text:
\section{Introduction}

In the past decade, there has been a great deal of interest in studying the physical properties of graphene both theoretically and experimentally\cite{BeenakkerPRL2006,BeenakkerRMP2008,Castro NetoRMP2009}. In graphene, the low-energy excitations are massless and chiral Dirac fermions with linear energy dispersion near the Dirac point. Because of the unique energy dispersion in graphene, there are some special phenomena such as the unconventional quantum Hall effect, Klein paradox, etc\cite{Castro NetoRMP2009,BaiPRB2009,BaiPRB2007,BaiSuperMicros2011,KatsnelsonNature2006}. Particularly relevant to the present work, the presence of the Dirac Fermions in the graphene-based superconductor junctions results in the specular Andreev reflection\cite{BeenakkerPRL2006,BeenakkerRMP2008}.

The Andreev reflection\cite{AndreevJETP1964} describes the tunneling phenomenon of electron excitation converting into hole excitation by the superconducting pair potential. The interface between a metal and a superconductor can reflect a negatively charged electron incident from the metal into a positively charged hole, while the missing charge enters the superconductor forming a Cooper pair. Usually the hole is reflected back along the path of the incident electron in the conventional materials, which is called ``retro-Andreev reflection''. However, the Andreev reflection in undoped graphene is specular---the so-called ''specular Andreev reflection'', in which the reflected angle is inverted\cite{BeenakkerPRL2006,BeenakkerRMP2008,BaiSuperMicros2011,BenjaminPRB2008}. Because the graphene needs to be described by the Dirac-like equation rather than the usual Schrodinger equation, the superconductive graphene needs to be described by the Dirac--Bogoliubov--de Gennes equation rather than the Bogoliubov--de Gennes equation for usual superconductors\cite{BeenakkerPRL2006,BeenakkerRMP2008,BhattacharjeePRL2006,BaiPLA2010,BaiPLA2008,BaiAPL2008}. However, graphene is not a natural superconductor. Recent research has shown that superconductivity can be induced in a graphene layer in the presence of a superconducting electrode by means of the proximity effect\cite{BeenakkerPRL2006,TitovPRB2006,BhattacharjeePRL2006}. So far, the physical properties of the graphene-based superconductor junctions have been extensively studied and many important results have been obtained. Most of them focus on the transport properties of the graphene/superconductive graphene interface and only the conductance is considered\cite{BeenakkerPRL2006,BaiSuperMicros2011,BhattacharjeePRL2006,BaiPLA2010,JongPRB1994,BeenakkerPRB1992,
BlonderPRB1982}, which prompts us to investigate the transport properties through a superconducting barrier in graphene and especially focus on the shot noise spectrum.
In this work, based on the Dirac--Bogoliubov--de Gennes equation and the Blonder, Tinkham, and Klapwijk (BTK) formulation, we derive the shot noise formula and provide numerical results of the transport properties in the graphene/superconductive graphene/graphene (G/SG/G) heterostructure. Dependence of the shot noise spectrum on the structure parameters and its physical mechanisms are discussed.

\section{Theory and Model}
 We consider transmission through the G/SG/G heterostructure occupying the $x$-$y$ plane, the schematic of which is shown in Fig. 1. The growth direction of the graphene is taken to be the $x$-axis. The left G region extends from $x=-\infty$ to $x=0$ and the right G region extends from $x=a$ to $x=+\infty$. The superconductive region occupies the $0<x<a$ region. The electron and hole excitations are described by the Dirac--Bogoliubov--de Gennes equation\cite{BeenakkerPRL2006,BaiSuperMicros2011,BhattacharjeePRL2006,BaiPLA2010,JongPRB1994,BeenakkerPRB1992,
 BlonderPRB1982}:
 \begin{equation}
 \left( {\begin{array}{*{20}{c}}
   {H_\alpha-E_F} & {\Delta\left( r\right)}  \\
   {\Delta{^*\left( r\right)}} & {E_F-H_\alpha}  \\
\end{array}} \right)\Psi_\alpha = E\Psi_\alpha.
\label{DBG-equation}
\end{equation}
Here, $\Psi_\alpha=\left( \psi_{A\alpha},\psi_{B\alpha},\psi_{A\bar{\alpha}}^*,-\psi_{B\bar{\alpha}}^* \right)$ are the four-component wave functions for the electron and hole spinors. The index $\alpha$ denotes $K$ or $K'$ for the electrons or holes near the Dirac $K$ and $K'$ points. $\bar{\alpha}$ takes values $K'\left( K \right)$ for $\alpha=K\left( {K'} \right)$. $E_F$ denotes the Fermi energy. $A$ and $B$ denote the two inequivalent sites in the hexagonal lattice of graphene. The Hamiltonian $H_\alpha$ is given by
\begin{equation}
H_\alpha = -i\hbar v_F\left[ \sigma_x\partial_x + \rm{sgn} \left(\alpha \right)\sigma_y\partial_y \right]+U \left( r \right),
\end{equation}
where $v_F$ denotes the Fermi velocity of the quasiparticles in graphene and $\rm{sgn} \left( \alpha \right)$ takes the values of $\pm 1$ for $\alpha=K\left( K ' \right)$.

For $0<x<a$, the superconducting electrode on the top of the graphene layer induces a nonzero pair potential $\Delta\left( \bf{r} \right)$ via the proximity effect. We model the pair potential as
\begin{equation}
\Delta\left( \bf{r} \right) = \{{\begin{array}{l}
 \ 0, {\kern 50pt}  \rm{others},  \\
 \ \Delta_0 e^{i\phi}, {\kern 20pt}  0 < x < a,  \\
 \end{array}}
\end{equation}
where $\Delta_0$ and $\phi$ are the amplitude and the phase of the induced superconductive order parameter. The electrostatic potential $U\left( \bf{r} \right)$ in the G and SG regions can be tuned independently by a gate voltage or by doping. We take
\begin{equation}
U\left( \bf{r} \right) = \{{\begin{array}{l}
\ 0, {\kern 50pt}  \rm{others},  \\
\ -U_0, {\kern 25pt}  0 < x < a.  \\
 \end{array}}
\end{equation}

Eq. (\ref{DBG-equation}) can be solved straightforwardly to yield the wave functions $\Psi$ in the G and SG regions, respectively. In the G region, for the electrons and holes traveling in the $\pm x$ directions with a transverse momentum $k_y=q$ and energy $\varepsilon$, the wave functions are given by
\begin{equation}
\begin{array}{l}
 \begin{array}{*{20}c}
  { \Psi _N ^{e\pm}  = \frac{exp\left( {iqy\pm ikx} \right)}{\sqrt{\cos \alpha}}\left({ \begin{array}{l}
 \begin{array}{*{20}c}
 {exp\left( {\mp i\alpha/2} \right)}\\
   \end{array}\\
   \begin{array}{*{20}c}
   {\pm exp\left( {\pm i\alpha/2} \right)}\\
   \end{array}\\
   \begin{array}{*{20}c}
   {\kern 30pt}{0}\\
   \end{array}\\
   \begin{array}{*{20}c}
   {\kern 30pt}{0}\\
   \end{array}\\
   \end{array}} \right),}\\
\end{array}\\
 \begin{array}{*{20}c}
  { \Psi _N ^{h\pm}  = \frac{exp\left( {iqy\pm ik'x} \right)}{\sqrt{\cos \alpha'}}\left( {\begin{array}{l}
 \begin{array}{*{20}c}
 {\kern 30pt}{0}\\
 \end{array}\\
   \begin{array}{*{20}c}
 {\kern 30pt}{0}\\
 \end{array}\\
   \begin{array}{*{20}c}
 {exp\left( {\mp i\alpha'/2} \right)}\\
 \end{array}\\
   \begin{array}{*{20}c}
 {\mp exp\left( {\pm i\alpha'/2} \right)}\\
 \end{array}\\
   \end{array}} \right),}\\
   \end{array}\\
 \end{array}
\end{equation}
with
\begin{equation}
\begin{array}{l}
 \begin{array}{*{20}c}
 {\sin \alpha  =\frac{\hbar v_F q}{\varepsilon + E_F}},\\
\end{array}\\
 \begin{array}{*{20}c}
 {\sin \alpha'  =\frac{\hbar v_F q}{\varepsilon - E_F}},\\
\end{array}\\
 \begin{array}{*{20}c}
 {k = \frac{\varepsilon + E_F}{\hbar v_F}\cos \alpha },\\
\end{array}\\
 \begin{array}{*{20}c}
 {k' = \frac{\varepsilon - E_F}{\hbar v_F}\cos \alpha' }.\\
\end{array}\\
 \end{array}
\end{equation}
$\alpha$ is the incident angle of the electron and $\alpha'$ is the reflection angle of the hole. Note that for the Andreev process to take place, the maximum incident angle of the electron is given by
\begin{equation}
\alpha_c =\arcsin\left( {\frac{\left| {\varepsilon - E_F} \right|}{\varepsilon + E_F}}\right).
\end{equation}

