# Exact time-evolving scattering states
in open quantum-dot systems with an interaction:
Discovery of time-evolving resonant states

Akinori Nishino1 and Naomichi Hatano2 1 Faculty of Engineering, Kanagawa
University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama, Kanagawa, 221-8686,
Japan  2 Institute of Industrial Science, The University of Tokyo, 5-1-5
Kashiwanoha, Kashiwa, Chiba 277-8574, Japan<EMAIL_ADDRESS>
###### Abstract

We study exact time-evolving many-electron states of an open double quantum-
dot system with an interdot Coulomb interaction. A systematic construction of
the time-evolving states for arbitrary initial conditions is proposed. For any
initial states of one- and two-electron plane waves on the electrical leads,
we obtain exact solutions of the time-evolving scattering states, which
converge to known stationary scattering eigenstates in the long-time limit.
For any initial states of localized electrons on the quantum dots, we find
exact time-evolving states of a new type, which we refer to as time-evolving
resonant states. In contrast to stationary resonant states, whose wave
functions spatially diverge and not normalizable, the time-evolving resonant
states are normalizable since their wave functions are restricted to a finite
space interval due to causality. The exact time-evolving resonant states
enable us to calculate the time-dependence of the survival probability of
electrons on the quantum dots for the system with the linearized dispersions.
It decays exponentially in time on one side of an exponential point of
resonance energies while, on the other side, it oscillates during the decay as
a result of the interference of the two resonance energies.

††:

  * November 2023

Keywords: open quantum systems, quantum dots, initial-value problems,
scattering states, resonant states, exceptional points, exact solutions

##  1 Introduction

Quantum transport in open quantum-dot systems with interacting electrons has
been intensively studied in recent years. One of the pioneering experiments is
a measurement of zero-bias resonance peaks of electrical conductance across a
quantum dot at low temperatures, which is understood as the nonequilibrium
Kondo effect [1, 2, 3]. On the other hand, in order to analyze the quantum
transport in open quantum-dot systems theoretically, it is necessary to
consider nonequilibrium states realized in the systems. The Landauer formula,
which was first developed in noninteracting cases, assumes that an electric
current in coherent electron transport is carried by quantum-mechanical
scattering states [4, 5, 6]. The non-equilibrium Green’s-function method has
been successful as an extension of the Landauer formula to interacting cases
[7, 8, 9, 10].

However, explicit many-electron wave functions realized in open quantum-dot
systems had not been obtained in interacting cases, until we constructed exact
many-electron scattering eigenstates for an open single quantum-dot system
described by the interacting resonant-level model [11, 12, 13]. We found it
remarkable that incident free-electron plane waves are partially scattered to
many-body bound states around the quantum-dot due to the Coulomb interaction.
By using the exact solution, we calculated the average electric current in the
system under finite bias voltages, which agrees with the results by other
methods [14, 15, 16, 17]. Besides the interacting resonant-level model, we
also constructed exact scattering eigenstates for the Anderson model [18] and
the open double quantum-dot systems [19, 20].

In open quantum systems, resonant states with complex energy eigenvalues have
attracted renewed interest. The concept of the resonant state was originally
introduced for the study of decaying states of unstable nuclei; the imaginary
part of the complex energy eigenvalue corresponds to the lifetime of the
decaying state [21, 22, 23]. However, the resonant state is often regarded as
“unphysical” since its wave function diverges in space and is not
normalizable. Normalization and probabilistic interpretation of the resonant
state have been discussed for many years [24, 25, 26, 27, 28, 29]. Recently,
resonant states are employed for understanding the resonant transport in open
quantum-dot systems [29, 30, 31]. The two-body bound state appearing in the
exact many-electron scattering states [11, 12, 13, 18, 19, 20] is regarded as
a pair of two resonant states, where the imaginary part of the complex energy
eigenvalue characterizes the binding strength of the two-body bound states.

