text:
\section{Introduction}

In this note we seek to prove Lipschitz stability estimates for the fractional Calder\'on problem with a finite number of measurements under structural a priori information on the potential. More precisely, we consider the following
direct problem
\begin{align}
\label{eq:frac_Cal}
\begin{split}
((-\D)^s+ q)u &= 0 \mbox{ in } \Omega, \\
u& = f \mbox{ in } \Omega_e,
\end{split}
\end{align}
where $\Omega \subset \R^n$ with $n\geq 1$ is an open, bounded domain and $\Omega_e=\R^n \setminus \overline{\Omega}$.
Assuming that zero is not a Dirichlet eigenvalue, i.e., that the following condition holds
\begin{align}
\label{eq:not_zero}
[u \in H^{s}(\R^n), \ ((-\D)^s + q) u = 0 \mbox{ in } \Omega, \ u = 0 \mbox{ in } \Omega_e] \Rightarrow u \equiv 0 \mbox{ in } \R^n,
\end{align}
it is possible to formally define the (generalized) Dirichlet-to-Neumann map (c.f. \eqref{eq:DtN} for a weak formulation of it)
\begin{align}
\label{eq:DtN}
\Lambda_q: H^{s}(\Omega_e) \rightarrow \widetilde{H}^{-s}(\Omega_e), \
f \mapsto (-\D)^s u|_{\Omega_e}.
\end{align}
In \cite{GSU16} it was shown that the associated inverse problem -- the fractional Calder\'on problem -- in which one seeks to recover the potential $q$ in a suitable function space (e.g. $q \in L^{\infty}(\Omega)$) from the knowledge of the Dirichlet-to-Neumann map $\Lambda_q$, is injective. This uniqueness result was extended to (almost) scaling critical spaces, $q\in Z^{-s}(\Omega)$, in \cite{RS17}. Motivated by the formal well-definedness in terms of degrees of freedom versus unknowns, in \cite{GRSU18} it was proved that a single measurement suffices to reconstruct the potential for the fractional Calder\'on problem.
In \cite{RS17} it was also shown that for the full infinite dimensional problem with infinite measurements conditional stability with a logarithmic modulus holds. By virtue of the constructions in \cite{RS17d}, this dependence is optimal. We refer to \cite{CL18, GLX17, HL17, LL17, RS17, S17} and the references therein for further developments on the fractional Calder\'on and related problems.

In the present note, we show that as in the case of the \emph{classical} Calder\'on problem, the a priori information of the finite dimensionality of the potential $q$ allows one to improve the modulus of continuity in the stability estimate to a \emph{Lipschitz continuous} dependence. Moreover, as in the classical setting the Lipschitz constant however diverges exponentially with the increasing ``complexity" of the setting (c.f. Section \ref{sec:opti} for a more precise explanation of this). First results of this kind were proven in \cite{AV05, Ron06} for the classical Calder\'on problem, where a Lipschitz stability estimate and the exponential behavior of the Lipschitz constant were achieved for piecewise constant conductivities. This line of research was extended to many different directions (c.f. \cite{BF11, AdHGS16, GS15, AdHGS17, BdHQ13, AS18}). In \cite{AdHGS17} a conditional Lipschitz stability estimate for a piecewise linear potential in the classical Schr\"{o}dinger equation by means of local Cauchy data was achieved. To some extent our main results represent the nonlocal analogue of the latter as we also deal with Cauchy data which allow us to remove spectral conditions. We also prove that a finite number of measurements $\tilde{f}_j^{(k)}$ (see Theorem \ref{thm:Lip_frac_Runge} and Proposition \ref{prop:finite_Cauchy} for a precise definition), suffice to recover the potential in the finite dimensional case in a Lipschitz stable way.

Originally, these improved stability properties were derived by using two main ingredients (c.f. \cite{AV05}):
\begin{itemize}
\item Propagation of smallness estimates;
\item Asymptotic estimates at the discontinuity interfaces for singular solutions.
\end{itemize}
The first ingredient allows one to propagate the information from the boundary into the interior of the domain. It is based on quantitative unique continuation results as for instance in \cite{ARV09}. In order to combine this with Alessandrini's identity which yields control on the potentials/conductivities, the second ingredient, namely singular solutions, is used. The singular solutions are related to localized solutions in the sense of for instance \cite{G08, HPS17} and provide very concentrated information around the domains of interest and in particular near the discontinuity interfaces.

In the setting of the \emph{fractional} Calder\'on problem the argument simplifies considerably. Due to the strong approximation properties which are valid in this setting (and which are dual to the strong uniqueness properties for the fractional Laplacian) \cite{GSU16, DSV14}, we do not need to construct singular solutions in this set-up. It suffices to use quantitative Runge approximation properties (c.f. Theorems \ref{thm:Runge_frac} and \ref{thm:quant_Runge_frac}, which were first deduced in \cite{RS17}). This is in the same spirit as the results in \cite{GSU16} and \cite{RS17}, where it was not necessary to construct CGO solutions but where these were replaced by the very strong Runge approximation properties that hold in the fractional set-up. Exploiting the strong Runge approximation property for the fractional Laplacian, our argument is of a ``perturbative flavour", which is reminiscent
of the very recent work \cite{AS18} on the classical Calder\'on problem.

In the sequel we discuss the Runge approximation approach in the setting in which the potential $q$ will be assumed to be contained in a finite dimensional subspace of $L^{\infty}(\Omega)$. With only small modifications and by using the results from \cite{RS17}, it is possible to extend this to critical function spaces such as $L^{\frac{n}{2s}}(\Omega)$. We address both the situation in which zero is not a Dirichlet eigenvalue of the problem \eqref{eq:frac_Cal} (c.f. Theorem \ref{thm:Lip_frac_Runge}) and the setting in which zero is allowed to be a Dirichlet eigenvalue (c.f. Theorem \ref{thm:Lip_frac_Runge_eigen}). We derive Runge approximation properties in the latter, which might be interesting in their own right.

\subsection{Organization of the article}
After recalling some auxiliary properties from \cite{GSU16} in Section \ref{sec:pre}, we first consider the case in which the potential $q\in L^{\infty}(\Omega)$ is such that zero is \emph{not} a Dirichlet eigenvalue. In this case the main result on the improved stability estimates (c.f. Theorem \ref{thm:Lip_frac_Runge} and Corollary \ref{thm:Lip_frac_Runge_fulldata}) is a quite direct consequence of the quantitative Runge approximation properties which were deduced in \cite{RS17}. It is stated and proved in Section \ref{sec:non_zero}.

In Section \ref{sec:zero} we extend Theorem \ref{thm:Lip_frac_Runge} to the situation, in which $q\in L^{\infty}(\Omega)$ is such that it \emph{may} lead to a zero Dirichlet eigenvalue for the fractional Schrödinger operator \eqref{eq:frac_Cal}. In order to prove Theorem \ref{thm:Lip_frac_Runge_eigen} and Proposition \ref{prop:finite_Cauchy}, the main results in this set-up, we first extend the Runge approximation properties for fractional Schrödinger operators without zero eigenvalues to Schrödinger operators with potentials which allow for zero Dirichlet eigenvalues. As these are still Fredholm operators of index zero, we only have to deal with a finite dimensional additional subspace, which can be treated ``by hand", c.f. Sections \ref{sec:qual_Runge_zero}, \ref{sec:quant_Runge_zero}. In particular this allows us to argue similarly as in the case without zero eigenvalue, c.f. Section \ref{sec:Proof_zero}.

Finally, in Section \ref{sec:opti}, we show that as in the classical case, in general the constants in the Lipschitz estimates depend exponentially on the ``complexity" of the problem, in that for the piecewise constant setting the constant depends exponentially on the number of underlying domains. This is based on a combination of ideas from \cite{Ron06} and \cite{RS17d}.

\section{Preliminaries}
\label{sec:pre}

In this section, we briefly recall some auxiliary properties from \cite{GSU16}, which will be used throughout the article. We first define the relevant function spaces and then discuss the well-posedness of the forward problem and the weak definition of the fractional Dirichlet-to-Neumann map in the absence of zero eigenvalues.

\subsection{Function spaces}

In the sequel, we will use the following $L^2$ based fractional Sobolev spaces (c.f. Chapter 3 in \cite{McLean}). Let $W \subset \R^n$ be open. Then, for $s\in \R$ we set:
\begin{align*}
&H^{s}(W)= \{y : \mbox{ there exists } Y\in H^{s}(\R^n) \mbox{ such that } y= Y|_{W} \},\\
&\|y\|_{H^{s}(W)}:= \inf\{\|Y\|_{H^{s}(\R^n)}: Y|_{W}=y\},\\
&\widetilde{H}^{s}(W)= \mbox{ closure of } C_c^{\infty}(W) \mbox{ in } H^s(W),\\
&H^s_{\overline{W}}= \{y \in H^{s}(\R^n): \supp(y) \subset \overline{W}\}.
\end{align*}
We have
\begin{align*}
(H^{s}(W))^{\ast} = \widetilde{H}^{-s}(W), \  (\widetilde{H}^{-s}(W))^{\ast} = (H^{s}(W)).
\end{align*}
If $W$ is Lipschitz and $s\geq 0$, then also
$\widetilde{H}^{s}(W) = H^{s}_{\overline{W}}$.

\subsection{Well-posedness and the Dirichlet-to-Neumann map}

We recall the well-posedness results and the weak definition of the Dirichlet-to-Neumann map for fractional Schrödinger operators without zero eigenvalues. This follows the lines of \cite{GSU16}.

\begin{lem}[Lemma 2.3 in \cite{GSU16}]
\label{lem:well_posed}
Let $s\in (0,1)$ and $n\in \N$. Let $\Omega \subset \R^n$ be open and let $q\in L^{\infty}(\Omega)$. Consider for $u,v \in H^s(\R^n)$ the bilinear form
\begin{align*}
B_q(u,v):=((-\D)^{\frac{s}{2}}u, (-\D)^{\frac{s}{2}}v)_{\R^n} + (q u, v)_{\Omega}.
\end{align*}
Then, there exists a countable set $\Sigma \subset \R$ such that for $\lambda \in \R \setminus \Sigma$, any $F\in (\widetilde{H}^s(\Omega))^{\ast}$ and any $f\in H^{s}(\R^n)$ there exists a unique $u\in H^{s}(\R^n)$ such that
\begin{align*}
B_{q}(u,w) - \lambda(u,w) = F(w)
\end{align*}
for $w \in \widetilde{H}^s(\Omega)$ and $u-f \in \widetilde{H}^s(\Omega)$. Moreover,
\begin{align*}
\|u\|_{H^s(\R^n)} \leq C (\|F\|_{(\tilde{H}^s(\Omega))^{\ast}} + \|f\|_{H^s(\R^n)}).
\end{align*}
\end{lem}

The solution from Lemma \ref{lem:well_posed} also agrees with the distributional solution to \eqref{eq:frac_Cal}.

With the bilinear form from Lemma \ref{lem:well_posed} at hand, it is possible to define the weak form of the generalized \emph{Dirichlet-to-Neumann map}
\begin{align*}
\Lambda_{q}: H^s(\R^n) / \widetilde{H}^s(\Omega) \rightarrow (H^s(\R^n) / \widetilde{H}^s(\Omega))^{\ast}
\end{align*}
by
\begin{align}
\label{eq:DtNa}
(\Lambda_q [f], [g]) = B_q(u_f, g).
\end{align}
Here $f,g \in H^{s}(\R^n)$ and $u_f$ is a solution to \eqref{eq:frac_Cal} such that $u-f \in \widetilde{H}^s(\Omega)$. We note that for a bounded Lipschitz domain $\Omega$, we can identify $H^s(\R^n) / \widetilde{H}^s(\Omega)$ with $H^s(\Omega_e)$. If $\Omega$ is bounded, open and Lipschitz, we will mostly drop the quotient space notation and simply write $\Lambda_q f$ instead of $\Lambda_q [f]$ (c.f. Lemma 2.4 in \cite{GSU16} for more details).

A further ingredient which will be used in the proof of Theorem \ref{thm:Lip_frac_Runge} is a generalized Alessandrini type identity for the fractional Calder\'on problem (c.f. Lemma 2.5 in \cite{GSU16}), which we briefly recall.

\begin{lem}[\cite{GSU16}, Lemma 2.5]
\label{lem:Aless}
Let $s\in (0,1)$, $n\geq 1$. Assume that $\Omega \subset \R^n$ is an open, bounded domain and that $W \subset \Omega_e$ is open. Assume that \eqref{eq:not_zero} holds. Then for two solutions $u_1, u_2$ of \eqref{eq:frac_Cal} with potentials $q_1, q_2 \in L^{\infty}(\Omega)$ and with exterior data $f_1, f_2$ we have
\begin{align*}
((q_1-q_2)u_1,u_2)_{\Omega} = ((\Lambda_{q_1}-\Lambda_{q_2})f_1, f_2)_{W}.
\end{align*}
\end{lem}

Finally, we recall the weak unique continuation property for the fractional Laplacian, which due to the antilocality of the fractional Laplacian is a global result:

\begin{lem}[\cite{GSU16}, Theorem 1.2]
\label{lem:WUCP}
Let $s\in(0,1)$ and $u\in H^{s}(\R^n)$ for $n\geq 1$. Assume that for $\Omega \subset \R^n$ open
\begin{align*}
u= 0 = (-\D)^s u \mbox{ in } \Omega.
\end{align*}
Then $u \equiv 0$.
\end{lem}

This result was proved in \cite{GSU16} based on Carleman estimates from \cite{Rue15}, c.f. also \cite{FF14} and \cite{Yu16} for unique continuation results for the fractional Laplacian.