In the SG region, the quasiparticles are mixtures of the electrons and holes. The wave functions of these quasiparticles moving along the $\pm x$-direction with the transverse momentum $q$ and energy $\varepsilon$ has the form of
\begin{equation}
\begin{array}{l}
 \begin{array}{*{20}c}
  { \Psi _S ^{e\pm}  = e^{iqy\pm i\left( {k_0 - i\kappa} \right)}\left({\begin{array}{l}
 \begin{array}{*{20}c}
  {{\kern 10pt}exp\left( {-i\beta} \right)}\\
  \end{array}\\
 \begin{array}{*{20}c}
  {\pm exp\left( {-i\beta \pm i\gamma} \right)}\\
  \end{array}\\
 \begin{array}{*{20}c}
  {{\kern 10pt}exp\left( {-i\phi} \right)}\\
  \end{array}\\
 \begin{array}{*{20}c}
  {\pm exp\left({ \pm i\gamma -i\phi} \right)}\\
  \end{array}\\
 \end{array} }\right), }\\
\end{array}\\
 \begin{array}{*{20}c}
  { \Psi _S ^{h\pm}  = e^{iqy\mp i\left( {k_0 + i\kappa} \right)}\left({\begin{array}{l}
 \begin{array}{*{20}c}
  {{\kern 10pt}exp\left( {i\beta} \right)}\\
  \end{array}\\
 \begin{array}{*{20}c}
  {\mp exp\left( {i\beta \mp i\gamma} \right)}\\
  \end{array}\\
 \begin{array}{*{20}c}
  {{\kern 10pt}exp\left( {-i\phi} \right)}\\
  \end{array}\\
 \begin{array}{*{20}c}
  {\mp exp\left( {\mp i\gamma -i\phi} \right)}\\
  \end{array}\\
 \end{array} }\right). } \\
 \end{array}\\
 \end{array}
 \label{SGWaveFunctions}
\end{equation}
The parameters $\beta$, $\gamma$, $k_0$, and $\kappa$ are defined by
\begin{equation}
\begin{array}{l}
 \begin{array}{*{20}c}
{\beta =\{ \begin{array}{l}
 \arccos\left( {\varepsilon/\Delta_0} \right), {\kern 50pt}  \varepsilon < \Delta_0 ,\\
 -i arcosh\left( {\varepsilon/\Delta_0}\right), {\kern 33pt}  \varepsilon > \Delta_0 , \\
 \end{array}}\\
\end{array}\\
 \begin{array}{*{20}c}
 {\gamma =\arcsin\left[ {\hbar v_F q/\left( U_0 + E_F \right)}\right],}\\
\end{array}\\
 \begin{array}{*{20}c}
 {k_0 = \sqrt{\left( E_F + U_0 \right)^2/\left( \hbar v_F\right)^2 - q^2},}\\
\end{array}\\
 \begin{array}{*{20}c}
 {\kappa = \frac{\left( E_F +U_0 \right)\Delta_0}{\left( \hbar v_F \right)^2 k_0}\sin\beta ,}\\
\end{array}\\
 \end{array}
\end{equation}
Taking into account both the Andreev and normal reflection processes, the wave functions in the left G, SG, and right G regions can be written as
\begin{equation}
\begin{array}{l}
 \begin{array}{*{20}c}
 {\Psi_1 = \Psi_N^{e +}+r_c\Psi_N^{e -}+r_{Ac}\Psi_N^{h-},}
\end{array}\\
 \begin{array}{*{20}c}
 {\Psi_2 = A\Psi_S^{e +}+B\Psi_S^{e -}+C\Psi_S^{h+}+D\Psi_S^{h-},}\\
\end{array}\\
 \begin{array}{*{20}c}
 {\Psi_3 = t_c\Psi_N^{e +}+t_{Ac}\Psi_N^{h+},}\\
\end{array}\\
 \end{array}
 \label{WaveFunctions}
\end{equation}
respectively. Here, $r_c$ and $r_{Ac}$ are the amplitudes of the normal and Andreev reflections, respectively; $t_c$ and $t_{Ac}$ are the amplitudes of the normal and Andreev transmissions, respectively. $A$, $B$, $C$, and $D$ are the amplitudes of electronlike and holeslike quasiparticles in the SG region. All the amplitudes in Eq. (\ref{WaveFunctions}) can be determined by demanding wave function continuity at the interfaces. These boundary conditions are given by
\begin{equation}
\Psi_1\left( 0 \right)=\Psi_2\left( 0 \right), {\kern 10pt} \Psi_2\left( a \right)=\Psi_3\left( a \right).
\end{equation}
The scattering amplitudes can be obtained by numerically solving these continuity equations.

The electron and hole operators of the outgoing states are related to the electron and hole operators of the incoming states via the scattering matrix\cite{BlanterRep2000,MuzykantskiiPRB1994},
\begin{equation}
\left( {\begin{array}{*{20}{c}}
   {b_{Le}}  \\
   {b_{Re}}  \\
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
   {S_{LL}^{e e}} & {S_{RL}^{e e}}  \\
   {S_{LR}^{e e}} & {S_{RR}^{e e}}  \\
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
   {a_{Le}}  \\
   {a_{Re}}  \\
\end{array}} \right),{\kern 10pt}\\
\left( {\begin{array}{*{20}{c}}
   {b_{Lh}}  \\
   {b_{Rh}}  \\
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
   {S_{LL}^{e h}} & {S_{RL}^{e h}}  \\
   {S_{LR}^{e h}} & {S_{RR}^{e h}}  \\
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
   {a_{Lh}}  \\
   {a_{Rh}}  \\
\end{array}} \right),
\label{ScatteringMatrix}
\end{equation}
where the element $S^{e e}$ gives the outgoing electron current amplitude in response to an incoming electron current amplitude and $S^{e h}$ gives the outgoing hole current amplitude in response to an incoming electron current amplitude. The generalized current operator for the electrons and holes in the left electrode can be written as\cite{BlanterRep2000,MuzykantskiiPRB1994,AnantramPRB1996}
\begin{equation}
I_L\left( t \right)  = \frac{e}{2\pi \hbar}\int_0^\infty dEdE' e^{i\left( {E-E'}\right)t/\hbar}\left[ {\begin{array}{*{20}{c}}{<a_{Le}^+\left( E \right) a_{Le}\left( {E'} \right)> - <b_{Le}^+\left( E \right) b_{Le}\left( {E'} \right)>}\\ {+ <b_{Lh}^+\left( {-E} \right) b_{Lh}\left( {-E'} \right)>}\\
\end{array}}\right].
\end{equation}
Substituting the scattering matrix in Eq. (\ref{ScatteringMatrix}), we can obtain
\begin{equation}
I_\alpha \left( t \right)  = \frac{e}{2\pi \hbar}\sum_{\mu m\gamma p}\int_0^\infty dEdE' e^{i\left( {E-E'}\right)t/\hbar}\left[ {\begin{array}{*{20}{c}}{a_{\mu e m}^+\left( E \right) A_{\mu\gamma}^{m p}\left( {\alpha,E,E'}\right) a_{\gamma e p}\left( {E'}\right)}\\
{ + \sum_{n} a_{\mu h m}^+\left( E \right) S_{\alpha\mu n m}^h\left( {E'}\right) a_{\gamma h p}\left({E'} \right)}\\
\end{array}}\right],
\end{equation}
where
\begin{equation}
A_{\mu\gamma}^{m p}\left( {\alpha,E,E'}\right) = \delta_{\mu\alpha}\delta_{\gamma\alpha}\delta_{m n}\delta_{p n} - \sum_n S_{\alpha\mu n m}^{e+}\left( E \right) S_{\alpha\gamma n p}^e \left( {E'} \right).
\end{equation}

The general expression for the current fluctuations between contacts $\alpha$ and $\beta$ is
\begin{equation}
S_{\alpha\beta}\left( {t-t'} \right) = \frac{1}{2} <\Delta I_\alpha\left( t \right)\delta I_\beta\left( {t'} \right) + \delta I_\beta\left( {t'}\right)\Delta I_\alpha\left( t \right)>,
\end{equation}
with its Fourier transform
\begin{equation}
2\pi\delta\left( {\omega - \omega'} \right)S_{\alpha\beta}\left( \omega \right) =  <\Delta I_\alpha\left( \omega \right)\delta I_\beta\left( {\omega'} \right) + \delta I_\beta\left( {\omega'}\right)\Delta I_\alpha\left( \omega \right)> .
\end{equation}
We restrict our consideration to coherent tunneling and neglect the Coulomb interaction. In the zero-frequency limit with $\omega=0$, providing all the information above, we can express the noise power as
\begin{equation}
S_{\alpha\beta}=\frac{e^2}{2\pi\hbar}\sum_{\mu m \gamma p}\int_0^\infty dE\times \left[ {\begin{array}{*{20}{c}}
   {\begin{array}{*{20}{c}}{A_{\mu\gamma}^{m p}\left( {\alpha,E,E}\right) A_{\gamma\mu}^{p m}\left( {\beta,E,E}\right)}\\
   {\times{f_{\mu e}\left( E \right)\left[ {1-f_{\eta e}\left( E\right)} \right] + f_{\eta e}\left( E \right)\left[ {1-f_{\mu e}\left( E \right)} \right]}}\\
   \end{array} }  \\
  {\begin{array}{*{20}{c}}{+\sum_{nl} S_{\alpha\mu n m}^{h*}\left( E \right) S_{\alpha\gamma n p}^h\left( E\right) S_{\beta\gamma l p}^{h*}\left( E \right) S_{\beta\mu l m}^h\left( E \right)}\\
   {\times{f_{\mu e}\left( {-E} \right)\left[ {1-f_{\eta e}\left( {-E}\right)} \right] + f_{\eta e}\left( {-E} \right)\left[ {1-f_{\mu e}\left( {-E} \right)} \right]}}\\
 \end{array}   }  \\
\end{array}} \right].
\end{equation}
By introducing the distribution function for the electrons $f_e\left( E \right)=\left[ {exp\left[ {\left( {E-eV}\right)/k_B T}\right]+1}\right]^{-1}$ and that for the holes $f_h\left( E \right)=\left[ {exp\left[ {\left( {E+eV}\right)/k_B T}\right]+1}\right]^{-1}$, the zero-temperature conductance can be obtained from the current operator and from the usual quantum statistical assumptions for the averages and correlations of the electron and hole operators in the normal reservoirs as
\begin{equation}
G\left( {eV} \right) = G_0\int_0^{\alpha_c}\left( {1-\left| r_c \right|^2 + \left| r_{Ac}\right|^2\frac{\cos\alpha'}{\cos\alpha}} \right)\cos\alpha d\alpha .
\end{equation}
The shot noise power can be obtained as
\begin{equation}
S\left( 0 \right)=4eG_0\int_0^{\alpha_c}\left[ {\left|r_c \right|^2 \left|t_c \right|^2 + \left|r_{Ac} \right|^2 \left|t_{Ac} \right|^2\left( \frac{\cos\alpha'}{\cos\alpha} \right)^2}\right]\cos\alpha d\alpha ,
\end{equation}
where $G_0=4e^2 N\left( {eV}\right)/h$ is the ballistic conductance of metallic graphene, $V$ is the bias voltage, and $N\left( \varepsilon \right)= \left( {\varepsilon+E_F} \right) w/\left( {\pi\hbar v_F}\right)$ denotes the number of available channels for a graphene sample of width $w$.