In the present article, we study time-evolving many-electron states of an open
double quantum-dot system of spinless electrons with an interdot Coulomb
interaction. We analyze a general double quantum-dot system of arbitrary
arrangements of two quantum dots including serial, parallel, and T-shaped
double quantum-dot systems [32]. Quite recently, for the open quantum systems
with a single quantum dot, exact time-evolving many-electron scattering states
have been constructed for initial states of free-electron plane waves under an
assumption on the form of wave functions [33, 34, 35]. We develop a systematic
construction of the time-evolving states for the open double quantum-dot
system under arbitrary initial conditions without assuming the form of wave
functions as was used in Refs. [33, 34, 35]. For demonstrations, we explicitly
construct exact solutions of the time-evolving states for two types of initial
states: i) the one- and two-electron plane-wave states on the electrical
leads; ii) the localized states on the two quantum dots. In the case i), we
obtain the time-evolving scattering states, which exponentially converge to
the known stationary scattering eigenstates [20] in the long-time limit. In
particular, at exceptional points of the resonance energies, we find
exponential functions multiplied by a linear term of time in the wave
functions. Here we remark that the purely exponential behavior without any
deviations in the form of power-law behavior [36, 37, 38, 39, 40, 41, 42] is
due to the unbounded linear dispersion relation that we assume for the
electrical leads. We also remark that the two initial states are related to
the manipulation of charge-qubit devices: the case i) can be utilized as
“initialization” of the qubits and the case ii) may correspond to “coherent
manipulation” [43].

In the one-electron case of the case ii), we find a new time-evolving state,
which we refer to as a time-evolving resonant state. The wave function of the
time-evolving resonant state increases exponentially only inside a finite
space interval and hence is normalizable, in contrast to the “unphysical”
resonant states with spatially diverging wave functions [29]. The restriction
of the wave function to the space interval is originated from causality, which
was also observed in the Friedrichs model [39]. We show through the
construction of exact solutions that the wave function coincides with the
resonant state inside the space interval. Furthermore, in the two-electron
case of the case ii), we obtain a time-evolving two-body resonant state that
corresponds to the two-body resonance pole in the interacting case.

The explicit wave functions of time-evolving resonant states enable us to
calculate exactly the time-dependence of the existence probability of
electrons on the leads and the survival probability of electrons on the
quantum dots under the initial states of localized electrons on the quantum
dots. The exponential behavior of the wave functions leads to that of both the
existence and the survival probabilities. It is remarkable that their behavior
changes around exceptional points of the resonance energies. In a parameter
region on one side of an exceptional point, we observe a pure exponential
increase of the existence probability in space and a pure exponential decay of
the survival probability in time. In a region on the other side of the
exceptional point, we observe oscillation during the exponential behavior,
which are results of the interference of two one-body resonance poles.

The paper is organized as follows: In Section 2, an open double quantum-dot
system with an interdot Coulomb interaction is introduced. In Section 3, we
propose a systematic construction of time-evolving states under arbitrary
initial conditions. We construct exact solutions of the time-evolving
scattering states for any initial states of one- and two-electron plane waves
on the electrical leads. In Section 4, we obtain exact time-evolving resonant
states for the initial states of one and two electrons localized on the
quantum dots. By using the exact solutions, we investigate the existence
probability of electrons on the electrical leads and the survival probability
of electrons on the quantum dots. Section 5 is devoted to concluding remarks.
Appendix provides a concise review of resonant states.

##  2 Open double quantum-dot systems

We introduce an open double quantum-dot system with an interdot Coulomb
interaction [20]. Let us consider an open quantum system which consists of two
quantum dots and two one-dimensional electrical leads of noninteracting
spinless electrons as is illustrated in Fig. 1. We assume a single energy
level for each quantum dot and linearize the dispersion relation in the
vicinity of the Fermi energy for each lead. The two quantum dots are connected
to each other and to the origin $x=0$ of each lead. The Hamiltonian of the
system is given by

$\displaystyle H=$ $\displaystyle\sum_{m=1,2}\int dx\,c^{\dagger}_{m}(x)v_{\rm
F}\frac{1}{\mathrm{i}}\frac{d}{dx}c_{m}(x)+\sum_{m=1,2\atop\alpha=1,2}\big{(}v_{m\alpha}c^{\dagger}_{m}(0)d_{\alpha}+v_{m\alpha}^{\ast}d^{\dagger}_{\alpha}c_{m}(0)\big{)}$
(2.1)
$\displaystyle+v^{\prime}d^{\dagger}_{1}d_{2}+v^{\prime\ast}d^{\dagger}_{2}d_{1}+\sum_{\alpha=1,2}\epsilon_{{\rm
d}\alpha}n_{{\rm d}\alpha}+U^{\prime}n_{{\rm d}1}n_{{\rm d}2}.$

