\documentclass[reqno]{amsart} \usepackage{amssymb,amsmath} \usepackage{hyperref} \usepackage{graphicx} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 79, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/79\hfil Three species Gross--Pitaevskii systems] {Spatial patterns for the three species Gross--Pitaevskii system in the plane} \author[M. Caliari, M. Squassina\hfil EJDE-2008/79\hfilneg] {Marco Caliari, Marco Squassina} % in alphabetical order \address{Marco Caliari \newline Dipartimento di Informatica \\ Universit\`a di Verona \newline C\`a Vignal 2, Strada Le Grazie 15, 37134 Verona, Italy} \email{marco.caliari@univr.it} \address{Marco Squassina \newline Dipartimento di Informatica \\ Universit\`a di Verona \newline C\`a Vignal 2, Strada Le Grazie 15, 37134 Verona, Italy} \email{marco.squassina@univr.it} \thanks{Submitted March 14, 2008. Published May 28, 2008.} \subjclass[2000]{35B40, 35Q55, 81V05, 81V45} \keywords{Mixtures of Bose-Einstein condensates; spatial segregation; \hfill\break\indent Gross-Pitaevskii vector equation; ground state solutions, Thomas-Fermi approximation} \begin{abstract} In this paper we highlight some particular spatial patterns of ground state solutions for the three species Gross--Pitaevskii system in the plane having physical coefficients with particular attention to the cases where the inter-species coefficients become large. The solutions models least energy stationary states of a mixture of three Bose--Einstein condensates. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Although Bose--Einstein condensates were predicted by Einstein \cite{einstein} around 1925, their successful experimental realization for atomic gases was firstly achieved in 1995, see \cite{BecR}. Next, in 1997, the condensation for a mixture of two interacting species with the same mass was realized, see \cite{BecSR}. Finally, around 2003, triplet species states were observed in \cite{triplet}. In two recent papers \cite{CORT08,CalSqu} we investigated the numerical approximation (via spectral methods) and the large interaction patterns (via variational arguments) of ground state solutions for a class of vector Gross--Pitaevskii equations in $\mathbb{R}^2$ modelling a binary mixture of Bose--Einstein condensates \cite{Bigpap,PiTaBook}. As known, depending upon the anisotropy of the trapping potentials, there are various situations where the full physical model in $\mathbb{R}^3$ can be reduced, with a good approximation, to the planar case (see \cite[Section 2.2]{Baobab}), which, therefore, is physically meaningful. The aim of this paper is to complete the work initiated in \cite{CalSqu}, by showing some particular spatial patterns, in the large interaction regime, for the ground state solutions of the three species 2D Gross--Pitaevskii system \begin{gather*} \hbar i\partial_t\psi_1=-\frac{\hbar^2}{2m_1}\Delta \psi_1 +V_1(x_1,x_2)\psi_1+\theta_{11}\hbar^2|\psi_1|^2\psi_1 +\sum_{j\neq 1}^3\theta_{1j}\hbar^2|\psi_j|^2\psi_1, \\ \hbar i\partial_t\psi_2=-\frac{\hbar^2}{2m_2}\Delta \psi_2 +V_2(x_1,x_2)\psi_2+\theta_{22}\hbar^2|\psi_2|^2\psi_2 +\sum_{j\neq 2}^3\theta_{2j}\hbar^2|\psi_j|^2\psi_2, \\ \hbar i\partial_t\psi_3=-\frac{\hbar^2}{2m_3}\Delta \psi_3 +V_3(x_1,x_2)\psi_3+\theta_{33}\hbar^2|\psi_3|^2\psi_3 +\sum_{j\neq 3}^3\theta_{3j}\hbar^2|\psi_j|^2\psi_3, \end{gather*} for the unknown $\psi_i:\mathbb{R}^2\to\mathbb{C}$, where $\hbar$ denotes the reduced Planck constant and $m_i$ are the masses of the atomic species composing the Bose--Einstein triple mixture. The coefficients of the coupling matrix $(\theta_{ij})$, which is symmetric so as to give the system a variational structure, are positive and play the role of intra-species $(\theta_{ii})$ and inter-species $(\theta_{i\neq j})$ coefficients respectively and can be represented as $$ \theta_{ij}=2\pi\frac{\sigma_{ij}}{m_{ij}},\quad \frac{1}{m_{ij}}=\frac{1}{m_i}+\frac{1}{m_j},\quad \sigma_{ij}=\sigma_{ji}, \quad i,j=1,2,3, $$ where the constants $\sigma_{ij}$ are related to the scattering lengths for the $i$-$j$ species, depending on the interaction potential between atoms. We point out that, due to Feshbach resonance, the interspecies scattering lengths can be made large, by applying a suitable external magnetic field (see \cite{feshbach}). Concerning the potential functions, we consider the general harmonic off-centered case, that is there are three centers $(x_1^i,x_2^i)$ and six positive constants $\omega_{ix},\omega_{iy}$, $i=1,2,3$, such that $$ V_i(x_1,x_2)={\tfrac{m_i}{2}}\big(\omega_{ix}^2(x_1-x_1^i)^2 +\omega_{iy}^2(x_2-x_2^i)^2\big). $$ The potential $V_i$s are often taken with the same centers, typically, without loss of generality, the origin. On the other hand, there are some relevant physical situations which lead to consider the off-centered case (see e.g.