\section{Numerical results and discussion}

Now we present numerical results for the Andreev reflection coefficients and the tunneling conductance for the $G/SG/G$ junction with $U_0\neq 0$. In this condition, there is a large mismatch of Fermi surfaces on the G and SG sides. Such a mismatch is well known to act as an effective barrier\cite{BaiSuperMicros2011,BhattacharjeePRL2006,BaiPLA2010}. The transport properties of the two cases of $E_F\gg \Delta_0$ ($E_F=0$) and $E_F\ll \Delta_0 $ are significantly different. In the case of $E_F\gg \Delta_0$, i.e., the incident electron and the reflected hole both lie in the conduction band, which results in the ``retro-Andreev reflection'', while in the case of $E_F\ll \Delta_0 $, only the ``specular Andreev reflection'' takes place, since the incident electron in the conduction band is converted into the reflected hole in the valence band\cite{BeenakkerPRL2006,BaiSuperMicros2011,BhattacharjeePRL2006,BaiPLA2010}. In general, it is very difficult to reach the regime $E_F\ll \Delta_0$ in experiment\cite{BeenakkerPRL2006}. So we only consider the case of $E_F\gg \Delta_0$ and the condition of comparable $E_F$ and $\Delta_0$, in the latter of which both normal Andreev reflection and specular Andreev reflection play roles and we can see that the retroreflection crosses over to specular Andreev reflection.

Firstly, we consider dependence of the transport properties on the SG-barrier thickness. The tunneling conductance through the SG-barrier as a function of the thickness for different potential strengths $U_0$ is shown in Fig. 2. Dimensionless thickness $k_0 a$ is used. The solid and dashed lines correspond to $U_0/E_F = 2$ and 10, respectively. When the bias voltage is small, the conductance exhibits oscillation features. Their oscillation amplitudes decay with increase of the thickness of the SG layer. The oscillation period of the shot noise power is the same as that of the conductance. The shot noise characterises correlations of the current. It can be seen from the curves that the growing trend of the shot noise is opposite to that of the conductance. When the conductance reaches the maximum value, the shot noise approaches the minimum value. This can be interpreted by the relation between the shot noise and the properties of the scatterer. A coherent conductor with all the transmission channels open (The open channel means that the transmission probability is close to one.) has minimal shot noise with the Fano factor approaching 0. A coherent conductor with all closed channels (The closed channel means that the transmission probability is close to zero.) has maximal shot noise with the Fano factor approaching 1. The strength of the shot noise is in the middle of the two limits when a conductor has open and closed channels coexistent. Therefore large transmission probabilities enhance the conductance and suppress the shot noise. It can also be seen in the panels (b) and (d) that for the case of $E_F=\Delta_0$ the conductance and shot noise decrease with the increase of the thickness of the SG layer. This is because that the parameter $\kappa$ is proportional to $\left( {\Delta/\hbar v_F}\right)\sin\beta$ for identical $U_0$ and it is in the exponential form of $e^{\pm\kappa a}$ in the wave function as shown in Eq. (\ref{SGWaveFunctions}). It is larger for the case of $E_F=\Delta_0$ than for the case of $E_F\gg\Delta_0$, which results in the quick decrease of the shot noise and the conductance. On the other hand, we can see from Fig. 2 that with the increase of the potential strength, the tunneling conductance decreased, which illustrates that the Fermi surface mismatch between the normal and superconducting regions suppresses transmission. We also considered the dependence of the tunneling conductance on the value of $U_0$ for small bias voltages. As expected, we found that the oscillation amplitude decreases monotonically with the increase of $U_0$ in the case of $E_F=\Delta_0$ and finally approaches a constant value.

In Figs. 3 and 4, we provide numerical results of the tunneling conductance as a function of the bias voltage $V$. Similar results to Beenakker's\cite{BeenakkerPRL2006} are obtained. In the limit of $V\rightarrow 0$, all the conductances have the same value of $4/3$. For the case of $\Delta_0 \geq E_F$, a sharp change in the conductance occurs at $eV = \Delta_0$ and all the conductances vanish at $eV=E_F$, which is shown in Fig. 3.  This is because of that no Andreev reflection occurs for all the incident angles (the critical angle of incidence $\alpha_c=0$) when $\varepsilon=E_F$. For the small bias voltages before the turning point $eV=E_F$, the conductance decreases with the increase of the SG energy gap. The conductance curves exhibit oscillatory behavior in the region of $eV>\Delta_0$, which is different from the condition considered by Beenakker. By analyzing the transmission coefficients of the system, we found that in the condition of $eV>\Delta_0$ the transmission spectrum demonstrates oscillatory behavior and the oscillating period increases with the increase of the SG energy gap. The oscillatory behavior originates from the effect of the quantum-mechanical interference between the electron-like and hole-like quasiparticles\cite{McMillanandPRL1966,PasanaiPhysica.C2014,DongPRB2003} in the SG barrier. This effect gives rise to oscillations in the reflection and transmission probabilities for the incident energies larger than the gap energy.

In Fig. 4, numerical results of the conductance for different SG-barrier thickness $a$ are provided. It can be seen that the conductance is sensitive to the SG-barrier thickness, especially in the condition of $eV>\Delta_0$. In this condition, the oscillation period increases and the oscillation amplitude decreases sharply with the increase of the thickness. This is also a result of the exponential term $e^{\pm\kappa a}$ in the wave function. It can be interpreted by the $ a \rightarrow \infty $ limit. In this limit, the components of the wave function with the exponential term $e^{+ \kappa a}$ are nonphysical, therefore only two of the components of wave function in the SG-region are physical. The model reduces to that of the the G/SG-junction, in other words, the model proposed by Beenakker\cite{BeenakkerPRL2006} is obtained.

We also considered the shot noise properties of the conductance through the G/SG/G structure. Numerical results of the shot noise and the Fano factor are provided in Fig. 5. The parameters of panels (a) and (c) are the same with Fig. 4 and teh parameters of (b) and (b) are the same with Fig. 3. We can see from the curves that the shot noise oscillates greatly in the region of $eV\geq \Delta_0$, while it increases monotonically in the region $eV<\Delta_0$. These behaviors are similar to the tunneling conductance. In tunneling through the G/SG/G-structure, transmission is enhanced by the active hole channels. As a result, the values of the shot noise are small in comparison with the Poisson value, which is $2eG$ corresponding to the uncorrelated transport. When the hole channels in addition to the electron channels contribute to the transport, the interference effect is strong and the shot noise is significantly suppressed. In the region of $eV<\Delta_0$, the impact of the proximity effect in the SG-region is strong giving rise to strong conductance and the values of $S$ and $F$ approach 0. In the case of $eV\geq \Delta_0$, the amplitude of oscillation decreases with the increase of $eV$; in the case of $eV<\Delta_0$, the amplitude of oscillation increases with the increase of $eV$, which originates from the same reason as the conductance.

\section{Conclusions}

Based on the Dirac--Bogoliubov--de Gennes equation and the scattering theory, we investigated the transport properties of the relativistic electrons and holes through the G/SG/G junction. We have deduced the analytical formulas of the tunneling conductance and the shot noise. Numerical results of the tunneling conductance and shot noise in the system are provided. We compared the two cases, one of which is in the presence of the specular Andreev reflection and the other of which is in the absence of the specular Andreev reflection. The physical results can be summarized as follows. Firstly, the conductance increases with the increase of the thickness of the SG-layer and the shot noise is suppressed by the conductance in the case of $eV=E_F$. Secondly, the potential strength significantly affects the transport properties. It suppresses the conductance and enhances the shot noise. Thirdly, we obtained similar results with the model of the G/SG junction in the condition $eV\leq \Delta_0$. In the limit of $a\rightarrow \infty$, the results of Beenakker for a G/SG junction can be reproduced. Fourthly, the thickness of the SG-layer affects the conductance and the shot noise more prominently in the condition $eV\geq \Delta_0$, causing the decrease of the oscillation amplitudes and the characteristic features of the specular Andreev reflection. In conclusion, the conductance is a combined result of the Andreev reflection and the specular Andreev reflection, which can be tuned by the system parameters; the shot noise is suppressed by the SG-barrier because of the contribution of the hole channels in addition to the electron channels.

