\_{q\_2}\right)g\right\|\_{H\_\bullet^{-1/2}}^2$$

$$= \left|\frac{f\_0(b\_1\sqrt{\gamma\_1}) - b\_1^2 f\_2(b\_1\sqrt{\gamma\_1})}{f\_0(b\_1\sqrt{\gamma\_1}) + b\_1^2 f\_2(b\_1\sqrt{\gamma\_1})} - \frac{f\_0(b\_2\sqrt{\gamma\_2}) - b\_2^2 f\_2(b\_2\sqrt{\gamma\_2})}{f\_0(b\_2\sqrt{\gamma\_2}) + b\_2^2 f\_2(b\_2\sqrt{\gamma\_})}\right|^2 \tag{A11}$$

$$\ge \frac{4\Pi^2}{\left(1 + \frac{b\_1^4 \gamma\_1}{8}\right)^2 \left(1 + \frac{b\_1^4 \gamma\_2}{8}\right)^2}{\left(1 + \frac{b\_1^4 \gamma\_1}{8}\right)^2 \left(1 + \frac{b\_1^4 \gamma\_2}{8}\right)^2}\tag{A12}$$

where:

$$\mathbf{II} = \left| b\_2^2 f\_0(b\_1 \sqrt{\gamma\_1}) f\_2(b\_2 \sqrt{\gamma\_2}) - b\_1^2 f\_0(b\_2 \sqrt{\gamma\_2}) f\_2(b\_1 \sqrt{\gamma\_1}) \right|.$$

and we used (A2) for *<sup>n</sup>* <sup>=</sup> 0, 2. On the contrary:

$$\mathbf{H} \ge \frac{1}{8} \left| b\_2^4 \gamma\_2 - b\_1^4 \gamma\_1 \right| - \mathbf{J}\_1 - \mathbf{J}\_2 - \mathbf{J}\_{3\prime} \tag{A12}$$

where:

$$\mathbf{J}\_1 = \left| b\_2^2 \mathbb{S}\_2(b\_2 \sqrt{\gamma\_2}) - b\_1^2 \mathbb{S}\_2(b\_1 \sqrt{\gamma\_1}) \right|,\tag{A13}$$

$$\mathbf{J}\_2 = \frac{1}{8} \left| b\_2^4 \gamma\_2 \mathbf{S}\_0(b\_1 \sqrt{\gamma\_1}) - b\_1^4 \gamma\_1 \mathbf{S}\_0(b\_2 \sqrt{\gamma\_2}) \right|,\tag{A14}$$

and:

$$\mathbf{J}\_3 = \left| b\_2^2 \mathbf{S}\_0(b\_1 \sqrt{\gamma\_1}) \mathbf{S}\_2(b\_2 \sqrt{\gamma\_2}) - b\_1^2 \mathbf{S}\_0(b\_2 \sqrt{\gamma\_2}) \mathbf{S}\_2(b\_1 \sqrt{\gamma\_1}) \right|. \tag{A15}$$

To estimate **<sup>J</sup>***i*, *<sup>i</sup>* <sup>=</sup> 1, 2, 3, we use (A2), (A4), (A5), and Lemma A2. We get:

$$\mathbf{J}\_1 \le \frac{b\_2^4 \gamma\_2^2 |b\_1^2 - b\_2^2|}{30\pi} + \frac{b\_1^2 \left(b\_2^2 \gamma\_2 + b\_1^2 \gamma\_1\right) |b\_2^2 \gamma\_2 - b\_1^2 \gamma\_1|}{96}.\tag{A16}$$

$$\mathcal{J}\_2 \le \frac{b\_1^2 \gamma\_1 \left| b\_2^4 \gamma\_2 - b\_1^4 \gamma\_1 \right|}{32} + \frac{b\_1^4 \gamma\_1 \left| b\_2^2 \gamma\_2 - b\_1^2 \gamma\_1 \right|}{32}. \tag{A17}$$

$$\begin{split} \mathcal{J}\_3 \leq \frac{b\_1^2 b\_2^4 \gamma\_1 \gamma\_2^2 |b\_2^2 - b\_1^2|}{120\pi} + \frac{b\_1^6 \gamma\_1^2 |b\_1^2 \gamma\_1 - b\_2^2 \gamma\_2|}{36\pi^{\frac{3}{2}}} + \frac{b\_1^4 b\_2^4 \gamma\_1 \gamma\_2 |b\_1^2 \gamma\_1 - b\_2^2 \gamma\_2|}{36\pi^{\frac{5}{2}}} \\ &+ \frac{b\_1^2 b\_2^4 \gamma\_1 \gamma\_2^2 |b\_1^2 \gamma\_1 - b\_2^2 \gamma\_2|}{480\pi^{\frac{3}{2}}}. \end{split} \tag{A18}$$