\ \cite{ribolimod}). We will prove that, when one of the inter-species coefficients, say $\theta_{i_0j_0}$, becomes very large, then phase separation behaviour between the wave densities $\psi_{i_0}$ and $\psi_{j_0}$ tends to appear. We shall highlight analytically (see Proposition \ref{Gsspatseg}) and numerically (see Figures \ref{as1}, \ref{as2}, \ref{as3}, \ref{as4}, \ref{as5} within Section \ref{NumSim}) the spatial segregation of the solution components $\phi_i$ of the ground state. In general, this phenomenon can appear by two possibly coexisting causes, that is the separation of the trapping potential centers on one side (see Section \ref{TFapp} and Figure~\ref{as2}) and the large interaction regime on the other (see Section \ref{SIr} and Figure~\ref{as3}), the second one persisting also in absence of external potentials. \section{Functional setting} Let $\mathcal{H}$ be the Hilbert subspace of $H^1(\mathbb{R}^2,\mathbb{C}^3)$ defined by $$ \mathcal{H}=\Big\{(\psi_1,\psi_2,\psi_3)\in H^1(\mathbb{R}^2,\mathbb{C}^3): \int_{\mathbb{R}^2} V_i(x_1,x_2)|\psi_i|^2<\infty,\; i=1,2,3\Big\}, $$ which is the natural framework for bound state solutions, endowed with the norm $$ \|(\psi_1,\psi_2,\psi_3)\|_{\mathcal{H}}^2=\sum_{i=1}^3\int_{\mathbb{R}^2} \frac{\hbar^2}{2m_i}|\nabla \psi_i|^2+V_i(x_1,x_2)|\psi_i|^2, $$ and consider the total energy associated with the system, given by the Hamiltonian $E=E_\infty+J$, where $E_\infty,J:\mathcal{H}\to\mathbb{R}$ are defined by \begin{gather*} E_\infty(\psi_1,\psi_2,\psi_3) =\sum_{i=1}^3 E_\infty^i(\psi_i), \\ J(\psi_1,\psi_2,\psi_3) =\sum_{i\neq j}^3 J^{ij}(\psi_i,\psi_j), \end{gather*} being, for any $i,j=1,2,3$, \begin{gather*} E_\infty^i(\psi_i) =\int_{\mathbb{R}^2} \frac{\hbar^2}{2m_i}|\nabla\psi_i|^2+ V_i(x_1,x_2)|\psi_i|^2+\frac{\theta_{ii}\hbar^2}{2}|\psi_i|^4, \\ J^{ij}(\psi_i,\psi_j) =\theta_{ij}\hbar^2\int_{\mathbb{R}^2} |\psi_i|^2|\psi_j|^2. \end{gather*} We assume that $\theta_{ij}>0$ for any $i,j$. It is standard to see that, along a solution, the energy map $$ \{t\mapsto E(\psi_1(\cdot,t),\psi_2(\cdot,t),\psi_3(\cdot,t))\},\quad t\geq 0 $$ is a constant and that the total particle numbers are time independent, \begin{equation} \label{totmasses} \int_{\mathbb{R}^2} |\psi_i(\cdot,t)|^2=N_i,\quad t\geq 0,\; i=1,2,3. \end{equation} The \emph{ground state} solution (also called \emph{least energy} solution) of the Gross--Pitaevskii system is a solution $(\psi_1,\psi_2,\psi_3)\in \mathcal{H}$ with ansatz \begin{equation} \label{ansatz} \psi_i(x_1,x_2,t)=e^{-{\rm i}\frac{\mu_it}{\hbar}}\phi_i(x_1,x_2),\quad (x_1,x_2)\in\mathbb{R}^2,\; t\geq 0,\; i=1,2,3 \end{equation} where $(\phi_1,\phi_2,\phi_3)\in \mathcal{H}$ is real valued and minimizes the functional $E$ constrained to the normalization conditions \eqref{totmasses} (with $\phi_i$ in place of $\psi_i$). Consequently, the functions $\phi_i$s solve the nonlinear eigenvalue problem \begin{equation} \label{GPSv} \begin{gathered} -\frac{\hbar^2}{2m_1}\Delta \phi_1 +V_1(x_1,x_2)\phi_1+\theta_{11}\hbar^2|\phi_1|^2\phi_1+\sum_{j\neq 1}^3 \theta_{1j}\hbar^2|\phi_j|^2\phi_1=\mu_1\phi_1, \\ -\frac{\hbar^2}{2m_2}\Delta \phi_2 +V_2(x_1,x_2)\phi_2+\theta_{22}\hbar^2|\phi_2|^2\phi_2+\sum_{j\neq 2}^3\theta_{2j} \hbar^2|\phi_j|^2\phi_2=\mu_2\phi_2, \\ -\frac{\hbar^2}{2m_3}\Delta \phi_3 +V_3(x_1,x_2)\phi_3+\theta_{33}\hbar^2|\phi_3|^2\phi_3+\sum_{j\neq 3}^3\theta_{3j} \hbar^2|\phi_j|^2\phi_3=\mu_3\phi_3, \\ {\int_{\mathbb{R}^2} \phi_i^2=N_i},\quad i=1,2,3. \end{gathered} \end{equation} A direct computation yields the representation formula for the eigenvalues \begin{equation} \label{formulautov} N_i\mu_i=E^i_\infty(\phi_i)+\frac{\theta_{ii}\hbar^2}{2}\int_{\mathbb{R}^2}|\phi_i|^4+\sum_{j\neq i}^3 J^{ij}(\phi_i,\phi_j), \end{equation} for any $i=1,2,3$. The existence of nontrivial solutions of the nonlinear system \eqref{GPSv} is straightforward as all the coupling constants are positive, which makes the Hamiltonian $E$ coercive and weakly lower semicontinuous on the $L^2\times L^2$ sphere \eqref{totmasses} in $\mathcal{H}$. In