Here $c^{\dagger}_{m}(x)$ and $c_{m}(x)$ are the creation and annihilation
operators of electrons at the position $x$ on the lead $m$, respectively, and
$d^{\dagger}_{\alpha}$ and $d_{\alpha}$ are those on the quantum dot $\alpha$.
We also define the number operator $n_{{\rm
d}\alpha}=d^{\dagger}_{\alpha}d_{\alpha}$ of electrons on the quantum dot
$\alpha$. The parameter $v_{m\alpha}$ is the transfer integral between the
lead $m$ and the quantum dot $\alpha$, $v^{\prime}$ is that between the two
quantum dots, $\epsilon_{{\rm d}\alpha}$ is the energy level on the quantum
dot $\alpha$ and $U^{\prime}$ expresses the strength of the interdot Coulomb
interaction. Here and hereafter, we set $\hbar=1$. In what follows, the Fermi
velocity $v_{\rm F}$ of the leads is also set to unity: $v_{\rm F}=1$.

Figure 1:  A schematic diagram of the double quantum-dot system.

We remark that actual energy levels on the quantum dots are affected by the
electrons contained in the quantum dots. In order to investigate the electron
distribution within a quantum dot of finite size, we would have to solve the
Schrödinger equations and the Poisson equation self-consistently [43], which,
however, is out of scope of the present paper.

The Hamiltonian in Eq. (2.1) includes various arrangements of the two quantum
dots [32]. For example, in the case of $v^{\prime}=0$, we have a parallel-
coupled double quantum-dot system; in the case of $v_{12}=v_{21}=0$, we have a
serial-coupled double quantum-dot system; in the case of $v_{12}=v_{22}=0$, we
have a T-shaped double quantum-dot system. The arrangement of the two quantum
dots affects the quantum interference of electrons traveling through the
quantum dots.

Let us derive the time-dependent Schrödinger equation for the open double
quantum-dot system in Eq. (2.1). First, we consider the one-electron case,
which is equivalent to the noninteracting case. The time-dependent one-
electron state is given in the form

$\displaystyle|\Psi(t)\rangle=\Big{(}\int\\!dx\sum_{m=1,2}g_{m}(x,t)c^{\dagger}_{m}(x)+\sum_{\alpha=1,2}e_{\alpha}(t)d^{\dagger}_{\alpha}\Big{)}|0\rangle.$
(2.2)

Here $|0\rangle$ is the vacuum state. The wave functions $g_{m}(x,t)$ for
$m=1,2$ and $e_{\alpha}(t)$ for $\alpha=1,2$ are determined by the time-
dependent Schrödinger equation
$\mathrm{i}\partial_{t}|\Psi(t)\rangle=H|\Psi(t)\rangle$, which is explicitly
written in the form of the following coupled partial-differential equations:

$\displaystyle\mathrm{i}\partial_{t}g_{m}(x,t)=\frac{1}{\mathrm{i}}\partial_{x}g_{m}(x,t)+\delta(x)\sum_{\alpha}v_{m\alpha}e_{\alpha}(t),$
(2.3a) $\displaystyle\mathrm{i}\partial_{t}e_{\alpha}(t)=\epsilon_{{\rm
d}\alpha}e_{\alpha}(t)+\sum_{m}v_{m\alpha}^{\ast}g_{m}(0,t)+v^{\prime}_{\alpha}e_{\overline{\alpha}}(t).$
(2.3b)

Here $\delta(x)$ is Dirac’s delta function. We expressed the partial
differential operators $\partial/\partial t$ and $\partial/\partial x$ by
$\partial_{t}$ and $\partial_{x}$, respectively, and set
$\overline{\alpha}=3-\alpha$, $v_{1}^{\prime}=v^{\prime}$ and
$v_{2}^{\prime}=v^{\prime\ast}$. We remark that the time-dependent states in
noninteracting cases were studied for the single-quantum-dot systems with
general dispersion relations in Ref. [44].