\section{Absence of zero Dirichlet eigenvalue}\label{sec:non_zero}

In this section, we discuss our Lipschitz stability result in the setting in which zero is not a Dirichlet eigenvalue. It is obtained by sampling the Dirichlet-to-Neumann map \emph{finitely} many times.

\begin{thm}
\label{thm:Lip_frac_Runge}
Let $s\in(0,1)$ and $n,m\in \N$. Let $\Omega \subset \R^n$ be an open, bounded and smooth domain. Suppose that $W_1, W_2 \subset \Omega_e$ are open Lipschitz domains with $\overline{W}_k \cap \overline{\Omega} = \emptyset$ for $k\in\{1,2\}$.
Assume that for some orthonormal (w.r.t. the $L^2(\Omega)$ scalar product) functions $g_1,\dots,g_m \in L^{\infty}(\Omega)$ the potentials
$q_1,q_2 \in L^{\infty}(\Omega)$ are such that
\begin{align*}
q_1,q_2 \in \spa\{g_1,\dots,g_m\} \mbox{ and } \|q_1\|_{L^{\infty}(\Omega)}, \|q_2\|_{L^{\infty}(\Omega)}\leq C_0<\infty.
\end{align*}
Furthermore, suppose that \eqref{eq:not_zero} holds for $q_1, q_2$.

Then there exist
\begin{itemize}
\item a constant $C_1>0$ which depends only on the geometry of the domains $W_k$, $k\in\{1,2\}$, on the functions $g_1,\dots, g_m$, on the constant $C_0$ and on $m$,
\item and
data $\{\tilde{f}_{1}^{(k)},\dots \tilde{f}_{m}^{(k)}\} \subset \widetilde{H}^{s}(W_k)$ with $\| \tilde{f}_j^{(k)} \|_{\widetilde{H}^{s}(\Omega)} = 1$,
\end{itemize}
such that
\begin{align*}
\|q_1-q_2\|_{L^{\infty}(\Omega)} \leq C_1 \inf \left\{ \|(\Lambda_{q_1} - \Lambda_{q_2})\tilde{f}^{(1)}\|_{H^{-s}(W_{2})},  \|(\Lambda_{q_1} - \Lambda_{q_2})\tilde{f}^{(2)}\|_{H^{-s}(W_{1})} \right\}.
\end{align*}
Here $\tilde{f}^{(k)}:=(\tilde{f}^{(k)}_1, \dots, \tilde{f}^{(k)}_m)$ and $(\Lambda_{q_1}-\Lambda_{q_2}) \tilde{f}^{(k)} := ((\Lambda_{q_1}-\Lambda_{q_2})\tilde{f}^{(k)}_1, \dots, (\Lambda_{q_1}-\Lambda_{q_2}) \tilde{f}^{(k)}_m) $ for $k\in \{1,2\}$.
\end{thm}

\begin{rmk}
\label{rmk:ortho_linind}
With no loss of generality and by arguing with a Gram-Schmidt orthonormalization process we can relax the orthonormality requirement on the basis $g_1, \dots, g_m$ into a linearly independence one.
\end{rmk}

\begin{rmk}
\label{rmk:dependence}
We emphasize that the dependences of the constant $C_1>0$ can be made explicit in terms of the quantities $W_k$, $k\in\{1,2\}$, on $C_0>0$ and in terms of the functions $g_1,\dots, g_m$ (c.f. Section \ref{sec:non_zero}). It depends exponentially on the ``complexity" of the setting, c.f. Section \ref{sec:opti}.
\end{rmk}

\begin{rmk}
\label{rmk:finite}
We remark that the functions $f_1^{(k)}, \dots, f_m^{(k)}$ with $k \in \{1,2\}$ are obtained as ``control functions" for the Runge approximation of the basis to $g_1, \dots, g_m$ by solutions to a fractional Schrödinger equation with potential $q_k$. Thus, in our argument, the functions $f_1^{(k)}, \dots, f_m^{(k)}$ with $k \in \{1,2\}$ \emph{depend} on the choice of the potentials $q_k$.

In particular, we emphasize that although the stability estimate is formulated in terms of a \emph{finite} sample of the full Dirichlet-to-Neumann map, this should be mainly viewed as of theoretical interest (rather than of practical relevance), since in general the functions $\tilde{f}^{(k)}$ with $k\in \{1,2\}$ are \emph{unknown}. Hence, although one could easily turn the stability argument leading to Theorem \ref{thm:Lip_frac_Runge} into a recovery algorithm, we do not formulate it explicitly here.

The Lipschitz stability property itself however \emph{should} be regarded as of practical relevance, since it improves the very slow logarithmic stability from the general set-up to much better Lipschitz bounds in the finite dimensional setting (at the price of large constants, which depend on the complexity of the problem).
\end{rmk}

For the sake of completeness, we formulate below the Lipschitz stability estimate from full boundary data, i.e. from the knowledge of the (generalized) Dirichlet-to-Neumann map. The proof is a straightforward consequence of Theorem \ref{thm:Lip_frac_Runge}.

\begin{cor}
\label{thm:Lip_frac_Runge_fulldata}
Under the hypotheses of Theorem \ref{thm:Lip_frac_Runge} we also have that
\begin{align*}
\|q_1-q_2\|_{L^{\infty}(\Omega)} \leq C_1 \|\Lambda_{q_1} - \Lambda_{q_2}\|_{*}.
\end{align*}
Here $\|\Lambda_{q_1} - \Lambda_{q_2}\|_{*}:=
\sup\left\{\int\limits_{\R^n} ((\Lambda_{q_1}- \Lambda_{q_2}) f_1) f_2 dx: \ \|f_j\|_{\widetilde{H}^{s}(W_k)}=1\right\}$.
\end{cor}

We note that the proof of Theorem \ref{thm:Lip_frac_Runge} is more direct than in the case of the classical Calder\'on problem. Instead of constructing singular solutions, it here suffices to argue by Runge approximation only.

In order to formulate the auxiliary results on Runge approximation in Section \ref{sec:quant_runge}, we first recall that in the setting in which zero is not a Dirichlet eigenvalue of the equation \eqref{eq:frac_Cal}, i.e., in the case that \eqref{eq:not_zero} holds, we can in particular consider the Poisson operator
\begin{align}
\label{eq:Poisson}
P_q: H^{s}_{\overline{W}} \rightarrow H^s(\R^n), \
f \mapsto u.
\end{align}
Here $u$ denotes the solution of \eqref{eq:frac_Cal} with exterior data $f$.

In Section \ref{sec:finite} we then present the proof of Theorem \ref{thm:Lip_frac_Runge} and discuss several examples.

\subsection{Quantitative Runge approximation}
\label{sec:quant_runge}

The article \cite{RS17} established stability for the fractional Calder\'on problem by quantifying the uniqueness results of \cite{GSU16}. Here a crucial ingredient was a quantitative Runge approximation property which we will also rely on in the sequel:

\begin{thm}[\cite{RS17}, Theorem 1.4]
\label{thm:Runge_frac}
Let $s\in(0,1)$ and $n\in \N$.
Let $\Omega \subset \R^n$ be an open, bounded, smooth domain. Let $W \subset \Omega_e$ be an open Lipschitz domain with $\overline{\Omega}\cap \overline{W}= \emptyset$. Assume that $q\in L^{\infty}(\Omega)$ is such that $0$ is not a Dirichlet eigenvalue of $(-\D)^s + q$ in $\Omega$.
Let $P_q$ denote the Poisson operator from \eqref{eq:Poisson} and let $r_{\Omega}$ denote the restriction operator to $\Omega$.

Then, there exist constants $C>0$, $\mu>0$ such that for any $\overline{v} \in H^{s}_{\overline{\Omega}}$ there exists $f\in H^s_{\overline{W}}$ with
\begin{align*}
\|\overline{v}- r_{\Omega} P_{s}f \|_{L^2(\Omega)} \leq \epsilon \|\overline{v}\|_{H^{s}_{\overline{\Omega}}}, \  \|f\|_{H^{s}_{\overline{W}}} \leq Ce^{C \epsilon^{-\mu}} \|\overline{v}\|_{L^2(\overline{\Omega})}.
\end{align*}
The constants $C>0$ and $\mu>0$ only depend on the geometries of $\Omega, W$ and on $n,s$, $\|q\|_{L^{\infty}(\Omega)}$.
\end{thm}

We recall that this result was obtained by duality to a quantitative unique continuation result for fractional Schrödinger equations. In the sequel, it will allow us to quantitatively approximate arbitrary functions by solutions to the fractional Schrödinger equation.

\subsection{Proof of Theorem \ref{thm:Lip_frac_Runge}}
\label{sec:finite}

With the Runge approximation result of Theorem \ref{thm:Runge_frac} at hand, in this section we address the proof of Theorem \ref{thm:Lip_frac_Runge}.