\section{Acknowledgements}

This project was supported by the National Natural Science
Foundation of China (No. 11004063), and the Fundamental Research
Funds for the Central Universities, SCUT (No. 2014ZG0044).

\clearpage


text:
\section*{Abstract}  The quantum probability flux of a particle integrated
over time and a distant surface gives the probability for the particle
crossing that surface at some time.
 We prove  the free flux-across-surfaces theorem , which was conjectured by Combes,  Newton and
Shtokhamer \cite{Combes}, and which relates the integrated  quantum flux to
the usual quantum mechanical formula for the cross section.  The integrated
quantum flux is equal to the probability of outward crossings of
surfaces by Bohmian trajectories in the scattering regime.

\section{Introduction}

Time-dependent scattering theory is concerned  with the long-time behavior
of wave packets $\psi_t$.
Dollard's scattering-into-cones theorem\ \cite{Dollard,Dollard2}
asserts that, assuming, say, asymptotic completeness, the probability of finding a particle with a wave function
$\psi\in\mathchoice{ \hbox{${\cal H}$} _{ac}(H)$,  the absolutely continuous subspace for the
Hamiltonian $H$,
in the far future in a
given cone $C\subset \mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3$ (with vertex at the origin) equals the
probability that the quantum mechanical momentum of $\Omega_-^\dagger\psi$
lies in  the same cone, \begin{equation} \label{SICT}  \lim_{t \to \infty}  \int_C d^3x |\psi_t({\bf x})|^2 =
\int_C d^3v |\widehat{\Omega_-^\dagger\psi}({\bf v})|^2, \end{equation} where $\Omega_-:=
 {\mbox{\rm s-}} \lim_{t\to\infty} e^{iHt}e^{-iH_0t}$ is the   wave operator, $H = H_0 +V$
with the free Hamiltonian $H_0 = -\Delta/2$  (we choose units such that $\hbar=m=1$) and the interaction potential $V$. $\widehat{\ }$
denotes the Fourier transform.  The scattering-into-cones theorem\ is regarded as fundamental,  from
which  the expression for the differential
 cross section $\frac{d\sigma}{d\Omega}=
|f(\theta,\phi)|^2$ from the time-independent theory is to be derived
from the r.h.s.\ of (\ref{SICT})
(e.g.\ \cite{ReedSimon3}, p.\ 356,
\cite{Enss2}).

Combes, Newton and Shtokhamer \cite{Combes} observed however
that what is relevant for  scattering theory  is
 a formula for the probability that the particle  crosses some distant
surface  at some time during the scattering process, since the detectors
click at some {\it random} time, which is {\it not} chosen by the
experimenter. Heuristically, this probability should be given by integrating
the quantum mechanical probability  flux over the relevant time interval and
this surface. (The flux is often used that way in  textbooks.)
 Combes, Newton and  Shtokhamer hence conjectured the ``flux-across-surfaces theorem''  \begin{equation}
\label{FAST}  \lim_{R \to \infty}  \int_0^\infty dt \int_{C\cap{\partial B_R}} {\bf j}^{\psi_t} \cdot {\bf n}
d\sigma= \int_C d^3v |\widehat{\Omega_-^\dagger\psi}({\bf v})|^2, \end{equation} where
$B_R$ is the ball with radius $R$ and outward normal ${\bf n}$. To our knowledge
there exists no proof of this theorem. A simpler statement, also not
previously proven, is the ``free flux-across-surfaces theorem ,'' for freely evolving $\psi_t$, \begin{equation}
\label{FASFREET}  \lim_{R \to \infty}  \int_0^\infty dt \int_{C\cap{\partial B_R}} {\bf j}^{\psi_t} \cdot
{\bf n} d\sigma= \int_C d^3v |\hat{\psi}({\bf v})|^2 \end{equation} which in a sense is physically
good enough, because the scattered wave packet will  move almost freely
after the scattering has essentially been completed (see also
\cite{Combes}).  We shall prove the ``free flux-across-surfaces theorem '' in this paper, commenting
at the end on the  general flux-across-surfaces theorem .

We want first to give the heuristic argument for (\ref{FASFREET}). The flux
should contribute to the integral in (\ref{FASFREET}) only for large times,
because the packet has to travel a long time before it reaches the  distant
sphere ${\partial B_R}$, so that we may use the long-time asymptotics  of the free
evolution. Writing \begin{equation} \label{FREEHEU} \psi_t({\bf x}) = (e^{-iH_0t} \psi)({\bf x})
= \int d^3y \frac{ e^{i\frac{|{\bf x}-{\bf y}|^2}{2t}} }{(2\pi it)^{3/2}} \psi({\bf y})
\end{equation} and expanding the exponent of the propagator, we obtain \begin{equation} \psi_t({\bf x})
= \frac{e^{i\frac{x^2}{2t}}}{(it)^{3/2}} \hat{\psi}(\frac{{\bf x}}{t}) +
\frac{e^{i\frac{x^2}{2t}}}{(it)^{3/2}} \int \frac{d^3y}{(2\pi)^{3/2}}e^{-i\frac{{\bf x}\cdot{\bf y}}{t}}
(e^{i\frac{y^2}{2t}} - 1) \psi({\bf y}) \end{equation} so that for large times (the second term
should be negligible since $|(e^{i\frac{y^2}{2t}} - 1)| \to  0$ as $t\to\infty$) \begin{equation}
\label{ASYMP} \psi_t({\bf x}) \approx (it)^{-3/2}e^{i\frac{x^2}{2t}} \hat{\psi}(\frac{{\bf x}}{t}). \end{equation}
The importance of this asymptotics  for scattering theory has long been
recognized, see e.g.\ \cite{Brenig} and \cite{Dollard}.

Consider now a cone $C$. Substituting ${\bf v} := \frac{{\bf x}}{t}$ one readily
obtains the  scattering-into-cones theorem\  \begin{equation} \label{SIC}  \lim_{t \to \infty}  \int_{C} d^3x |\psi_t({\bf x})|^2 =
\int_{C} d^3v |\hat{\psi}({\bf v})|^2. \end{equation} But the l.h.s.\ of (7) should be
unaffected if $C$ is replaced by the truncated cone $C_R=C\cap B_R^c,
B_R^c := \mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3 \setminus B_R$, for
any $R>0$. Thus writing $\int_{C_R} d^3x  |\psi_t({\bf x})|^2 = \int_0^t dt'
\int_{C_R} d^3x \frac{\partial}{\partial t'} |\psi_{t'}({\bf x})|^2 + \int_{C_R}
d^3x  |\psi_0({\bf x})|^2$ and using the quantum flux equation
$\frac{\partial}{\partial t} |\psi_t|^2 + \nabla \cdot \bj^{\psi_t} = 0$ together
with
 Gauss' theorem and taking $R\to\infty$ provides a heuristic argument for
the free flux-across-surfaces theorem . Unfortunately, because of the difficulty in controlling the
relevant approximations, this  argument cannot be readily turned into a
rigorous proof (see also \cite{Combes}).

Instead we may more directly compute the flux using
(\ref{ASYMP}), from which  we find for  $t\to\infty$
\begin{equation}
\label{JASYMPHEU}
\bj^{\psi_t}({\bf x}) = {\mbox{\rm Im}}  \/ \psi_t^*({\bf x}) \nabla \psi_t({\bf x}) \approx
 t^{-3} |\hat{\psi}(\frac{{\bf x}}{t})|^2\frac{{\bf x}}{t}.  \end{equation} Noting  that the flux is purely
outgoing for large times, i.e.\ parallel to the outward normal ${\bf n}$ of
${\partial B_R}$, we then find upon substituting ${\bf v} := \frac{{\bf x}}{t}$ that   \begin{eqnarray} \int_0^\infty
dt \int_{{C\cap{\partial B_R}}} \bj^{\psi_t} \cdot {\bf n} d\sigma  &\approx&  \int_0^\infty dt
\int_{{C\cap{\partial B_R}}} t^{-3}|\hat{\psi}(\frac{{\bf x}}{t})|^2   \frac{{\bf x}}{t} \cdot {\bf n}({\bf x}) d\sigma \nonumber \\
&=& \int_C d^3v |\hat{\psi}({\bf v})|^2. \end{eqnarray}
(Note that the middle term above does
not in fact depend upon $R$.)
This calculation  can smoothly be
turned into a rigorous proof, to which we now turn.