Next, we consider the two-electron case. The time-dependent two-electron state
is given in the form

$\displaystyle|\Psi(t)\rangle$
$\displaystyle=\Big{(}\int_{x_{1}<x_{2}}dx_{1}dx_{2}\sum_{m_{1},m_{2}=1,2}g_{m_{1}m_{2}}(x_{1},x_{2},t)c^{\dagger}_{m_{1}}(x_{1})c^{\dagger}_{m_{2}}(x_{2})$
(2.3d)
$\displaystyle\quad+\sum_{m=1,2}\sum_{\alpha=1,2}e_{m,\alpha}(t)c^{\dagger}_{m}(x)d^{\dagger}_{\alpha}+f_{12}(t)d^{\dagger}_{1}d^{\dagger}_{2}\Big{)}|0\rangle,$

where we assume the following Fermionic anti-symmetries for the wave functions
$g_{m_{1}m_{2}}(x_{1},x_{2},t)$ and $f_{12}(t)$:

$\displaystyle
g_{m_{1}m_{2}}(x_{1},x_{2},t)=-g_{m_{2}m_{1}}(x_{2},x_{1},t),\qquad
f_{12}(t)=-f_{21}(t).$ (2.3e)

The time-dependent Schrödinger equation
$\mathrm{i}\partial_{t}|\Psi(t)\rangle=H|\Psi(t)\rangle$ gives the following
coupled partial-differential equations for the wave functions
$g_{m_{1}m_{2}}(x_{1},x_{2},t)$, $e_{m,\alpha}(x,t)$ and
$f_{\alpha\overline{\alpha}}(t)$:

$\displaystyle\mathrm{i}\partial_{t}g_{m_{1}m_{2}}(x_{1},x_{2},t)=\frac{1}{\mathrm{i}}(\partial_{x_{1}}\\!+\\!\partial_{x_{2}})g_{m_{1}m_{2}}(x_{1},x_{2},t)$
$\displaystyle+\sum_{\alpha}\big{(}v_{m_{2}\alpha}\delta(x_{2})e_{m_{1},\alpha}(x_{1},t)\\!-\\!v_{m_{1}\alpha}\delta(x_{1})e_{m_{2},\alpha}(x_{2},t)\big{)},$
(2.3fa)
$\displaystyle\mathrm{i}\partial_{t}e_{m,\alpha}(x,t)=\Big{(}\frac{1}{\mathrm{i}}\partial_{x}+\epsilon_{{\rm
d}\alpha}\Big{)}e_{m,\alpha}(x,t)$
$\displaystyle+\sum_{n}v_{n\alpha}^{\ast}g_{mn}(x,0,t)+v^{\prime}_{\alpha}e_{m,\overline{\alpha}}(x,t)-v_{m\overline{\alpha}}\delta(x)f_{\alpha\overline{\alpha}}(t),$
(2.3fb)
$\displaystyle\mathrm{i}\partial_{t}f_{\alpha\overline{\alpha}}(t)=(2\overline{\epsilon}_{\rm
d}+U^{\prime})f_{\alpha\overline{\alpha}}(t)+\sum_{m}\big{(}v^{\ast}_{m\alpha}e_{m,\overline{\alpha}}(0,t)-v^{\ast}_{m\overline{\alpha}}e_{m,\alpha}(0,t)\big{)}.$
(2.3fc)

Here we introduced $\overline{\epsilon}_{\rm d}=(\epsilon_{{\rm
d}1}+\epsilon_{{\rm d}2})/2$.

The many-electron scattering eigenstates have been constructed as the
stationary solutions of the set of Schrödinger equations in Eqs. (2.3f),
(2.3f) and (2.3fc) under the scattering boundary conditions [19, 20]. As an
effect of the interdot interaction, the incident plane waves are partially
scattered to two- and three-body bound states at the quantum dots. The binding
strength of the many-body bound states corresponds to the imaginary part of
the resonance poles of the scattering wave functions.