\begin{proof}[Proof of Theorem \ref{thm:Lip_frac_Runge}]
We choose $2 m$ functions $h_{1}^{(k)},\dots,h_{m}^{(k)} \in \widetilde{H}^s(\Omega)$, $k\in\{1,2\}$, such that the matrix
\begin{align*}
M:=
\begin{pmatrix} ( g_1, h_{1}^{(1)} h_{1}^{(2)}  )_{\Omega}& \dots & ( g_m, h_{1}^{(1)} h_{1}^{(2)} )_{\Omega}\\
\vdots & \vdots & \vdots\\
 ( g_1, h_{m}^{(1)}h_{m}^{(2)} )_{\Omega} & \dots & ( g_m, h_{m}^{(1)} h_{m}^{(2)} )_{\Omega}
\end{pmatrix}
\end{align*}
is invertible. This can always be ensured by for example defining $h_{1}^{(1)}h_{1}^{(2)},\dots, h_{m}^{(1)} h_{m}^{(2)}$ to be a (regularized) approximation of the  basis $g_{1},\dots,g_m$ itself.
Then, by the Runge approximation result of Theorem \ref{thm:Runge_frac}, there exist functions $u_j^{(1)}, u_j^{(2)}$ with $j\in \{1,\dots,m\}$ such that
the following quantitative approximation property holds: For $k \in \{1,2\}$ there exist boundary data $f_{j}^{(k)} \in H^s_{\overline{W}_k}$ and solutions $u_{j}^{(k)}= h_j^{(k)} + r_{j}^{(k)}$ of the fractional Schrödinger equations
\begin{align*}
((-\D)^s + q_k)u_{j}^{(k)} & = 0 \mbox{ in } \Omega, \ u_{j}^{(k)} = f_{j}^{(k)} \mbox{ in } \Omega_e
\end{align*}
with
\begin{align}
\label{eq:approx}
\|u_{j}^{(k)} - h_j^{(k)}\|_{L^2(\Omega)}\leq \epsilon \|h_j^{(k)}\|_{H^s_{\overline{\Omega}}}, \ \|f_{j}^{(k)}\|_{H^{s}_{\overline{W}_k}} \leq C e^{C \epsilon^{-\mu}} \|h_{j}^{(k)}\|_{L^2(\Omega)}, \ j \in \{1,\dots, m\},
\end{align}
for some constants $C>1$ and $\mu>0$ (which have the dependences which were explained in Theorem \ref{thm:Runge_frac}). \\
Inserting the functions $u_j^{(k)}$ into the generalized  Alessandrini's identity from Lemma \ref{lem:Aless} then yields
\begin{align}
\label{eq:Al}
\begin{split}
((q_1 - q_2) h_j^{(1)}, h_j^{(2)})_{\Omega}
&= -((q_1-q_2) r_j^{(1)}, h_j^{(2)})_{\Omega}
-((q_1-q_2) r_j^{(2)}, h_j^{(1)})_{\Omega}
-((q_1-q_2) r_j^{(2)}, r_j^{(1)})_{\Omega} \\
& \quad + ((\Lambda_{q_1}- \Lambda_{q_2})f_j^{(1)}, f_j^{(2)})_{\Omega_e}.
\end{split}
\end{align}
Thus,
\begin{align*}
\left| ((q_1 - q_2) h_j^{(1)}, h_j^{(2)})_{\Omega}  \right|
&\leq \left| ((q_1-q_2) r_j^{(1)}, h_j^{(2)})_{\Omega}  \right|
+ \left| ((q_1-q_2) r_j^{(2)}, h_j^{(2)})_{\Omega} \right|
+ \left| ((q_1-q_2) r_j^{(1)}, r_j^{(2)})_{\Omega} \right| \\
& \quad +  \| (\Lambda_{q_1}-\Lambda_{q_2}) f_j^{(1)}\|_{H^{-s}(W_2)} \|f_j^{(2)}\|_{H^{s}_{\overline{W}_2}}.
\end{align*}
Using this together with the assumption that
\begin{align*}
q_1-q_2 = \sum\limits_{j=1}^{m} a_j g_j,
\end{align*}
and the estimates from \eqref{eq:approx}, leads to $m$ inequalities of the form
\begin{align}
\label{eq:est_f}
\begin{split}
\left| \sum\limits_{j=1}^{m} a_j ( g_j ,h_{l}^{(1)}h_{l}^{(2)} )_{\Omega} \right|
&\leq C \|(\Lambda_{q_1}-\Lambda_{q_2})\tilde{f}_l^{(1)}\|_{H^{-s}(W_2)} e^{C \epsilon^{-\mu}} \|h_{l}^{(1)}\|_{L^2(\Omega)} \|h_{l}^{(2)}\|_{L^2(\Omega)} \\
& \quad + C \epsilon \|q_1-q_2\|_{L^{\infty}(\Omega)} \|h_{l}^{(1)}\|_{H^{s}_{\overline{\Omega}}}\|h_{l}^{(2)}\|_{H^{s}_{\overline{\Omega}}},\quad l \in\{1,\dots,m\},
\end{split}
\end{align}
where $\tilde{f}_l^{(1)}(x):= \frac{f_l^{(1)}(x)}{\|f_l^{(1)}\|_{\tilde{H}^s(W_1)}}$.
Abbreviating $h_{l}:= h_{l}^{(1)} h_{l}^{(2)}$ for $l\in \{1,\dots,m\}$, we observe that for each $l\in\{1,\dots,m\}$ the estimate \eqref{eq:est_f} can be rewritten as the following vector valued system of equations
\begin{align*}
\begin{pmatrix} ( g_1, h_{1} )_{\Omega}& \dots & ( g_m, h_{1} )_{\Omega}\\
\vdots & \vdots & \vdots\\
(g_1, h_{m} )_{\Omega} & \cdots & ( g_m, h_{m} )_{\Omega}
\end{pmatrix}
\begin{pmatrix}
a_1 \\ \vdots\\ a_m
\end{pmatrix} = v_1(\epsilon) + v_{2}(\epsilon),
\end{align*}
where $v_1(\epsilon), v_2(\epsilon) \in \R^n$ obey the bounds
\begin{align*}
|v_1(\epsilon)| &\leq C e^{C \epsilon^{-\mu}}  \sup\limits_{l \in \{1,\dots,m\}} \|(\Lambda_{q_1}-\Lambda_{q_2}) \tilde{f}_l^{(1)}\|_{H^{-s}(W_2)} \|h_{l}^{(1)}\|_{L^2(\Omega)} \|h_{l}^{(2)}\|_{L^2(\Omega)}\\
|v_2(\epsilon)| &\leq C \epsilon \|q_1-q_2\|_{L^{\infty}(\Omega)} \sup\limits_{l \in \{1,\dots,m\}}\|h_{l}^{(1)}\|_{H^{s}_{\overline{\Omega}}}\|h_{l}^{(2)}\|_{H^{s}_{\overline{\Omega}}}.
\end{align*}
By virtue of the assumed invertibility of the matrix $M$, we infer that for $a=(a_1,\dots, a_m)$
\begin{align}
\label{eq:gen}
\begin{split}
\|a\|
&\leq C_M e^{C \epsilon^{-\mu}} \sup\limits_{l \in \{1,\dots,m\}} \|(\Lambda_{q_1}-\Lambda_{q_2}) \tilde{f}_l^{(1)}\|_{H^{-s}(W_2)}  \|h_{l}^{(1)}\|_{L^2(\Omega)} \|h_{l}^{(2)}\|_{L^2(\Omega)} \\
& \quad + C_M \epsilon \sup\limits_{l\in \{1,\dots,m\}} \|h_{l}^{(1)}\|_{H^{s}_{\overline{\Omega}}}\|h_{l}^{(2)}\|_{H^{s}_{\overline{\Omega}}}\|a\|.
\end{split}
\end{align}
Here $\|a\|:= \|a\|_{\ell^{\infty}}$; the subscript $M$ in the constant $C_M>0$ emphasizes that this constant in particular depends on the invertibility properties of the matrix $M$.
We note the last term on the right hand side of \eqref{eq:gen} can be absorbed into the left hand side of \eqref{eq:gen} provided that $C_M \epsilon \sup\limits_{l\in\{1,\dots,m\}}  \|h_{l}^{(1)}\|_{H^{s}_{\overline{\Omega}}}\|h_{l}^{(2)}\|_{H^{s}_{\overline{\Omega}}}  < 1$.
As a consequence, setting
\begin{align*}
L_{0}:=\sup\limits_{l\in\{1,\dots,m\}}  \|h_{l}^{(1)}\|_{H^{s}_{\overline{\Omega}}}\|h_{l}^{(2)}\|_{H^{s}_{\overline{\Omega}}}, \
L_{1}:=\sup\limits_{l\in\{1,\dots,m\}} \|h_{l}^{(1)}\|_{L^2(\Omega)} \|h_{l}^{(2)}\|_{L^2(\Omega)} ,
\end{align*}
 and choosing $\epsilon = \frac{1}{2}(C_M L_0)^{-1} $, we have
\begin{align*}
\|a\| \leq C_M L_{1} e^{C C_M^{\mu} L_{0}^{\mu}}\|(\Lambda_{q_1}-\Lambda_{q_2}) \tilde{f}^{(1)}\|_{H^{-s}(W_2)} .
\end{align*}
Noting that $\Lambda_q$ is a symmetric operator, we observe that in \eqref{eq:Al}, we could also have reversed the roles of $f_j^{(1)}, f_j^{(2)}$, which would have led to
\begin{align*}
\|a\| \leq C_M L_{1} e^{C C_M^{\mu} L_{0}^{\mu}}\|(\Lambda_{q_1}-\Lambda_{q_2}) \tilde{f}^{(2)}\|_{H^{-s}(W_1)}.
\end{align*}
This concludes the argument for Theorem \ref{thm:Lip_frac_Runge}.
\end{proof}

\begin{rmk}
\label{rmk:dependences_1}
The above proof keeps quite explicitly track of the dependences of the constants on the various parameters. It is an interesting question, to optimize the choice of the basis functions $g_j$ and the dual test functions $h_{l}^{(k)}$ for $k\in\{1,2\}$.
\end{rmk}

\begin{example}
\label{ex:pc}
As an example of Theorem \ref{thm:Lip_frac_Runge}, one may consider the case of piecewise constant potentials, which has been studied in \cite{BdHQ13} for the classical Schrödinger case. In this case we have a (up to null sets) disjoint covering $\Omega = \bigcup\limits_{j=1}^{N} D_j$ by bounded Lipschitz sets $D_j \subset \Omega$ and $g_j:= \chi_{D_j}$, where $\chi_{D_j}$ are the characteristic functions of the sets $D_j$, i.e.,
\begin{align*}
\chi_{D_j}(x):=\left\{ \begin{array}{ll} 1 \mbox{ if } x \in D_j, \\ 0 \mbox{ else}. \end{array} \right.
\end{align*}
If $s< \frac{1}{2}$ we can then choose the functions $h_j^{(k)}$ from the proof of Theorem \ref{thm:Lip_frac_Runge} to be the (normalized) characteristic functions of the domains $D_j$, while for $s\geq \frac{1}{2}$, we can for instance choose smoothed out versions of these. Choosing for $\Omega = [0,1]^n$ the sets $D_j $ to be translates of the interval $[0,N^{-1/n}]^n$ then shows that the constant $L_0^{\mu}$ is proportional to $N^{\tilde{\mu}}$ for some constant $\tilde{\mu}>0$. This is consistent with the results of \cite{Ron06} (c.f. also Section \ref{sec:opti}).
\end{example}

\begin{example}
\label{ex:pl}
From an applications point of view another interesting example of a system of basis functions $\{g_1,\dots, g_m\}$ corresponds to piecewise affine basis functions, as discussed in \cite{AdHGS17} for the classical Schrödinger case. Considering a partition of $\Omega$ as in Example \ref{ex:pc}, we here have $m=Nn$ and
\begin{align*}
\{g_1, \dots, g_m\}= \{\chi_{D_1}, \dots, \chi_{D_N}, x_1 \chi_{D_1}, \dots, x_1 \chi_{D_N}, x_2 \chi_{D_2}, \dots, x_2 \chi_{D_2}, \dots, x_n \chi_{D_N}, \dots, x_n \chi_{D_N}\}.
\end{align*}
Another canonical set of basis functions $\{g_1,\dots, g_m\}$ for instance consists of a subset of the trigonometric functions.
\end{example}

\begin{rmk}
\label{rmk:rough}
We emphasize that the choice of the space $q\in L^{\infty}(\Omega)$ does not play a major role in our arguments. Instead of using the $L^{\infty}$ based Runge approximation results from above, it would also have been possible to work with the space $Z^{-s}_0(\R^n)$ and the approximation results from Lemmas 8.1 and 8.2 in \cite{RS17}. This would for instance have allowed us to treat potentials $q$ in finite dimensional subspaces of $L^{\frac{n}{2s}}(\Omega)$. Since there are no major changes in our arguments (except for having to rely on the stronger results of Lemmas 8.1 and 8.2 from \cite{RS17} instead of Theorem \ref{thm:Runge_frac}) and since such a more general set-up would introduce additional technicalities, we have opted against formulating the corresponding results.
\end{rmk}

\section{Presence of Zero Eigenvalue}
\label{sec:zero}

In the case in which zero is a Dirichlet eigenvalue of the problem \eqref{eq:frac_Cal}, i.e. in the case in which \eqref{eq:not_zero} is violated, we have to modify the functional analytic set-up of the argument from Section \ref{sec:non_zero} slightly. In particular, we
can no longer work with the Dirichlet-to-Neumann map, but have to consider the Cauchy data associated with the problem.

Let us make this more precise: For $\Omega \subset \R^n$ open, bounded and smooth and $W \subset \Omega_e$ open and Lipschitz, we consider the inhomogeneous Dirichlet problem
\begin{align}
\label{eq:inhom_Dir}
\begin{split}
((-\D)^s + q) u & = F \mbox{ in } \Omega,\\
u & = f \mbox{ in } \Omega_e,
\end{split}
\end{align}
with $F \in (\widetilde{H}^{s}(\Omega))^{\ast}=H^{-s}(\Omega)$, $f\in \widetilde{H}^s(W)$.
By virtue of the results of Grubb \cite{Gr15} (which are formulated for smooth domains, hence our restriction to this class of domains), the inhomogeneous, zero Dirichlet data problem
\begin{align}
\label{eq:inhom_Dir_a}
\begin{split}
((-\D)^s + q) \tilde{u} & = \tilde{F} \mbox{ in } \Omega,\\
\tilde{u} & = 0 \mbox{ in } \Omega_e,
\end{split}
\end{align}
with $\tilde{F} \in H^{-s}(\Omega)$ gives rise to a Fredholm operator
\begin{align*}
\tilde{T}: \widetilde{H}^s(\Omega) \ni \tilde{u} \mapsto \tilde{F} \in H^{-s}(\Omega).
\end{align*}
Due to the observations in Lemma 2.3 in \cite{GSU16}, this operator is of index zero (indeed, if $q=0$, it is of index zero and multiplication with $q \in L^{\infty}(\Omega)$ is a compact perturbation, which does not change the index of $\tilde{T}$). We note that the problem \eqref{eq:inhom_Dir} can be reduced to \eqref{eq:inhom_Dir_a} by extending $f$ by zero and defining $\tilde{u}:= u- e_0 f$, which leads to $\tilde{F}=F-(-\D)^s (e_0 f)$. Here $e_0$ denotes the extension operator by zero. We have used that $\widetilde{H}^s(W)=H^s_{\overline{W}}$ for the Lipschitz domain $W \subset \Omega_e$.

The reduction of \eqref{eq:inhom_Dir} to \eqref{eq:inhom_Dir_a} and the results from \cite{Gr15} allow us to invoke the Fredholm alternative (c.f. for instance Theorem 2.27 in \cite{McLean}). In particular, this entails that there exists a finite dimensional space
\begin{align}
\label{eq:eigen}
Z_2:=\{z \in \widetilde{H}^s(\Omega): \ ((-\D)^s + q) z = 0 \mbox{ in } \Omega\} \subset \widetilde{H}^s(\Omega) \subset L^2(\Omega),
\end{align}
such that the problem \eqref{eq:inhom_Dir} is solvable, if and only if
\begin{align}
\label{eq:solvability}
(F,z)_{\Omega} + (f,(-\D)^s z)_{W} = 0 \mbox{ for all } z \in Z_2.
\end{align}
Here we used that $\supp(f) \subset \overline{W}$ and view the identity \eqref{eq:solvability} as a duality pairing (in the usual sense, c.f. for instance Theorem 3.14 in \cite{McLean}). We remark that since $\widetilde{H}^{s}(\Omega)= H^s_{\overline{\Omega}}$, we can also interpret $Z_2$ as a subset of $H^{s}(\R^n)$ (and not only of $H^{s}(\Omega)$).
Seeking to define the operator $A:= \iota r_{\Omega}( P_{q}-Id)$ in analogy to the operator from \cite{RS17}, where
\begin{itemize}
\item $r_{\Omega}$ denotes the restriction operator onto $\Omega$,
\item $\iota$ is the (compact) inclusion operator of $H^{s}(\R^n)$ into $L^2(\Omega)$,
\item $P_q$ is a Poisson type operator for \eqref{eq:inhom_Dir} (which is defined in \eqref{eq:Pq} below),
\item and $Id$ denotes the identity operator,
\end{itemize}
the following functional analytic set-up arises naturally: Motivated by the solvability properties of \eqref{eq:inhom_Dir} for a fixed open Lipschitz set $W \subset \Omega_e$, we introduce the following orthogonal decompositions
\begin{align*}
\widetilde{H}^{s}(W)= Z_1 \perp H_1, \ L^2(\Omega) = Z_2 \perp H_2.
\end{align*}
Here $Z_2$ is the function space from \eqref{eq:eigen}, while $Z_1$ is defined as the orthogonal complement of
\begin{align}
\label{eq:eigen_1}
H_{1}= \{y \in \widetilde{H}^{s}(W): \ (y,(-\D)^s z)_{W}=0, \ z \in Z_2 \} \subset \widetilde{H}^s(W).
\end{align}
By the Fredholm alternative, both $Z_1$ and $Z_2$ are finite dimensional.

