\label{Th-SynthesExistenceSW}
    Let $d\in\{1,2\}$, $\lambda,\omega\in\mathbb R$ and $\gamma>0$. Denote by $\mu_0$ the lowest eigenvalue of $-A$. In each of the following situation, there exists a positive solution of \cref{Eq-StationaryAGP} in $\mathrm{D}\left(\sqrt{|A|}\right)$,
    \begin{itemize}
        \item $\lambda>0$ and $\omega>-\mu_0$,
        \item $\lambda = 0$ and $\omega=\mu_0$,
        \item $\lambda<0$ and $\omega<-\mu_0$.
    \end{itemize}
    In every other cases, there is no non-trivial non-negative solution in $\mathrm{D}\left(\sqrt{|A|}\right)$.

\begin{cases}\complexI\p_t u+ H u +\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases} \label{Eq-GP}
\end{equation}
by a white noise potential. In the study of Bose-Einstein condensation, \cref{Eq-GP} appears as a mean-field limit of the many body Schrödinger equation, the nonlinear term appearing as a consequence of particle interactions \cite{BEC_Pitaevskii}. Adding a random potential in \cref{Eq-GP} can be seen as a way of modeling spatial inhomogeneities (see for example \cite{PhysRevLett.104.193901,PhysRevLett.109.243902} and references therein), thus one can see \cref{Eq-AGP} as a toy model of Bose-Einstein condensation in inhomogeneous environment with zero correlation length. In this context, at least formally, \cref{Eq:AGP} has two conserved quantities physically relevant, the mass
$$\mathcal{M}(u)=\frac{1}{2}\int_{\R^d}|u|^2\d x$$
and the energy
$$\mathcal{E}(u)=\frac{1}{2}\langle-(H+\xi)u,u\rangle - \frac{\lambda}{2\gamma+2}\int_{\R^d}|u|^{2\gamma+2}\d x.$$
Standing waves of \cref{Eq:AGP} then appear naturally as solutions of minimal energy at fixed mass. Formally, if $\phi$ minimize the energy at fixed mass, by a Lagrange multiplier argument, there exists $\omega\in\R$ such that
$$-(H+\xi)\phi-\lambda|\phi|^{2\gamma}\phi+\omega \phi = 0$$
and $u(t,x)=\e^{i\omega t}\phi(x)$ is solution of \cref{Eq:AGP}.\\

In dimension 2, it is known from \cite{Mackowiak_2025,MackowiakStrichartzConfAnderson} that \cref{Eq:AGP} should be interpreted in a renormalized sense. This renormalization has to be taken into account in the study of standing waves. The construction of the renormalized operator has been conducted in \cite{MackowiakStrichartzConfAnderson} and will be recall in \cref{Sec-AndersonOp2d}. The renormalized operator is defined through an exponential transform inspired by the one introduced in \cite{Haire_Labbe} to study the continuous parabolic Anderson model, and later used in \cite{debussche_weber_T2,Tzvetkov_2023,tzvetkov2020dimensional} to solve the Anderson nonlinear Schrödinger equation on the torus $\T^2$ and in \cite{debussche2017solution,debussche2023global} to solve the same equation but on the full space $\R^2$. Let $Y=(-H)^{-1}\xi$, it is almost surely of regularity $1-$, so one can define $\rho=\e^Y$ and search for a formula for $(H+\xi)u$ when $u$ is of the form $u=\rho v$. A formal computation gives
$$(H+\xi)u=\rho\left(Hv+2\nabla Y\cdot \nabla v + xY\cdot xv + |\nabla Y|^2v\right).$$
Unfortunately, $\nabla Y$ is of regularity $0-$, thus the product $|\nabla Y|^2$ barely fails to exist. In \cite{Mackowiak_2025}, the author proved that the wick product $\wick{|\nabla Y|^2}$ can be defined as the almost sure limit, in suitable distribution spaces of regularity $0-$, of
$$|\nabla Y_N|^2-\E[|\nabla Y_N|^2],$$
where $Y_N$ is a suitable regularisation of $Y_N$.  Thus, the renormalized operator can be formally defined as
$$(H+\xi)u=\rho\left(Hv+2\nabla Y\cdot \nabla v + xY\cdot xv + \wick{|\nabla Y|^2}v\right).$$
The rigorous construction of $H+\xi$ rely on the associated quadratic form.

Even in dimension 1, the construction of $H+\xi$ is not straightforward as $\xi$ is not a function. On compact interval with no potential, $\p_x^2+\xi$ has first been constructed in \cite{FukushimaNakao} using its quadratic form and has been studied in \cite{DumazLabbeAndersonLoc1D} using a twisted Sturm-Liouville approach, allowing to construct $\p_x^2+\xi$ on unbounded interval too. In \cref{Sec-AndersonOp1d}, we propose a construction of $H+\xi$ as a self-adjoint unbounded operator through its quadratic form. \cref{Prop-IPP1d} shows $\mathrm{D}(H+\xi)=\mathrm{D}(\p_x^2+\xi)\cap \mathrm{D}(|x|^2)$, where $\p_x^2+\xi$ is the unbounded self-adjoint operator on $\L^2(\R)$ constructed in \cite{DumazLabbeAndersonLoc1D} and $\mathrm{D}(|x|^2)=\{u\in\L^2(\R),xu\in\L^2(\R)\}$.\\

We are then interested in standing waves for
\begin{equation}{~}
    \begin{cases}\complexI\p_t u+ Au+\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases}

\begin{cases}\complexI\p_t u+ H u + \xi u+\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases} \label{Eq:AGP}
\end{equation}
of the form $u(t,x)=\e^{i\omega t}\phi(x)$, where $H=\Delta-|x|^2$ is the Hermite operator, $\xi$ is a real spatial white noise, $\lambda\in\R$ and $\gamma>0$. Such a solution is called a standing wave of \cref{Eq:AGP} of frequency $\omega\in\R$ and profile $\phi$. The study of standing waves is motivated by the modeling of Bose-Einstein condensation.
In fact, \cref{Eq:AGP} can be seen as a perturbation of the Gross-Pitaevskii equation
\begin{equation}{~}
    \begin{cases}\complexI\p_t u+ H u +\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases}

\begin{theo}\label{Th-EnergyGS}

    Let $d\in\{1,2\}$, $\lambda,\omega\in\R$, $\gamma>0$ and $A$ be the self-adjoint operator constructed in \cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\xi$ a spatial white noise, $\D^{1,2}=\mathrm{D}(\sqrt{|A|})$ and $\D^{-1,2}$ its dual. For $u\in\D^{1,2}$, we define the mass as
    $$\mathcal{M}(u)=\frac{1}{2}\int_{\R^d}|u|^2\d x$$
    and the energy as
    $$\mathcal{E}(u)=\frac{1}{2}\langle -Au,u\rangle_{\D^{-1,2},\D^{1,2}}-\frac{\lambda}{2\gamma+2}\int_{\R^d}|u|^{2\gamma+2}\d x.$$
    Then, it holds
    \begin{enumerate}
        \item For $\lambda\leqslant0$, for any $m>0$, there exists $\phi_m\in\D^{1,2}$ such that $\phi_m>0$ and
        $$\argmin_{\substack{u\in\D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u) = \left\{\e^{\complexI\theta}\phi_m,\; \theta\in\R\right\}.$$
        \item For $\lambda>0$ and $\gamma<\frac{2}{d}$, for any $m>0$, there exists $\phi_m\in\displaystyle\argmin_{u\in\D^{1,2},\;\mathcal{M}(u)=m}\mathcal{E}(u)$ such that $\phi_m>0$.
        \item For $\lambda>0$ and $\gamma=\frac{2}{d}$, there exists $m_{*,\lambda}(\Xi)>0$ such that for all $m\in(0,m_{*,\lambda}(\Xi))$, there exists $\phi_m\in\displaystyle\argmin_{u\in\D^{1,2},\;\mathcal{M}(u)=m}\mathcal{E}(u)$ such that $\phi_m>0$. Moreover, there exists $m_{*,\lambda}(\Xi)\leqslant m^*_\lambda(\Xi)\leqslant m^*_\lambda$ such that
        $$\forall m>m^*_\lambda(\Xi), \inf_{\substack{u\in\D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty,$$
        where $m^*_\lambda$ is the critical mass of the classical nonlinear Schrödinger equation (i.e.\ without any potential).
        \item For $\lambda>0$ and $\gamma>\frac{2}{d}$, for any $m>0$,
        $$\inf_{\substack{u\in\D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty.$$
    \end{enumerate}

\end{theo}

\label{Eq-AGP}
\end{equation}
in dimension $d\in\{1,2\}$, where $A=H+\xi$. As $A$ is independent of time, standing waves $u(t,x)=\e^{\complexI\omega t}\phi(x)$ solves \cref{Eq-AGP} if and only if $\p

In this paper, we are interested in particular solutions of the Anderson-Gross-Pitaevskii
\begin{equation}{~}
    \begin{cases}\mathrm{i}\p_t u+ H u + \xi u+\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases} \label{Eq:AGP}
\end{equation}
of the form $u(t,x)=\mathrm{e}^{i\omega t}\phi(x)$, where $H=\Delta-|x|^2$ is the Hermite operator, $\xi$ is a real spatial white noise, $\lambda\in\mathbb R$ and $\gamma>0$. Such a solution is called a standing wave of \cref{Eq:AGP} of frequency $\omega\in\mathbb R$ and profile $\phi$. The study of standing waves is motivated by the modeling of Bose-Einstein condensation.
In fact, \cref{Eq:AGP} can be seen as a perturbation of the Gross-Pitaevskii equation
\begin{equation}{~}
    \begin{cases}\mathrm{i}\p_t u+ H u +\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases} \label{Eq-GP}
\end{equation}
by a white noise potential. In the study of Bose-Einstein condensation, \cref{Eq-GP} appears as a mean-field limit of the many body Schrödinger equation, the nonlinear term appearing as a consequence of particle interactions \cite{BEC_Pitaevskii}. Adding a random potential in \cref{Eq-GP} can be seen as a way of modeling spatial inhomogeneities (see for example \cite{PhysRevLett.104.193901,PhysRevLett.109.243902} and references therein), thus one can see \cref{Eq-AGP} as a toy model of Bose-Einstein condensation in inhomogeneous environment with zero correlation length. In this context, at least formally, \cref{Eq:AGP} has two conserved quantities physically relevant, the mass
$$\mathcal{M}(u)=\frac{1}{2}\int_{\mathbb R^d}|u|^2\mathrm{d} x$$
and the energy
$$\mathcal{E}(u)=\frac{1}{2}\langle-(H+\xi)u,u\rangle - \frac{\lambda}{2\gamma+2}\int_{\mathbb R^d}|u|^{2\gamma+2}\mathrm{d} x.$$
Standing waves of \cref{Eq:AGP} then appear naturally as solutions of minimal energy at fixed mass. Formally, if $\phi$ minimize the energy at fixed mass, by a Lagrange multiplier argument, there exists $\omega\in\mathbb R$ such that
$$-(H+\xi)\phi-\lambda|\phi|^{2\gamma}\phi+\omega \phi = 0$$
and $u(t,x)=\mathrm{e}^{i\omega t}\phi(x)$ is solution of \cref{Eq:AGP}.\\

In dimension 2, it is known from \cite{Mackowiak_2025,MackowiakStrichartzConfAnderson} that \cref{Eq:AGP} should be interpreted in a renormalized sense. This renormalization has to be taken into account in the study of standing waves. The construction of the renormalized operator has been conducted in \cite{MackowiakStrichartzConfAnderson} and will be recall in \cref{Sec-AndersonOp2d}. The renormalized operator is defined through an exponential transform inspired by the one introduced in \cite{Haire_Labbe} to study the continuous parabolic Anderson model, and later used in \cite{debussche_weber_T2,Tzvetkov_2023,tzvetkov2020dimensional} to solve the Anderson nonlinear Schrödinger equation on the torus $\mathbb T^2$ and in \cite{debussche2017solution,debussche2023global} to solve the same equation but on the full space $\mathbb R^2$. Let $Y=(-H)^{-1}\xi$, it is almost surely of regularity $1-$, so one can define $\rho=\mathrm{e}^Y$ and search for a formula for $(H+\xi)u$ when $u$ is of the form $u=\rho v$. A formal computation gives
$$(H+\xi)u=\rho\left(Hv+2\nabla Y\cdot \nabla v + xY\cdot xv + |\nabla Y|^2v\right).$$
Unfortunately, $\nabla Y$ is of regularity $0-$, thus the product $|\nabla Y|^2$ barely fails to exist. In \cite{Mackowiak_2025}, the author proved that the wick product $\wick{|\nabla Y|^2}$ can be defined as the almost sure limit, in suitable distribution spaces of regularity $0-$, of
$$|\nabla Y_N|^2-\mathbb E[|\nabla Y_N|^2],$$
where $Y_N$ is a suitable regularisation of $Y_N$.  Thus, the renormalized operator can be formally defined as
$$(H+\xi)u=\rho\left(Hv+2\nabla Y\cdot \nabla v + xY\cdot xv + \wick{|\nabla Y|^2}v\right).$$
The rigorous construction of $H+\xi$ rely on the associated quadratic form.