##  3 Time-evolving scattering states

###  3.1 One-electron scattering states

Let us construct time-evolving scattering states of the open double quantum-
dot system by solving initial-value problems for the time-dependent
Schrödinger equation given in Sec. 2. We propose a systematic construction of
exact solutions for arbitrary initial states at time $t=0$ without assuming
the form of wave functions as was used in Refs. [33, 34, 35]. In this
subsection, we consider the one-electron case. The Schrödinger equation (2.3a)
tells us that, in each region of $x>0$ or $x<0$, the general solution
$g_{m}(x,t)$ is given by an arbitrary function $F(x-t)$ of the variable $x-t$.
In other words, for the real parameter $a$ under the condition $x+a>0$ if
$x>0$ or the condition $x+a<0$ if $x<0$, the solution $g_{m}(x,t)$ is
invariant under the translation $(x,t)\mapsto(x+a,t+a)$. By combining the two
conditions as $x(x+a)>0$, we have

$\displaystyle g_{m}(x+a,t+a)=g_{m}(x,t),\quad\mbox{if }x(x+a)>0.$ (2.3fa)

Because of the delta-function term in Eq. (2.3a), the wave function
$g_{m}(x,t)$ is discontinuous at $x=0$. Hence, at $x=0$, we match in the
following form the wave functions that are given in the two regions $x>0$ and
$x<0$. We obtain the matching condition of $g_{m}(x,t)$ at $x=0$ by
integrating Eq. (2.3a) over an infinitesimal region around $x=0$, which
results in

$\displaystyle
g_{m}(0+,t)-g_{m}(0-,t)=-\mathrm{i}\sum_{\alpha}v_{m\alpha}e_{\alpha}(t).$
(2.3fb)

Since the value of $g_{m}(x,t)$ at $x=0$, which appears in Eq. (2.3b), is not
determined by the Schrödinger equations (2.3a) and (2.3b), we assume

$\displaystyle g_{m}(0,t)=\frac{1}{2}\big{(}g_{m}(0+,t)+g_{m}(0-,t)\big{)}.$
(2.3fc)

We remark that the difference of the value $g_{m}(0,t)$ should be absorbed
into the system parameters by the renormalization-group method [45].

By inserting Eqs. (2.3fc) and (2.3fb) into Eq. (2.3b), we obtain

$\displaystyle\mathrm{i}\partial_{t}e_{\alpha}(t)$
$\displaystyle=(\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha})e_{\alpha}(t)+\big{(}v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}}\big{)}e_{\overline{\alpha}}(t)+\sum_{m}v_{m\alpha}^{\ast}g_{m}(0-,t).$
(2.3fd)

Here we introduced the level width
$\Gamma_{\alpha\beta}=\sum_{m}v^{\ast}_{m\alpha}v_{m\beta}/2$. By introducing
an auxiliary function
$\tilde{e}_{\alpha}(t)=\mathrm{e}^{\mathrm{i}(\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha})t}e_{\alpha}(t)$, the differential
equation (2.3fd) is rewritten as

$\displaystyle\mathrm{i}\partial_{t}\tilde{e}_{\alpha}(t)$
$\displaystyle=(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})\mathrm{e}^{\mathrm{i}(\Delta\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Delta\Gamma_{\alpha})t}\tilde{e}_{\overline{\alpha}}(t)+\sum_{m}v_{m\alpha}^{\ast}\mathrm{e}^{\mathrm{i}(\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha})t}g_{m}(0-,t),$ (2.3fe)

where $\Delta\epsilon_{{\rm d}\alpha}=\epsilon_{{\rm d}\alpha}-\epsilon_{{\rm
d}\overline{\alpha}}$ and
$\Delta\Gamma_{\alpha}=\Gamma_{\alpha\alpha}-\Gamma_{\overline{\alpha}\,\overline{\alpha}}$.