addition, by the standard gradient inequality $\int_{\mathbb{R}^2}|\nabla |\phi||^2\leq \int_{\mathbb{R}^2}|\nabla\phi |^2$, it follows that the ground state solutions can be sought among nonnegative functions, so the $\phi_i$ are positive. \section{Large inter-species parameters} \label{SIr} Let $\mathcal{H}\subset H^1(\mathbb{R}^2)\times H^1(\mathbb{R}^2)$ be the realization of the Hilbert subspace given in the introduction. For any index pair $i\neq j$, we set \begin{gather*} {\mathcal S} =\Big\{(\phi_1,\phi_2,\phi_3)\in \mathcal{H}:\,\int_{\mathbb{R}^2}\phi_i^2=N_i,\;\text{$\forall i=1,2,3$}\Big\}, \\ {\mathcal S}_{ij}^\infty =\Big\{(\phi_1,\phi_2,\phi_3)\in {\mathcal S}:\, \int_{\mathbb{R}^2}\phi_i^2\phi_j^2=0\Big\}, \\ {\mathcal S}^\infty =\bigcap_{i\neq j,\; i,j=1}^3{\mathcal S}_{ij}^\infty. \end{gather*} Assume now that one of the inter-species parameter, say $\theta_{i_0j_0}$ with $i_0\neq j_0$, gets very large, say $\theta_{i_0j_0}=\kappa\to\infty$ while the other remain bounded, say $\theta_{lm}\in(0,1]$ for any other $l,m=1,2,3$ with $l\neq m$. The least energy level of the ground state solutions is then defined and denoted as follows $$ c_{\kappa}^{i_0j_0}=\inf_{(\phi_1,\phi_2,\phi_3)\in {\mathcal S}} \big[E_\infty(\phi_1,\phi_2,\phi_3)+J_\kappa(\phi_1,\phi_2,\phi_3)\big], $$ where the Hamiltonian is $E_\infty+J_\kappa=E_\kappa=E:\mathcal{H}\to\mathbb{R}$, with $$ J_\kappa(\phi_1,\phi_2,\phi_3)=\kappa\hbar^2\int_{\mathbb{R}^2} |\phi_{i_0}|^2|\phi_{j_0}|^2 +\sum_{n\neq m,\,n\neq i_0,\,m\neq j_0}^3 J^{nm}(\phi_n,\phi_m). $$ We also define the candidate for the limiting (as $\kappa\to\infty$) energy $c_\infty^{i_0j_0}$, \begin{equation} \label{specialenerlevel} c_\infty^{i_0j_0}=\!\!\!\inf_{(\phi_1,\phi_2,\phi_3)\in {\mathcal S}_{i_0j_0}^\infty} \Big[E_\infty(\phi_1,\phi_2,\phi_3) +\sum_{n\neq m,\,n\neq i_0,\,m\neq j_0}^3 J^{nm}(\phi_n,\phi_m)\Big] . \end{equation} With obvious modifications one can define the energy levels corresponding to the case where more than one parameter diverges. In the case where $\theta_{ij}\to\infty$ for \emph{all} $i\neq j$ then the limiting energy is $c_\infty$, \begin{equation} \label{specialenerlevelTT} c_\infty=\!\!\!\inf_{(\phi_1,\phi_2,\phi_3)\in{\mathcal S}^\infty}E_\infty(\phi_1,\phi_2,\phi_3). \end{equation} As ${\mathcal S}^\infty\subset {\mathcal S}_{i_0j_0}^\infty\subset {\mathcal S}$, taking into account the definition of $c_{\kappa}^{i_0j_0}$, $c_\infty^{i_0j_0}$ and $c_\infty$ it holds \begin{equation} \label{controlenergg} c_{\kappa}^{i_0j_0}\leq c_\infty^{i_0j_0}\leq c_\infty, \end{equation} for any $\kappa>0$. In this setting the following result holds. \begin{proposition} \label{Gsspatseg} As $\kappa$ goes to infinity, the sequence of ground state solutions $(\phi_1^\kappa,\phi_2^\kappa,\phi_3^\kappa)\subset {\mathcal S}$ converges in $\mathcal{H}$ to a function $(\phi_1^\infty,\phi_2^\infty,\phi_3^\infty)\in {\mathcal S}_{i_0j_0}^\infty$ at energy level $c_\infty^{i_0j_0}$. Moreover, there exist $\mu_i^\infty>0$ such that the variational inequalities hold \begin{equation} \label{varineqqq} -\frac{\hbar^2}{2m_i}\Delta \phi_i^\infty+V_i(x_1,x_2)\phi_i^\infty +\theta_{ii}\hbar^2|\phi_i^\infty|^2\phi_i^\infty \leq \mu_i^\infty\phi_i^\infty \quad \text{in }\mathbb{R}^2, \end{equation} for all $i=1,2,3$. \end{proposition} \begin{remark} \label{rmk3.2} \rm It is natural to wonder if the limit function $\phi_i^\infty$ solves the equation \begin{equation*} -\frac{\hbar^2}{2m_i}\Delta \phi_i^\infty+V_i(x_1,x_2)\phi_i^\infty+\theta_{ii}\hbar^2(\phi_i^\infty)^3 =\mu_i^\infty\phi_i^\infty \quad \text{in $\Omega_i=\{\phi_i^\infty>0\}$} \end{equation*} when $\Omega_i\subset\mathbb{R}^2$ is an open set. In other words, taken any positive and compactly supported function $\varphi$ with support in $\Omega_i$, do we have $\kappa\int_{\mathbb{R}^2} |\phi_{j}^\kappa|^2\phi_{i}^\kappa\varphi\to 0$ when $\kappa\to\infty$? We believe this is true. \end{remark} \begin{proof}[Proof of Proposition~\ref{Gsspatseg}] In light of the first inequality of \eqref{controlenergg}, if $(\phi_1^\kappa,\phi_2^\kappa,\phi_3^\kappa)\in \mathcal{H}$, $\phi_i^\kappa\not\equiv 0$ for any $i$ is the ground state solution, we have $E_\kappa(\phi_1^\kappa,\phi_2^\kappa,\phi_3^\kappa)=c_\kappa^{i_0j_0}$ and \begin{align*} \label{BddkappaInt} \kappa\hbar^2\int_{\mathbb{R}^2} |\phi_{i_0}^\kappa|^2|\phi_{j_0}^\kappa|^2 &\leq \sum_{n\neq m}^3\theta_{nm}\hbar^2\int_{\mathbb{R}^2} |\phi_n|^2|\phi_m|^2 \\ & \leq (J_\kappa+E_\infty)(\phi_1^\kappa,\phi_2^\kappa,\phi_3^\kappa)\\ &=c_\kappa^{i_0j_0}\leq c_\infty^{i_0j_0}, \end{align*} for every $\kappa>0$. As a consequence, we obtain $\int_{\mathbb{R}^2} |\phi_{i_0}^\kappa|^2|\phi_{j_0}^\kappa|^2\to 0$ as $\kappa\to\infty$. In addition, we have $\|(\phi_1^\kappa,\phi_2^\kappa,\phi_3^\kappa)\|_\mathcal{H}^2\leq E_\kappa(\phi_1^\kappa,\phi_2^\kappa,\phi_3^\kappa)\leq c_\infty^{i_0j_0}$ for any $\kappa>0$. Hence, the sequences $(\phi_1^\kappa,\phi_2^\kappa,\phi_3^\kappa)$ is bounded in $\mathcal{H}$, with respect to $\kappa$. In particular there exist $(\phi_1^\infty,\phi_2^\infty,\phi_3^\kappa)$ in $\mathcal{H}$ such that, up to a subsequence, $(\phi_1^\kappa,\phi_2^\kappa,\phi_3^\kappa)\rightharpoonup (\phi_1^\infty,\phi_2^\infty,\phi_3^\infty)$ in $\mathcal{H}$ and $\phi_i^\kappa(x_1,x_2)\to\phi_i^\infty(x_1,x_2)$ for a.e.\ $(x_1,x_2)\in\mathbb{R}^2$ as $\kappa\to\infty$. Hence, by Fatou's Lemma, we get $\int_{\mathbb{R}^2} (\phi_{i_0}^\infty)^2(\phi_{j_0}^\infty)^2=0$, namely $\phi_{i_0}^\infty\phi_{j_0}^\infty=0$ a.e.\ in $\mathbb{R}^2$. Since by definition $\int_{\mathbb{R}^2}|\phi_i^\kappa|^2=N_i$ for any $\kappa>0$ and $\mathcal{H}$ in compactly embedded into the space $L^r(\mathbb{R}^2)\times L^r(\mathbb{R}^2)$ for any $r\geq 2$ (combining an inequality like \eqref{embedkeyineq} with the Gagliardo--Nirenberg inequalities), up to a further subsequence, we have $\int_{\mathbb{R}^2}|\phi_i^\infty|^2=N_i$ for $i=1,2,3$. Whence \begin{equation} \label{appartenenzaclass} (\phi_1^\infty,\phi_2^\infty,\phi_3^\infty)\in {\mathcal S}_{i_0j_0}^\infty. \end{equation} Observe also that $E^i_\infty(\phi_i^\kappa)\leq c_\infty^{i_0j_0}$ for any $i=1,2,3$ and \begin{equation*} \sup_{\kappa\geq 1}\mu_i^\kappa=\frac{1}{N_i}\sup_{\kappa\geq 1}\Big\{E^i_\infty(\phi_i^\kappa)+\frac{\theta_{ii}\hbar^2}{2} \int_{\mathbb{R}^2} |\phi_i^\kappa|^4+ \sum_{m\neq i}^3\theta_{im}\hbar^2\int_{\mathbb{R}^2} |\phi_m^\kappa|^2|\phi_i^\kappa|^2 \Big\}<\infty, \end{equation*} denoting $\mu_i^\kappa$ the eigenvalues corresponding to $\phi_i^\kappa$. Hence, up to a subsequence, $\mu_i^\kappa\to \mu_i^\infty$ as $\kappa\to\infty$. By testing the equations of the system by an arbitrary compactly supported positive function $\eta$, we get \begin{equation*} \frac{\hbar^2}{2m_i}\int_{\mathbb{R}^2}\nabla \phi_i^\kappa\cdot\nabla\eta+\int_{\mathbb{R}^2} V_i(x_1,x_2)\phi_i^\kappa\eta+\theta_{ii}\int_{\mathbb{R}^2}|\phi_i^\kappa|^2 \phi_i^\kappa\eta \leq \mu_i^\kappa\int_{\mathbb{R}^2} \phi_i^\kappa\eta, \end{equation*} for all $\kappa>0$. Hence, letting $\kappa\to\infty$, it turns out that $\phi_i^\infty$ satisfies the variational inequalities \eqref{varineqqq}. Notice that, by Fatou's lemma and the first inequality of \eqref{controlenergg}, we have \begin{align*} & \sum_{i=1}^3\frac{\hbar^2}{2m_i}\int_{\mathbb{R}^2} |\nabla \phi_i^\infty|^2+ \sum_{i=1}^3\int_{\mathbb{R}^2} V_i|\phi_i^\infty|^2+ \sum_{i=1}^3\frac{\theta_{ii}\hbar^2}{2} \int_{\mathbb{R}^2} |\phi_i^\infty|^4 \\ &\quad+\lim_{\kappa\to\infty}\kappa\hbar^2\int_{\mathbb{R}^2} |\phi_{i_0}^\kappa|^2|\phi_{j_0}^\kappa|^2 +\!\!\!\sum_{n\neq m,\,n\neq {i_0},\,m\neq {j_0}}^3 J^{nm}(\phi_n^\infty,\phi_m^\infty)\\ & \leq \sum_{i=1}^3\frac{\hbar^2}{2m_i}\liminf_{\kappa\to\infty}\int_{\mathbb{R}^2} |\nabla \phi_i^\kappa|^2+ \sum_{i=1}^3\liminf_{\kappa\to\infty}\int_{\mathbb{R}^2} V_i|\phi_i^\kappa|^2+ \sum_{i=1}^3 \frac{\theta_{ii}\hbar^2}{2}\liminf_{\kappa\to\infty}\int_{\mathbb{R}^2} |\phi_i^\kappa|^4 \\ &\quad +\lim_{\kappa\to\infty}\kappa\hbar^2\int_{\mathbb{R}^2} |\phi_{i_0}^\kappa|^2|\phi_{j_0}^\kappa|^2 +\!\!\!\sum_{n\neq m,\,n\neq {i_0},\,m\neq {j_0}}^3 \liminf_{\kappa\to\infty}J^{nm}(\phi_n^\kappa,\phi_m^\kappa)\\ & \leq \liminf_{\kappa\to\infty} E_\kappa(\phi_1^\kappa,\phi_2^\kappa,\phi_3^\kappa) =\liminf_{\kappa\to\infty} c_\kappa^{i_0j_0} \leq c_\infty^{i_0j_0}. \end{align*} Recalling formula \eqref{appartenenzaclass}, by the definition of $c_\infty^{i_0j_0}$ the above inequalities rewrite as \begin{align*} & E_\infty(\phi_1^\infty,\phi_2^\infty,\phi_3^\infty) +\lim_{\kappa\to\infty}\kappa\hbar^2\int_{\mathbb{R}^2} |\phi_{i_0}^\kappa|^2|\phi_{j_0}^\kappa|^2 +\!\!\!\sum_{n\neq m,\,n\neq {i_0},\,m\neq {j_0}}^3 J^{nm}(\phi_n^\infty,\phi_m^\infty)\\ & \leq c_\infty^{i_0j_0} \leq E_\infty(\phi_1^\infty,\phi_2^\infty,\phi_3^\infty) +\!\!\!