Even in dimension 1, the construction of $H+\xi$ is not straightforward as $\xi$ is not a function. On compact interval with no potential, $\p_x^2+\xi$ has first been constructed in \cite{FukushimaNakao} using its quadratic form and has been studied in \cite{DumazLabbeAndersonLoc1D} using a twisted Sturm-Liouville approach, allowing to construct $\p_x^2+\xi$ on unbounded interval too. In \cref{Sec-AndersonOp1d}, we propose a construction of $H+\xi$ as a self-adjoint unbounded operator through its quadratic form. \cref{Prop-IPP1d} shows $\mathrm{D}(H+\xi)=\mathrm{D}(\p_x^2+\xi)\cap \mathrm{D}(|x|^2)$, where $\p_x^2+\xi$ is the unbounded self-adjoint operator on $\mathrm L^2(\mathbb R)$ constructed in \cite{DumazLabbeAndersonLoc1D} and $\mathrm{D}(|x|^2)=\{u\in\mathrm L^2(\mathbb R),xu\in\mathrm L^2(\mathbb R)\}$.\\

We are then interested in standing waves for
\begin{equation}{~}
    \begin{cases}\mathrm{i}\p_t u+ Au+\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases} \label{Eq-AGP}
\end{equation}
in dimension $d\in\{1,2\}$, where $A=H+\xi$. As $A$ is independent of time, standing waves $u(t,x)=\mathrm{e}^{\mathrm{i}\omega t}\phi(x)$ solves \cref{Eq-AGP} if and only if $\phi$ is solution of
\begin{equation}
    -A\phi-\lambda|\phi|^{2\gamma}\phi+\omega \phi = 0. \label{Eq-StationaryAGP}
\end{equation}
This paper aim to construct standing waves in presence of a white noise potential and present some of their properties.

When $\xi=0$ and $H=\Delta$, standing waves are well-known object. See for example \cite{NehariExistence,BerestyckiLions,StraussExistenceSolitary} for existence, \cite{BerestyckiLions} for non-existence, \cite{GidasSymmetry,KwongUniqueness} for uniqueness and \cite{BerestyckiInstabilité,CazenaveLionsStability,CazenaveStabilityInstability,WeinsteinSharpInterpolation,GrillakisStability,GrillakisStabilityII} for results on stability. Let us stress out many of these results relies on the particular structure of \cref{Eq-StationaryAGP} in this case, which is not preserved in presence of potentials. When $\xi=0$ and $H=\Delta-V$ with $V$ a regular and potentially unbounded potential, we refer the reader to \cite{KavianSelfSimilar,FukuizumiStabilityInstability,HiroseStructure,HiroseUniqueness,FukuizumiExponentialDecay,MaedaSymmetry,SelemBifurcation,GuoSeringerMassConcentration,PelinovskyGSEnergyCriticalGP} and references therein for many results on existence, uniqueness, localization and stability.

Nonlinear elliptic equations in presence of white noise potential on compact manifold have been studied in \cite{EulryVariational} using mountain pass methods and in \cite{ZhangConstraintMinAnderson} using constraint minimization. In both \cite{EulryVariational,ZhangConstraintMinAnderson}, the Anderson operator $\Delta+\xi$ is constructed using paracontrolled calculus. We will present here the construction of solution of \cref{Eq-StationaryAGP} through constraint minimization, namely energy minimization at fixed mass and action minimization. This allow to construct a positive solution of \cref{Eq-StationaryAGP} whenever such a solution exists.

In dimension 2, it is known from \cite{Mackowiak_2025,MackowiakStrichartzConfAnderson} that \cref{Eq:AGP} should be interpreted in a renormalized sense. This renormalization has to be taken into account in the study of standing waves. The construction of the renormalized operator has been conducted in \cite{MackowiakStrichartzConfAnderson} and will be recall in \cref{Sec-AndersonOp2d}. The renormalized operator is defined through an exponential transform inspired by the one introduced in \cite{Haire_Labbe} to study the continuous parabolic Anderson model, and later used in \cite{debussche_weber_T2,Tzvetkov_2023,tzvetkov2020dimensional} to solve the Anderson nonlinear Schrödinger equation on the torus $\mathbb T^2$ and in \cite{debussche2017solution,debussche2023global} to solve the same equation but on the full space $\mathbb R^2$. Let $Y=(-H)^{-1}\xi$, it is almost surely of regularity $1-$, so one can define $\rho=\mathrm{e}^Y$ and search for a formula for $(H+\xi)u$ when $u$ is of the form $u=\rho v$. A formal computation gives
$$(H+\xi)u=\rho\left(Hv+2\nabla Y\cdot \nabla v + xY\cdot xv + |\nabla Y|^2v\right).$$
Unfortunately, $\nabla Y$ is of regularity $0-$, thus the product $|\nabla Y|^2$ barely fails to exist. In \cite{Mackowiak_2025}, the author proved that the wick product $\wick{|\nabla Y|^2}$ can be defined as the almost sure limit, in suitable distribution spaces of regularity $0-$, of
$$|\nabla Y_N|^2-\mathbb E[|\nabla Y_N|^2],$$
where $Y_N$ is a suitable regularisation of $Y_N$. Thus, the renormalized operator can be formally defined as
$$(H+\xi)u=\rho\left(Hv+2\nabla Y\cdot \nabla v + xY\cdot xv + \wick{|\nabla Y|^2}v\right).$$
The rigorous construction of $H+\xi$ rely on the associated quadratic form.

Even in dimension 1, the construction of $H+\xi$ is not straightforward as $\xi$ is not a function. On compact interval with no potential, $\p_x^2+\xi$ has first been constructed in \cite{FukushimaNakao} using its quadratic form and has been studied in \cite{DumazLabbeAndersonLoc1D} using a twisted Sturm-Liouville approach, allowing to construct $\p_x^2+\xi$ on unbounded interval too. In \cref{Sec-AndersonOp1d}, we propose a construction of $H+\xi$ as a self-adjoint unbounded operator through its quadratic form. \cref{Prop-IPP1d} shows $\mathrm{D}(H+\xi)=\mathrm{D}(\p_x^2+\xi)\cap \mathrm{D}(|x|^2)$, where $\p_x^2+\xi$ is the unbounded self-adjoint operator on $\mathrm L^2(\mathbb R)$ constructed in \cite{DumazLabbeAndersonLoc1D} and $\mathrm{D}(|x|^2)=\{u\in\mathrm L^2(\mathbb R),xu\in\mathrm L^2(\mathbb R)\}$.\\

We are then interested in standing waves for
\begin{equation}{~}
 \begin{cases}\mathrm{i}\p_t u+ Au+\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases} \label{Eq-AGP}
\end{equation}
in dimension $d\in\{1,2\}$, where $A=H+\xi$. As $A$ is independent of time, standing waves $u(t,x)=\mathrm{e}^{\mathrm{i}\omega t}\phi(x)$ solves \cref{Eq-AGP} if and only if $\phi$ is solution of
\begin{equation}
 -A\phi-\lambda|\phi|^{2\gamma}\phi+\omega \phi = 0. \label{Eq-StationaryAGP}
\end{equation}
This paper aim to construct standing waves in presence of a white noise potential and present some of their properties.

When $\xi=0$ and $H=\Delta$, standing waves are well-known object. See for example \cite{NehariExistence,BerestyckiLions,StraussExistenceSolitary} for existence, \cite{BerestyckiLions} for non-existence, \cite{GidasSymmetry,KwongUniqueness} for uniqueness and \cite{BerestyckiInstabilité,CazenaveLionsStability,CazenaveStabilityInstability,WeinsteinSharpInterpolation,GrillakisStability,GrillakisStabilityII} for results on stability. Let us stress out many of these results relies on the particular structure of \cref{Eq-StationaryAGP} in this case, which is not preserved in presence of potentials. When $\xi=0$ and $H=\Delta-V$ with $V$ a regular and potentially unbounded potential, we refer the reader to \cite{KavianSelfSimilar,FukuizumiStabilityInstability,HiroseStructure,HiroseUniqueness,FukuizumiExponentialDecay,MaedaSymmetry,SelemBifurcation,GuoSeringerMassConcentration,PelinovskyGSEnergyCriticalGP} and references therein for many results on existence, uniqueness, localization and stability.

Nonlinear elliptic equations in presence of white noise potential on compact manifold have been studied in \cite{EulryVariational} using mountain pass methods and in \cite{ZhangConstraintMinAnderson} using constraint minimization. In both \cite{EulryVariational,ZhangConstraintMinAnderson}, the Anderson operator $\Delta+\xi$ is constructed using paracontrolled calculus. We will present here the construction of solution of \cref{Eq-StationaryAGP} through constraint minimization, namely energy minimization at fixed mass and action minimization. This allow to construct a positive solution of \cref{Eq-StationaryAGP} whenever such a solution exists.

Nonlinear elliptic equations in presence of white noise potential on compact manifold have been studied in \cite{EulryVariational} using mountain pass methods and in \cite{ZhangConstraintMinAnderson} using constraint minimization. In both \cite{EulryVariational,ZhangConstraintMinAnderson}, the Anderson operator $\Delta+\xi$ is constructed using paracontrolled calculus. We will present here the construction of solution of \cref{Eq-StationaryAGP} through constraint minimization, namely energy minimization at fixed mass and action minimization. This allow to construct a positive solution of \cref{Eq-StationaryAGP} whenever such a solution exists.

In order to do so, we use constrained minimization methods. In \cref{Sec-EnergyGS}, we study the minimization of energy at fixed mass and prove the following result.