The coupled first-order differential equations for $\tilde{e}_{\alpha}(t)$ and
$\tilde{e}_{\overline{\alpha}}(t)$ are transformed to a second-order
differential equation for $\tilde{e}_{\alpha}(t)$ of the form

$\displaystyle\partial^{2}_{t}\tilde{e}_{\alpha}(t)-\mathrm{i}(\Delta\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Delta\Gamma_{\alpha})\partial_{t}\tilde{e}_{\alpha}(t)+(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})(v^{\prime}_{\overline{\alpha}}-\mathrm{i}\Gamma_{\overline{\alpha}\alpha})\tilde{e}_{\alpha}(t)=G_{\alpha}(t),$
(2.3ff)

where the inhomogeneous term is given by

$\displaystyle G_{\alpha}(t)=-\sum_{m}\mathrm{e}^{\mathrm{i}(\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha})t}\bigg{[}v_{m\alpha}^{\ast}(\mathrm{i}\partial_{t}-\epsilon_{{\rm
d}\overline{\alpha}}+\mathrm{i}\Gamma_{\overline{\alpha}\,\overline{\alpha}})+v_{m\overline{\alpha}}^{\ast}(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})\bigg{]}g_{m}(0-,t).$
(2.3fg)

In order to solve Eq. (2.3ff), we investigate the characteristic equation of a
homogeneous linear differential equation associated with Eq. (2.3ff) as

$\displaystyle\lambda^{2}-\mathrm{i}(\Delta\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Delta\Gamma_{\alpha})\lambda+(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})(v^{\prime}_{\overline{\alpha}}-\mathrm{i}\Gamma_{\overline{\alpha}\alpha})=0,$
(2.3fh)

which gives the two characteristic roots as

$\displaystyle\lambda_{\alpha,\pm}=\frac{1}{2}\big{(}\mathrm{i}(\Delta\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Delta\Gamma_{\alpha})+\eta_{\pm}\big{)},$ (2.3fi)
$\displaystyle\eta_{\pm}=\pm\eta=\pm\sqrt{(\mathrm{i}\Delta\epsilon_{{\rm
d}\alpha}+\Delta\Gamma_{\alpha})^{2}-4(v^{\prime}-\mathrm{i}\Gamma_{12})(v^{\prime\ast}-\mathrm{i}\Gamma_{21})}.$

We notice that the two roots $\lambda_{\alpha,\pm}$ merge into one at the
parameter points giving $\eta=0$.

The general solution of Eq. (2.3ff) is obtained as

$\displaystyle\tilde{e}_{\alpha}(t)$
$\displaystyle=\cases{\sum_{s=\pm}C_{\alpha,s}\mathrm{e}^{\lambda_{\alpha,s}t}+\sum_{s=\pm}\frac{1}{\lambda_{\alpha,s}-\lambda_{\alpha,\overline{s}}}\int^{t}_{-\infty}dt^{\prime}\,\mathrm{e}^{\lambda_{\alpha,s}(t-t^{\prime})}G_{\alpha}(t^{\prime})&for
$\eta\neq 0$,\\\
(D_{\alpha,0}+tD_{\alpha,1})\mathrm{e}^{\lambda^{0}_{\alpha}t}+\int_{-\infty}^{t}dt^{\prime}\,(t-t^{\prime})\mathrm{e}^{\lambda^{0}_{\alpha}(t-t^{\prime})}G_{\alpha}(t^{\prime})&for
$\eta=0$,\\\ }$ (2.3fj)

where $C_{\alpha,s}$ for $s=\pm$ and $D_{\alpha,r}$ for $r=0,1$ are
integration constants, and
$\lambda^{0}_{\alpha}=\lambda_{\alpha,s}|_{\eta=0}$. For $\eta=0$ in Eq.
(2.3fj), we find terms linear in time $t$ inside and outside the integral with
respect to the variable $t^{\prime}$. We shall elucidate in A.1 that the
parameter points satisfying $\eta=0$ correspond to exceptional points of a
non-Hermite effective Hamiltonian.

Now we propose a systematic construction of the time-evolving one-electron
states under arbitrary initial conditions as is shown by the flow chart in
Fig. 2:

Step 1)

The wave function $g_{m}(x,t)$ for $x<0$ in (i) of Fig. 2 is obtained by the
translation invariance in Eq. (2.3fa) with the wave function $g_{m}(x,0)$ of
given initial states.