In this setting the Poisson operator to the problem \eqref{eq:inhom_Dir} with $F=0$ is now defined as the mapping
\begin{align}
\label{eq:Pq}
P_q: H_1 \rightarrow H^s(\R^n)/Z_2, \ f \mapsto [u].
\end{align}
Here $u$ solves \eqref{eq:inhom_Dir} with $F=0$ and boundary data $f\in H_1$ and $H^s(\R^n)/Z_2$ denotes the quotient space of $H^{s}(\R^n)$ and $Z_2$ (for whose definition we use that $Z_2 \subset H^s(\R^n)$). The notation $[u]$ denotes the equivalence class in $H^s(\R^n)/Z_2$ containing the element $u$. By an argument as in the proof of Lemma 2.3 in \cite{GSU16} (now formulated in the quotient space $H^s(\R^n)/Z_2$) the operator $P_q$ is bounded and linear.

Having established this notation, by Sobolev embedding, we define the compact, linear operator
\begin{align*}
A: H_1  \subset \widetilde{H}^{s}(W) \rightarrow H_2 \subset L^2(\Omega), \
f \mapsto \iota r_{\Omega} P_q f,
\end{align*}
which by the previous considerations is a well-defined operator. With slight abuse of notation we have here identified $H_2$ with $L^2(\Omega)/Z_2$ (in particular, with this identification, we do not use the notation of equivalence classes any longer but simply write $u$ instead of $[u]$). We remark that by the disjointness of the sets $\Omega, W$ the operator $A$ can also be written as $A f = \iota r_{\Omega} (P_qf -f)$. Its Banach space adjoint $A^t$ can be computed to be $A^t v = (-\D)^s w|_{W}$, where $v \in H_2$ and $w \in H_2$ are related through the dual equation
\begin{align}
\label{eq:dual}
\begin{split}
((-\D)^s + q) w & = v \mbox{ in } \Omega,\\
w & = 0 \mbox{ in } \Omega_e.
\end{split}
\end{align}
We remark that the dual problem is solvable, since $v \in H_2$. In order to have well-posedness, as in the definition of $P_q$, we have also chosen to define the solution map to $\eqref{eq:dual}$ to involve the orthogonal projection onto $H_2$, whence $w\in H_2$. We will always use this implicitly in the sequel.

Hence, by general functional analysis the Hilbert space adjoint of $A$ is given by
\begin{align*}
A^{\ast}: H_2 \rightarrow H_1 ,\
v \mapsto R_{s}^{-1}(-\D)^s w|_{W},
\end{align*}
where $R_s$ denotes the Riesz isomorphism from $\widetilde{H}^{s}(W)$ to $H^{-s}(W)$.

Due to the presence of zero as an eigenvalue to \eqref{eq:frac_Cal}, we can no longer define the Dirichlet-to-Neumann operator as in \eqref{eq:DtN}. Instead we consider the \emph{Cauchy data} set associated with the problem \eqref{eq:frac_Cal}. For the potentials $q_i \in L^{\infty}(\Omega)$ with $i\in\{1,2\}$ and a set $W \subset \Omega_e$ this amounts to considering the sets
\begin{align*}
\mathcal{C}_i
& := \{(f,g) \in \widetilde{H}^{s}(W)\times H^{-s}(W): f=u_i|_{W}, \ g= (-\D)^s u_i|_{W} \\
& \quad \quad \mbox{ for some solution $u_i \in H^{s}(\R^n)$ to \eqref{eq:frac_Cal} with potential $q_i$} \}.
\end{align*}
Equipped with the $\widetilde{H}^{s}(W) \times H^{-s}(W)$ norm, which for convenience we abbreviate by a subindex $H$, the Cauchy data form a closed subspace of the Hilbert space $\widetilde{H}^{s}(W) \times H^{-s}(W)$. We define the distance of the two Cauchy data sets to be
\begin{align*}
d(\mathcal{C}_1, \mathcal{C}_2)
= \max\left\{ \sup\limits_{h\in \mathcal{C}_2, h \neq 0} \inf\limits_{k \in \mathcal{C}_1} \frac{\|h-k\|_{H}}{\|h\|_{H}},  \sup\limits_{h\in \mathcal{C}_1, h \neq 0} \inf\limits_{k \in \mathcal{C}_2} \frac{\|h-k\|_{H}}{\|h\|_{H}} \right\} .
\end{align*}
We recall that if zero is not an eigenvalue of the fractional Schrödinger operator, this distance is comparable to the norm of the difference of the Dirichlet-to-Neumann maps $\Lambda_{q_1}$, $\Lambda_{q_2}$.

In presenting the analogue of the results of Section \ref{sec:non_zero} in the outlined more general set-up in which zero is allowed to be an eigenvalue of \eqref{eq:frac_Cal}, we begin by  formulating the full data analogue of Theorem \ref{thm:Lip_frac_Runge}.

\begin{thm}
\label{thm:Lip_frac_Runge_eigen}
Let $n,m \in \N$ and $s\in(0,1)$.
Let $\Omega \subset \R^n$ be an open, bounded, smooth domain. Suppose that $W_1, W_2 \subset \Omega_e$ are Lipschitz with $\overline{W_j} \cap \overline{\Omega} = \emptyset$ for $j\in \{1,2\}$.
Let $q_1,q_2 \in L^{\infty}(\Omega)$ with
\begin{align*}
q_1, q_2 \in \spa\{g_1,\dots,g_m\} \mbox{ and } \|q_1\|_{L^{\infty}(\Omega)}, \|q_{2}\|_{L^{\infty}(\Omega)} \leq C_0 <\infty,
\end{align*}
for some orthonormal (w.r.t. the $L^2(\Omega)$ scalar product) functions $g_1,\dots, g_m \in L^{\infty}(\Omega)$.
Then there exists a constant $C_2>0$, which depends only on the geometry of the domains $W_1, W_2$, on the functions $g_1,\dots,g_m$ and on the constant $C_0>0$,
such that
\begin{align*}
\|q_1-q_2\|_{L^{\infty}(\Omega)} \leq C_2 d(\mathcal{C}_1, \mathcal{C}_2).
\end{align*}
\end{thm}

\begin{rmk}
\label{rmk:dependences_3}
As in the case of Theorem \ref{thm:Lip_frac_Runge}, the dependence of $C_2$ on $N$ can be made quite explicit. It is essentially determined by the geometry of the domains $W_1, W_2$, by the constant $C_0>0$ and by the choice of the functions $g_1,\dots,g_m$.
\end{rmk}

Similarly, as in Theorem \ref{thm:Lip_frac_Runge}, it is also possible to formulate a \emph{finite} measurement result for the setting in which zero is allowed to be an eigenvalue of the fractional Schrödinger equation \eqref{eq:frac_Cal}. In order to formulate this, we introduce a suitable \emph{finite measurement Cauchy data set}, which is a finite subset of the full Cauchy data set:
For given functions $p_1,\dots, p_m \in \widetilde{H}^s(W)$ and $m\in \N$ we set $p:=(p_1,\dots, p_m) \in (\widetilde{H}^s(W))^n$ and define
\begin{align*}
\mathcal{C}_i(p)
&:=\{(p_l,g) \in \{p_1,\dots,p_m\} \times H^{-s}(W) \subset \widetilde{H}^s(W) \times H^{-s}(W), \ (p_l,g)\in \mathcal{C}_i \}
 \subset \mathcal{C}_i.
\end{align*}
As in \cite{AdHGS17} we measure the distance between finite sets of Cauchy data by the following quantity:
\begin{align*}
d(\mathcal{C}_1(p), \mathcal{C}_2(p))
= \max\left\{ \sup\limits_{h\in \mathcal{C}_2(p), h \neq 0} \inf\limits_{k \in \mathcal{C}_1(p)} \frac{\|h-k\|_{H}}{\|h\|_{H}},  \sup\limits_{h\in \mathcal{C}_1(p), h \neq 0} \inf\limits_{k \in \mathcal{C}_2(p)} \frac{\|h-k\|_{H}}{\|h\|_{H}} \right\}.
\end{align*}

With this notation at hand, the finite Cauchy data result can be formulated as follows:

\begin{prop}
\label{prop:finite_Cauchy}
Under the hypotheses of Theorem \ref{thm:Lip_frac_Runge_eigen} there exist
\begin{itemize}
\item a constant $C_2>0$, which depends only on the geometry of the domains $W_1, W_2$, on the functions $g_1,\dots,g_m$ and on the constant $C_0>0$,
\item functions $f_1^{(k)},\dots, f_m^{(k)} \in \tilde{H}^s(W_k)$ with $k\in \{1,2\}$,
\end{itemize}
such that
\begin{align*}
\|q_1-q_2\|_{L^{\infty}(\Omega)} \leq C_2 d(\mathcal{C}_1(f^{(1)}), \mathcal{C}_2(f^{(2)})).
\end{align*}
Here $f^{(k)}:=(f_1^{(k)}, \dots, f_m^{(k)})$, where $k\in \{1,2\}$.
\end{prop}

In the sequel, we derive these results by slight adaptations of the arguments from Section \ref{sec:non_zero}. In particular, this necessitates derivations of analogues of the qualitative and quantitative Runge approximation property, which are derived in Sections \ref{sec:qual_Runge_zero} and \ref{sec:quant_Runge_zero}. Relying on these auxiliary results, which might be interesting on their own right, we then present the argument for Theorem \ref{thm:Lip_frac_Runge_eigen} in Section \ref{sec:Proof_zero}.

\subsection{Runge approximation}

In order to derive the stability properties claimed above, we extend the Runge approximation property from Theorem \ref{thm:Runge_frac} to the case in which the potential $q$ is allowed to have zero as a Dirichlet eigenvalue for \eqref{eq:frac_Cal}, i.e. the condition \eqref{eq:not_zero} is violated.

\subsubsection{Qualitative Runge approximation}
\label{sec:qual_Runge_zero}
Before turning to the quantitative Runge approximation property, we reprove its qualitative variant.
The main result here reads:

\begin{lem}
\label{lem:qual_Runge_zero}
Let $n\in \N$, $s\in (0,1)$.
Let $\Omega \subset \R^n$ be an open, bounded and smooth domain. Assume that $W \subset \Omega_e$ is open with $\overline{\Omega}\cap \overline{W}= \emptyset$. Let $q\in L^{\infty}(\Omega)$.
Then, the set
\begin{align*}
\mathcal{R}:=\{r_{\Omega} P_q f  + z: \ f \in H_1 , \ z \in Z_2 \}
\end{align*}
is dense in $L^2(\Omega)$.
\end{lem}

\begin{proof}
In order to show the density result, by Hahn-Banach, it suffices to prove that any function $h \in L^2(\Omega)$ with $(h,g)_{\Omega}=0$ for all $g \in \mathcal{R}$ has to vanish identically. Given such a function $h$, we may immediately assume that $(h,z)_{\Omega}=0$ for all $z\in Z_2$, as $f=0 \in H_1$. In particular this entails that the dual problem
\begin{align*}
((-\D)^s + q) w &= h \mbox{ in } \Omega, \\
w & = 0 \mbox{ in } \Omega_e,
\end{align*}
is solvable with $w\in H_2$. Hence,
\begin{align*}
0 & = (h,P_q f)_{\Omega} = (((-\D)^s + q)w, P_q f)_{\Omega}
= ((-\D)^s w|_{W}, f)_{W} \mbox{ for all } f \in H_1.
\end{align*}
As a consequence, $(-\D)^s w|_{W} \in (H_1)^a$, where $(H_1)^a$ denotes the annihilator of $H_1$. By definition of the space $H_1$ this however entails that there exist $(\lambda_j, z_j) \in \R \times Z_2$, $j\in \{1,\dots,m\}$ and $m\in \N$, such that
\begin{align*}
(-\D)^s w|_{W} = \sum\limits_{j=1}^{m} \lambda_j (-\D)^s z_j|_{W}.
\end{align*}
Since also $w=0= \sum\limits_{j=1}^{m}\lambda_j z_j$ on $W$, by the weak unique continuation property (c.f. Lemma \ref{lem:WUCP}), this entails that $w= \sum\limits_{j=1}^{m} \lambda_j z_j$, whence $w  \in Z_2 \cap H_2$. From this we however infer $w\equiv 0$ and thus $h\equiv 0$, which concludes the argument.
\end{proof}

\begin{rmk}
\label{rmk:dense}
By orthogonality of the spaces $H_2, Z_2$, Lemma \ref{lem:qual_Runge_zero} also directly implies that for $W \subset \Omega_e$ with $\overline{\Omega}\cap \overline{W}= \emptyset$ the set
\begin{align*}
\mathcal{R}:= \{r_{\Omega} P_q f: \ f\in H_1 \}
\end{align*}
is dense in $H_2$.
\end{rmk}