Let $d\in\{1,2\}$, $\lambda,\omega\in\R$, $\gamma>0$ and $A$ the self-adjoint operator constructed in \cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\xi$ a spatial white noise. Let $\D^{1,2}=\mathrm{D}(\sqrt{|A|})$ and $\D^{-1,2}$ its dual. Denote by $\mu_0$ the smallest eigenvalue of $-A$. Let $\omega>-\mu_0$ and define $$\mathcal{S}_\omega = \mathcal{E} + \omega \mathcal{M}$$ and $$\mathcal{N}_\omega =\left\{u\in\D^{1,2}\backslash\{0\},\; \langle \nabla\mathcal{S}_\omega u,u \rangle_{\D^{-1,2},\D^{1,2}}=0\right\}.$$
 Then, there exists $\psi_\omega\in\displaystyle\argmin_{\mathcal{N}_\omega}\mathcal{S}_\omega$ such that $\psi_\omega>0$.

\begin{itemize}
 \item Assume $\gamma>\frac{2}{d}$, let $v\in\W^{1,2}\backslash\{0\}$ and $u=\rho v$, then for any $\alpha>0$, \cref{Eq-EnergyVside} implies
 \begin{align*}
 \mathcal{E}(u_\alpha) &= - \frac{\lambda\alpha^{d\gamma}}{2\gamma+2}\int_{\R^d}|v(x)|^{2\gamma+2}\rho(x/\alpha)^{2\gamma+2}\d x &(I)\\
 &+ \frac{\alpha^2}{2}\int_{\R^d}\rho(x/\alpha)^2|\nabla v(x)|^2\d x &(II)\\
 &+ \frac{1}{2\alpha^2}\int_{\R^d} (1-Y(x/\alpha))\rho(x/\alpha)^2|xv(x)|^2\d x &(III)\\
 &-\frac{1}{2}\langle Z,|u_\alpha|^2\rangle. &(IV)
 \end{align*}
 \cref{Lem-EstimScalTransf,Lem-ProductRule} imply $(IV)=O(\alpha^s)$ as $\alpha$ goes to infinity, for any $s\in(0,2)$, as $Z\in\W^{0-,\infty}$. By dominated convergence, $(III)=O(\alpha^{-2})$ as $\alpha$ goes to infinity. Finally, $(II)=O(\alpha^2)$ and
 $$(I)\sim - \frac{\lambda\alpha^{d\gamma}\rho(0)^{2\gamma+2}}{2\gamma+2}\int_{\R^d}|v(x)|^{2\gamma+2}\d x.$$
 Chose $v = \sqrt{2m}\rho(0)^{-1}h_0$ ($m>0$), then $\mathcal{M}(u_\alpha)$ converges to $m$ whereas $\mathcal{E}(u_\alpha)$ goes to $-\infty$, because $d\gamma>2$. Thus, \cref{Lem-AsympEnergy} implies
 $$\forall m>0,\; \inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty.$$
 \item Assume $\gamma=\frac{2}{d}$ and let $Q$ be the unique positive and radial solution of
 $$\Delta Q + \lambda Q^{2\gamma+1} = Q.$$
 Recall $\mathcal{M}(Q)=m_\lambda^*=\frac{1}{2}\left(\frac{(\gamma+1)J}{\lambda}\right)^{\frac{1}{\gamma}}$. For $c>0$, let $u=c\rho Q$. For any $\alpha>0$, it holds
 \begin{align*}
 \mathcal{E}(u_\alpha) &= \frac{\alpha^2}{2}\int_{\R^d}\rho(x/\alpha)^2|c\nabla Q(x)|^2\d x - \frac{\lambda\alpha^{2}}{2\gamma+2}\int_{\R^d}|cQ(x)|^{2\gamma+2}\rho(x/\alpha)^{2\gamma+2}\d x &(I)\\
 &+ \frac{c^2}{2\alpha^2}\int_{\R^d} (1-Y(x/\alpha))\rho(x/\alpha)^2|xQ(x)|^2\d x -\frac{1}{2}\langle Z,|u_\alpha|^2\rangle. &(II)
 \end{align*}
 As previously $(II)=O(\alpha^s)$ as $\alpha$ goes to infinity, for any $s\in(0,2)$, and for $c\rho(0)\neq 1$,
 $$(I)\sim\frac{\alpha^2\lambda(c\rho(0))^2}{2\gamma+2}\left(1-(c\rho(0))^{2\gamma}\right)\int_{\R^d}|Q(x)|^{2\gamma+2}\d x$$
 as $\alpha$ goes to infinity, using \cref{Eq-PohozaevQ}. Thus, for $c\rho(0)>1$, \cref{Lem-AsympEnergy} implies
 $$\inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=c\rho(0)m^*_\lambda}}\mathcal{E}(u)=-\infty,$$
 that is
 $$\forall m>m^*_\lambda,\; \inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty.$$
 Let $m'>m>0$ and $\phi\in\D^{1,2}$ such that $\mathcal{M}(\phi)=m$. Let $\psi=\sqrt{\frac{m'}{m}}\phi$. Then,
 \begin{align*}
 \mathcal{E}(\phi) &= \frac{m}{m'}\left[\frac{1}{2}\langle -A\psi,\psi\rangle - \frac{\lambda}{2\gamma+2}\left(\frac{m}{m'}\right)^{\gamma}\int_{\R^d}|\psi|^{2\gamma+2}\d x\right]\\
 &=\frac{m}{m'}\mathcal{E}(\psi) + \frac{m\lambda}{m'(2\gamma+2)}\left[1 - \left(\frac{m}{m'}\right)^{\gamma}\right]\int_{\R^d}|\phi|^{2\gamma+2}\d x\\
 &\geqslant \frac{m}{m'}\mathcal{E}(\psi).
 \end{align*}
 Thus,
 $$m'\inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=m}}\mathcal{E}(u) \geqslant m \inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=m}'}\mathcal{E}(u).$$
 In particular, if $\displaystyle\inf_{u\in\D^{1,2},\; \mathcal{M}(u)=m}\mathcal{E}(u)=-\infty$, then $\displaystyle\inf_{u\in\D^{1,2},\; \mathcal{M}(u)=m'}\mathcal{E}(u)=-\infty$. Similarly, if $\displaystyle\inf_{u\in\D^{1,2},\; \mathcal{M}(u)=m'}\mathcal{E}(u)>-\infty$, then $\displaystyle\inf_{u\in\D^{1,2},\; \mathcal{M}(u)=m}\mathcal{E}(u)>-\infty$. Hence, we can define
 \begin{align*}
 m^*_\lambda(\Xi) &= \inf\left\{m>0,\; \inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty\right\}\\
 &= \sup\left\{m>0,\; \inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=m}}\mathcal{E}(u)>-\infty\right\} \in [0,m^*_\lambda]
 \end{align*}
 To conclude, we only have to prove a lower bound for $m^*_\lambda(\Xi)$.

Our aim is now to prove \cref{Th-ACtionGS}. Showing directly that the minimization problem
\begin{equation}
 \inf_{\mathcal{N}_{\omega,\lambda}}\mathcal{S}_{\omega,\lambda}\label{Eq-MinAction}
\end{equation}
has non-negative solutions may be a bit technical as it is not straightforward that
$$u\in\argmin_{\mathcal{N}_{\omega,\lambda}}\mathcal{S}_{\omega,\lambda}\Rightarrow |u|\in\argmin_{\mathcal{N}_{\omega,\lambda}}\mathcal{S}_{\omega,\lambda}.$$
Thus, we will introduce two other, but equivalent, minimization problems which are easier to study. Remark if $u$ solves \cref{Eq-StationaryAGP} for some $\lambda>0$, then $\alpha u$ solves \cref{Eq-StationaryAGP} for $\lambda'=\frac{\lambda}{|\alpha|^{2\gamma}}$. Thus, the parameter $\lambda$ can be seen as a Lagrange multiplier, and solving \cref{Eq-StationaryAGP} for some $\lambda>0$ is equivalent to solving \cref{Eq-StationaryAGP} for all $\lambda>0$. Following an idea of \cite{ROSE1988207}, we define for $u\in\D^{1,2}$
\begin{equation}
 \mathcal{P}_\omega(u)=\langle (-A+\omega) u, u\rangle,\label{Eq-POmega}
\end{equation}
\begin{equation}
 \mathcal{Q}(u) = \int_\R |u|^{2(\gamma+1)}\d x\label{Eq-Q(u)}
\end{equation}
and if $u\neq0,$
\begin{equation}
 \mathcal{J}_\omega(u) = \frac{\mathcal{P}_\omega(u)}{\mathcal{Q}(u)^{1/(\gamma+1)}}.\label{Eq-JOmega}
\end{equation}
Let us stress out $\mathcal{J}_\omega(\alpha u)=\mathcal{J}_\omega(u)$ for any $\alpha\in\C^*$, thus if $u$ minimizes $\mathcal{J}_\omega$, $\alpha u$ minimizes $\mathcal{J}_\omega$ too. Using standard argument of variational calculus, one can show minimizers of both
\begin{equation}
 \inf_{\Atop{u\in\D^{1,2}}{\mathcal{Q}(u)=q}}\mathcal{P}_\omega(u)\label{Eq-MinP/Q=q}
\end{equation}
and
\begin{equation}
 \inf_{\D^{1,2}\backslash\{0\}}\mathcal{J}_\omega\label{Eq-MinJ}
\end{equation}
are solutions of \cref{Eq-StationaryAGP} for some $\lambda>0$. For $\lambda>0$, one can rewrite
$$\mathcal{S}_{\omega,\lambda}(u)=\frac{1}{2}\mathcal{P}_\omega(u)-\frac{\lambda}{2(\gamma+1)}\mathcal{Q}(u)$$
and
\begin{equation}
 \mathcal{N}_{\omega,\lambda} = \left\{u\in\D^{1,2}\backslash\{0\},\; \mathcal{J}_\omega(u) = \lambda \mathcal{Q}(u)^{\gamma/(\gamma+1)} \right\}.\label{Eq-NehariJQ}
\end{equation}
As remarked in \cite{ROSE1988207}, minimizers of \cref{Eq-MinP/Q=q,Eq-MinJ} are essentially the same, that is up to a multiplicative factor, due to the the homogeneity of $\mathcal{J}_\omega$. The following proposition shows minimization problems \cref{Eq-MinAction,Eq-MinP/Q=q,Eq-MinJ} are equivalent, up to a multiplicative factor (see also \cite{LiuRotatingActionGS}).

In order to do so, we use constrained minimization methods. In \cref{Sec-EnergyGS}, we study the minimization of energy at fixed mass and prove the following result.

\begin{theo}\label{Th-EnergyGS}

    Let $d\in\{1,2\}$, $\lambda,\omega\in\mathbb R$, $\gamma>0$ and $A$ be the self-adjoint operator constructed in \cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\xi$ a spatial white noise, $\mathcal D^{1,2}=\mathrm{D}(\sqrt{|A|})$ and $\mathcal D^{-1,2}$ its dual. For $u\in\mathcal D^{1,2}$, we define the mass as
    $$\mathcal{M}(u)=\frac{1}{2}\int_{\mathbb R^d}|u|^2\mathrm{d} x$$
    and the energy as
    $$\mathcal{E}(u)=\frac{1}{2}\langle -Au,u\rangle_{\mathcal D^{-1,2},\mathcal D^{1,2}}-\frac{\lambda}{2\gamma+2}\int_{\mathbb R^d}|u|^{2\gamma+2}\mathrm{d} x.$$
    Then, it holds
    \begin{enumerate}
        \item For $\lambda\leqslant0$, for any $m>0$, there exists $\phi_m\in\mathcal D^{1,2}$ such that $\phi_m>0$ and
        $$\argmin_{\substack{u\in\mathcal D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u) = \left\{\mathrm{e}^{\mathrm{i}\theta}\phi_m,\; \theta\in\mathbb R\right\}.$$
        \item For $\lambda>0$ and $\gamma<\frac{2}{d}$, for any $m>0$, there exists $\phi_m\in\displaystyle\argmin_{u\in\mathcal D^{1,2},\;\mathcal{M}(u)=m}\mathcal{E}(u)$ such that $\phi_m>0$.
        \item For $\lambda>0$ and $\gamma=\frac{2}{d}$, there exists $m_{*,\lambda}(\Xi)>0$ such that for all $m\in(0,m_{*,\lambda}(\Xi))$, there exists $\phi_m\in\displaystyle\argmin_{u\in\mathcal D^{1,2},\;\mathcal{M}(u)=m}\mathcal{E}(u)$ such that $\phi_m>0$. Moreover, there exists $m_{*,\lambda}(\Xi)\leqslant m^*_\lambda(\Xi)\leqslant m^*_\lambda$ such that
        $$\forall m>m^*_\lambda(\Xi), \inf_{\substack{u\in\mathcal D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty,$$
        where $m^*_\lambda$ is the critical mass of the classical nonlinear Schrödinger equation (i.e.\ without any potential).
        \item For $\lambda>0$ and $\gamma>\frac{2}{d}$, for any $m>0$,
        $$\inf_{\substack{u\in\mathcal D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty.$$
    \end{enumerate}