\newpage

\section{The Flux-Across-Surfaces Theorem}

First we fix the following  notation,
illustrated also in the figure.

\begin{figure}[h]
\label{STREUNG}
\begin{center}
\leavevmode%
\epsfxsize=10cm
\epsffile{cone2.eps}
\end{center}
\caption{The initial wave packet evolves under the influence of the
scatterer at the origin. In Bohmian mechanics (see remark 3.11)
the flow lines of the corresponding flux
represent the possible trajectories of the particle.}
 \end{figure}

For $R>0$ let
$B_R := \{{\bf x}\in\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3: x \le R \}$ and ${\partial B_R} = \{{\bf x}\in\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3: x = R \}$,
with $x=|{\bf x}|$.
 Further let ${\bf n} : {\partial B_R} \to \mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3$,
${\bf n}({\bf x}) := \frac{{\bf x}}{x}$  be the outward normal of the sphere
  ${\partial B_R}$. The cone spanned by the subset
 $\Sigma \subset \partial B_1$ of the unit sphere    is
$C := \{\lambda {\bf x} \in \mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3: {\bf x} \in \Sigma, \lambda \ge 0 \}$ and its
intersection with the sphere
  ${\partial B_R}$
is $R\Sigma := C \cap {\partial B_R} = \{R{\bf x} \in \mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3: {\bf x} \in \Sigma \}$.
Another characterization of cones is provided by
the unit vector ${\bf n}_C$, $\|{\bf n}_C\|=1$ and the opening angle
  $\theta_C \in [0,\pi]$, namely  $C:= \{{\bf x} \in \mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3:
{\bf x} \cdot {\bf n}_C > x \cos \theta_C \}$.
We chose polar coordinates
 $(r,\theta, \phi), r\ge 0, \theta \in [0,\pi], \phi \in [0,2\pi)$ centered
at the origin,  ${\bf x}(r,\theta,\phi) = (r\sin \theta \cos \phi,
r\sin\theta\sin\phi, r\cos\theta),$ with the $z$-direction ${\bf n}_C$. In these
polar coordinates $B_R = \{(r, \theta, \phi): r \le R \}$, ${\partial B_R} = \{(r,
\theta, \phi): r=R \}$ and  $C = \{(r,\theta, \phi): \theta < \theta_C \}$.
The intersection of the cone  $C$ with the sphere  ${\partial B_R}$  is now $C\cap
{\partial B_R} = \{(r,\theta,\phi): r=R, \theta < \theta_C \}$ with outward normal
 ${\bf n}(\theta,\phi) = R^{-1} {\bf x}(R,\theta,\phi)$. $d\Omega =
\sin\theta d\theta d\phi$ denotes the solid angle.

\begin{theorem}
\label{FASFREE}

Let $\psi \in {\cal S}(\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3)$ and $\psi_t := e^{-iH_0t}\psi$.
Then for all $T\in \mathchoice{ \hbox{${\rm I}\!{\rm R}$} $ and any cone ~$C$
\begin{equation}
\label{FASFREEF}
 \lim_{R \to \infty}  \int_T^\infty dt \int_{C\cap{\partial B_R}} \bj^{\psi_t}({\bf x}) \cdot {\bf n} d\sigma
=  \lim_{R \to \infty}  \int_T^\infty dt \int_{C\cap{\partial B_R}} |\bj^{\psi_t}({\bf x}) \cdot {\bf n} |d\sigma
=\int_C d^3v |\hat{\psi}({\bf v})|^2.\nonumber
\end{equation}
\end{theorem}

\begin{rmrem}
The condition
  $\psi\in {\cal S}(\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3)$, the Schwarz space,
 is introduced for the sake of simplicity. The  proof may be
performed with milder assumptions. Note, however, that  ${\cal S}(\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3)$
is a time invariant domain under the free evolution.
\end{rmrem}
\begin{rmrem}
The reason for formulating the theorem as slightly stronger than (\ref{FASFREET}),
including information also about the modulus of $\bj^{\psi_t} \cdot {\bf n}$,
is that  in Bohmian mechanics (see remark \ref{BOHMREM}) the first (second)
flux integral in (\ref{FASFREEF})
gives simply the expected value of the number of signed crossings
(the total number of crossings) by the
Bohmian trajectories of the surface.
If they both agree it is an easy consequence that
(\ref{FASFREEF}) equals the  asymptotic probability that the particle crosses
$C\cap {\partial B_R}$ at some time in $[0,\infty)$.
\end{rmrem}

It will be convenient to introduce a notion of closeness of
fluxes.

\begin{rmdef}
\label{FASCLOSE}
Two smooth functions ${\bf j}_1,{\bf j}_2 : \mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3\times\mathchoice{ \hbox{${\rm I}\!{\rm R}$}  \to \mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3$ are said to
be  ``close in the sense of the asymptotic flux across surfaces,'' or
${\bf j}_1 \stackrel{FAS}{\sim} {\bf j}_2$, if for some $T>0$
\begin{equation}
\label{CONDFASCLOSE}
 \lim_{R \to \infty}  \int_T^\infty dt \int_{\partial B_R} |({\bf j}_1-{\bf j}_2)\cdot {\bf n}|d\sigma = 0.
\end{equation}
\end{rmdef}

\begin{lemma} \label{CLASSICFLUX} Suppose that  for ${\bf j}: \mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3\times\mathchoice{ \hbox{${\rm I}\!{\rm R}$}  \to
\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3$ and ${\bf j}^\Phi_0({\bf x},t) :=  t^{-3} |\Phi(\frac{{\bf x}}{t})|^2\frac{{\bf x}}{t}$ with smooth $\Phi
\in L^2(\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3)$, we have ${\bf j} \stackrel{FAS}{\sim} {\bf j}^\Phi_0$. Then for all
cones  $C\subset \mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3$ and some $T>0$  \begin{equation} \label{ALL}  \lim_{R \to \infty}  \int_T^\infty
dt \int_{C\cap{\partial B_R}} {\bf j} ({\bf x},t) \cdot {\bf n}({\bf x}) d\sigma =  \lim_{R \to \infty}  \int_T^\infty
dt \int_{C\cap{\partial B_R}} |{\bf j}({\bf x},t) \cdot {\bf n}({\bf x})| d\sigma =  \int_C  d^3v
|\Phi({\bf v})|^2. \end{equation} \end{lemma}

{\bf Proof:}
By  definition (\ref{FASCLOSE}) it is sufficient to establish   (\ref{ALL}) for
${\bf j}$ replaced by ${\bf j}^\Phi_0$.

Using spherical coordinates  ${\bf x}(r,\theta,\phi) =  (r\sin \theta \cos \phi,
r\sin\theta\sin\phi, r\cos\theta)$ we compute \begin{eqnarray} \int_T^\infty dt \int_{C\cap{\partial B_R}}
{\bf j}^\Phi_0 ({\bf x}) \cdot {\bf n}({\bf x}) d\sigma &=& \int_T^\infty dt\int_{C\cap{\partial B_R}} t^{-3}
 |\Phi(\frac{{\bf x}}{t})|^2 \frac{{\bf x}}{t} \cdot {\bf n}({\bf x}) d\sigma \nonumber \\ &=& \int_T^\infty
dt\int_\Sigma d\Omega R^2 t^{-3}  |\Phi(\frac{{\bf x}(R,\theta,\phi)}{t})|^2
\frac{{\bf x}(R,\theta,\phi)}{t} \cdot {\bf n}(\theta,\phi). \nonumber \end{eqnarray} Observing that
 $\frac{{\bf x}(R,\theta,\phi)}{t} = {\bf x}(\frac{R}{t},\theta,\phi)$ and
substituting  $v := \frac{R}{t}$ we obtain \begin{eqnarray}  \lim_{R \to \infty}  \int_T^\infty dt
\int_{C\cap{\partial B_R}} {\bf j}^\Phi_0 ({\bf x}) \cdot {\bf n}({\bf x}) d\sigma &=&  \lim_{R \to \infty}  \int_0^{R/T} dv
v^2 \int_\Sigma d\Omega |\Phi(v,\theta,\phi)|^2 \nonumber \\ &=& \int_C
d^3v |\Phi({\bf v})|^2.   \end{eqnarray} The observation that ${\bf x} \cdot {\bf n}({\bf x}) =  |{\bf x}
\cdot {\bf n}({\bf x})|$ finally shows that all equalities in (\ref{ALL}) hold.

\proofend

\begin{lemma} \label{FREELEMMA} Let $\psi \in {\cal S}(\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3), \psi_t :=
e^{-iH_0t}\psi$ and $\bj^{\psi_t} = {\rm Im} \psi_t^* \nabla \psi_t$. Then  \begin{equation}
\label{CONDFREELEMMA} \bj^{\psi_t}({\bf x},t) \stackrel{FAS}{\sim} \frac{{\bf x}}{t} t^{-3}
|\hat{\psi}(\frac{{\bf x}}{t})|^2. \end{equation}

\end{lemma}

{\bf Proof:}  We verify the conditions in definition (\ref{FASCLOSE}). For
$t>0$ we may write \begin{eqnarray} \label{FREEASYMPTOTICS} \psi_t({\bf x}) &=& (e^{-iH_0t}
\psi)({\bf x}) \nonumber \\ &=&\int d^3y \frac{ e^{i\frac{|{\bf x}-{\bf y}|^2}{2t}}
}{(2\pi it)^{3/2}} \psi({\bf y})
 \nonumber \\ &=& \frac{e^{i\frac{x^2}{2t}}}{(it)^{3/2}} \hat{\psi}(\frac{{\bf x}}{t}) +
\frac{e^{i\frac{x^2}{2t}}}{(it)^{3/2}} \int \frac{d^3y}{(2\pi)^{3/2}}e^{-i\frac{{\bf x}\cdot{\bf y}}{t}}
(e^{i\frac{y^2}{2t}} - 1) \psi({\bf y}). \end{eqnarray} Since \begin{equation} \label{EKLEIN}
 |e^{i\frac{y^2}{2t}} - 1| \le 2 \end{equation} for all ${\bf y}\in\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3, t>0$, we obtain that \begin{equation}
\label{FDEF} f({\bf v},t) := \int \frac{d^3y}{(2\pi)^{3/2}} e^{-i{\bf v}\cdot{\bf y}}
(e^{i\frac{y^2}{2t}} - 1)\psi({\bf y})   \end{equation} is well defined for all ${\bf v}\in\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3$.
Because  $\psi \in {\cal S}(\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3)$  we may interchange differentiation and
integration to further  obtain that $f$ is differentiable on $\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3 \times
[T,\infty)$.