\subsubsection{Quantitative Runge approximation}
\label{sec:quant_Runge_zero}
Relying on the qualitative Runge approximation from above,
as the main result of this subsection, we prove the following quantitative Runge approximation property.

\begin{thm}
\label{thm:quant_Runge_frac}
Let $s\in(0,1)$ and $n\in \N$.
Let $\Omega \subset \R^n$ be an open, bounded, smooth domain. Let $W \subset \Omega_e$ be a Lipschitz domain with $\overline{\Omega}\cap \overline{W}= \emptyset$. Assume that $q\in L^{\infty}(\Omega)$. Then there exist constants $C,\mu>0$ such that for any $\overline{v} \in H^{s}_{\overline{\Omega}}$ there are $f\in H^{s}_{\overline{W}}$ and $z\in Z_2$ with
\begin{align*}
\|\overline{v}- P_{s}f - z\|_{L^2(\Omega)} \leq \epsilon \|\overline{v}\|_{H^{s}_{\overline{\Omega}}}, \  \|f\|_{H^{s}_{\overline{W}}} \leq C e^{C \epsilon^{-\mu}} \|\overline{v}\|_{L^2(\Omega)}.
\end{align*}
The constants $C,\mu$ only depend on $s,n, \Omega, W, \|q\|_{L^{\infty}(\Omega)}$.
\end{thm}

The quantitative Runge approximation property will follow from a modification of Lemma 2.3 in \cite{RS17} and the quantitative unique continuation result from \cite{RS17}. Indeed, the finite dimensional additional part, will not play any role for the cost of controlability (c.f. the arguments below, in particular Lemma \ref{lem:control}).\\

As an auxiliary step towards the proof of Theorem \ref{thm:quant_Runge_frac}, we rely on the singular value decomposition of the operators $A, A^{\ast}$.

\begin{lem}
\label{lem:zero_eig_spec}
Let $n\in \N$, $s\in (0,1)$.
Let $\Omega \subset \R^n$ be an open, bounded and smooth domain. Let $W \subset \Omega_e$ be open with $\overline{\Omega}\cap \overline{W}= \emptyset$.
Let $A:H_1 \subset \widetilde{H}^s(W)\rightarrow H_2 \subset L^2(\Omega)$ be as above. Then $A$ is a compact, injective linear operator whose image in $H_2$ is dense. Moreover, there exists a singular value decomposition with $\{\varphi_k\}_{k \in \N} \subset H_1$, $\{\psi_k\}_{k\in \N} \subset H_2$, $\{\sigma_k\}_{k\in \N} \subset \R_+$ and $\sigma_k \geq \sigma_{k+1}\geq \dots >0$. In particular,
\begin{align*}
A \varphi_k = \sigma_k \psi_k, \ A^{\ast} \psi_k = \sigma_k \varphi_k.
\end{align*}
\end{lem}

\begin{proof}
The compactness of $A$ follows from the regularity properties of the Poisson operator $P_q$, which maps $H_1$ into a subspace of $H^s(\R^n)$, and the compactness of the embedding $\iota :\widetilde{H}^{s}(\Omega) \rightarrow L^2(\Omega)$. The injectivity is a consequence of the unique continuation principle for the fractional Laplacian. The density of the image in $H_2$ follows from the qualitative Runge approximation of Lemma \ref{lem:qual_Runge_zero} and of Remark \ref{rmk:dense}.\\
Hence, by the spectral theorem for compact, self-adjoint operators, the operator $A^{\ast} A: H_1 \rightarrow H_1$ has an orthonormal eigenvalue basis $\{\varphi_k\}_{k\in \N}$ with positive, decreasing eigenvalues $\{\mu_k\}_{k\in \N}$. For $\sigma_k:= \sqrt{\mu_k}$ we set $\psi_k:= \frac{1}{\sigma_k} A \varphi_k$ and claim that this forms a complete, orthonormal system in $H_2$. As orthonormality follows from the definition, it suffices to check completeness. This however follows from the density of the span of $\{\varphi_k\}_{k\in \N}$ in $H_1$ and the unique continuation properties of the fractional Laplacian: By Hahn-Banach, it suffices to prove that if for some $v\in H_2$ it holds that
\begin{align}
\label{eq:HB}
(\psi_k, v)_{\Omega} = 0 \mbox{ for all } k \in \N,
\end{align}
then $v\equiv 0$. But as \eqref{eq:HB} is equivalent to the condition that $(\varphi_k, A^{\ast}v)_{\widetilde{H}^s(W)}=0$ for all $k\in \N$, we infer that on the one hand $A^{\ast} v \in Z_1$. By definition of $A^{\ast}$, this implies that $(\varphi_k,(-\D)^s w|_{W})_{W}=0$ for all $k \in \N$. Here $w\in H_2$ denotes the solution of the dual problem \eqref{eq:dual} with inhomogeneity $v\in H_2$.
Thus, there exist $\alpha_i \in \R$, $z_i \in Z_2$, $i\in\{1,\dots,m\}$ such that
\begin{align*}
(-\D)^s w|_{W} = \sum\limits_{i=1}^{m} \alpha_i (-\D)^s z_i|_{W}.
\end{align*}
As also $w= 0=\sum\limits_{i=1}^{m} \alpha_i  z_i$ on $W$, the (weak) unique continuation property of the fractional Laplacian (c.f. Lemma \ref{lem:WUCP}) implies that on the one hand $w\equiv \sum\limits_{i=1}^{m} \alpha_i  z_i \in Z_2$. However, on the other hand the assumption that $v\in H_2$ yields $w \in H_2$. Since thus $w \in Z_2 \cap H_2$, we arrive at the conclusion $w\equiv 0$ and hence $v \equiv 0$, which concludes the completeness proof.
\end{proof}

Based on the singular value decomposition and following the lines of the argument presented in Lemma 3.3  in \cite{RS17}, we have the following auxiliary result:

\begin{lem}
\label{lem:control}
Let $n\in \N$, $s\in (0,1)$.
Let $\Omega \subset \R^n$ be an open, bounded and smooth domain. Let $W \subset \R^n$ be open with $\overline{\Omega}\cap \overline{W}= \emptyset$. Let $w \in H^{s}_{\overline{\Omega}}\cap H_2$ be a solution of the dual equation \eqref{eq:dual} with inhomogeneity $v \in H_2$ and potential $q \in L^{\infty}(\Omega)$.
Assume that for some $C>0, \mu >0$ we have
\begin{align*}
\|v\|_{H^{-s}(\Omega)} \leq C \left(\log \left( \frac{\|v\|_{L^2(\Omega)}}{\|(-\D)^s w\|_{H^{-s}(W)}}\right) \right)^{-\mu} \|v\|_{L^2(\Omega)}.
\end{align*}
Then for any $\overline{v}\in H^{s}_{\overline{\Omega}}$ and any $\epsilon>0$ there exist $f\in H_1$ and $z \in Z_2$ such that
\begin{align*}
\|\overline{v}- P_{s}f - z\|_{L^2(\Omega)} \leq \epsilon \|\overline{v}\|_{H^{s}_{\overline{\Omega}}}, \  \|f\|_{H^{s}_{\overline{W}}} \leq C \exp(C \epsilon^{-\mu}) \|\overline{v}\|_{L^2(\overline{\Omega})}.
\end{align*}
\end{lem}

\begin{proof}
The argument essentially follows as in \cite{RS17}, since the additional finite dimensional subspace $Z_2$ can be dealt with ``by hand". More precisely, given $\overline{v}\in H^{s}_{\overline{\Omega}} \subset L^2(\Omega)$, the singular value decomposition of Lemma \ref{lem:zero_eig_spec} implies that
\begin{align*}
\overline{v}= \sum\limits_{j=1}^{\infty} \beta_j \psi_j + \sum\limits_{l=1}^m \tilde{\beta}_l z_l,
\end{align*}
where $z_l \in Z_2$.
Considering the projection $\tilde{v}=\sum\limits_{j=1}^{\infty} \beta_j \psi_j$ of $\overline{v}$ onto $H_2$ first,
we define
\begin{align*}
R_{\alpha} \tilde{v} = \sum\limits_{\sigma_j \geq \alpha} \frac{\beta_j}{\sigma_j} \varphi_j \in H_1 \subset H^{s}_{\overline{W}}.
\end{align*}
Then by orthogonality (of the functions $\varphi_j$ but also of the spaces $H_2$ and $Z_2$) we obtain that
\begin{align*}
\|R_{\alpha} \tilde{v}\|_{H^s_{\overline{W}}} \leq \frac{1}{\alpha} \|\tilde{v}\|_{L^2(\Omega)} \leq \frac{1}{\alpha} \|\overline{v}\|_{L^2(\Omega)}.
\end{align*}
Furthermore, we set $r_{\alpha}:= \sum\limits_{\sigma_j < \alpha} \beta_j \psi_j \in H_2$.
Observing that $A R_{\alpha} \tilde{v} = \sum\limits_{\sigma_j \geq \alpha} \beta_j \psi_j $, relying on Lemma \ref{lem:zero_eig_spec} and using orthogonality of $H_2$ and $Z_2$, then yields
\begin{align*}
\|A R_{\alpha}(\tilde{v})-\tilde{v}\|_{L^2(\Omega)}^2
&\leq |(\tilde{v},r_{\alpha})_{L^2(\Omega)}|
= |(\overline{v},r_{\alpha})_{L^2(\Omega)}|
\leq \|\overline{v}\|_{\widetilde{H}^{s}(\Omega)}\|r_{\alpha}\|_{H^{-s}(\Omega)}
\leq \|\overline{v}\|_{H^{s}(\R^n)}\|r_{\alpha}\|_{H^{-s}(\Omega)}\\
&\leq \|\overline{v}\|_{H^{s}_{\overline{\Omega}}}\|r_{\alpha}\|_{H^{-s}(\Omega)}
\leq \|\overline{v}\|_{H^{s}_{\overline{\Omega}}} \left(\log\left( \frac{\|r_{\alpha}\|_{L^2(\Omega)}}{\|A^{\ast} r_{\alpha}\|_{H^{s}_{\overline{W}}}} \right) \right)^{-\mu} \|r_{\alpha}\|_{L^2(\Omega)}\\
& \leq \|\overline{v}\|_{H^{s}_{\overline{\Omega}}} \left( \log \left(1/\alpha\right) \right)^{-\mu} \|r_{\alpha}\|_{L^2(\Omega)}
 \leq \|\overline{v}\|_{H^{s}_{\overline{\Omega}}} \left(\log \left(1/\alpha\right) \right)^{-\mu} \|A R_{\alpha}(\tilde{v})-\tilde{v}\|_{L^2(\Omega)}.
\end{align*}
Hence, by orthogonality we have that for $f:= R_{\alpha} \tilde{v} $
\begin{align*}
&\|R_{\alpha} \tilde{v}\|_{H^s_{\overline{W}}} \leq \frac{1}{\alpha} \|\overline{v}\|_{L^2(\Omega)}, \\
& \|A R_{\alpha}(\tilde{v})+ \sum\limits_{l=1}^{m}\tilde{\beta}_l z_l -\overline{v}\|_{L^2(\Omega)}
= \|A R_{\alpha}(\tilde{v})-\tilde{v}\|_{L^2(\Omega)}
 \leq \|\overline{v}\|_{H^{s}_{\overline{\Omega}}}\left(\log\left(1/\alpha\right)\right)^{-\mu} .
\end{align*}
Optimizing in $\alpha$ concludes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:quant_Runge_frac}]
By combining the observations from Lemma \ref{lem:control} with the unique continuation properties from \cite{RS17} (which hold as soon as one has a pair $(w,v)$ solving \eqref{eq:dual}; in particular, it does not require a spectral assumption), we in particular infer the quantitative Runge property.
\end{proof}

\subsection{Proofs of Theorem \ref{thm:Lip_frac_Runge_eigen} and of Proposition \ref{prop:finite_Cauchy}}
\label{sec:Proof_zero}

The proofs of Theorem \ref{thm:Lip_frac_Runge_eigen} and of Proposition \ref{prop:finite_Cauchy} follow similarly as the proof of Theorem \ref{thm:Lip_frac_Runge}. However, instead of using Alessandrini's identity as formulated in Lemma \ref{lem:Aless}, we rely on the following modification of it:

\begin{lem}
\label{lem:Aless_1}
Let $s\in (0,1)$, $n \geq 1$ and assume that $\Omega \subset \R^n$ is an open, bounded domain. Let further $W \subset \Omega_e$ be open.
Let $u_1, u_2 \in H^{s}(\R^n)$ be solutions to \eqref{eq:frac_Cal} with potentials $q_1, q_2 \in L^{\infty}(\Omega)$. Then for any further solution $v_2$ to \eqref{eq:frac_Cal} with potential $q_2$ we have
\begin{align}
\label{eq:Aless_1}
((q_1-q_2)u_1, u_2)_{\Omega} = ((-\D)^s u_2, u_1 - v_2)_{W} - ((-\D)^s u_1 - (-\D)^s v_2, u_2)_{W}.
\end{align}
In particular,
\begin{align}
\label{eq:Cauchy}
\left| ((q_1 - q_2)u_1, u_2)_{\Omega} \right|
\leq d(\mathcal{C}_1, \mathcal{C}_2)\|u_2\|_H \|u_1\|_H,
\end{align}
with $\|w\|_H:= \|w\|_{\widetilde{H}^s(W)} + \|(-\D)^s w\|_{H^{-s}(W)}$.
\end{lem}

\begin{proof}
The generalized Alessandrini's identity is a consequence of the same ideas as the proof of Lemma \ref{lem:Aless}. Indeed, by using the equations for $u_1, u_2$, we infer
\begin{align*}
((q_1-q_2)u_1, u_2)_{\Omega} = ((-\D)^s u_2, u_1 )_{W} - ((-\D)^s u_1, u_2)_{W}.
\end{align*}
Next we note that if $v_2$ is also a solution of the equation \eqref{eq:frac_Cal} with potential $q_2$, then
\begin{align*}
0 = ((-\D)^s u_2, v_2 )_{W} - ((-\D)^s v_2, u_2)_{W}.
\end{align*}
Subtracting the previous two identities implies the claim of \eqref{eq:Aless_1}.