\end{theo}

It is classical that the set of minimal energy states of fixed mass is stable in $\mathcal D^{1,2}$ for \cref{Eq-AGP}. In particular, \cref{Th-EnergyGS} implies the orbital stability of the ground-state $\phi_m$ in the defocusing case ($\lambda<0$). \cref{Prop-ExistenceEnergyGS,Prop-FrequencyEnergyGS} show that, for $\lambda\leqslant 0$ or $\gamma<\frac{2}{d}$, energy minimization at fixed mass provide positive solutions of \cref{Eq-StationaryAGP} for any possible value of $\omega$. We also prove the small mass asymptotic of $\phi_m$ and its frequency $\omega_m$.\\

In \cref{Sec-ActionGS}, we investigate the minimization of action $\mathcal{S}_\omega = \mathcal{E} + \omega \mathcal{M}$. In the focusing case ($\lambda> 0$), one easily verifies the action is not lowerbounded. We thus introduce the Nehari manifold and show the following result.

\begin{theo}\label{Th-ACtionGS}

    Let $d\in\{1,2\}$, $\lambda,\omega\in\mathbb R$, $\gamma>0$ and $A$ the self-adjoint operator constructed in \cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\xi$ a spatial white noise. Let $\mathcal D^{1,2}=\mathrm{D}(\sqrt{|A|})$ and $\mathcal D^{-1,2}$ its dual. Denote by $\mu_0$ the smallest eigenvalue of $-A$. Let $\omega>-\mu_0$ and define $$\mathcal{S}_\omega = \mathcal{E} + \omega \mathcal{M}$$ and $$\mathcal{N}_\omega =\left\{u\in\mathcal D^{1,2}\backslash\{0\},\; \langle \nabla\mathcal{S}_\omega u,u \rangle_{\mathcal D^{-1,2},\mathcal D^{1,2}}=0\right\}.$$
    Then, there exists $\psi_\omega\in\displaystyle\argmin_{\mathcal{N}_\omega}\mathcal{S}_\omega$ such that $\psi_\omega>0$.

\end{theo}

Let us stress out an important property of solution of \cref{Eq-StationaryAGP} compared to the noiseless case. In presence of white noise, standing waves have a limited Sobolev regularity.
\begin{theo}\label{Th-MaxRegSW}
    Let $d\in\{1,2\}$, $\lambda,\omega\in\mathbb R$ and $\gamma>0$. Almost surely any solution $u$ of \cref{Eq-StationaryAGP} belongs to $\mathcal W^{2-\frac{d}{2}-,q}$ for all $q\in[2,+\infty)$, where $\mathcal W^{s,q}$ spaces are defined in \cref{Sec-ConfiningSobolev}. Moreover, almost surely any solution $u$ of \cref{Eq-StationaryAGP} which belongs to $\mathrm{H}^{2-\frac{d}{2},q}(U)$ for some open set $U\subset\mathbb R^d$ and $q\in[2,+\infty]$ vanishes everywhere in $U$.
\end{theo}

In order to prove Theorem~\ref{Th-SynthesExistenceSW}, the text says they "use constrained minimization methods".

- In \cref{Sec-EnergyGS}, they "study the minimization of energy at fixed mass" (Theorem~\ref{Th-EnergyGS}). It is then stated that \cref{Prop-ExistenceEnergyGS,Prop-FrequencyEnergyGS} "show that, for $\lambda\leqslant 0$ or $\gamma<\frac{2}{d}$, energy minimization at fixed mass provide positive solutions of \cref{Eq-StationaryAGP} for any possible value of $\omega$".

- In \cref{Sec-ActionGS}, they instead "investigate the minimization of action $\mathcal{S}_\omega=\mathcal{E}+\omega\mathcal{M}$". In the focusing case ($\lambda>0$), they note that "one easily verifies the action is not lowerbounded", so they "introduce the Nehari manifold" and obtain an action minimizer on it (Theorem~\ref{Th-ACtionGS}) for $\omega>-\mu_0$.

- They also "stress out" that, in the presence of white noise, standing waves have "limited Sobolev regularity" (Theorem~\ref{Th-MaxRegSW}), giving the maximal regularity and a unique-continuation-type statement (if a solution lies in $\mathrm{H}^{2-\frac{d}{2},q}(U)$ on some open set $U$, then it "vanishes everywhere in $U$").

In order to prove the main theorem, the text says they "use constrained minimization methods".

Next, they "study the minimization of energy at fixed mass" and use the following result.

\begin{theo}\label{Th-EnergyGS}

    Let $d\in\{1,2\}$, $\lambda,\omega\in\R$, $\gamma>0$ and $A$ be the self-adjoint operator constructed in \cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\xi$ a spatial white noise, $\D^{1,2}=\mathrm{D}(\sqrt{|A|})$ and $\D^{-1,2}$ its dual. For $u\in\D^{1,2}$, we define the mass as
    $$\mathcal{M}(u)=\frac{1}{2}\int_{\R^d}|u|^2\d x$$
    and the energy as
    $$\mathcal{E}(u)=\frac{1}{2}\langle -Au,u\rangle_{\D^{-1,2},\D^{1,2}}-\frac{\lambda}{2\gamma+2}\int_{\R^d}|u|^{2\gamma+2}\d x.$$
    Then, it holds
    \begin{enumerate}
        \item For $\lambda\leqslant0$, for any $m>0$, there exists $\phi_m\in\D^{1,2}$ such that $\phi_m>0$ and
        $$\argmin_{\substack{u\in\D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u) = \left\{\e^{\complexI\theta}\phi_m,\; \theta\in\R\right\}.$$
        \item For $\lambda>0$ and $\gamma<\frac{2}{d}$, for any $m>0$, there exists $\phi_m\in\displaystyle\argmin_{u\in\D^{1,2},\;\mathcal{M}(u)=m}\mathcal{E}(u)$ such that $\phi_m>0$.
        \item For $\lambda>0$ and $\gamma=\frac{2}{d}$, there exists $m_{*,\lambda}(\Xi)>0$ such that for all $m\in(0,m_{*,\lambda}(\Xi))$, there exists $\phi_m\in\displaystyle\argmin_{u\in\D^{1,2},\;\mathcal{M}(u)=m}\mathcal{E}(u)$ such that $\phi_m>0$. Moreover, there exists $m_{*,\lambda}(\Xi)\leqslant m^*_\lambda(\Xi)\leqslant m^*_\lambda$ such that
        $$\forall m>m^*_\lambda(\Xi), \inf_{\substack{u\in\D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty,$$
        where $m^*_\lambda$ is the critical mass of the classical nonlinear Schrdinger equation (i.e.\ without any potential).
        \item For $\lambda>0$ and $\gamma>\frac{2}{d}$, for any $m>0$,
        $$\inf_{\substack{u\in\D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty.$$
    \end{enumerate}

\end{theo}

It is then stated that \cref{Prop-ExistenceEnergyGS,Prop-FrequencyEnergyGS} show that, for $\lambda\leqslant 0$ or $\gamma<\frac{2}{d}$, energy minimization at fixed mass provides positive solutions of
\begin{equation}
    -A\phi-\lambda|\phi|^{2\gamma}\phi+\omega \phi = 0. \label{Eq-StationaryAGP}
\end{equation}
for any possible value of $\omega$.

Later, they instead "investigate the minimization of action $\mathcal{S}_\omega=\mathcal{E}+\omega\mathcal{M}$". In the focusing case ($\lambda>0$), they note that "one easily verifies the action is not lowerbounded", so they "introduce the Nehari manifold" and obtain an action minimizer on it (Theorem~\ref{Th-ACtionGS}) for $\omega>-\mu_0$.

They also "stress out" that, in the presence of white noise, standing waves have "limited Sobolev regularity" (Theorem~\ref{Th-MaxRegSW}), giving the maximal regularity and a unique-continuation-type statement (if a solution lies in $\mathrm{H}^{2-\frac{d}{2},q}(U)$ on some open set $U$, then it "vanishes everywhere in $U$").

\label{Th-SynthesExistenceSW}
    Let $d\in\{1,2\}$, $\lambda,\omega\in\mathbb R$ and $\gamma>0$. Denote by $\mu_0$ the lowest eigenvalue of $-A$. In each of the following situation, there exists a positive solution of
\begin{equation}
    -A\phi-\lambda|\phi|^{2\gamma}\phi+\omega \phi = 0. \label{Eq-StationaryAGP}
\end{equation}
 in $\mathrm{D}\left(\sqrt{|A|}\right)$,
    \begin{itemize}
        \item $\lambda>0$ and $\omega>-\mu_0$,
        \item $\lambda = 0$ and $\omega=\mu_0$,
        \item $\lambda<0$ and $\omega<-\mu_0$.
    \end{itemize}
    In every other cases, there is no non-trivial non-negative solution in $\mathrm{D}\left(\sqrt{|A|}\right)$.,

Question:
Let \(d\in\{1,2\}\), let \(A=H+\xi\) be the Anderson Hamiltonian in dimension \(d\), and let \(\mu_0\) denote the lowest eigenvalue of \(-A\). For \(\lambda,\omega\in\mathbb R\) and \(\gamma>0\), consider the stationary equation
\[
-A\phi-\lambda|\phi|^{2\gamma}\phi+\omega\phi=0.
\]
Which statement about positive or nonnegative solutions \(\phi\in \mathrm D(\sqrt{|A|})\) is valid?

A positive solution \(\phi\in \mathrm D(\sqrt{|A|})\) exists in each of the following three cases: \(\lambda>0\) and \(\omega>-\mu_0\); \(\lambda=0\) and \(\omega=\mu_0\); \(\lambda<0\) and \(\omega<-\mu_0\). In every other case, there is no non-trivial non-negative solution in \(\mathrm D(\sqrt{|A|})\).

Choices:
B. A positive solution \(\phi\in \mathrm D(\sqrt{|A|})\) exists in each of the following three cases: \(\lambda>0\) and \(\omega\ge -\mu_0\); \(\lambda=0\) and \(\omega=\mu_0\); \(\lambda<0\) and \(\omega\le -\mu_0\). In every other case, there is no non-trivial non-negative solution in \(\mathrm D(\sqrt{|A|})\).
C. If \(\lambda>0\) and \(\omega>-\mu_0\), or if \(\lambda=0\) and \(\omega=\mu_0\), or if \(\lambda<0\) and \(\omega<-\mu_0\), then there exists a non-trivial non-negative solution \(\phi\in \mathrm D(\sqrt{|A|})\).
D. A positive solution \(\phi\in \mathrm D(\sqrt{|A|})\) exists whenever \(\lambda>0\) and \(\omega\in\mathbb R\), whenever \(\lambda=0\) and \(\omega=\mu_0\), and whenever \(\lambda<0\) and \(\omega< -\mu_0\). In every other case, there is no non-trivial non-negative solution in \(\mathrm D(\sqrt{|A|})\).
E. A positive solution \(\phi\in \mathrm D(\sqrt{|A|})\) exists in each of the following three cases: \(\lambda>0\) and \(\omega>-\mu_0\); \(\lambda=0\) and \(\omega\ge \mu_0\); \(\lambda<0\) and \(\omega<-\mu_0\). In every other case, there is no non-trivial non-negative solution in \(\mathrm D(\sqrt{|A|})\).


\label{thm:main}
    Let $\Lambda_\alpha$ be a Delone set associated with an $\alpha$-Kakutani tiling of $\R$. Then $\Lambda_\alpha$ is uniformly spread if and only if
    \begin{equation}\label{eq:special values}
            r_\alpha:=\frac{\log(\min\{\alpha,1-\alpha\})}{\log(1-\min\{\alpha,1-\alpha\})}\in\left\{1,\frac{3}{2},2,3,4  \right\}.
    \end{equation}

A set $\Lambda\subset\R^d$ is \emph{Delone} if it is \emph{uniformly discrete} and \emph{relatively dense}, that is, if there exist constants $0<r,R<\infty$ so that $B(x,r)\cap\Lambda=x$ and $B(y,R)\cap\Lambda\neq\emptyset$ for all $x\in\Lambda$ and $y\in \R^d$, respectively. clearly lattices and their translations are Delone sets in $\R^d$, and there are numerous non-lattice examples that are not periodic in any way. Given such a non-lattice Delone set, a natural question is to ask how ``close'' it is to a lattice in $\R^d$. One natural interpretation of such a question involves the concept of \emph{bounded displacement (BD) equivalence}, where two Delone sets $\Lambda,\Gamma\subset\R^d$ are BD equivalent if there exists a bijection $\varphi:\Lambda\rightarrow \Gamma$ satisfying $\sup_{x\in\Lambda}\|x-\varphi(x)\|<\infty$.
A Delone set $\Lambda\subset\R^d$ is \emph{uniformly spread} if it is BD equivalent to some lattice in $\R^d$, or equivalently, by a simple application of Hall's marriage theorem, if there is some $c>0$ for which $\Lambda$ and $c\cdot\Z^d$ are BD equivalent. See \cite{SmilanskySolomon-2025} for proofs and a comprehensive discussion.