**Proof of (22).** We suppose that *<sup>b</sup>*<sup>1</sup> <sup>=</sup> *<sup>b</sup>*<sup>2</sup> <sup>=</sup> *<sup>b</sup>* <sup>&</sup>gt; 0. We obtain:

$$\begin{aligned} \mathbf{J}\_1 &\le \frac{b^b}{96} |\gamma\_1 - \gamma\_2| \le 0.01041b^4 ||q\_1 - q\_2||\_{L^\infty(B(0,1))},\\ \mathbf{J}\_2 &\le \left(\frac{b^b}{32} + \frac{b^b}{32}\right) |\gamma\_1 - \gamma\_2| \le 0.0625b^4 ||q\_1 - q\_2||\_{L^\infty(B(0,1))},\\ \mathbf{J}\_3 &\le \left(\frac{b^8}{36\pi^{\frac{5}{2}}} + \frac{b^{10}}{36\pi^{\frac{5}{2}}} + \frac{b^8}{480\pi^{\frac{5}{2}}}\right) |\gamma\_1 - \gamma\_2| \le 0.01004b^4 ||q\_1 - q\_2||\_{L^\infty(B(0,1))}.\end{aligned}$$

and from (A11) and the above estimates, we get that:

$$\Pi \ge 0.042b^4 \|q\_1 - q\_2\|\_{L^\infty}.$$

Since *γ*1, *γ*2, and *b* are less than 1, (5.11) and the above estimate gives us:

$$\|\Lambda\_{q\_1} - \Lambda\_{q\_2}\|\_{\mathcal{L}(H^{1/2}\_{\theta}; H^{-1/2}\_{\theta})}^2 \ge 4\frac{8^4}{9^4} (0, 042)^2 b^8 \|q\_1 - q\_2\|\_{L^\infty}^2 = 0,\\ 0044b^8 \|q\_1 - q\_2\|\_{L^\infty}^2$$

this implies (22).

**Proof of (23).** Now *<sup>γ</sup>*<sup>1</sup> <sup>=</sup> *<sup>γ</sup>*2. Let us define:

$$M(\gamma, b\_1, b\_2) = \gamma \left( b\_1^3 + b\_1^2 b\_2 + b\_1 b\_2^2 + b\_2^3 \right).$$

It is easy to check that:

$$\begin{aligned} \frac{1}{8} |b\_2^4 \gamma\_2 - b\_1^4 \gamma\_1| &= \frac{1}{8} M(\gamma, b\_1, b\_2) |b\_2 - b\_1|, \\ \mathbf{J}\_1 &\le \left(\frac{1}{30\pi} + \frac{1}{9\pi^{\frac{3}{2}}}\right) M(\gamma, b\_1, b\_2) |b\_2 - b\_1|, \\ \mathbf{J}\_2 &\le \left(\frac{1}{32} + \frac{1}{256\pi^{\frac{1}{2}}}\right) M(\gamma, b\_1, b\_2) |b\_2 - b\_1|, \\ \mathbf{J}\_3 &\le \left(\frac{1}{120} + \frac{1}{18\pi^{\frac{3}{2}}} + \frac{1}{420\pi^{\frac{3}{2}}}\right) M(\gamma, b\_1, b\_2) |b\_2 - b\_1|, \end{aligned}$$

therefore:

$$\|\Lambda\_{q\_1} - \Lambda\_{q\_2}\|\_{\mathcal{L}(H\_\theta^{1/2}; H\_\theta^{-1/2})} \ge \frac{2}{\left(1 + \frac{1}{8}\right)^2} \left(\frac{\gamma}{8} \left|b\_1^4 - b\_2^4\right| - \mathbf{J}\_1 - \mathbf{J}\_2 - \mathbf{J}\_3\right),$$

$$\ge \frac{2}{\left(1 + \frac{1}{8}\right)^2} 0, 04216 M(\gamma, b\_1, b\_2) |b\_2 - b\_1| \ge 0, 2665 \gamma b^3 |b\_2 - b\_1|,$$

and we obtain (23).

#### **References**