The estimate \eqref{eq:Cauchy} follows from \eqref{eq:Aless_1} by the definition of the aperture between two metric spaces. More precisely, we have
\begin{align*}
|((q_1-q_2)u_1, u_2)_{\Omega}|
&\leq |((-\D)^s u_2, (u_1-u_2))_{W}| + |((-\D)^s u_1 - (-\D)^s v_2, u_2)_{W}|\\
& \leq \|(-\D)^s u_2\|_{H^{-s}(W)} \|u_1 - v_2 \|_{\widetilde{H}^s(W)} + \|u_2\|_{\widetilde{H}^s(W)}\|(-\D)^s (u_1 -v_2)\|_{H^{-s}(W)}\\
&\leq (\|(-\D)^s u_2\|_{H^{-s}(W)} + \|u_2\|_{\widetilde{H}^s(W)})\|u_1 - v_2\|_{H}.
\end{align*}
Moreover, since the above inequality holds for any solution $v_2$ to \eqref{eq:frac_Cal} with potential $q_2$ it follows that
\begin{align*}
|((q_1-q_2)u_1, u_2)_{\Omega}|\leq \left (\inf_{v_2\in \mathcal{C}_2}\frac{ \|u_1-v_2\|_{H}}{\| u_1\|_{H}} \right) \|u_1\|_H \|u_2\|_H \leq d(\mathcal{C}_1, \mathcal{C}_2) \|u_1\|_H \|u_2\|_H.
\end{align*}
\end{proof}

\begin{rmk}
\label{rmk:Cauchy_data}
If it is known that $(u_1, (-\D)^s u_1) \in \mathcal{C}_1(p^{(1)})$ and $ (u_2, (-\D)^s u_2)$, $(v_2, (-\D)^s v_2) \in \mathcal{C}_2(p^{(2)})$ for some functions $p^{(1)}, p^{(2)}$ as in the definition of the finite Cauchy data, then it is possible to replace the term $d(\mathcal{C}_1, \mathcal{C}_2)$ in \eqref{eq:Cauchy} by the quantity $d(\mathcal{C}_1(p^{(1)}), \mathcal{C}_2(p^{(2)}))$.
\end{rmk}

With Lemma \ref{lem:Aless_1} at hand, the proofs of Theorem \ref{thm:Lip_frac_Runge_eigen} and Proposition \ref{prop:finite_Cauchy} follow analogously as the proof of Theorem \ref{thm:Lip_frac_Runge}: It suffices to replace the use of the Runge approximation result from \cite{RS17}, which was recalled in Theorem \ref{thm:Runge_frac}, by the Runge approximation from Theorem \ref{thm:quant_Runge_frac}. We discuss some of the details of this. As there are no major differences in the arguments for the proofs of Proposition \ref{prop:finite_Cauchy} and for Theorem \ref{thm:Lip_frac_Runge_eigen}, we only present the details for the proof of Proposition \ref{prop:finite_Cauchy}.

\begin{proof}[Proof of Proposition \ref{prop:finite_Cauchy}]
As in the proof of Theorem \ref{thm:Lip_frac_Runge} we choose functions $h_{1}^{(k)}, \dots, h_{m}^{(k)}$, $k\in \{1,2\}$, such that the matrix
\begin{align*}
M:= \begin{pmatrix}
(g_1, h_{1}^{(1)} h_{1}^{(2)})_{\Omega} & \dots & (g_m, h_{1}^{(1)} h_{1}^{(2)})_{\Omega}\\
\vdots & \vdots & \vdots\\
(g_1, h_{m}^{(1)} h_{m}^{(2)})_{\Omega} & \dots & (g_m, h_{m}^{(1)} h_{m}^{(2)})_{\Omega}
\end{pmatrix}
\end{align*}
is invertible (by choosing $h_{l}^{(k)}$ suitably, we can even ensure that the matrix $M$ is arbitrarily close to the identity matrix). By the Runge approximation result of Theorem \ref{thm:quant_Runge_frac}, we obtain solutions $u_{l}^{(k)} \in H_2$ to the Schrödinger equation \eqref{eq:frac_Cal} and kernel elements $z_{l}^{(k)} \in Z_2$ such that
\begin{align*}
h_{l}^{(k)} = u_{l}^{(k)} + z_{l}^{(k)} + r_{l}^{(k)} \mbox{ in } \Omega, \ u_l^{(k)} = f_l^{(k)} \mbox{ in } \Omega_e,
\end{align*}
where $k\in \{1,2\}$, and
\begin{align*}
\|r_{l}^{(k)}\|_{L^2(\Omega)} \leq \epsilon \|h_{l}^{(k)}\|_{H^{s}_{\overline{\Omega}}}, \
\|u_{l}^{(k)}\|_{H^{s}_{\overline{W}}} \leq C e^{C \epsilon^{-\mu}} \|h_{l}^{(k)}\|_{L^2(\Omega)}.
\end{align*}
Relying on the estimates from Lemma \ref{lem:Aless_1}, we obtain
\begin{align}
\label{eq:eigen_bda}
\begin{split}
\left|  ((q_1 - q_2) h_l^{(1)}, h_l^{(2)}) \right|
&\leq
\left|  ((q_1 - q_2) (u_l^{(1)} + z_l^{(1)}), (u_l^{(2)} + z_l^{(2)})) \right|\\
& \quad + |((q_1 - q_2) r_j^{(1)}, h_j^{(2)})| + |((q_1 - q_2) r_j^{(2)}, h_j^{(1)})| + |((q_1 - q_2) r_j^{(1)}, r_j^{(2)})|\\
&\leq  d(\mathcal{C}_1(f^{(1)}), \mathcal{C}_2(f^{(2)})) \|u_j^{(1)} + z_j^{(1)}\|_H \|u_j^{(2)} + z_j^{(2)}\|_H\\
& \quad + |((q_1 - q_2) r_j^{(1)}, h_j^{(2)})| + |((q_1 - q_2) r_j^{(2)}, h_j^{(1)})| + |((q_1 - q_2) r_j^{(1)}, r_j^{(2)})|\\
& \leq  d(\mathcal{C}_1(f^{(1)}), \mathcal{C}_2(f^{(2)})) \|u_j^{(1)} + z_j^{(1)}\|_H \|u_j^{(2)} + z_j^{(2)}\|_H \\
& \quad + \epsilon \|q_1 - q_2\|_{L^{\infty}(\Omega)} (\|h_j^{(1)}\|_{H^s_{\overline{\Omega}}} + \|h_j^{(2)}\|_{H^s_{\overline{\Omega}}}).
\end{split}
\end{align}
Next we estimate the term $\|u_j^{(1)} + z_j^{(1)}\|_H $.
First we note that
\begin{align}
\label{eq:u1}
\begin{split}
\|u_j^{(1)} + z_j^{(1)}\|_{H}
&= \|u_j^{(1)} + z_j^{(1)}\|_{\widetilde{H}^s(W)} + \|(-\D)^s (u_j^{(1)}+z_j^{(1)})\|_{H^{-s}(W)}\\
&= \|f_j^{(1)}\|_{\widetilde{H}^s(W)} +  \|(-\D)^s (u_j^{(1)}+z_j^{(1)})\|_{H^{-s}(W)},
\end{split}
\end{align}
where we have used that $u_j^{(1)}+z_j^{(1)}= f_j^{(1)}$ in $W$. Next, we have
\begin{align}
\label{eq:u2}
\begin{split}
\|(-\D)^s (u_j^{(1)} + z_j^{(1)})\|_{H^{-s}(W)}
&\leq \|(-\D)^s (u_j^{(1)} + z_j^{(1)}-f_j^{(1)})\|_{H^{-s}(W)} + \|(-\D)^s f_j^{(1)}\|_{H^{-s}(W)} \\
&\leq \|(-\D)^s (u_j^{(1)}+ z_j^{(1)}-f_j^{(1)})\|_{H^{-s}(W)} + C \|f_j^{(1)}\|_{\widetilde{H}^{s}(W)}.
\end{split}
\end{align}
We have used $f_j^{(1)} \in H^{s}_{\overline{W}}$ in order to deal with the second contribution on the right hand side.
Since $z_j^{(1)}, u_j^{(1)}+ z_j^{(1)}-f_j^{(1)} \in H^{s}_{\overline{\Omega}}$, we infer for $x\in W$ that
\begin{align}
\label{eq:u3}
\begin{split}
|(-\D)^s (u_j^{(1)}+ z_j^{(1)}-f_j^{(1)})(x) |
&= \left| c_{s,n}\int\limits_{\Omega} \frac{(u_j^{(1)}+ z_j^{(1)}-f_j^{(1)})(y)}{|x-y|^{n+2s}} dy \right|\\
&\leq c_{n,s} |\Omega|^{1/2}(|x|+ C_{\Omega, W})^{-n-2s} \|(u_j^{(1)}+ z_j^{(1)}-f_j^{(1)})\|_{L^2(\Omega)}\\
&= c_{n,s} |\Omega|^{1/2}(|x|+ C_{\Omega, W})^{-n-2s} \|u_j^{(1)}+ z_j^{(1)}\|_{L^2(\Omega)}.
\end{split}
\end{align}
Hence, integrating this over $x\in W$ we obtain
\begin{align}
\label{eq:u4}
\|(-\D)^s (u_j^{(1)}+ z_j^{(1)}-f_j^{(1)})\|_{H^{-s}(W)} \leq
\|(-\D)^s (u_j^{(1)}+ z_j^{(1)}-f_j^{(1)})\|_{L^{2}(W)}
\leq C \|u_j^{(1)}+ z_j^{(1)}\|_{L^2(\Omega)}.
\end{align}
In order to bound the right hand side of \eqref{eq:u4}, we use the decomposition $h^{(1)}_j = u_j^{(1)} + z_j^{(1)} + r_j^{(1)}$ and the $L^2(\Omega)$ orthogonality of $u_j^{(1)}$ and $z_j^{(1)}$. We obtain
\begin{align}
\label{eq:z1}
 \|z_j^{(1)} + u_j^{(1)}\|_{L^2(\Omega)} \leq \|h_j^{(1)}\|_{L^2(\Omega)} + \|r_j^{(1)}\|_{L^2(\Omega)}
\leq 2 \|h_j^{(1)}\|_{L^2(\Omega)}.
\end{align}
Here we used that we can bound $ \|r_j^{(1)}\|_{L^2(\Omega)} \leq \epsilon \|h_j^{(1)}\|_{H^{s}_{\overline{\Omega}}} \leq \|h_j^{(1)}\|_{L^2(\Omega)}$, if $\epsilon \leq \frac{\|h_j^{(1)}\|_{L^2(\Omega)}}{\|h_j^{(1)}\|_{H^{s}_{\overline{\Omega}}}}$ (c.f. Lemma \ref{lem:control}).
Summing up the estimates \eqref{eq:u1}-\eqref{eq:z1} and using that $\|f_j^{(1)}\|_{\widetilde{H}^{s}(W)}\leq C e^{C \epsilon^{-\mu}} \|h_j^{(1)}\|_{L^2(\Omega)}$, we thus arrive at
\begin{align}
\label{eq:combi}
\|u^{(1)}_j+ z_j^{(1)}\|_H \leq (C e^{C \epsilon^{-\mu}} + 2) \|h_j^{(1)}\|_{L^2(\Omega)}).
\end{align}

Assuming that
\begin{align*}
q_1 - q_2 = \sum\limits_{j=1}^{m} a_j g_j,
\end{align*}
and using the estimate \eqref{eq:Cauchy} from Lemma \ref{lem:Aless_1} and Remark \ref{rmk:Cauchy_data} in combination with \eqref{eq:combi} and \eqref{eq:eigen_bda}, we infer
\begin{align*}
\left| \sum\limits_{j=1}^{m} a_j (g_j, h_{l}^{(1)} h_{l}^{(2)})_{\Omega} \right|
&\leq C d(\mathcal{C}_1(f^{(1)}), \mathcal{C}_2(f^{(2)})) e^{C \epsilon^{-\mu}}\|h_l^{(1)}\|_{H^s_{\overline{\Omega}}}\|h_l^{(2)}\|_{H^s_{\overline{\Omega}}}\\
& \quad + \epsilon \|q_1 - q_2\|_{L^{\infty}(\Omega)} \|h_l^{(1)}\|_{H^s_{\overline{\Omega}}}\|h_l^{(2)}\|_{H^s_{\overline{\Omega}}}.
\end{align*}
The proof of Proposition \ref{prop:finite_Cauchy} is now concluded by the same absorption argument as in the proof of Theorem \ref{thm:Lip_frac_Runge}.
\end{proof}