The Delone sets considered in this manuscript are those associated with $\alpha$-Kakutani tilings for $\alpha\in(0,1)$, which form a simple class of multiscale substitution tilings \cite{SmilanskySolomon-2021}. These tilings are defined by a substitution rule on a closed interval $I\subset\R$ of unit length that splits $I$ into two subintervals, a left one of length $\alpha$ and a right one of length $1-\alpha$. Starting with a copy of $I$, which we think of as a \emph{tile}, we apply a uniform inflation and substitute any tile of length greater than $1$. This results in a growing family of patches that exhausts the real line, and limits taken in an appropriate way define a space of tilings of $\R$, all made of interval-tiles of lengths between $\min\{\alpha,1-\alpha\}$ and $1$. Given such a tiling, an \emph{associated} Delone set is obtained by selecting one point from each tile, with the requirements that distinct tiles yield distinct points and that the resulting set is uniformly discrete. A natural choice is to take the left endpoint of each tile. However, since all Delone sets associated with a fixed tiling are BD equivalent, the particular choice of points is immaterial.

Our main result is the following classification.

Our main result is the following classification.

\subsection{Commensurable tilings as primitive substitution tilings}\label{subsec:construction of rho_alpha} Recall our assumption that $\alpha=\min\{\alpha,1-\alpha\}\in(0,1/2]$, and consider a commensurable $\alpha$-Kakutani rule. Let $n\geq m>0$ be coprime integers so that
\begin{equation}\label{eq:rational ratio}
 \frac{n}{m}=r_\alpha=\frac{\log\alpha}{\log(1-\alpha)}.
\end{equation}
If $n=m$ then $\alpha=1/2$, and any $\alpha$-Kakutani tiling is a translation of a lattice tiling in $\R$, with all associated Delone sets uniformly spread. Assume from now on that $n> m$, and consider the associated graph $G_\alpha$. By \eqref{eq:rational ratio} we have
$$
g_\alpha :=\frac{1}{n}\log\frac{1}{\alpha}=\frac{1}{m}\log\frac{1}{1-\alpha}.
$$
Then by adding vertices to $G_\alpha$, the loop of length $\log(1/\alpha)$ and the loop of length $\log(1/(1-\alpha))$ can be partitioned into $n$ and $m$ edges of equal length $g_\alpha$, respectively. This defines a directed graph $G'_\alpha$ with $n+m-1$ vertices and $n+m$ edges of equal length. Label the original vertex with $1$, and assign labels $2,\ldots,n$ and $n+1,\ldots,n+m-1$ to the $(n-1)+(m-1)$ new vertices that are added along the two loops.

\begin{prop}\label{prop:generating patches}
 Consider a commensurable $\alpha$-Kakutani substitution rule with $r_\alpha=n/m$ for $n,m\in\N$ coprime. Let $\ell\in\Z_{\geq 0}$ and put
 \begin{equation}\label{eq:t_l}
 t_\ell:= \ell\cdot g_\alpha=\ell\cdot \frac{1}{n}\log\frac{1}{\alpha}.
 \end{equation}
 Then the (Kakutani) generating patch $F_{t_\ell}(I)$ and the (primitive substitution) generating patch $(\xi\rho_\alpha)^\ell(I)$ are geometrically the same, where $\xi=\alpha^{-1/n}$.
\end{prop}
\begin{proof}
 We prove by induction on $\ell\geq0$. For $\ell=0$, both patches contain only the single tile $I$ and so are geometrically the same. Assume the claim holds for some $\ell\geq0$. Tiles in $F_{t_\ell}(I)$ and $(\xi\rho_\alpha)^\ell(I)$ are either intervals of unit length, that is, copies of $I$, or intervals of length at most $|T_n|=|T_{n+m-1}|=\alpha^{1/n}$. By construction, the patch $F_{t_{\ell+1}}(I)$ is defined by inflating the patch $F_{t_\ell}(I)$ by $\alpha^{-1/n}$ and then substituting every tile of length strictly greater than $1$ according to the $\alpha$-Kakutani rule. The tiles that are substituted are exactly the unit intervals from $F_{t_\ell}(I)$. On the other hand, the tiles in $(\xi\rho_\alpha)^{\ell+1}(I)$ are the result of one application of $\xi\rho_\alpha$ on the tiles in $(\xi\rho_\alpha)^\ell(I)$. For copies of $I$ this amounts to an application of the $\alpha$-Kakutani rule once and inflation by $\xi=\alpha^{-1/n}$, while all other tiles in the patch are simply inflated by the same constant, and so $F_{t_{\ell+1}}(I)$ and $(\xi\rho_\alpha)^{\ell+1}(I)$ are the same.
\end{proof}

We focus here on Delone sets in $\R$.
For a detailed discussion about equivalence relations on Delone sets in $\R^d$ we refer to the book chapter \cite{SmilanskySolomon-2025}.
Let $|U|$ denote the Lebesgue measure of a bounded measurable set $U\subset \R$. A Delone set $\Lambda\subset\R$ has \emph{asymptotic density} $d_\Lambda$ if
$$
\lim_{n\rightarrow\infty}\frac{\#(\Lambda\cap U_n)}{|U_n|}
$$
exists and is equal to $d_\Lambda$ for any sequence \emph{van Hove sequence} $(U_n)$, that is, a sequence of bounded measurable sets so that $|(\partial U_n)^{+\varepsilon}|/|U_n|\rightarrow0$ for all $\varepsilon>0$, where $A^{+\varepsilon}$ is the $\varepsilon$ neighbourhood of $A\subset\R$. The \emph{discrepancy} of $\Lambda$ with respect to the parameter $\beta>0$ and a measurable set $B\subset\R$ is defined as
$$
\disc{\Lambda}{\beta}{U}:=\left|\#(\Lambda \cap U)-\beta\cdot|U|\right|.
$$
A criterion established by Laczkovich \cite{Laczkovich-1992} provides a necessary and sufficient condition for a Delone set in $\R^d$ to be uniformly spread. In the case of a Delone set $\Lambda\subset \R$ with asymptotic density $d_\Lambda$, Laczkovich's criterion can be stated in the following way.
\begin{thm}\label{thm:Laczkovich}
 A Delone set $\Lambda\subset \R$ with asymptotic density $d_\Lambda$ is uniformly spread if and only if
 $$
 \disc{\Lambda}{d_\Lambda}{U}\leq C\cdot|\partial U|^{+1}
 $$
 for all bounded measurable sets $U\subset\R$.
\end{thm}

\begin{thm}\label{thm:BD_substitution}
 Let $\rho$ be a primitive substitution rule in $\R^d$ on a set of prototiles that are biLipschitz homeomorphic to closed balls, and let $\Lambda$ be a Delone set associated with a tiling generated by $\rho$. Let $\ell\ge 2$ be the minimal index such that the corresponding eigenspace satisfies $W_{\lambda_\ell} \nsubseteq \mathbf{1}^\perp$, where $\mathbf{1}=(1,\ldots,1)\in \R^k$. Then
\begin{enumerate}
\item
If $\absolute{\lambda_\ell}<\lambda_1^{\frac{d-1}{d}}$ then $\Lambda$ is uniformly spread.
\item
If $\absolute{\lambda_\ell}>\lambda_1^{\frac{d-1}{d}}$ then $\Lambda$ is not uniformly spread.
\end{enumerate}
\end{thm}
We note that if $\absolute{\lambda_\ell}=\lambda_1^{\frac{d-1}{d}}$ then $\Lambda$ may exhibit either property, for an example see \cite{FrettlohSmilanskySolomon-2021}.
In the case $d=1$, checking the conditions in Theorem \ref{thm:BD_substitution} is reduced to determining if $\lambda_\ell$ is in the interior of the closed unit disk or in its complement.

The discrepancy can be large however, and by \cite[\S 6]{SmilanskySolomon-2025} there exist arbitrarily large intervals $U$ for which
\begin{equation}\label{eq:incomm_discrepancy}
 \disc{\Lambda_\alpha}{d_{\Lambda_\alpha}}{U}=\Omega\left(\frac{|U|}{\log(|U|)}\right).
\end{equation}
By Laczkovich's criterion as stated in Theorem \ref{thm:Laczkovich}, if $\Lambda_\alpha$ is uniformly spread then there exists a constant $C>0$ so that for any interval $U$ the discrepancy is bounded by $\disc{\Lambda_\alpha}{d_{\Lambda_\alpha}}{U}\leq 2C$. By \eqref{eq:incomm_discrepancy} this is clearly not the case, and so $\Lambda_\alpha$ is not uniformly spread. In fact, for any fixed incommensurable $\alpha$-Kakutani rule there are continuously many associated Delone sets that are pairwise BD non-equivalent, see \cite{SmilanskySolomon1-2022}.

Assume now that $r_\alpha=n/m$ with $n>m>0$ coprime integers. By Proposition \ref{prop:primitive cover}, any $\alpha$-Kakutani tiling is also generated by a primitive substitution rule $\rho_\alpha$ with substitution matrix $M_\alpha$ as in \eqref{eq:substitution matrix}. All tiles are closed interval and are therefore closed balls in $\R$, and so we can apply Solomon's criterion for uniform spreadness, stated above as Theorem \ref{thm:BD_substitution}, with $d=1$. By
Proposition \ref{prop:eigenspacesnotperp} and the preceding discussion, the eigenvalue $\lambda_2$ of $M_\alpha$ is non-zero and satisfies $W_{\lambda_2}\nsubseteq{\bf 1}^\perp$, and so $\ell=2$. Since $\gcd(n-m,n)=\gcd(m,n)=1$, Proposition \ref{prop:nounitroots} applies to $f_\alpha(x)$ as defined in \eqref{eq:charpoly}, and so $|\lambda_2|\neq1$. Combining the above, we deduce that an associated Delone set $\Lambda_\alpha$ is uniformly spread if and only if $|\lambda_2|<1$. By Proposition \ref{prop:PVpolynomials}, for this to hold $f_\alpha(x)$ must be one of the four polynomials listed in \eqref{eq:PVpoly}. We thus conclude that the values of $\alpha\in(0,1/2)$ for which $\Lambda_\alpha$ is uniformly spread are given by
\begin{equation}\label{eq:PVvalues}
 r_\alpha=\frac{n}{m}\in\left\{\frac32,\frac21,\frac31,\frac41\right\},
\end{equation}
and the proof is complete.
\end{proof}

For simplicity of presentation, we will assume throughout that $\alpha\in(0,1/2]$, and so $\alpha=\min\{\alpha,1-\alpha\}$. We note that if $r_\alpha=1$ then $\alpha=1/2$, and the left endpoints of tiles in any corresponding tiling are simply translated lattices in $\R$. When $r_\alpha=2$ then $\alpha=1/\phi^2$, where $\phi$ is the \emph{golden ratio}. The other values of $\alpha$ are harder to present explicitly, though the cases $r_\alpha=3/2$ and $r_\alpha=3$ are related to the \emph{supergolden ratio} and the \emph{plastic ratio}, respectively. The latter is the smallest \emph{Pisot–Vijayaraghavan (PV) number}, and the case $r_\alpha=4$ is related to the second smallest PV number, see \S \ref{sec:remarks}. For all other values of $\alpha$ the corresponding Delone sets are never uniformly spread.