It is useful to introduce \begin{equation} \label{GDEF} {\bf g}({\bf v},t) := \nabla f({\bf v},t) =
-i \int \frac{d^3y}{(2\pi)^{3/2}} e^{-i{\bf v}\cdot{\bf y}} (e^{i\frac{y^2}{2t}} - 1) {\bf y}
\psi({\bf y}).  \end{equation} Further we put  \begin{equation} \alpha({\bf x},t) :=
\frac{e^{i\frac{x^2}{2t}}}{(it)^{3/2}} \hat{\psi}(\frac{{\bf x}}{t}) \end{equation} and \begin{equation} \beta({\bf x},t) :=
\frac{e^{i\frac{x^2}{2t}}}{(it)^{3/2}} f(\frac{{\bf x}}{t},t), \end{equation} i.e.\ $\psi_t({\bf x}) =
\alpha({\bf x},t) + \beta({\bf x},t)$, and \begin{eqnarray} \nabla \alpha({\bf x},t) &=&
\frac{e^{i\frac{x^2}{2t}}}{(it)^{3/2}} \left( i\frac{{\bf x}}{t} \hat{\psi}(\frac{{\bf x}}{t}) + \frac{1}{t} (\nabla
\hat{\psi})(\frac{{\bf x}}{t}) \right) \\ \nabla \beta({\bf x},t) &=&
\frac{e^{i\frac{x^2}{2t}}}{(it)^{3/2}} \left( i\frac{{\bf x}}{t} f(\frac{{\bf x}}{t},t) +  \frac{1}{t} {\bf g}(\frac{{\bf x}}{t},t)
\right). \end{eqnarray} We may thus write \begin{eqnarray} \label{CURRENT} \bj^{\psi_t}({\bf x}) &=& {\mbox{\rm
Im}} (\psi_t^*({\bf x}) \nabla \psi_t({\bf x}) ) \nonumber \\ &=& {\mbox{\rm Im}}
(\alpha^*({\bf x},t) \nabla \alpha({\bf x},t) + \beta^*({\bf x},t) \nabla \alpha({\bf x},t)+
\alpha^*({\bf x},t)  \nabla \beta({\bf x},t) + \beta^*({\bf x},t) \nabla \beta({\bf x},t))
\nonumber \\ &=& \frac{{\bf x}}{t} t^{-3} |\hat{\psi}(\frac{{\bf x}}{t})|^2 + N({\bf x},t),
\end{eqnarray} with \begin{equation} \label{NDEF} N({\bf x},t) := {\mbox{\rm Im}} \Bigl( t^{-4}
\hat{\psi}^*(\frac{{\bf x}}{t})\nabla\hat{\psi}(\frac{{\bf x}}{t}) + \beta^*({\bf x},t) \nabla \alpha({\bf x},t) +
\alpha^*({\bf x},t) \nabla \beta({\bf x},t) + \beta^*({\bf x},t) \nabla
\beta({\bf x},t)\Bigr). \end{equation}

Thus to obtain (\ref{CONDFREELEMMA})  we need only  show that
(\ref{CONDFASCLOSE}) is satisfied for some  $T>0$
and ${\bf j}_1-{\bf j}_2$ given by (\ref{NDEF}).
We shall make use of the bounds
\begin{eqnarray}
\label{FGLEMMA}
\sup_{{\bf v}\in\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3, t>0}  |f({\bf v},t)|  \le 2(2\pi)^{-3/2} \|\psi\|_1 =: c_f, \\
\label{GLEMMA}
\sup_{{\bf v}\in\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3, t>0}  |g({\bf v},t)|  \le 2(2\pi)^{-3/2}\|y\psi({\bf y})\|_1 =: c_g
\end{eqnarray}
($\| \cdot \|_1$ denotes the norm in $L_1$)
and the fact  that
\begin{equation}
\label{FTOZERO}
 \lim_{R \to \infty}  f({\bf v},\frac{R}{v}) = 0 \quad \forall {\bf v}\in\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3.
\end{equation}
(Note that $f({\bf v},\frac{R}{v})$ is well defined even for ${\bf v}=0$ by
(\ref{FDEF}).)
(\ref{FGLEMMA}) and (\ref{GLEMMA})
hold since  $|e^\frac{iy^2}{2t} - 1| \le 2$ for all ${\bf v},{\bf y}\in\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3, t>0$.
Since
$\psi \in L_1(\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3)$ and $ \lim_{R \to \infty}  |e^{i\frac{y^2v}{2R}} -1| = 0$ for all
${\bf v},{\bf y} \in \mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3$, (\ref{FTOZERO}) follows by dominated convergence.

  We analyze the contribution
of the expressions on the r.h.s.\ of
(\ref{NDEF}) term by term. For the first term we obtain, using
 $|{\mbox{\rm Im}} z| \le |z|$, the substitution  $v = \frac{R}{t}$, and the
Schwarz inequality \begin{eqnarray} \label{TERM1} |\mbox{\rm Im}\,  \int_T^\infty dt \int_{\partial B_R} t^{-4}
\hat{\psi}^*(\frac{{\bf x}}{t}) (\nabla \hat{\psi})(\frac{{\bf x}}{t}) \cdot {\bf n} d{\bf \sigma} | &\le& \int_T^\infty dt
\int_{\partial B_1} d\Omega R^2 t^{-4}  |\hat{\psi}(\frac{{\bf x}}{t})||(\nabla \hat{\psi})(\frac{{\bf x}}{t})|
\nonumber \\&\le& \int_0^\infty dv v^2\int_{\partial B_1} d\Omega R^{-1}
|\hat{\psi}({\bf v})||(\nabla \hat{\psi})({\bf v})| \nonumber \\ &\le& R^{-1} \|\hat{\psi}\|_2
\||\nabla \hat{\psi}| \|_2  \to 0 \end{eqnarray} as $R\to\infty$, since $\hat{\psi} \in {\cal
S}$.

For the second term
\begin{eqnarray}
\label{TERM2A}
\mbox{\rm Im}\,  \beta^*\nabla\alpha &=& \mbox{\rm Im}\,  t^{-3} f^*(\frac{{\bf x}}{t},t) \left(i\frac{{\bf x}}{t} \hat{\psi}(\frac{{\bf x}}{t}) +
\frac{1}{t} (\nabla \hat{\psi})(\frac{{\bf x}}{t}) \right)
\end{eqnarray}
we obtain, similarly using
 (\ref{FGLEMMA}), \begin{eqnarray} \label{TERM2} |\mbox{\rm Im}\,  \int_T^\infty dt \int_{\partial B_R} \beta^*
\nabla \alpha \cdot {\bf n}  d\sigma|  &\le& \int_T^\infty dt\int_{\partial B_1}
d\Omega R^2t^{-3} |f^*(\frac{{\bf x}}{t},t)|\bigl(|\frac{R}{t}\hat{\psi}(\frac{{\bf x}}{t})| + \frac{1}{t}
|(\nabla \hat{\psi})(\frac{{\bf x}}{t}) | \bigr)\nonumber \\ &\le& \int_0^\infty dv
\int_{\partial B_1} d\Omega v^2 |f^*({\bf v},\frac{R}{v})|\bigl( |\hat{\psi}({\bf v})| +
\frac{1}{R} |(\nabla \hat{\psi})({\bf v}) |\bigr) \nonumber \\ &\le& \int_0^\infty dv
\int_{\partial B_1} d\Omega v^2 |f^*({\bf v},\frac{R}{v})| |\hat{\psi}({\bf v})| +
\frac{1}{R} c_f \||\nabla \hat{\psi}| \|_1. \nonumber \end{eqnarray} The second term tends
to zero as   $R\to\infty$, and the first term  also vanishes: using
(\ref{FGLEMMA})
 and the fact that $\hat{\psi}\in L^1(\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3)$ we see that the integrand is
dominated by an integrable function uniformly in $R$, so that with
(\ref{FTOZERO}) the integral vanishes for $R\to\infty$ by dominated
convergence.

For  \begin{eqnarray} \mbox{\rm Im}\,  \alpha^*\nabla\beta &=& \mbox{\rm Im}\,  t^{-3} \hat{\psi}^*(\frac{{\bf x}}{t}) \left( i\frac{{\bf x}}{t}
f(\frac{{\bf x}}{t},t) + \frac{1}{t} g(\frac{{\bf x}}{t},t) \right)  \end{eqnarray} we may proceed in an analogous
manner and obtain \begin{eqnarray} \label{TERM3} |\mbox{\rm Im}\,  \int_T^\infty dt\int_{\partial B_R}
\alpha^*\nabla\beta \cdot {\bf n} d\sigma|  &\le& \int_T^\infty dt\int_{\partial B_1}
d\Omega R^2t^{-3} |\hat{\psi}^*(\frac{{\bf x}}{t})| (|\frac{R}{t} f(\frac{{\bf x}}{t},t)| +
\frac{1}{t}|{\bf g}(\frac{{\bf x}}{t},t)|) \nonumber \\ &\le& \int_0^\infty dv \int_{\partial B_1}
d\Omega v^2 |\hat{\psi}^*({\bf v})| (|f({\bf v},\frac{R}{v})| +
\frac{1}{R}|{\bf g}({\bf v},\frac{R}{v})|) \nonumber \\ &\le& \int_0^\infty dv
\int_{\partial B_1} d\Omega v^2 |\hat{\psi}^*({\bf v})| |f({\bf v},\frac{R}{v})| + c_g
\frac{1}{R}\| \hat{\psi} \|_1 \nonumber \\ &\to& 0 {{\rm ~ as} } \quad R \to
\infty. \end{eqnarray}