\section{Necessity of an Exponential Dependence in the Case of Piecewise Constant Potentials}
\label{sec:opti}

In Example \ref{ex:pc} we dealt with the setting of piecewise constant potentials associated with a partitioning of $\Omega$ into $N$ Lipschitz subdomains $D_1, \dots, D_N$. In this context we saw that by precisely keeping track of the dependences in the proof of Theorem \ref{thm:Lip_frac_Runge}, the constant $C_1>1$ from Theorem \ref{thm:Lip_frac_Runge} could be bounded by $e^{C N^{\tilde{\mu}}}$ for some $\tilde{\mu}>0$. In this section we show that, as in the case of the classical Calder\'on problem (c.f. \cite{Ron06}), (up to the precise choice of the exponent) such an exponential dependence on the constant $N>1$ is also necessary in general.
In order to observe this, we combine the strategy introduced by Rondi \cite{Ron06}, which is based on an adaptation of the instability argument of Mandache \cite{M01} with the constructions given in \cite{RS17d}.

\begin{thm}
\label{thm:exp_dep}
Let $s\in (0,1)$, let $\Omega =B_1(0)$ be the unit ball in $\R^n$ for some $n \in \N$ and let $W:= B_3(0) \setminus \overline{B_2(0)}$. Let $\lambda_1$ denote the first Dirichlet eigenvalue of $(-\D)^s$ on $\Omega$. Then there exist $N_0 \in \N$, $\mu>0$ and $\bar{C}\geq 1$ such that for all $N\geq N_0$ there is a partitioning $D_1, \dots, D_N$ of $\Omega$ into Lipschitz sets,
such that the Lipschitz constant $C_1$ from Theorem \ref{thm:Lip_frac_Runge} associated with the choice
\begin{align*}
q_1, q_2 \in \spa \{\chi_{D_1}, \dots, \chi_{D_N}\}, \ \|q_1\|_{L^{\infty}(\Omega)}, \|q_2\|_{L^{\infty}(\Omega)} \leq \frac{ \lambda_1}{2}
\end{align*}
necessarily satisfies
\begin{align*}
C_1 \geq \exp(\bar{C} N^{\mu}).
\end{align*}
\end{thm}

\begin{proof}
Let $\tilde{C}_1>1$ be the Lipschitz constant in the estimate of Theorem \ref{thm:Lip_frac_Runge} for the knowledge of the \emph{full}, infinite measurement Dirichlet-to-Neumann map with potentials $q_1, q_2$ and a set $\{g_1, \dots, g_N\}$ as in Theorem \ref{thm:exp_dep}, i.e. assume that for all $q_1, q_2 \in \spa \{\chi_{D_1}, \dots, \chi_{D_N}\}$ it holds
\begin{align}
\label{eq:full_est}
\|q_1 - q_2\|_{L^{\infty}(\Omega)} \leq \tilde{C}_1 \|\Lambda_{q_1}-\Lambda_{q_2}\|_{\ast},
\end{align}
where $\|\Lambda_{q_1}-\Lambda_{q_2}\|_{\ast}:= \sup\{((\Lambda_{q_1}-\Lambda_{q_2})f_1, f_2)_{W}: \|f_1\|_{\widetilde{H}^{s}(W)}=1=\|f_2\|_{\widetilde{H}^s(W)} \}$.
Since $\|\Lambda_{q_1}-\Lambda_{q_2}\|_{\ast} \geq \|(\Lambda_{q_1}-\Lambda_{q_2})\tilde{f}_1\|_{H^{-s}(W)}$ for $\tilde{f_1} \in \widetilde{H}^s(W)$ with $\|\tilde{f}_1\|_{\widetilde{H}^{s}(W)}=1$, we note that $C_1 \geq \tilde{C}_1$. Hence it suffices to prove the claimed lower bound for the constant $\tilde{C}_1$.

In order to prove this, we follow the strategy of Rondi \cite{Ron06}. Thus, on the one hand we construct an exponentially large $\delta$-discrete set $X \subset \mathcal{Q}_N$, where
\begin{align*}
\mathcal{Q}_N:=\left\{q\in L^{\infty}(\Omega): \ q \mbox{ is piecewise constant}, \ \|q\|_{L^{\infty}(\Omega)}\leq \frac{\lambda_1}{2}  \right\}, \ d_{\mathcal{Q}_N}(q_1, q_2)= \|q_1-q_2\|_{L^{\infty}(\Omega)}.
\end{align*}
We recall that a set is said to be $\delta$-discrete, if all its elements are at a mutual distance larger or equal to $\delta$.
On the other hand, we show that the set of Dirichlet-to-Neumann maps
\begin{align*}
\mathcal{L}:=\left\{\Lambda_{q}: q \in \mathcal{Q}_N, \ \|q\|_{L^{\infty}(\Omega)} \leq \frac{\lambda_1}{2} \right\}
\end{align*}
equipped with the operator norm $d_{\mathcal{L}}(\Lambda_{q_1},\Lambda_{q_2}):= \|\Lambda_{q_1}-\Lambda_{q_2}\|_{\widetilde{H}^{s}(W) \rightarrow H^{-s}(W)}$ has a relatively small $\epsilon$-net, i.e. there exists a subset of the set of $\mathcal{L}$ such that all points in $\mathcal{L}$ are at most $\epsilon$ far from this subset. Combining these two observations with the estimate from Theorem \ref{thm:Lip_frac_Runge} then yields the claimed lower bound on the constant $\tilde{C}_1>0$ (and hence also on the constant $C_1>0$ from Theorem \ref{thm:Lip_frac_Runge}).

Let us be more precise. We first cover $\Omega$ by (up to null-sets) disjoint cubes of side length $N^{-1}$ and a remainder (e.g. by covering $\Omega$ by a suitable coordinate grid of the desired lengths scales). We consider the first $N$ of these cubes. This allows us to construct a large $\delta$-discrete set on $\Omega$ for an arbitrary choice of $\delta\in (0,\min\{\lambda_1/2,1\})$: Indeed, by considering the piecewise constant functions, which on the chosen $N$ cubes attain any of the values $\pm \delta, 0$ and vanish outside of the $N$ cubes, we obtain a $\delta$-discrete set consisting of $3^N$ elements.

Using the strong smoothing properties of the Dirichlet-to-Neumann map for the fractional Laplacian, which were studied in \cite{RS17d}, it is possible to construct a small $\epsilon$-net for $\mathcal{L}$. Indeed, an orthonormal basis with smoothing properties in our geometry was constructed in Lemma 2.1 in \cite{RS17d}. Based on that, for any $\epsilon \in (0,1)$ an $\epsilon$-net of $\mathcal{L}$ with respect to the distance $
\tilde{d}_{\mathcal{L}}(\Lambda_{q_1},\Lambda_{q_2}):= \|
\Lambda_{q_1}-\Lambda_{q_2}\|_{L^2(W)\rightarrow L^2(W)}$ was
constructed in Lemma 3.2 in \cite{RS17d}. Its cardinality was estimated by $
\exp(\eta(-\log(\epsilon))^{2n-1})$.

We seek to exploit these two observations. To this end, we first relate the $L^2$ and $H^{s}$ based operator norms of the fractional Dirichlet-to-Neumann maps. We infer that
\begin{align*}
\sup\limits_{\|f_1\|_{\widetilde{H}^{s}(W)}=1=\|f_2\|_{\widetilde{H}^s(W)}} \int\limits_{W}((\Lambda_{q_1}-\Lambda_{q_2}) f_1, f_2) dx
&\leq \|\Lambda_{q_1}-\Lambda_{q_2}\|_{L^2(W) \rightarrow L^2(W)} \|f_1\|_{L^2(W)}\|f_2\|_{L^2(W)}\\
&\leq C_p\|\Lambda_{q_1}-\Lambda_{q_2}\|_{L^2(W)\rightarrow L^2(W)} ,
\end{align*}
where in the last line we used Poincar\'e's inequality $\|f_1\|_{L^2(W)} \leq C_p \|f_1\|_{\widetilde{H}^s(W)}$. Hence, in particular, $d_{\mathcal{L}}(f_1, f_2) \leq C_p \tilde{d}_{\mathcal{L}}(f_1,f_2)$.

Next we observe that for each $\epsilon>0$ there exists $N_0=N_0(\epsilon)$ such that for all $N\geq N_0>0$ we have $3^N \geq \exp(\eta(-\log(\epsilon))^{2n-1})$, where $\eta>0$ is a constant which is independent of $\epsilon$.

We now combine all these auxiliary results: For each $\delta \in (0, \min\{\lambda_1/2,1\})$ and $\epsilon \in (0,1) $, there exist $q_1, q_2 \in \mathcal{Q}_N$ which are $\delta$-separated with respect to $d_{\mathcal{Q}_N}(\cdot, \cdot)$ and such that $\Lambda_{q_1}, \Lambda_{q_2}$ are at distance with respect to $\tilde{d}_{\mathcal{L}}(\cdot, \cdot)$ at most $2\epsilon$ apart, such that for some constant $C>0$ (depending on $\eta$, but independent of $N$)
\begin{align}
\label{eq:final}
\delta \leq d_{\mathcal{Q}_N}(q_1,q_2)
\leq C_1 d_{\mathcal{L}}(\Lambda_{q_1},\Lambda_{q_2})
\leq C_1 C_p \tilde{d}_{\mathcal{L}}(\Lambda_{q_1},\Lambda_{q_2})
\leq 2 C_1 C_p \epsilon
\leq 2 C_1 \exp(-C N^{1/(2n-1)}).
\end{align}
Here we have first made use of the fact that the functions $q_1, q_2$ are $\delta$-separated, then we used our Lipschitz bound, the estimate $d_{\mathcal{L}} \leq C_p \tilde{d}_{\mathcal{L}}$ and finally our choice of $N$ and $\epsilon$.
Rearranging \eqref{eq:final}, choosing for instance $\delta=\min\{\lambda_1/4, 1/2\}$ and noting that $C_p>1$ does not depend on $N$ then implies the claim.
\end{proof}

\section{\normalsize{Acknowledgments}}
The research carried out by  E. Sincich for the preparation of this paper has been supported by FRA 2016 {\it{Problemi inversi, dalla stabilit\`{a} alla ricostruzione}} funded by Universit\`{a} degli Studi di Trieste. E.S  acknowledges the support of Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`{a} e le loro Applicazioni (GNAMPA) by the grant {\it{Analisi di problemi inversi:  stabilit\`{a} e
ricostruzione} }.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