\subsection{Key steps of the proof}

A complete proof requires some further preparation, but  we can aready outline the main steps of the argument. First, in the \emph{incommensurable} case $r_\alpha\notin\Q$, the discrepancy in $\alpha$-Kakutani tilings, that is, the fluctuation of the number of tiles that appear in large patches relative to their expected number (namely, the density), can be shown to be large. This, when combined with Laczkovich's criterion for uniform spreadness \cite{Laczkovich-1992}, is enough to establish that any associated Delone set is not BD equivalent to a lattice. This is a special case of \cite[Theorem 8.2]{SmilanskySolomon-2021}, which states that any Delone set associated with an incommensurable multiscale substitution tiling in $\R^d$ is not uniformly spread.

The case $r_\alpha\in\Q$, namely the \emph{commensurable} case, is rather different, and is the new result contained in this contribution.
The first step is to observe that, by introducing labelled prototiles and modifying the substitution rule in a suitable way, one can construct a standard primitive substitution that ``covers'' the commensurable multiscale substitution construction, in the sense that it generates substitution tilings that are geometrically identical to the corresponding $\alpha$-Kakutani tilings described above.
 The problem of uniform spreadness of Delone set associated with primitive substitution tilings can be approached using Solomon's criterion \cite{Solomon-2014}. This criterion, which is a careful application of the aforementioned Laczkovich criterion, is given in terms of a comparison between the leading Perron-Frobenius eigenvalue of the associated substitution matrix, and the subsequent eigenvalues. In the special case considered here, this becomes a question of classification of certain \emph{Pisot–Vijayaraghavan (PV) polynomials} \cite{DubickasJankauskas-2014}, from which the special values that appear in \eqref{eq:special values} can be extracted.

We complete our discussion by showing that such scarcity of uniformly spread constructions does not necessarily carry over when moving away from $\alpha$-Kakutani tilings to more complicated multiscale substitution tilings. For example, when considering $(\alpha,\beta)$-Kakutani tilings, generated by a substitution rule on $I$, now split into three tiles of lengths $\alpha,\beta,1-\alpha-\beta$, we present a countable set of parameters $(\alpha,\beta)$ for which the associated Delone sets are uniformly spread.

A complete proof requires some further preparation, but the post-theorem introduction outlines these main steps.

1. **Assume** $\alpha\in(0,1/2]$, so $\alpha=\min\{\alpha,1-\alpha\}$.

2. **Incommensurable case** ($r_\alpha\notin\Q$): show that the *discrepancy* in $\alpha$-Kakutani tilings—“the fluctuation of the number of tiles that appear in large patches relative to their expected number (namely, the density)”—“can be shown to be large.” Combine this with “Laczkovich's criterion for uniform spreadness” to conclude that any associated Delone set “is not BD equivalent to a lattice,” hence is not uniformly spread. This is noted as a special case of \cite[Theorem 8.2]{SmilanskySolomon-2021}.

3. **Commensurable case** ($r_\alpha\in\Q$): this is described as “rather different” and the “new result.”
   - **Cover by a standard primitive substitution**: “by introducing labelled prototiles and modifying the substitution rule in a suitable way,” construct “a standard primitive substitution that ‘covers’ the commensurable multiscale substitution construction,” meaning it generates substitution tilings “geometrically identical” to the $\alpha$-Kakutani tilings.
   - **Apply Solomon's criterion**: study uniform spreadness for the resulting primitive substitution tilings via “Solomon's criterion,” formulated as “a comparison between the leading Perron-Frobenius eigenvalue of the associated substitution matrix, and the subsequent eigenvalues.”
   - **Reduce to PV polynomial classification**: “In the special case considered here, this becomes a question of classification of certain Pisot–Vijayaraghavan (PV) polynomials,” and “from [this] the special values that appear in \eqref{eq:special values} can be extracted,” yielding exactly the listed set $\left\{1,\frac{3}{2},2,3,4\right\}$.

4. **Further remark beyond the theorem**: they add that this “scarcity” need not persist for more general multiscale substitution tilings (e.g. $(\alpha,\beta)$-Kakutani tilings), where they “present a countable set of parameters $(\alpha,\beta)$ for which the associated Delone sets are uniformly spread.”

A complete proof requires some further preparation, but the post-theorem introduction outlines these main steps.

1. **Assume** $\alpha\in(0,1/2]$, so $\alpha=\min\{\alpha,1-\alpha\}$.

2. **Incommensurable case** ($r_\alpha\notin\Q$): show that the *discrepancy* in $\alpha$-Kakutani tilings—“the fluctuation of the number of tiles that appear in large patches relative to their expected number (namely, the density)”—“can be shown to be large.” Combine this with “Laczkovich's criterion for uniform spreadness” to conclude that any associated Delone set “is not BD equivalent to a lattice,” hence is not uniformly spread. This is noted as a special case of Smilansky–Solomon, *[title and year not provided]*, Theorem 8.2.

3. **Commensurable case** ($r_\alpha\in\Q$): this is described as “rather different” and the “new result.”
   - **Cover by a standard primitive substitution**: “by introducing labelled prototiles and modifying the substitution rule in a suitable way,” construct “a standard primitive substitution that ‘covers’ the commensurable multiscale substitution construction,” meaning it generates substitution tilings “geometrically identical” to the $\alpha$-Kakutani tilings.
   - **Apply Solomon's criterion**: study uniform spreadness for the resulting primitive substitution tilings via “Solomon's criterion,” formulated as “a comparison between the leading Perron-Frobenius eigenvalue of the associated substitution matrix, and the subsequent eigenvalues.”
   - **Reduce to PV polynomial classification**: “In the special case considered here, this becomes a question of classification of certain Pisot–Vijayaraghavan (PV) polynomials,” and “from [this] the special values that appear in
\begin{equation}\label{eq:special values}
            r_\alpha:=\frac{\log(\min\{\alpha,1-\alpha\})}{\log(1-\min\{\alpha,1-\alpha\})}\in\left\{1,\frac{3}{2},2,3,4  \right\}.
    \end{equation}
can be extracted,” yielding exactly the listed set $\left\{1,\frac{3}{2},2,3,4\right\}$.

4. **Further remark beyond the theorem**: they add that this “scarcity” need not persist for more general multiscale substitution tilings (e.g. $(\alpha,\beta)$-Kakutani tilings), where they “present a countable set of parameters $(\alpha,\beta)$ for which the associated Delone sets are uniformly spread.”

Biconditional or Equivalence
Classification or Bijection

Question:
Let \(\Lambda_\alpha\subset \mathbb{R}\) be a Delone set associated with an \(\alpha\)-Kakutani tiling of \(\mathbb{R}\), where an \(\alpha\)-Kakutani tiling is built from repeatedly splitting a unit interval into a left subinterval of length \(\alpha\) and a right subinterval of length \(1-\alpha\). A Delone set is called uniformly spread if it is bounded-displacement equivalent to a lattice in \(\mathbb{R}\), equivalently, if there exist \(c>0\) and a bijection \(\varphi:\Lambda_\alpha\to c\mathbb{Z}\) such that \(\sup_{x\in\Lambda_\alpha}|x-\varphi(x)|<\infty\). Define
\[
r_\alpha:=\frac{\log(\min\{\alpha,1-\alpha\})}{\log(1-\min\{\alpha,1-\alpha\})}.
\]
Which of the following statements is equivalent to \(\Lambda_\alpha\) being uniformly spread?

\(r_\alpha\in\left\{1,\tfrac{3}{2},2,3,4\right\}.\)

Choices:
B. \(r_\alpha\in\mathbb{Q}.\)
C. \(r_\alpha\in\left\{1,\tfrac{3}{2},2,3,4\right\}\cup (\mathbb{R}\setminus\mathbb{Q}).\)
D. \(r_\alpha\in\left\{1,\tfrac{3}{2},2,3,4\right\}\text{ or }|\lambda_2|\le 1\text{ for the substitution matrix arising in the commensurable case}.\)
E. \(r_\alpha\in\left\{1,\tfrac{3}{2},2,3,4,\tfrac{5}{2}\right\}.\)

special-values classification replaced by mere commensurability

iff replaced by a sufficient condition plus all incommensurable parameters

strict eigenvalue inequality in commensurable case weakened to boundary-inclusive condition

finite PV-derived list enlarged by a nearby rational value


\label{Theorem ID of alpha Cauchy}
The $\alpha$-Cauchy variable $\mathcal{C}_\alpha$ is infinitely divisible if  $1 < \alpha \leq 6/5$.

\begin{theorem}\label{Kristiansen's Theorem}{\cite{Kri94}}
    The independent product $\G_2 \times \W $ is infinitely divisible for any non-negative random variable $\W$. See the beginning of section \ref{section Gamma type} for the definition of $\G_2 $.
\end{theorem}

\begin{equation}\label{Introduction X}
        \mathbb{E}(X^s) =  \frac{\sin(\pi/\a)}{\pi}\frac{ \Gamma\left(\frac{1}{\a} +\frac{ \varepsilon p s}{\a}\right)\Gamma\left(1 - \frac{1}{\a} - \frac{ \varepsilon p s}{\a}\right)}{ \Gamma(2 + s) },
    \end{equation}

\begin{theorem}\label{Theorem ID of alpha Cauchy}
The $\alpha$-Cauchy variable $\mathcal{C}_\a$ is infinitely divisible if  $1 < \a \leq 6/5$.
\end{theorem}

\begin{theorem}\label{thm Gamma type}
    For positive $a,b,c,d$, we consider the random variable $\X_{a,b,c,d}$ satisfying
       \begin{equation}\label{def Xabcd}
        \mathbb{E}(\X_{a,b,c,d}^s) = \frac{ \Gamma(c) }{ \Gamma(a)\Gamma(b)} \frac{ \Gamma(a + s)\Gamma(b-s)}{ \Gamma(c+ d s) }, \quad -a < s < b.
    \end{equation}
\begin{itemize}
    \item[(I)] $\X_{a,b,c,d}$ does not exist if one of the following conditions holds: \vspace{1mm}
    \begin{itemize}
        \item[(1)]  $d > 2$; \vspace{1mm}
        \item[(2)]   $c < ad$; \vspace{1mm}
        \item[(3)]  $3a + b > c, \; d=2$;  \vspace{1mm}
        \item[(4)] $1 < d \leq 2,\; a +b \geq 1, \;c < ad -d + f(d)$.
    \end{itemize}
    \vspace{1mm}
    \item[(II)] $\X_{a,b,c,d}$ exists if one of the following conditions holds: \vspace{1mm}
    \begin{itemize}
        \item[(1)]  $d \leq 1, c \geq ad$;\vspace{1mm}
        \item[(2)]  $1 < d \leq 2,\; a +b \leq 1, \;c \geq ad -d + f(d)$;\vspace{1mm}
        \item[(3)] $1 < d \leq 2,\; \frac{2c}{d} \geq 3a+b, \;2(\frac{c}{d}-a)(\frac{c}{d}+\frac{1}{2}-a)\geq a+b$.\vspace{1mm}
    \end{itemize}
\end{itemize}
 where $f$ is the function defined in Theorem \ref{Theorem admissible parameter domain}.
\end{theorem}

For $\alpha > 1$, the $\alpha$-Cauchy random variable, denoted by $\mathcal{C}_\alpha$, was introduced by Yano, Yano and Yor \cite{YYY09} in 2009. The density of $\mathcal{C}_\alpha$ is
\begin{equation}\label{eq density alpha Cauchy}
 f_{\mathcal{C}_\alpha}(t) = \frac{\sin(\pi/\alpha)}{2\pi/\alpha}\frac{1}{1+|t|^\alpha}, \quad \alpha > 1, \,\, t \in \mathbb{R}.
\end{equation}
In particular, when $\alpha = 2$, it is
the standard Cauchy distribution, also called the Lorentzian distribution or Lorentz distribution. It is the ratio of two independent normal random variables. It is a continuous distribution describing resonance behavior. It also describes the distribution of horizontal distances at which a line segment tilted at a random angle cuts the x-axis.
It is important in physics as it is the solution to the differential equation describing forced resonance, while in spectroscopy it is the description of the line shape of spectral lines. In this paper, we study the infinite divisibility of $\mathcal{C}_\alpha$.