It remains to show that for some $T>0$
\begin{equation}
 \lim_{R \to \infty}  \int_T^\infty dt \int_{\partial B_R} |\beta^* \nabla \beta \cdot {\bf n} | d\sigma = 0.
\end{equation}
Now,
\begin{eqnarray}
\beta^*\nabla \beta &=& it^{-3} f^*(\frac{{\bf x}}{t},t) \int \frac{d^3y}{(2\pi)^{3/2}}
e^{-i\frac{{\bf x}}{t} \cdot {\bf y}} (e^{i\frac{y^2}{2t}} -1 )(\frac{{\bf x}}{t} - \frac{{\bf y}}{t}) \psi({\bf y})    \\
&=&   -it^{-3} f^*(\frac{{\bf x}}{t},t)  \int \frac{d^3y}{(2\pi)^{3/2}}
e^{-i\frac{{\bf x}}{t} \cdot {\bf y}}(\frac{{\bf x}}{t} - \frac{{\bf y}}{t}) \psi({\bf y})  \nonumber \\
&+& it^{-3} f^*(\frac{{\bf x}}{t},t)  \int \frac{d^3y}{(2\pi)^{3/2}}
e^{-i\frac{{\bf x}}{t} \cdot {\bf y}} e^{i\frac{y^2}{2t}} (\frac{{\bf x}}{t} - \frac{{\bf y}}{t}) \psi({\bf y}) \nonumber  \\
\label{VORVORLETZTER}
&=& t^{-3} f^*(\frac{{\bf x}}{t},t)  \int \frac{d^3y}{(2\pi)^{3/2}} \Bigl( \nabla_{{\bf y}}
e^{-i\frac{{\bf x}}{t}\cdot {\bf y}} \Bigr) \psi({\bf y})   \\ \label{ZWISCHEN}
&-& t^{-4} f^*(\frac{{\bf x}}{t},t) (\nabla \hat{\psi})(\frac{{\bf x}}{t})    \\
\label{LETZTER}
&-& t^{-3}  f^*(\frac{{\bf x}}{t},t)
\int \frac{d^3y}{(2\pi)^{3/2}}
\nabla_{{\bf y}} \Bigl( e^{-i\frac{{\bf x}}{t} \cdot {\bf y}} e^{i\frac{y^2}{2t}} \Bigr)
\psi({\bf y}).
\end{eqnarray}
Treating (\ref{ZWISCHEN}) like   (\ref{TERM2A}) we see that (\ref{ZWISCHEN})
doesn't contribute. Partial  integration of  (\ref{VORVORLETZTER}) $+$
(\ref{LETZTER})
yields
\begin{eqnarray}
{\bf a}({\bf x},t) &:=& t^{-3}  f^*(\frac{{\bf x}}{t},t)
\int \frac{d^3y}{(2\pi)^{3/2}}
\Bigl( e^{-i\frac{{\bf x}}{t} \cdot {\bf y}} e^{i\frac{y^2}{2t}}
\nabla_{{\bf y}} \psi({\bf y}) -  e^{-i\frac{{\bf x}}{t} \cdot {\bf y}} \nabla_{{\bf y}} \psi({\bf y}) \Bigr)
\nonumber \\
&=& t^{-3} f^*(\frac{{\bf x}}{t},t) \int  \frac{d^3y}{(2\pi)^{3/2}}
e^{-i\frac{{\bf x}}{t}\cdot{\bf y}} (e^{i\frac{y^2}{2t}} - 1) \nabla_{{\bf y}} \psi({\bf y}) \nonumber \\
&=& -t^{-3} f^*(\frac{{\bf x}}{t},t) \int \frac{d^3y}{(2\pi)^{3/2}}
\frac{t^2}{x^2} (\nabla_{{\bf y}}^2 e^{-i\frac{{\bf x}}{t}\cdot{\bf y}})
(e^{i\frac{y^2}{2t}} - 1) \nabla_{{\bf y}} \psi({\bf y}) \nonumber \\
&=& -t^{-1} x^{-2} f^*(\frac{{\bf x}}{t},t) \int \frac{d^3y}{(2\pi)^{3/2}}
  e^{-i\frac{{\bf x}}{t}\cdot{\bf y}} \nabla_{{\bf y}}^2\Bigl(
(e^{i\frac{y^2}{2t}} - 1) \nabla_{{\bf y}} \psi({\bf y}) \Bigr),
\end{eqnarray}
with  two partial integrations in the last step.

Now
\begin{eqnarray}
\nabla_{{\bf y}} (e^{i\frac{y^2}{2t}} - 1) &=& it^{-1}{\bf y} e^{i\frac{y^2}{2t}}, \nonumber \\
\nabla_{{\bf y}}^2(e^{i\frac{y^2}{2t}} - 1)  &=& (-\frac{y^2}{t^2} + 3it^{-1})e^{i\frac{y^2}{2t}}
\end{eqnarray}
and
\begin{equation}
|e^{i\frac{y^2}{2t}} -1| \le \frac{y^2}{2t},
\end{equation}
so that for $t\ge T > 0$
\begin{equation}
| \nabla_{{\bf y}}^2(
(e^{i\frac{y^2}{2t}} - 1) \nabla_{{\bf y}} \psi({\bf y}) )| \le |h({\bf y})| t^{-1}
\end{equation}
with some $h\in {\cal S}(\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3)$ appropriately chosen.
Hence,
\begin{equation}
|\int \frac{d^3y}{(2\pi)^{3/2}}
  e^{-i\frac{{\bf x}}{t}\cdot{\bf y}} \Bigl(\nabla_{{\bf y}}^2( (e^{i\frac{y^2}{2t}} - 1) \nabla_{{\bf y}}
\psi({\bf y}) )\Bigr)|  \le  t^{-1} \int \frac{d^3y}{(2\pi)^{3/2}} |h({\bf y})| =:
c t^{-1}. \end{equation} Thus we arrive at  \begin{equation} |{\bf a}({\bf x},t)| \le  c t^{-2} x^{-2}
|f(\frac{{\bf x}}{t},t)|, \end{equation} and with ${\bf R}(\theta,\phi) := {\bf x}(R,\theta,\phi)$ we obtain
\begin{eqnarray} \label{NAJA} \int_T^\infty dt \int_{\partial B_R} |{\bf a} \cdot  {\bf n}|d\sigma
&\le&
 c \int_T^\infty dt t^{-2} \int_{\partial B_1} d\Omega  |f(\frac{{\bf R}}{t},t)|. \end{eqnarray}
On the one hand  (cf.\ (\ref{FGLEMMA})) \begin{equation} \sup_{t\ge T, R>0}
|f(\frac{{\bf R}}{t},t)| \le c_f, \end{equation} and on the other hand with the
Riemann-Lebesgue lemma  \begin{eqnarray}   \lim_{R \to \infty}   |f(\frac{{\bf R}}{t},t)| = 0 \ \forall \ t>0.
\end{eqnarray} Hence the r.h.s. of (\ref{NAJA}) tends to zero (dominated convergence)
as $R\to\infty$ and we have thus finished the  the proof of lemma
(\ref{FREELEMMA}). \proofend

\begin{corollary}
\label{FASTOK}
For some $T>0$,  Theorem (\ref{FASFREE}) holds .
\end{corollary}

The analysis so far actually establishes the theorem for any $T>0$. We now
show that the restriction $T>0$ can be removed.

\begin{lemma}
\label{FINITELEMMA}
For all $-\infty < T_1 < T_2 < \infty$
\begin{equation}
 \lim_{R \to \infty}  \int_{T_1}^{T_2} dt \int_{{\partial B_R}} |\bj^{\psi_t}({\bf x})\cdot{\bf n}|d\sigma = 0.
\end{equation}
\end{lemma}