Fifteen years ago, choreographer Anna Halprin opened the door to her mountainside home in Kentfield, buck naked. "Just a moment!" she said impishly to a startled reporter, and slipped on a kimono.
This time around, on a recent weekday, the 94-year-old Halprin is fully dressed in a turquoise tunic and silver necklace, but her uninhibited way of being is still obvious. Relaxed from a morning regimen of lap-swimming and hula hoops, she takes a seat overlooking the famous "dance deck" at her Marin County home, where in the '50s and '60s she essentially invented postmodern dance.
Of course, she quickly qualifies, a lot happens in that 10 seconds, a transformation of awareness that had profound implications for 20th century art. Before Halprin, American dance was cast in Graham's regal mold, presented formally onstage, and performed by highly trained bodies that acted out the choreographer's vision in a rarefied movement language. Halprin's rebellion was to declare that any movement, performed with presence and intention, could be a dance, and anybody could be a dancer.
"She really was the genesis of so much postmodern dance," says her biographer, Stanford Professor Janice Ross — though for many decades, the dance establishment in New York, a powerful world that tends to cast developments outside its sphere as "provincial," discounted her place in dance history.
That has changed. This summer, to mark Halprin's 95th birthday, fans and followers will host hundreds of events in 15 countries. Documentaries on Halprin will screen in Colombia and South Korea, and her works will be restaged in France and Israel. People in 46 nations, and in the Bay Area on Mount Tamalpais, will participate in her Planetary Dance on June 7. Tamalpa Institute, the training program based in Halprin's home studio, will hold a benefit tribute to its founder on July 12, the day before her birthday.
Even the New York establishment that has so long ignored Halprin is paying its respects. Last fall, avant-garde choreographer Stephen Petronio announced that his "Bloodlines" project, which traces the lineage of 20th century experimental dance, will include works by textbook luminaries Trisha Brown and Merce Cunningham — and Anna Halprin.
It's a remarkable resurgence for an artist who, raised by Russian immigrant Jews in the Chicago suburbs of Wilmette and Winnetka, made a life of art secluded in the woods with her husband, the influential landscape architect Lawrence Halprin.
Choreographer Robert Moses paces like a boxer preparing for the ring, breaking into spurts of movement that twitch and flow and transform a stocky 52-year-old body into an instrument of anxious grace.
"Get your rhythms in your weight," he calls sternly to his two dozen students after he demonstrates the phrase, a whirlwind of rippling torso and whiplash legs. "Questions? Going once, going twice? Fine, sold." A mordant smile cracks above his silver whiskers.
The dancers look relieved. Moses is a firm but reliable father figure to his troupe, Robert Moses' Kin, which will present its 20th anniversary season Thursday through next Sunday at San Francisco's Yerba Buena Center for the Arts Forum. He's also a pillar of Bay Area dance who's beginning to break out on the national scene with his work, a hybrid of ballet, modern and street-jive that vacillates between tension and tenderness.
His work reflects a richly observant and conflicted mind, forged in a North Philadelphia ghetto and impervious to compromise. Moses rarely discusses his life story. But when a visitor to his small office in the Flood Building asks, like his dances, it all comes pouring out.
A whole tale of unexpected survival through dance, as it happens.
On a recent Friday, ballet dancer Robert Dekkers sipped a cappuccino in the Hayes Valley sunshine, his left foot propped up on a bench, a metal crutch leaning against the table. He knew he was supposed to talk about the reason for the crutch, but he laughed shyly. "So many people have gone through so much worse," he said.
Dressed in a motorcycle jacket and boots, Dekkers had just taught one dance class and was about to hobble off to another before returning to the studio to work on "Do Be: Double Happiness," premiering this weekend at Diablo Ballet.
"Double Happiness" is part of an ambitious five-part suite created in collaboration with guitar-percussion duo the Living Earth Show, a yearlong project that has commissioned five composers, including recent Pulitzer nominee Christopher Cerrone, who wrote the music for this latest section. The full "Do Be" will premiere at Z Space in November, danced by the experimental company Dekkers founded, Post: Ballet. The dance meditates on Western civilization's frenzy for "doing" rather than "being"— a tension that Dekkers feels acutely right now.
In March, after three days of pain that he assumed was caused by a pulled muscle, Dekkers was rushed to the emergency room at California Pacific Medical Center and admitted to the intensive care unit. He had massive internal bleeding in his abdomen, and required a blood transfusion. The trigger for the bleeding remained mysterious, but the reason for its seriousness did not. Dekkers, now 30, was born with aortic stenosis; his mechanical aortic valve, implanted in 2004, requires him to take blood thinners.
Tina LeBlanc is getting teary, but not because of her impending farewell to the ballet stage.
"I was totally crushed," LeBlanc says in a quiet room at the San Francisco Ballet Building, remembering the day she auditioned for the summer training program at American Ballet Theatre in New York. She had made it into the school the summer before; even though she was just 15, she knew her dancing was strong.
Brenda Way is not the kind of woman you'd think of as flitting, but that's what she's doing this gray morning in the kitchen of her Oakland home. She twirls to put on the teakettle and reaches for sugar on a high shelf with an agility that belies a recent hip operation. She takes a seat at the table almost giddily, eager to share her reactions to Trisha Brown's latest dance at Cal Performances. But when the conversation turns to her own work, her blue eyes become serious and her makeup-free face assumes its usual expression of formidable thoughtfulness.
Way, 64, who founded the company now known as ODC/Dance 36 years ago, has been uncannily prescient in what she's asked her audiences to consider. In 2000, her "Crash" evoked the irrational exuberance that preceded 1929′s Black Tuesday — and the dot-com stock market faltered soon after. But Way's most arresting moment of topicality came in 2004, when her "On a Train Heading South" adorned the stage with hanging blocks of slowly melting ice — two years before Al Gore made us all acknowledge a certain inconvenient truth.
Normally, after such a socially charged piece, Way would retreat to pure movement invention, but last year she pressed onward with "Time Remaining," an allegory about religious extremism. Now she's unveiling what she conceives as the final installation of a trilogy. "A Pleasant Looking Woman in Sensible Clothes," premiering during ODC's annual home season this week, uses video by the Japanese artist Hiraki Sawa. In the early frames, a toy plane flies around a house. Soon more join it to form a horde.
The title comes from a New York Times story on Supreme Court confirmation hearings.
If Way takes the Times' phrase so personally, that might be because it evokes aspects of her. Way, who had two children before age 20, has always been domestic. And like a good wife and mother, she has often stood quietly in the background of great accomplishments — not only her children's but also her dance company's.


Creates supergene alignments given a set of bootstrapped gene trees. Weighted Statistical Binning is a simple modification to statistical binning, obtained by replicating each supergene tree by the number of genes in its bin. It uses a Condor cluster for running jobs on distributed systems in parallel. It improves species tree estimation using phylogenomic pipelines based on coalescent-based summary methods.
Weighted Statistical Binning: Enabling Statistically Consistent Genome-Scale Phylogenetic Analyses.
Supported by National Science Foundation (NSF) grant numbers 1461364, 0733029, and 1062335; and by Howard Hughes Medical Institutes (HHMI) graduate international student fellowship.


In the car world, M is a hallowed letter, thanks to BMW's M performance division and the many great cars that have worn M badges through the years. But now Volvo is launching a new "mobility brand" focusing on alternatives to car ownership, and it will also be called M. Does BMW know about this?
"Volvo Cars is becoming more than just a car company," CEO Håkan Samuelsson said in a statement. "We recognize that urban consumers are rethinking traditional car ownership. M is part of our answer." The new brand will debut in Sweden and the United States next year.
Volvo didn't offer many details on the services M will offer, but a company press release did say that M will provide "dependable, on-demand access to cars and services through an intuitive app." So it sounds like M will focus on car sharing, similar to General Motors' Maven mobility brand. BMW and Daimler also offer car-sharing services, branded ReachNow and Car2Go, respectively.
M will draw experience from a Volvo-operated car-sharing service in Sweden called Sunfleet, the company said. Sunfleet currently has a fleet of 1,700 cars and 500,000 annual transactions, according to Volvo. Sunfleet will be integrated with M in 2019, with all memberships incorporated into the new brand.
Volvo will also try to distinguish its car-sharing service from competitors by focusing on how people use their cars. The automaker said it is developing "proprietary technology" to track users' specific needs and preferences, and try to customize the experience based on that. It's unclear how this will work in practice, however.
Stockholm will be M's base of operations. Volvo plans to conduct a large-scale beta test of M services there this fall, ahead of the full launch next year. The automaker said more details about M and its services will be released in the coming months.
Volvo is one of several automakers trying to diversify beyond merely selling cars. The rise of car sharing and other mobility services presents a threat to new-car sales, but also an opportunity if car companies can successfully operate the services themselves. Volvo is also betting that subscription services, where customers pay a monthly fee much like they do for cell phones, will be the next big thing. These trends point to a future where relatively few people actually own the cars they are driving.


Uncrate is showing us off again! www.uncrate.com is probably the coolest site out there as far as gadgets go, and we worked with them on a feature of the Kombo Fish Tool. Check it out!
Hey Joshua. Send me the link to the videos. I'll definitely check them out! If we called it a skinning knife, it sometimes confuses people. With the word 'fillet', everyone associates that with fishing. But more knowledge is always welcomed! Cheers.
Hi Kombo, I wanted to share something with you I think you should see. The fillet knife you are selling is NOT a true fillet knife. THe Kombo is a skinning knife. I only know because I came across a series of cleaning videos by a fisherman from Savannah, GA. Please look up his videos and look at what a real fillet knife is. Captain Vince Russo is his name. He has been cleaning fish for over 50 years! If you watch his videos you will see.


In continuation with our post on http://aspoonfullofideas.com/blog/seven-tips-for-first-time-online-shoppers where we had mentioned tips for first time online shoppers, please read below for a review on Jabong.
Here is a short and crisp account of my experience with them.
Trendy, funky and cool stuff, Brand and budget friendly.
Clothes, Fragrances, Bags, Jewellery, Watches, Beauty essentials and Accessories.
What I would shop on Jabong personally and why?
Women's clothing had a lot of choice. (http://www.jabong.com/women/clothing/) Most of the brands are known to me so, I know my size to order. Some of them offer great discounts. For example, BIBA kurtis were at 40% discount.
Women's dresses were also trendy and right in my budget.
Shoes are something I always like to try on but, sometimes, a brand agrees with us and we tend to buy this same brand even if we go to shops. I am seriously tempted to buy some flats as there is so much choice.
Using the coupon, I bought sandals for my daughter from the brand Kittens as shown in the pic above. Aruja has used this brand before and I knew her size already. Some of these kids shoes were at 30% discount.
Sports shoes and accessories was something of a new category for me. I was happy to see this category with good brands. For eg. if you are a badminton player, you can buy a racket, shoes, shuttlecocks and a badminton bag all at Jabong.
A secure personal login account to keep track of your orders, to redeem your money in case of a return via Jabong credits, to track your orders and for easy transactions with your online shopping carts.
Regular payment options credit cards, Debit cards, COD with a convenience charge & net banking.
Good shipping and tracking procedure for your deliveries.
A good customer care team which answered the few queries I had promptly and POLITELY!
What is excellent is the way each of these is explained on the Jabong website. There were clear instructions in the form of a question and answer. Go to "our policies" on their website to know more.
There are three categories – shop by category, shop by collection and shop by brand. Shop by collection also has further options of below a certain price level or discount level. This helps if we already know our budget.
On the main home page, you can also directly go to SALE to see the latest offers.
When you browse through, there is an option on each image or selection, "save for later". This really helps as you can just go on adding everything you like and then shortlist according to your final requirement.
They are offering vouchers to join their newsletters. Check out below. Think of joining?? Also, they offer good offers from time to time.
It does have kid's products but, the offerings in the toy section needs to extend much beyond to really offer a choice or more options. Maybe they are not looking to make it a mom-baby site?
It should have more small value items. These can be added as an accessory or add-on. For example, If I am buying my girl a frock, I would love to add-on some hair clips or trendy necklaces which were not too expensive or matching socks with the shoes.
I was informed by the customer care team that I could not only shop for one item to qualify for the free shipping. I needed to add one more item. I am still not clear on what is their basis for free shipping. Is it minimum value like other websites or is it the number of products?
When the products, the delivery and the communication is great, who would not love to shop again. At every step of the order, I received communication on the status of my order.
The real test in most cases is when there arises a problem – either with the product or the delivery or the support team's communication.
I have faith that Jabong would pass this test and thus add more customers, more brands and more products to make our online shopping experiences much better.
There is ALWAYS scope for improvement. Good luck to Jabong team.


Home > Conference Reports > Warehousing Education and Research Council (WERC) 2009 Conference > How much do unmotivated employees cost you?
For years, business has tried to quantify the hard-dollar benefits of a motivated workforce and the liabilities of a workforce that's not engaged. At WERC, two speakers—Elijah Ray of UTi Logistics and Brian Childs of Kohler Co.—shed some light on the issue.
Ray and Childs cited statistics from The Gallup Group showing that organizations with four or more engaged employees for every one disengaged worker produced 2.6 times more earnings-per-share growth than companies with a one-to-one ratio. Childs, however, is not convinced of the accuracy of those findings. Based on his experience at Kohler, Childs thinks that the improvement in earnings-per-share growth could actually be much higher than industry data suggests.
One of the results of unmotivated employees is poor quality of work. The speakers noted that the costs of "poor quality" could account for between 15 and 40 percent of a company's total business costs. While this data is not specific to the supply chain, Ray said it is on par with what he has seen in the industry.


Prof. Isa Odidi, the erudite pharmacist, celebrated scientist, distinguished innovator, first-rate technocrat and consummate entrepreneur, whose foremost passion is bridging the worlds of design, science, technology, innovation and entrepreneurship, with a view to maximising the outcome for economic growth.
We say Congratulations to him as he just setup a world class pharmaceuticals manufacturing plant in China, the first by a Nigerian and African.
Congratulations Dr. Isa Odidi, who just setup a world class pharmaceuticals manufacturing plant in China, the first by a Nigerian and African.


text:
Women are the secret ingredient in Latin America's outsized returns
Claire Diaz-Ortiz	12 months
Claire Diaz-Ortiz Contributor
Claire Diaz-Ortiz is an angel investor and bestselling author of nine books that have been published in more than a dozen countries. An early employee at Twitter, she was called "The Woman Who Got the Pope on Twitter" by Wired and holds an MBA and other degrees from Stanford and Oxford.
The rise of the winged pink unicorn
Latin America takes the global lead in VC directed to female co-founders
Read the first part of this article, Latin America takes the global lead in VC directed to female co-founders, on Extra Crunch.
Latin America lags when it comes to female investors
Interestingly, these positive numbers for Latin America come in spite of a lack of female investing partners in Latin America.
New data shows that only 7% of VCs with check-writing ability in Latin America are women. This is just over half the current U.S. average of 12%.
See the full data set here.
This is a critical finding, given that studies indicate that investments in female founded or co-founded teams increase when more women sit across the table as investors.1 Specifically, 2019 research shows that women invest in female entrepreneurs at nearly three times the rate of male investors.2
This means that Latin America could be on the early side of a positive cycle. In other words, more female investors in Latin America would lead to more female co-founders, and faster exits at higher valuations.
Other parts of the world have seen the effects of this positive cycle. Sophia Bendz, a partner at leading European VC Atomico, has watched this happen: "I've seen first-hand the impact having female investment partners can have on increasing the amount of investment into female-led companies."

redpajama_set_name: RedPajamaCommonCrawl