A random variable is infinitely divisible (ID) if for any positive integer $n$, it is the sum of $n$ independent and identically distributed random variables. Typical examples of infinitely divisible distributions are normal, Cauchy, half Cauchy, Stable, Gamma, Poisson, and Student-t.
The theory of infinitely divisible distribution is a classic topic in probability theory. In 1929, de Finetti first introduced the concept of infinite divisibility, and then Kolmogorov, Lévy, and Khintchine further developed the theory. Infinitely divisible distributions are closely related to limit theorems and the theory of Lévy processes and have important applications in finance, insurance, biology, physics, and signal processing. Proving or disproving infinite divisibility of a certain distribution can sometimes be quite sophisticated. In some cases, it is more convenient to consider subclasses of infinitely divisible distributions, like self-decomposable (SD) distributions, generalized gamma convolutions (GGC) and hyperbolically completely monotone (HCM) distributions. We refer to Steutel and van Harn \cite{SH04}, Sato \cite{Sat99} and Bondesson \cite{Bon92} for abundant properties of these subclasses.

In 1987,
 Bondesson \cite{Bon87} proved that the half Cauchy variable $ |\mathcal{C}_2|$ is ID. In 1998, Di\'edhiou \cite{Die98} proved that $ |\mathcal{C}_2|$ is SD.
Yano, Yano, and Yor proved $ |\mathcal{C}_\alpha|$ is ID for any $1 < \alpha \leq 2$ by Bondesson's argument, see, Theorem 2.7 in \cite{YYY09}. Furthermore, they
proposed three open questions (\cite[Remark 2.9]{YYY09}):
\begin{enumerate}
 \item Is $\mathcal{C}_\alpha$ SD (or ID at least)?
 \item Is $|\mathcal{C}_\alpha|$ SD (or ID at least)?
 \item Is $|\mathcal{C}_\alpha|^{-p}$ SD (or ID at least) for $p > 0$?
\end{enumerate}

There are very few results on the above questions.
In the present paper, we study the infinite divisibility of
$ \mathcal{C}_\alpha$ and $ |\mathcal{C}_\alpha|^{p}, \, p \in \mathbb{R}$, and partially give positive answers to the questions of Yano, Yano, and Yor.

\subsection{Infinite divisibility of \texorpdfstring{$\alpha$}{}-Cauchy variables}
It is well-known that the Cauchy variable $\mathcal{C}_2$ is infinitely divisible and even stable. But when $\alpha \neq 2$, there is no known result on the infinite divisibility of $\mathcal{C}_\alpha$. The following theorem gives a positive answer to this question when $1 < \alpha \leq 6/5$.

\begin{abstract}
 We study the infinite divisibility of the $\alpha$-Cauchy variable $\mathcal{C}_\alpha$, $\alpha > 1$. The distribution of $\mathcal{C}_2$ is the well-known Cauchy distribution, which is infinitely divisible and even stable. But when $\alpha \neq 2$, there is no known result on the infinite divisibility of $\mathcal{C}_\alpha$.
 In this paper, we prove that $\mathcal{C}_\alpha$ is infinitely divisible if $1 < \alpha \leq 6/5$, and we give some sufficient conditions for $|\mathcal{C}_\alpha|^p, \, p\in \mathbb{R},$ to be infinitely divisible, which partially answers the open questions raised by Yano, Yano and Yor in 2009. In the proofs,
 a class of positive random variables having moments of Gamma type plays an important role, and we investigate the conditions for their existence.
\end{abstract}

\section{Introduction}
For $\alpha > 1$, the $\alpha$-Cauchy random variable, denoted by $\mathcal{C}_\a$, was introduced by Yano, Yano and Yor \cite{YYY09} in 2009. The density of $\mathcal{C}_\a$ is
\begin{equation}\label{eq density alpha Cauchy}
 f_{\mathcal{C}_\a}(t) = \frac{\sin(\pi/\a)}{2\pi/\a}\frac{1}{1+|t|^\a}, \quad \alpha > 1, \,\, t \in \mathbb{R}.
\end{equation}
In particular, when $\alpha = 2$, it is
the standard Cauchy distribution, also called the Lorentzian distribution or Lorentz distribution. It is the ratio of two independent normal random variables. It is a continuous distribution describing resonance behavior. It also describes the distribution of horizontal distances at which a line segment tilted at a random angle cuts the x-axis.
It is important in physics as it is the solution to the differential equation describing forced resonance, while in spectroscopy it is the description of the line shape of spectral lines. In this paper, we study the infinite divisibility of $\mathcal{C}_\a$.

\subsection{Infinite divisibility of \texorpdfstring{$\a$}{}-Cauchy variables}
It is well-known that the Cauchy variable $\mathcal{C}_2$ is infinitely divisible and even stable. But when $\alpha \neq 2$, there is no known result on the infinite divisibility of $\mathcal{C}_\a$. The following theorem gives a positive answer to this question when $1 < \a \leq 6/5$.

Yano, Yano and Yor introduced the $\alpha$-Cauchy variable to study the first hitting times of point $T_{\{a\}}(X_\alpha)$ for one-dimensional \textbf{symmetric} stable L\'evy process of index $\alpha$ ($1< \alpha \leq 2$) starting from zero, denoted by $X_\alpha = (X_\alpha(t): t\geq 0)$.
In 2019, Letemplier and Simon \cite{LS19} studied the first hitting time of zero $\tau(\alpha, \rho)$ for a real strictly $\alpha$-stable process (can be asymmetric) starting from one. The symmetric case $\tau(\alpha, \frac{1}{2})$ coincides with $T_{\{-1\}}(X_\alpha)$ mentioned above. They conjectured that $\tau(\alpha, \rho)$ is infinitely divisible for $\alpha \in (1,2]$.

We note that both $\mathcal{C}_\a$ and $|\mathcal{C}_\a|$ are not infinitely divisible when $\a$ is large enough. Indeed,
the fractional moments of the half $\alpha$-Cauchy variable $ |\mathcal{C}_\a|$ are of the form
\begin{equation*}
 \mathbb{E}\left[|\mathcal{C}_\a|^s\right] = \frac{\sin(\pi/\a)}{\pi} \Gamma\left(\frac{1}{\a} +\frac{s}{\a}\right)\Gamma\left(1 - \frac{1}{\a} - \frac{s}{\a}\right), \quad -1 < s < \a-1.
\end{equation*}
The right hand side tends to $1/(1+s)$ as $\a$ tends to infinity, thus
\begin{equation}
 |\mathcal{C}_\a| \claw \mathcal{U}, \qquad \text{as} \, \,\,\a \rightarrow \infty,
\end{equation}
where $\mathcal{U}$ is a uniform random variable on $(0,1)$. Since the support of $ \mathcal{U} $ is bounded, $\mathcal{U}$ is not ID; see Sato \cite[Corollary 24.4]{Sat99}. Similarly, as $\alpha$ tends to infinity, $\mathcal{C}_\a$ tends to a uniform random variable in $(-1,1)$ in distribution, which is not ID.
It is probable that $\mathcal{C}_\a$ is ID if and only if $1 < \a \leq 2$.

\begin{theorem}\label{Theorem ID of half alpha Cauchy}
Let $p \in (0, \infty)$.
The random variable $|\mathcal{C}_\a |^{ \varepsilon p}, \varepsilon = \pm 1,$ is infinitely divisible if
\begin{equation}\label{eq range p}
p \geq
 \begin{cases}
 (\a + 1)/3 ,& \quad \varepsilon = 1, \, 1 < \a \leq 2. \\
 \a/2 ,& \quad \varepsilon = 1, \, \a > 2. \\
 \a/2 ,& \quad \varepsilon = - 1, \, 1 < \a \leq 2. \\
 (2\a - 1)/3 ,& \quad \varepsilon = -1, \, \a > 2. \\
 \end{cases}
\end{equation}
\end{theorem}

\section{Proofs of Theorems \ref{Theorem ID of alpha Cauchy} and \ref{Theorem ID of half alpha Cauchy}}
\label{section proofs}
\subsection{Proof of Theorem \ref{Theorem ID of alpha Cauchy}}
We apply Proposition \ref{Proposition Steutel coro VI.4.8} to prove Theorem \ref{Theorem ID of alpha Cauchy}. Let $Y$ be a positive random variable with density $f$ given by
$$ f(x) = \frac{1}{2}(1+x)e^{-x}, \qquad x >0. $$
The fractional moments of $Y$ are
\begin{equation*}
 \mathbb{E}[Y^s] = \frac{1}{2}(s+2)\Gamma(s+1), \quad s > -1.
\end{equation*}
 Let $1 < \alpha \leq 2$.
Recall that
\begin{equation}\label{eq Mellin alpha Cauchy}
 \mathbb{E}\left[|\mathcal{C}_\a|^s\right] = \frac{\sin(\pi/\a)}{\pi} \Gamma\left(\frac{1}{\a} +\frac{s}{\a}\right)\Gamma\left(1 - \frac{1}{\a} - \frac{s}{\a}\right), \quad -1 < s < \a-1.
\end{equation}
Therefore, by Proposition \ref{Proposition Steutel coro VI.4.8}, if there exists a symmetric distribution
$V_\alpha$ having fractional moments,
\begin{equation}
 \mathbb{E} [|V_\alpha|^s] = \mathbb{E}\left[|\mathcal{C}_\a|^s\right] / \mathbb{E}[Y^s] = \frac{2\alpha \sin(\pi/\a)}{\pi} \frac{\Gamma\left(1+\frac{1}{\a} +\frac{s}{\a}\right)\Gamma\left(1 - \frac{1}{\a} - \frac{s}{\a}\right)}{\Gamma(3+s)},
\end{equation}
then $\mathcal{C}_\alpha$ is infinitely divisible. Compared with \eqref{def Xabcd}, we see that
\begin{equation}
 |V_\alpha| \elaw \X_{1+1/\alpha, 1-1/\alpha, 3, \alpha}^{1/\alpha}.
\end{equation}
By \eqref{fact equiv}, the existence of $V_\alpha$ is equivalent to the non-negativity of $E_{\alpha, 2+\alpha}^{2}(-t)$ for any $t>0$.
By formula (5.1.31) in \cite{GKMR20}, we have
\begin{equation}
\label{LT of 3ML}
 \int_0^{\infty} e^{-st} e^{-t} t^{\beta-1} E^\gamma_{\alpha, \beta}(-t^\alpha)dt = \frac{(s+1)^{-\beta}}{(1+(s+1)^{-\alpha})^\gamma}
\end{equation}
where $\alpha > 0, \beta > 0, \gamma > 0, s > 0$.
Two integrable functions have the same Laplace transform only if they differ on a set of Lebesgue measure zero, see, e.g. Proposition 1.2 in \cite{SSV12}. Moreover, a function is completely monotone if and only if it is the Laplace transform of positive measure on $[0, \infty)$ (see Theorem 1.4 in \cite{SSV12}), and we have the following equivalence
$$ \text{ $E_{\alpha, 2+\alpha}^{2}(-t) \geq 0$ for any $t>0$ $\Longleftrightarrow$
$ \frac{(s+1)^{-2-\alpha}}{(1+(s+1)^{-\alpha})^2} $ is completely monotone. } $$
The multiplication of completely monotone functions is still completely monotone, see \cite[Corollary 1.6]{SSV12}. Therefore, the
complete monotonicity of $\frac{(s+1)^{-1 - \alpha/2}}{1+(s+1)^{-\alpha} }$ implies that of $\frac{(s+1)^{-2-\alpha}}{(1+(s+1)^{-\alpha})^2}$.
 Using again \eqref{LT of 3ML}, we obtain that
$$ \text{$E_{\alpha, 1+\alpha/2}(-t) \geq 0$ for any $t>0$ } \text{$\Longleftrightarrow$
$\frac{(s+1)^{-1 - \alpha/2}}{1+(s+1)^{-\alpha} }$ is completely monotone.} $$
If $1< \alpha \leq 6/5$, then $1+\alpha/2 \geq 4\alpha/3$, by Theorem \ref{Theorem admissible parameter domain}, we have
$E_{\alpha, 1+\alpha/2}(-t) \geq 0$ for any $t>0$.
This completes the proof.