\sum_{n\neq m,\,n\neq {i_0},\,m\neq {j_0}}^3 J^{nm}(\phi_n^\infty,\phi_m^\infty) \end{align*} which yields \begin{equation} \label{strongerconcl} \lim_{\kappa\to\infty}\kappa\int_{\mathbb{R}^2} |\phi_{i_0}^\kappa|^2 |\phi_{j_0}^\kappa|^2=0. \end{equation} Therefore, the convergence of $\phi_i^\kappa$ to $\phi_i^\infty$ in $\mathcal{H}$ is strong, otherwise, assuming by contradiction that this is not the case, the previous inequalities would become strict, yielding immediately a contradiction with \eqref{strongerconcl}. Finally, as a further consequence, $c_\infty^{i_0j_0}=E_\infty(\phi_1^\infty,\phi_2^\infty,\phi_3^\infty)$, concluding the proof. \end{proof} \begin{remark}\rm In the assumptions and notations of Proposition~\ref{Gsspatseg}, not all the components become mutually phase segregated. If not, then, $\phi_i^\infty\phi_j^\infty=0$ a.e.\ for {\em any} index $i\neq j$, and so, by definition we would have $(\phi_1^\infty,\phi_2^\infty, \phi_3^\infty)\in{\mathcal S}^\infty$. In turn, by \eqref{specialenerlevelTT}, \eqref{strongerconcl} and Fatou's Lemma, we obtain $$ c_\infty\leq E_\infty(\phi_1^\infty,\phi_2^\infty,\phi_3^\infty) \leq\liminf_{\kappa\to\infty}E_\kappa(\phi_1^\kappa,\phi_2^\kappa,\phi_3^\kappa)\leq c_\infty^{i_0j_0}0}c_{\Lambda}=+\infty. $$ Indeed, assume by contradiction that this is not the case, namely $\Gamma<\infty$. Hence, the sequence of ground state solutions $(\phi_1^\Lambda,\phi_2^\Lambda,\phi_3^\Lambda)$ is bounded in $H^1(\mathbb{R}^2,\mathbb{R}^3)$. In particular, up to a subsequence, it converges weakly in $H^1(\mathbb{R}^2,\mathbb{R}^3)$ and pointwise a.e.\ to a function $(\phi_1^\infty,\phi_2^\infty,\phi_3^\infty)$. Moreover, since it holds \begin{equation} \label{embedkeyineq} \sup_{\Lambda\geq 1}\sup_{\varrho>0} \Big[\varrho^2\int_{\mathbb{R}^2\setminus B_\varrho(x_1^{i},x_2^{i})}\!\!\!|\phi_{i}^\Lambda|^2\Big]\leq\Gamma, \end{equation} for $i=1,2,3$, it follows that $(\phi_1^\Lambda,\phi_2^\Lambda,\phi_3^\Lambda)$ also converges, strongly, in $L^2(\mathbb{R}^2,\mathbb{R}^3)$. Since, for any $\Lambda>0$ and $i=1,2,3$, $$ \int_{\mathbb{R}^2} |\phi_{i}^\Lambda|^2=N_i, $$ taking the limit as $\Lambda\to+\infty$ entails $\int_{\mathbb{R}^2} |\phi_{i}^\infty|^2=N_i$. Whence $\phi_{i}^\infty\neq 0$ in $H^1(\mathbb{R}^2)$ for every $i=1,2,3$. On the other hand, we also have $$ \int_{\mathbb{R}^2} V_{i_0}^\Lambda(x_1,x_2)|\phi_{i_0}^\Lambda|^2\leq \Gamma, $$ for all $\Lambda>0$, yielding in particular $$ \int_{\mathbb{R}^2} \frac{m_{i_0}}{2} (x_1-x_1^{i_0})^2|\phi_{i_0}^\Lambda |^2\leq \frac{\Gamma}{\Lambda^2}. $$ By Fatou's lemma this entails $|x_1-x_1^{i_0}||\phi_{i_0}^\infty(x_1,x_2) |=0$ for a.e.\ $(x_1,x_2)\in\mathbb{R}^2$, namely $\phi_{i_0}^\infty=0$ in $H^1(\mathbb{R}^2)$, which produces a contradiction. \end{remark} \medskip \section{Location of components} \label{TFapp} In the so called Thomas--Fermi regime, a very good approximation of the ground state solutions of~\eqref{GPSv} which holds for sufficiently large values of the coupling constants $\theta_{ij}$, can be obtained by simply dropping the diffusion terms $-\Delta\phi_i$, the kinetic contributions, namely assuming the wave functions to be slowly varying (cf.\ \cite{fermi,thomas,lieb}). In turn, system \eqref{GPSv} reduces to the algebraic system ($\hbar=m_i=1$) \begin{equation} \label{systemGPGenTF} \begin{gathered} \theta_{11}|\phi_1|^2+\theta_{12}|\phi_2|^2+\theta_{13}|\phi_3|^2=\mu_1-V_1(x_1,x_2), \\ \noalign{\vskip3pt} \theta_{21}|\phi_1|^2+\theta_{22}|\phi_2|^2+\theta_{23}|\phi_3|^2=\mu_2-V_2(x_1,x_2), \\ \noalign{\vskip3pt} \theta_{31}|\phi_1|^2+\theta_{32}|\phi_2|^2+\theta_{33}|\phi_3|^2=\mu_3-V_3(x_1,x_2). \end{gathered} \end{equation} Let us denote by $\Theta=(\theta_{ij})$ the symmetric coupling matrix and set $\eta_i=\phi^2_i$ and $\chi_i(x_1,x_2)=\mu_i-V_i(x_1,x_2)$, where the eigenvalues $\mu_i$s should be computed through the normalization conditions \eqref{totmasses}. Moreover, assume that $|\Theta|>0$ (positive determinant). Then, we obtain \begin{gather} \label{kramers} |\Theta|\,\eta_1(x_1,x_2)= \begin{vmatrix} \chi_1(x_1,x_2) & \theta_{12} & \theta_{13} \\ \chi_2(x_1,x_2) & \theta_{22} & \theta_{23} \\ \chi_3(x_1,x_2) & \theta_{32} &\theta_{33} \end{vmatrix}, \\ |\Theta|\,\eta_2(x_1,x_2)= \begin{vmatrix} \theta_{11} & \chi_1(x_1,x_2) & \theta_{13} \\ \theta_{21} & \chi_2(x_1,x_2) & \theta_{23} \\ \theta_{31} & \chi_3(x_1,x_2) &\theta_{33} \end{vmatrix},\notag \\ |\Theta|\,\eta_3(x_1,x_2)= \begin{vmatrix} \theta_{11} & \theta_{12} & \chi_1(x_1,x_2) \\ \theta_{21} & \theta_{22} & \chi_2(x_1,x_2) \\ \theta_{31} & \theta_{32} & \chi_3(x_1,x_2) \end{vmatrix}.