{\bf Proof}:  First observe that \begin{equation} \label{DOLL} \int_{T_1}^{T_2} dt
\int_{{\partial B_R}} |\bj^{\psi_t}({\bf x})\cdot{\bf n}|d\sigma \le 4\pi  \int_{T_1}^{T_2} dt R^2
\sup_{{\bf x}\in {\partial B_R}} |\psi_t({\bf x})||\nabla\psi_t({\bf x})|. \end{equation} We want to apply
dominated convergence. With   \begin{equation} \label{PAIN} \psi_t({\bf x}) = (2\pi)^{-3/2}
\int d^3k
 e^{i{\bf k}\cdot{\bf x}}e^{-i\frac{k^2 t}{2}} \hat{\psi}({\bf k}),
\end{equation}
and
\begin{equation}
\nabla \psi_t({\bf x}) = i(2\pi)^{-3/2} \int d^3k
 e^{i{\bf k}\cdot{\bf x}}e^{-i\frac{k^2 t}{2}} {\bf k} \hat{\psi}({\bf k}),
\end{equation}
 we have that \begin{equation} \sup_{{\bf x}\in\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3, t\in\mathchoice{ \hbox{${\rm I}\!{\rm R}$} } |\nabla \psi_t({\bf x})| \le
(2\pi)^{-3/2} \|k\hat{\psi}({\bf k})\|_1. \end{equation} Since  $\psi\in {\cal S}(\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3)$ we
may perform $n$ partial integrations in (\ref{PAIN}) to obtain    \begin{eqnarray}
\psi_t({\bf x}) &=&  (2\pi)^{-3/2} x^{-n} \int d^3k
\Bigl[(\frac{1}{i}\nabla_k)^n e^{i{\bf k}\cdot{\bf x}}\Bigr] e^{-i\frac{k^2 t}{2}} \hat{\psi}({\bf k})
\nonumber \\ &=& (-1)^n (2\pi)^{-3/2} x^{-n}  \int d^3k  e^{i{\bf k}\cdot{\bf x}}
\Bigl[(\frac{1}{i}\nabla_k)^n e^{-i\frac{k^2 t}{2}} \hat{\psi}({\bf k})\Bigr]. \end{eqnarray} We
estimate  \begin{equation} |(\frac{1}{i}\nabla_k)^n e^{-i\frac{k^2 t}{2}} \hat{\psi}({\bf k})| \le |h({\bf k})|
(1+t^n) \end{equation} for some $h \in {\cal S}(\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3)$. For $n=2$ we thus have \begin{eqnarray}
 R^2 \sup_{{\bf x}\in {\partial B_R}} |\psi_t({\bf x})||\nabla\psi_t({\bf x})| &\le& (2\pi)^{-3}
\|k\hat{\psi}({\bf k})\|_1 \int d^3k |(\frac{1}{i}\nabla_k)^2  e^{-i\frac{k^2 t}{2}}
\hat{\psi}({\bf k})| \nonumber \\ &\le& c' (1+t^2)  \in L^1(T_1,T_2). \end{eqnarray} For $n=3$
and any fixed $t\in[T_1,T_2]$ we obtain \begin{eqnarray} R^2 \sup_{{\bf x}\in {\partial B_R}}
|\psi_t({\bf x})||\nabla\psi_t({\bf x})| &\le& (2\pi)^{-3}
R^{-1}\|k\hat{\psi}({\bf k})\|_1 \int d^3k |(\frac{1}{i}\nabla_k)^3  e^{-i\frac{k^2 t}{2}}
\hat{\psi}({\bf k})| \nonumber \\ &\le& c'' R^{-1} (1+t^3) \to 0 \end{eqnarray} for
$R\to\infty$.  Now we use dominated convergence in (\ref{DOLL}) and are
done.  \proofend

Theorem (\ref{FASFREE}) now follows directly from Cor.(3.7) and Lemma
(\ref{FINITELEMMA}).

\begin{rmrem}

The extension of our result to the free evolution of $N$ particles is
straightforward.  The extension to the interacting case, i.e.\ a proof of
(\ref{FAST}) (even for one-particle scattering), is open. The theory of
generalized eigenfunction expansions \cite{Ikebe} can be used to control the
 space-time behavior of $\psi_t({\bf x})$ and of the flux $\bj^{\psi_t}$. We may expand
$\psi_t({\bf x}) =  (2\pi)^{-3/2} \int d^3k e^{-i\frac{k^2 t}{2}} \phi({\bf x},{\bf k})
\widehat{\Omega_-^\dagger\psi}({\bf k})$,  where $\phi({\bf x},{\bf k})$ are solutions
of the Lippmann-Schwinger equation  \begin{equation} \label{LS1} \phi({\bf x},{\bf k}) =
e^{i{\bf k}\cdot{\bf x}} -  \frac{1}{2\pi} \int d^3y
\frac{e^{-i{k|{\bf x}-{\bf y}|}}}{|{\bf x}-{\bf y}|} V({\bf y}) \phi({\bf y},{\bf k}), \end{equation} (with {\it
incoming} spherical waves). The important connection between the wave
operators, generalized eigenfunctions and the Fourier transform is
expressed by
 $\widehat{\Omega_-^\dagger\psi}({\bf k}) = (2\pi)^{-3/2}\int d^3x
\phi^*({\bf x},{\bf k}) \psi({\bf x})$.  For a proof of (\ref{FAST}), relying
essentially on a stationary phase argument, we need additional smoothness
properties  of the eigenfunctions  which, to our knowledge, have not yet
been established. More precisely, we need to know that
$\phi({\bf x}, \cdot) \in C^\infty(\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3\setminus \{0\} )$ for all ${\bf x}\in\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3$,
 $\phi(\cdot,{\bf k}) \in C^\infty(\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3) $ for all ${\bf k}\in\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3\setminus \{0\}$,
and $\sup_{{\bf x}\in \mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3, {\bf k} \in
\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3\setminus \{0\} } \phi({\bf x},{\bf k}) < \infty$. The closest we could get
was, with \cite{Ikebe} and \cite{ReedSimon3}  Theorem XI.41 and XI.70,  that
for
 $V \in L^2(\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3)$ locally H\"older continuous with the possible exception
of finitely many  singularities and $|V({\bf x})| = O(x^{-2-h})$ for some $h>0$,
 $\phi({\bf x},{\bf k})$ is bounded and continuous for ${\bf x} \in \mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3$ and ${\bf k} \in D
\subset \mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3\setminus \{0\}$, where $D$ is {\it compact}.  It is well known
that for
 $V\in C^\infty(\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3)$  the solutions $\phi$ of the
stationary Schr\"odinger equation\
obey $\phi \in C^\infty(\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3)$ and thus the solutions $\phi(\cdot, {\bf k})$
of the Lippmann-Schwinger equation, which are special solutions of the
stationary Schr\"odinger equation\
parametrized by ${\bf k}$, are in $C^\infty(\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3)$
(see \cite{ReedSimon2}, Theorem IX.62). It remains to be shown that  for any
${\bf x}\in\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3$ both $\phi({\bf x},\cdot) \in C^\infty(\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3\setminus \{0\})$ and
$\sup_{{\bf x}\in\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3,{\bf k}\in\mathchoice{ \hbox{${\rm I}\!{\rm R}$} ^3\setminus \{0\}} |\phi({\bf x},{\bf k})| < \infty$.
This should be true for potentials which are sufficiently smooth and
have sufficiently strong decay at infinity \cite{DaumerPhD}. \end{rmrem}

\begin{rmrem} The mathematical physics of  scattering theory is mainly
concerned with the existence and asymptotic completeness of wave operators
$\Omega_\pm :=  {\mbox{\rm s-}} \lim_{t\to\mp \infty} e^{iHt}e^{-iH_0t}$. The wave operators
may be used to control the long-time behavior of wave packets $\psi_t :=
e^{-iHt}\psi$, in the sense of $\psi_t \stackrel{L^2}{\sim} e^{-iH_0t}
\Omega_-^\dagger \psi$,  i.e.\ the difference vanishes in $L^2$ as
$t\to\infty$.  Dollard's lemma implies  that for $\phi_t := e^{-iH_0t}\phi$
\begin{equation} \phi_t({\bf x}) \stackrel{L^2}{\sim} e^{i\frac{x^2}{2t}}(it)^{-3/2} \hat{\phi}(\frac{{\bf x}}{t}). \end{equation}
Asymptotic completeness of the wave operators implies, among other things,
that for any $\psi\in \mathchoice{ \hbox{${\cal H}$} _{ac}(H)$ there is a $\phi \in L^2$
such that $ \lim_{t \to \infty}  \|e^{-iHt}\psi - e^{-iH_0t}\phi\|_2 = 0$, where
$\phi = \Omega_-^\dagger \psi$ with  $\Omega_-$  unitary on
$\mathchoice{ \hbox{${\cal H}$} _{ac}(H)$
(see, e.g.,\ \cite{ReedSimon3}).
It then follows
 by the
triangle inequality that for any $\psi\in\mathchoice{ \hbox{${\cal H}$} _{ac}$
\begin{equation} \psi_t({\bf x}) \stackrel{L^2}{\sim}
e^{i\frac{x^2}{2t}}(it)^{-3/2} \widehat{\Omega_-^\dagger\psi}(\frac{{\bf x}}{t}). \end{equation}  From this the
general scattering-into-cones theorem\ (\ref{SICT}) follows easily (see. e.g. \cite{Dollard}).  This
is however not sufficient to prove the physically relevant flux-across-surfaces theorem . The
notion of closeness which should be used here is the closeness of fluxes in
the sense of the asymptotic flux across surfaces introduced in definition
(\ref{FASCLOSE}), and not the closeness of wave functions in $L^2$.

\end{rmrem}

\begin{rmrem} \label{BOHMREM} In the context of Bohmian mechanics
\cite{Albert,Bell,BOHI,DGZ,Holland}, a theory of point particles moving
along trajectories defined by an ODE arising from the wave function, with
velocity $\bj^{\psi_t}/|\psi_t|^2$, a theory that  can be shown to underly the
quantum  formalism (see. e.g.\ \cite{Bohm,OPAP}), it follows easily from
Theorem (\ref{FASFREE}) that  \begin{equation} \label{IMPO}  \lim_{R \to \infty}  \mathchoice{ \hbox{${\rm I}\!{\rm P}$} ^\psi({\bf x}_e^R \in
R\Sigma) =  \lim_{R \to \infty}  \int_0^\infty dt \int_{R\Sigma} \bj^{\psi_t} \cdot {\bf n} d\sigma \end{equation}
where ${\bf x}_e^R$ is the position at which the trajectory first  crosses the
sphere ${\partial B_R}$  and $\mathchoice{ \hbox{${\rm I}\!{\rm P}$} ^\psi$ is the quantum equilibrium measure, given by
the  density $|\psi|^2$.  This provides a natural definition of the
cross section measure. \end{rmrem}

\section*{Acknowledgments}
This work was supported in part  by
the DFG, by NSF Grant  No. DMS-9504556, and by the INFN.

redpajama_set_name: RedPajamaArXiv