Let $\T_{\nu}$ be the Student-t random variable with density
$$ \frac{\Gamma(\frac{\nu +1}{2})}{\sqrt{\pi \nu} \Gamma(\frac{\nu }{2})} \left( 1 + \frac{t^2}{\nu}\right)^{-\frac{\nu +1}{2}}, \quad \nu > 0, \; t \in \mathbb{R}.$$
Note that $|\T_{1}| \elaw |\mathcal{C}_2|$.
\begin{theorem}\label{thm half student}
 For $0 < \nu \leq 1$, the half Student-t random variable $|\T_\nu|$ is infinitely divisible.
\end{theorem}

Yano, Yano and Yor introduced the $\alpha$-Cauchy variable to study the first hitting times of point $T_{\{a\}}(X_\alpha)$ for one-dimensional \textbf{symmetric} stable L\'evy process of index $\alpha$ ($1< \alpha \leq 2$) starting from zero, denoted by $X_\alpha = (X_\alpha(t): t\geq 0)$.
In 2019, Letemplier and Simon \cite{LS19} studied the first hitting time of zero $\tau(\alpha, \rho)$ for a real strictly $\alpha$-stable process (can be asymmetric) starting from one. The symmetric case $\tau(\alpha, \frac{1}{2})$ coincides with $T_{\{-1\}}(X_\alpha)$ mentioned above. They conjectured that $\tau(\alpha, \rho)$ is infinitely divisible for $\alpha \in (1,2]$.

More precisely, by formula (5.12) in \cite{YYY09}, the relation between the first hitting time of point for stable processes and the alpha Cauchy distributions is as follows
$$\hat{X}_\alpha ( T_{\{a\}}(X_\alpha)) \stackrel{(d)}{=} |a| \mathcal{C}_\alpha, $$
where $\hat{X}_\alpha$ is an independent copy of $X_\alpha$. Using Bochner’s subordination -see e.g. Theorem 30.1 in \cite{Sat99}- the infinite divisibility of $T_{\{a\}}(X_\alpha)$ would entail the infinite divisibility of $ \mathcal{C}_\alpha$. Vise versa, the infinite divisibility of $\mathcal{C}_\alpha$ would support the conjecture on the infinite divisibility of $T_{\{a\}}(X_\alpha)$.

We note that both $\mathcal{C}_\alpha$ and $|\mathcal{C}_\alpha|$ are not infinitely divisible when $\alpha$ is large enough. Indeed,
the fractional moments of the half $\alpha$-Cauchy variable $ |\mathcal{C}_\alpha|$ are of the form
\begin{equation*}
    \mathbb{E}\left[|\mathcal{C}_\alpha|^s\right] =  \frac{\sin(\pi/\alpha)}{\pi} \Gamma\left(\frac{1}{\alpha} +\frac{s}{\alpha}\right)\Gamma\left(1 - \frac{1}{\alpha} - \frac{s}{\alpha}\right), \quad -1 < s < \alpha-1.
\end{equation*}
The right hand side tends to $1/(1+s)$ as $\alpha$ tends to infinity, thus
\begin{equation}
    |\mathcal{C}_\alpha| \stackrel{(d)}{\longrightarrow} \mathcal{U}, \qquad \text{as} \, \,\,\alpha \rightarrow \infty,
\end{equation}
where $\mathcal{U}$ is a uniform random variable on $(0,1)$. Since the support of $ \mathcal{U} $ is bounded, $\mathcal{U}$ is not ID; see Sato \cite[Corollary 24.4]{Sat99}. Similarly, as $\alpha$ tends to infinity, $\mathcal{C}_\alpha$ tends to a uniform random variable in $(-1,1)$ in distribution, which is not ID.
It is probable that $\mathcal{C}_\alpha$ is ID if and only if  $1 < \alpha \leq 2$.

\subsection{Infinite divisibility of powers of half \texorpdfstring{$\alpha$}{}-Cauchy variable}
We know from Bondesson \cite{Bon87} that $ |\mathcal{C}_\alpha|$ is ID for $1 < \alpha \leq 2$. We have seen that $|\mathcal{C}_\alpha|$ is not infinitely divisible when $\alpha$ is large enough. It is also probable that $|\mathcal{C}_\alpha|$ is ID if and only if  $1 < \alpha \leq 2$.

The following theorem describes the infinite divisibility of powers of half $\alpha$-Cauchy variable.

\begin{theorem}\label{Theorem ID of half alpha Cauchy}
Let $p \in (0, \infty)$.
The random variable $|\mathcal{C}_\alpha |^{ \varepsilon p}, \varepsilon = \pm 1,$ is infinitely divisible if
\begin{equation}\label{eq range p}
p \geq
    \begin{cases}
   (\alpha + 1)/3 ,& \quad \varepsilon = 1, \, 1 < \alpha \leq 2. \\
     \alpha/2 ,& \quad \varepsilon = 1, \, \alpha > 2. \\
     \alpha/2 ,& \quad \varepsilon = - 1, \, 1 < \alpha \leq 2. \\
     (2\alpha - 1)/3 ,& \quad \varepsilon = -1, \, \alpha > 2. \\
    \end{cases}
\end{equation}
\end{theorem}

In particular, this theorem recovers and extends the result of Bondesson mentioned above.

\subsection{Sketch of the proofs} We first sketch the proof of Theorem \ref{Theorem ID of alpha Cauchy}.
We will prove the infinite divisibility of $\alpha$-Cauchy ($1<\alpha \leq 6/5$) by proving that it is a sym-gamma(2) mixture; see Proposition \ref{Proposition Steutel coro VI.4.8}.
This motivates us to study the existence condition of a class of positive random variables having moments of Gamma type satisfying
       \begin{equation}\label{Introduction Xabcd}
        \mathbb{E}(\X_{a,b,c,d}^s) =  \frac{ \Gamma(c) }{ \Gamma(a)\Gamma(b)} \frac{ \Gamma(a + s)\Gamma(b-s)}{ \Gamma(c+ d s) }, \quad -a < s < b,
    \end{equation}
for positive $a,b,c,d$. We finish the proof by non-negativity conditions of the Mittag-Leffler function.

We then sketch the proof of Theorem \ref{Theorem ID of half alpha Cauchy}.
By Kristiansen's Theorem (see Theorem \ref{Kristiansen's Theorem}), the existence of a random variable $X$ satisfying
         \begin{equation}\label{Introduction X}
        \mathbb{E}(X^s) =  \frac{\sin(\pi/\alpha)}{\pi}\frac{ \Gamma\left(\frac{1}{\alpha} +\frac{ \varepsilon p s}{\alpha}\right)\Gamma\left(1 - \frac{1}{\alpha} - \frac{ \varepsilon p s}{\alpha}\right)}{ \Gamma(2 + s) },
    \end{equation}
for $s$ in some interval, will imply the infinite divisibility of $|\mathcal{C}_\alpha|^{\varepsilon p}, \varepsilon = \pm 1$. Therefore, we thoroughly investigate the conditions for the existence of random variables of the form \eqref{Introduction Xabcd}, and formulate our results as Theorem \ref{thm Gamma type}. Applying this theorem to \eqref{Introduction X}, we can prove Theorem \ref{Theorem ID of half alpha Cauchy}.

The rest of this paper is organized as follows.
 In Section \ref{section Preliminaries}, we introduce the three-parametric Mittag-Leffler function. We then state some
useful known propositions and theorems that will be used for the proofs of our theorems.
In Section \ref{section Gamma type},
we explicitly state and prove some necessary conditions and some sufficient conditions for the existence of some random variables having moments of Gamma type satisfying \eqref{Introduction Xabcd}, which serve as the foundation for proving the above two theorems. In Section \ref{section proofs}, we prove Theorems \ref{Theorem ID of alpha Cauchy} and \ref{Theorem ID of half alpha Cauchy}. In section \ref{section apllication of new Gamma type}, we give three other applications of Theorem \ref{thm Gamma type}.

To prove Theorem~\ref{Theorem ID of alpha Cauchy}, they "prove the infinite divisibility of $\alpha$-Cauchy ($1<\alpha \leq 6/5$) by proving that it is a sym-gamma(2) mixture; see Proposition~\ref{Proposition Steutel coro VI.4.8}." This leads them to "study the existence condition of a class of positive random variables having moments of Gamma type" with
\[
\mathbb{E}(\X_{a,b,c,d}^s) =  \frac{ \Gamma(c) }{ \Gamma(a)\Gamma(b)} \frac{ \Gamma(a + s)\Gamma(b-s)}{ \Gamma(c+ d s) }, \quad -a < s < b,
\]
for positive $a,b,c,d$. They "finish the proof by non-negativity conditions of the Mittag-Leffler function."

To prove the main theorem, they "prove the infinite divisibility of $\alpha$-Cauchy ($1<\alpha \leq 6/5$) by proving that it is a sym-gamma(2) mixture". We use the following proposition.

\begin{proposition}
    \label{Proposition Steutel coro VI.4.8}{\cite[Corollary VI.4.8]{SH04}}
    A random variable $X$ with $X \elaw Z \times Y$ is infinitely divisible
if $Z$ and $Y$ are independent, $Z$ has a symmetric distribution and $Y$ has the
density $f$ given by
$$ f(x) = \frac{1}{2}(1+x)e^{-x}, \qquad x >0.  $$
\end{proposition}

This leads them to "study the existence condition of a class of positive random variables having moments of Gamma type" with
\[
\mathbb{E}(\X_{a,b,c,d}^s) =  \frac{ \Gamma(c) }{ \Gamma(a)\Gamma(b)} \frac{ \Gamma(a + s)\Gamma(b-s)}{ \Gamma(c+ d s) }, \quad -a < s < b,
\]
for positive $a,b,c,d$. They "finish the proof by non-negativity conditions of the Mittag-Leffler function."

Question:
Let \(\alpha\) satisfy \(1<\alpha\le 6/5\), and let \(\mathcal C_\alpha\) denote the \(\alpha\)-Cauchy random variable with density
\[
f_{\mathcal C_\alpha}(t)=\frac{\sin(\pi/\alpha)}{2\pi/\alpha}\frac{1}{1+|t|^\alpha},\qquad t\in\mathbb R.
\]
Which of the following conclusions about \(\mathcal C_\alpha\) is valid under these hypotheses?

The random variable \(\mathcal C_\alpha\) is infinitely divisible; that is, for every positive integer \(n\), its distribution can be written as the distribution of a sum of \(n\) independent and identically distributed random variables.

Choices:
B. The random variable \(\mathcal C_\alpha\) is infinitely divisible for every \(\alpha>1\); that is, for every positive integer \(n\), its distribution can be written as the distribution of a sum of \(n\) independent and identically distributed random variables.
C. There exists at least one positive integer \(n\ge 2\) such that the distribution of \(\mathcal C_\alpha\) can be written as the distribution of a sum of \(n\) independent and identically distributed random variables.
D. The random variable \(\mathcal C_\alpha\) is self-decomposable; equivalently, for every \(c\in(0,1)\), there exists an independent random variable \(Y_c\) such that \(\mathcal C_\alpha\stackrel{d}=c\mathcal C_\alpha+Y_c\).
E. For each \(1<\alpha\le 6/5\), there exists a decomposition \(\mathcal C_\alpha\stackrel{d}=Z\times Y\) with \(Z\) and \(Y\) independent, \(Y\) having density \(\tfrac12(1+x)e^{-x}\) on \((0,\infty)\), and \(Z\) symmetric and itself infinitely divisible.

for-every-n quantifier in infinite divisibility weakened to existence of one n

ID conclusion replaced by stronger self-decomposability

Steutel product criterion only requires symmetric Z, not Z itself infinitely divisible