\notag \end{gather} As the coupling coefficients are positive, if we set $r_i=\sqrt{2\mu_i}$ for $i=1,2,3$, it is evident that system \eqref{systemGPGenTF} makes sense only if the right hand sides of each equation in it is positive, that is in the set $$ {\mathcal D}=\cap_{i=1}^3{\mathcal D}_i,\quad {\mathcal D}_i =\big\{(x_1,x_2)\in\mathbb{R}^2:\;\omega_{ix}^2(x_1-x_{i1})^2+\omega_{iy}^2(x_2-x_{i2})^2\leq r_i^2\big\}. $$ Furthermore, taking into account that, for any $i$, the $\eta_i$s are positive and are a combination of quadratic polynomials (due to the structure of $\chi_i$), there exist positive constants $\Omega_{ix}$, $\Omega_{iy}$ and $R_i$ and centers $(y_{i1},y_{i2})$ which allow to define the (possibly empty) overlap region of the components of the wave functions $$ {\mathcal O}=\cap_{i=1}^3{\mathcal O}_i,\quad {\mathcal O}_i =\big\{(x_1,x_2)\in{\mathcal D}:\;\Omega_{ix}^2(x_1-y_{i1})^2+\Omega_{iy}^2(x_2-y_{i2})^2\leq R_i^2\big\}. $$ Then, for ${\mathcal O}\neq\emptyset$, there is $\alpha_i>0$ such that a non-smooth approximation of the $i$-th component of the ground state is given by \begin{equation} \label{TFrappppree} \phi_i^2(x_1,x_2):= \begin{cases} \alpha_i(R_i^2-\Omega_{ix}^2(x_1-y_{i1})^2-\Omega_{iy}^2(x_2-y_{i2})^2), & \text{in ${\mathcal O}$}, \\[3pt] \frac{r_i^2-\omega_{ix}^2(x_1-x_{i1})^2-\omega_{iy}^2(x_2-x_{i2})^2} {2\theta_{ii}}, & \text{in ${\mathcal D}_i\setminus{\mathcal O}$}, \\[3pt] 0, & \text{in $\mathbb{R}^2\setminus {\mathcal D}_i$}. \end{cases} \end{equation} Thinking for instance to the case where the diagonal coefficients $\theta_{ii}$ are much larger than the $\theta_{ij}$s, i.e.\ $\theta_{ii}\gg\theta_{ij}\gg 1$, we have from \eqref{kramers}, e.g.\ for the component $\phi_1$, $$ \theta_{11}\theta_{22}\theta_{33}\,\phi_1^2(x_1,x_2)\approx \begin{vmatrix} \chi_1(x_1,x_2) & \theta_{12} & \theta_{13} \\ \chi_2(x_1,x_2) & \theta_{22} & \theta_{23} \\ \chi_3(x_1,x_2) & \theta_{32} & \theta_{33} \end{vmatrix} \approx\chi_1(x_1,x_2)\theta_{22}\theta_{33}. $$ This clarifies why it makes sense to extend to the set ${\mathcal D}_1\setminus{\mathcal O}$ the Thomas--Fermi approximation defined in ${\mathcal O}$ according to the second line of formula \eqref{TFrappppree} (for $i=1$). \medskip \section{Numerical computation of solutions} \label{NumSim} As done in \cite{CalSqu}, for the sake of self-containedness, we briefly describe the numerical algorithm used for the computation of the ground states. For further details, we refer the interested reader to \cite{CORT08}. It is sufficient to consider the single one-dimensional Gross--Pitaevskii equation. In fact the extension to the case with any number of equations is straightforward. Moreover, without loss of generality, we reduce to the case $\hbar=m=1$. The main idea is to directly minimize the energy $E(\phi)$ associated to a wave function $\psi(x)=e^{-i\mu t}\phi(x)$, discretized by Hermite functions, with a normalization constraint for the wave function. As it is known, the Hermite functions $(\mathcal{H}^\beta_l)_{l\in\mathbb{N}}$ are defined by $$ \mathcal{H}^\beta_l(x)=H^\beta_{l}(x)e^{-\frac12\beta^2x^2},\quad l\in\mathbb{N}, $$ where $(H_{l}^\beta)_{l\in\mathbb{N}}$ are the \emph{Hermite polynomials} \cite{boyd}, orthonormal in $L^2$ with respect to the weight $e^{-\beta^2x^2}$. The Hermite functions are the solutions (ground state, for $l=0$, and excited states, if else) to the eigenvalue problem for the linear Schr\"odinger equation with \emph{standard} harmonic potential $$ \frac{1}{2}\Big(-\frac{d^2}{dx^2}+(\beta^2x)^2\Big)\mathcal{H}_l= \lambda_l\mathcal{H}_l,\quad \lambda_l=\beta^2\big(l+\frac{1}{2}\big). $$ If we set $$ \phi=\sum_{l\in\mathbb{N}}\phi_l\mathcal{H}_l, $$ where $$ \phi_l=(\phi,\mathcal{H}_l)_{L^2}=\int_\mathbb{R} \phi\mathcal{H}_l, $$ the energy functional rewrites as \begin{equation*} E(\phi)=\sum_{l\in\mathbb{N}}\lambda_l\phi_l^2+ \int_\mathbb{R} \Big(V(x)-\frac{(\beta^2x)^2}{2}\Big) \Big(\sum_{l\in\mathbb{N}}\phi_l\mathcal{H}_l \Big)^2+ \frac12\theta\int_\mathbb{R}\Big(\sum_{l\in\mathbb{N}}\phi_l\mathcal{H}_l \Big)^4, \end{equation*} and the chemical potential turns into \begin{equation} \label{frapreMu} N\mu=E(\phi)+ \frac12\theta\int_\mathbb{R}\Big(\sum_{l\in\mathbb{N}}\phi_l\mathcal{H}_l \Big)^4 \end{equation} By minimizing $E$, under the constraint $\|\phi\|_{L^2}^2=N$, we look for local minima of \[ E(\phi;\lambda)=E(\phi)+\lambda\Big(N-\sum_{l\in\mathbb{N}}\phi_l^2\Big) \] which solve the system, with $k\in\mathbb{N}$, \begin{gather*} (\lambda_\kappa-\lambda)\phi_\kappa+ \int_\mathbb{R} \Big(V(x)-\frac{(\beta^2x)^2}{2}\Big)\mathcal{H}_k \Big(\sum_{l\in\mathbb{N}}\phi_l\mathcal{H}_l\Big)+ \theta\int_\mathbb{R} \mathcal{H}_k\Big(\sum_{l\in\mathbb{N}}\phi_l\mathcal{H}_l \Big)^3 =0,\\ \sum_{l\in\mathbb{N}}\phi_l^2=N. \end{gather*} We notice that, if $\phi$ is a solution of the above system, then it is immediately seen, by multiplying times $\phi_k$, summing up over $k$ and using \eqref{frapreMu}, that the Lagrange multiplier $\lambda$ equals the chemical potential $\mu$. Next, we truncate the Hermite series to degree $L-1$ and introduce an additional parameter $\rho=1$ in front of the first integral (its usage will be clear later), to obtain a corresponding truncated energy functional $E_L(\phi;\lambda;\rho)$, whose local minima solve the system, with $0\le k\le L-1$, \begin{gather*} (\lambda_\kappa-\lambda)\phi_\kappa+ \rho\int_\mathbb{R} \Big(V(x)-\frac{(\beta^2x)^2}{2}\Big)\mathcal{H}_k \Big(\sum_{l=0}^{L-1}\phi_l\mathcal{H}_l\Big)+ \theta\int_\mathbb{R} \mathcal{H}_k\Big(\sum_{l=0}^{L-1}\phi_l\mathcal{H}_l\Big)^3 =0,\\ \sum_{l=0}^{L-1}\phi_l^2=N. \end{gather*} To approximate the integrals, we used a Gauss--Hermite quadrature formula with $2L-1$ nodes relative to the weight $e^{-2\beta^2x^2}$. The system is solved by a modified Newton method with backtracking line-search, which guarantees global convergence to the ground states. We refer to \cite{Baoal,CalThal} and, in particular, to \cite{CORT08} for the details. Here we just mention that the initial guess for the Newton iteration is obtained by a continuation technique over $\rho$ and $\theta$, starting from the ground state of the Schr\"odinger equation with the standard harmonic potential, which corresponds to $\rho=\theta=0$. Using the tensor basis of the Hermite functions, the extension to the two-dimensional case is straightforward. In the following figures we show some typical spatial patters of the ground states solution triplet with respect some relevant features as: \begin{enumerate} \item the anisotropy of the trapping potentials (Figure~\ref{as1}); \item the phase separation via potential off-centering (Figure~\ref{as2}); \item the phase separation via large inter-atomic interactions (Figure~\ref{as3}); \item the shape of supports with respect to the number of atoms $N_i$ (Figure~\ref{as4}); \item the shape of supports with respect to the size of the masses $m_i$ (Figure~\ref{as5}). \end{enumerate} \begin{figure}[!ht] \includegraphics[scale=0.44]{figures/phi1AS1_1}\hfill \includegraphics[scale=0.44]{figures/phi2AS1_1}\hfill \includegraphics[scale=0.44]{figures/phi3AS1_1}\vskip5pt \includegraphics[scale=0.44]{figures/phi1AS1_2}\hfill \includegraphics[scale=0.44]{figures/phi2AS1_2}\hfill \includegraphics[scale=0.44]{figures/phi3AS1_2}\vskip5pt \includegraphics[scale=0.44]{figures/phi1AS1_3}\hfill \includegraphics[scale=0.44]{figures/phi2AS1_3}\hfill \includegraphics[scale=0.44]{figures/phi3AS1_3}\vskip5pt \includegraphics[scale=0.44]{figures/phi1AS1_4}\hfill \includegraphics[scale=0.44]{figures/phi2AS1_4}\hfill \includegraphics[scale=0.44]{figures/phi3AS1_4}\vskip5pt \includegraphics[scale=0.44]{figures/phi1AS1_5}\hfill \includegraphics[scale=0.44]{figures/phi2AS1_5}\hfill \includegraphics[scale=0.44]{figures/phi3AS1_5} \caption{(anisotropy) ground state $(\phi_1,\phi_2,\phi_3)$ (left to right); $\omega_{y1}$ and $\omega_{x2}$ assume values $\pi,1.1\pi,1.5\pi,2\pi,10\pi$ (top to bottom), other $\omega_{yi}=\pi$, $\omega_{xi}=\pi$, $m_i = 1.44\cdot 10^{-25}$, $N_i=10^7$, $\sigma_{11}=\sigma_{22}=\sigma_{33}=10^{-6}$ and $\sigma_{12}=\sigma_{23}=\sigma_{13}=10\sigma_{ii}$.} \label{as1} \end{figure} \clearpage \begin{figure}[!ht] \includegraphics[scale=0.42]{figures/phi1AS2_1}\hfill \includegraphics[scale=0.42]{figures/phi2AS2_1}\hfill \includegraphics[scale=0.42]{figures/phi3AS2_1}\vskip5pt \includegraphics[scale=0.42]{figures/phi1AS2_2}\hfill \includegraphics[scale=0.42]{figures/phi2AS2_2}\hfill \includegraphics[scale=0.42]{figures/phi3AS2_2}\vskip5pt \includegraphics[scale=0.42]{figures/phi1AS2_3}\hfill \includegraphics[scale=0.42]{figures/phi2AS2_3}\hfill \includegraphics[scale=0.42]{figures/phi3AS2_3}\vskip5pt \includegraphics[scale=0.42]{figures/phi1AS2_4}\hfill \includegraphics[scale=0.42]{figures/phi2AS2_4}\hfill \includegraphics[scale=0.42]{figures/phi3AS2_4}\vskip5pt \includegraphics[scale=0.42]{figures/phi1AS2_5}\hfill \includegraphics[scale=0.42]{figures/phi2AS2_5}\hfill \includegraphics[scale=0.42]{figures/phi3AS2_5} \caption{(off-centering) ground state $(\phi_1,\phi_2,\phi_3)$ (left to right); $V_2$ with center $(0,0)$; $V_1$ with centers $(-4,4)$, $(-3,3)$, $(-2,2)$, $(-1,1)$, $(-0.4,0.4)$ and $V_3$ with centers $(4,4)$, $(3,3)$, $(2,2)$, $(1,1)$, $(0.4,0.4)$ (up to $10^{-5}$, top to bottom); $\omega_{yi}=\omega_{xi}=\pi$, $m_i = 1.44\cdot 10^{-25}$, $N_i=10^7$, $\sigma_{11}=\sigma_{33}=2\cdot10^{-7}$, $\sigma_{22}=100\sigma_{11}$ and $\sigma_{ij}=50\sigma_{11}$.} \label{as2} \end{figure} \clearpage \begin{figure}[!ht] \includegraphics[scale=0.44]{figures/phi1AS3_1}\hfill \includegraphics[scale=0.44]{figures/phi2AS3_1}\hfill \includegraphics[scale=0.44]{figures/phi3AS3_1}\vskip5pt \includegraphics[scale=0.44]{figures/phi1AS3_2}\hfill \includegraphics[scale=0.44]{figures/phi2AS3_2}\hfill \includegraphics[scale=0.44]{figures/phi3AS3_2}\vskip5pt \includegraphics[scale=0.44]{figures/phi1AS3_3}\hfill \includegraphics[scale=0.44]{figures/phi2AS3_3}\hfill \includegraphics[scale=0.44]{figures/phi3AS3_3}\vskip5pt \includegraphics[scale=0.44]{figures/phi1AS3_4}\hfill \includegraphics[scale=0.44]{figures/phi2AS3_4}\hfill \includegraphics[scale=0.44]{figures/phi3AS3_4}\vskip5pt \includegraphics[scale=0.44]{figures/phi1AS3_5}\hfill \includegraphics[scale=0.44]{figures/phi2AS3_5}\hfill \includegraphics[scale=0.44]{figures/phi3AS3_5} \caption{(phase segregation) ground state $(\phi_1,\phi_2,\phi_3)$ (left to right); $\omega_{yi}=\omega_{xi}=\pi$, $m_i = 1.44\cdot 10^{-25}$, $N_i=10^7$, $\sigma_{11}=\sigma_{22}=\sigma_{33}=10^{-6}$ and $\sigma_{12}=0,0.3,0.8,1.4,2\cdot 10^{-6}$, $\sigma_{23}=0,0.5,1,1.8,5\cdot 10^{-6}$, $\sigma_{13}=0,0.7,1.8,5,50\cdot 10^{-6}$ (top to bottom).} \label{as3} \end{figure} \clearpage \begin{figure}[!ht] \includegraphics[scale=0.44]{figures/phi1AS4_1}\hfill \includegraphics[scale=0.44]{figures/phi2AS4_1}\hfill \includegraphics[scale=0.44]{figures/phi3AS4_1}\vskip5pt \includegraphics[scale=0.44]{figures/phi1AS4_2}\hfill \includegraphics[scale=0.44]{figures/phi2AS4_2}\hfill \includegraphics[scale=0.44]{figures/phi3AS4_2}\vskip5pt \includegraphics[scale=0.44]{figures/phi1AS4_3}\hfill \includegraphics[scale=0.44]{figures/phi2AS4_3}\hfill \includegraphics[scale=0.44]{figures/phi3AS4_3}\vskip5pt \includegraphics[scale=0.44]{figures/phi1AS4_4}\hfill \includegraphics[scale=0.44]{figures/phi2AS4_4}\hfill \includegraphics[scale=0.44]{figures/phi3AS4_4}\vskip5pt \includegraphics[scale=0.44]{figures/phi1AS4_5}\hfill \includegraphics[scale=0.44]{figures/phi2AS4_5}\hfill \includegraphics[scale=0.44]{figures/phi3AS4_5} \caption{(numbers of atoms) ground state $(\phi_1,\phi_2,\phi_3)$ (left to right); $\omega_{yi}=\omega_{xi}=\pi$, $m_i = 1.44\cdot 10^{-25}$, $\sigma_{ii}=10^{-6}$, $\sigma_{ij}=4\sigma_{ii}$, $N_1=1,0.8,0.4,0.3,0.1\cdot 10 ^7$ and $N_3=1,1.3,2,4,6\cdot 10^7$ (top to bottom).} \label{as4} \end{figure} \clearpage \begin{figure}[!ht] \includegraphics[scale=0.44]{figures/phi1AS5_1}\hfill \includegraphics[scale=0.44]{figures/phi2AS5_1}\hfill \includegraphics[scale=0.44]{figures/phi3AS5_1}\vskip5pt \includegraphics[scale=0.44]{figures/phi1AS5_2}\hfill \includegraphics[scale=0.44]{figures/phi2AS5_2}\hfill \includegraphics[scale=0.44]{figures/phi3AS5_2}\vskip5pt \includegraphics[scale=0.44]{figures/phi1AS5_3}\hfill \includegraphics[scale=0.44]{figures/phi2AS5_3}\hfill \includegraphics[scale=0.44]{figures/phi3AS5_3}\vskip5pt \includegraphics[scale=0.44]{figures/phi1AS5_4}\hfill \includegraphics[scale=0.44]{figures/phi2AS5_4}\hfill \includegraphics[scale=0.44]{figures/phi3AS5_4}\vskip5pt \includegraphics[scale=0.44]{figures/phi1AS5_5}\hfill \includegraphics[scale=0.44]{figures/phi2AS5_5}\hfill \includegraphics[scale=0.44]{figures/phi3AS5_5} \caption{(atomic masses) ground state $(\phi_1,\phi_2,\phi_3)$ (left to right); $\omega_{yi}=\omega_{xi}=\pi$, $N_i = 10^{7}$, $\sigma_{ii}=10^{-6}$, $\sigma_{ij}=4\sigma_{ii}$, $m_1=1.44,1,0.8,0.5,0.3\cdot 10^{-25}$, $m_2=1.44\cdot 10^{-25}$ and $m_3=1.44,1.8,1.9,2,2.1\cdot 10^{-25}$ (top to bottom).} \label{as5} \end{figure} \clearpage \begin{thebibliography}{00} \bibitem{BecR} M. 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