Step 2)

In the process (i)$\to$(ii) in Fig. 2, we use the general solution of
$e_{\alpha}(t)$ in Eq. (2.3fj). The integration constant $C_{\alpha,s}$ or
$D_{\alpha,r}$ is determined by the initial conditions for $e_{\alpha}(t)$.

Step 3)

In the process (i), (ii)$\to$(iii) in Fig. 2, we use the matching condition of
$g_{m}(x,t)$ in Eq. (2.3fb).

We find that the construction above provides an exact solution if we obtain
the integration in Eq. (2.3fj), whose integrand is given by the initial state.

Figure 2: The flow chart of the construction of the time-evolving one-electron
states.

Let us now specifically consider the time-evolving scattering states for the
initial conditions

$\displaystyle
g_{m}(x,0)=\frac{1}{\sqrt{2\pi}}\delta_{m,\ell}\mathrm{e}^{\mathrm{i}kx}\;\mbox{
for }m=1,2,\quad e_{\alpha}(0)=0\;\mbox{ for }\alpha=1,2,$ (2.3fk)

which stand for the one-electron plane-wave state with the wave number $k$ on
the entire lead $\ell$ along with the absence of electrons on the quantum
dots.

###### Proposition 3.1

The set of Schrödinger equations (2.3a) and (2.3b) for the initial conditions
(2.3fk) is solved as follows: i) For $\eta\neq 0$, the solution is given by

$\displaystyle
g^{(\ell)}_{m,k}(x,t)=\frac{1}{\sqrt{2\pi}}\delta_{m,\ell}\mathrm{e}^{\mathrm{i}k(x-t)}$
$\displaystyle\hskip
60.0pt\\!+\\!\sum_{\alpha,s}e^{(\ell)}_{\alpha,k}\,y_{m\overline{\alpha},\overline{s}}\Big{[}\mathrm{e}^{\mathrm{i}k(x-t)}\\!-\\!\mathrm{e}^{\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)(x-t)}\Big{]}\tilde{\theta}(t\\!-\\!x)\theta(x),$
(2.3fla) $\displaystyle
e^{(\ell)}_{\alpha,k}(t)=e^{(\ell)}_{\alpha,k}\mathrm{e}^{-\mathrm{i}kt}+\sum_{s=\pm}\frac{1}{\eta_{s}}\bigg{[}\lambda_{\alpha,\overline{s}}e^{(\ell)}_{\alpha,k}+\mathrm{i}(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})e^{(\ell)}_{\overline{\alpha},k}\bigg{]}\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)t},$ (2.3flb)

where we used

$\displaystyle
e^{(\ell)}_{\alpha,k}=\frac{1}{\sqrt{2\pi}}\frac{(k-\epsilon_{{\rm
d}\overline{\alpha}}+\mathrm{i}\Gamma_{\overline{\alpha}\,\overline{\alpha}})v_{\ell\alpha}^{\ast}+(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})v_{\ell\overline{\alpha}}^{\ast}}{(k-\overline{\epsilon}_{{\rm
d}}+\mathrm{i}\overline{\Gamma}-\mathrm{i}\eta/2)(k-\overline{\epsilon}_{{\rm
d}}+\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta/2)},$ $\displaystyle
y_{m\alpha,\pm}=\frac{1}{\eta_{s}}(\mathrm{i}\lambda_{\alpha,\mp}v_{m\overline{\alpha}}\\!+\\!(v_{\alpha}^{\prime}\\!\\!-\\!\mathrm{i}\Gamma_{\alpha\overline{\alpha}})v_{m\alpha})$
(2.3flm)

and the following two types of step functions:

$\theta(t)=\cases{1&if $t>0$,\\\ 1/2&if $t=0$,\\\ 0&if $t<0$,\\\
}\quad\tilde{\theta}(t)=\cases{1&if $t>0$,\\\ 0&if $t\leq 0$.\\\ }$ (2.3fln)

ii) For $\eta=0$, the solution is given by

$\displaystyle
g^{(\ell)}_{m,k}(x,t)=\frac{1}{\sqrt{2\pi}}\delta_{m,\ell}\mathrm{e}^{\mathrm{i}k(x-t)}-\mathrm{i}\sum_{\alpha}v_{m\alpha}\Big{[}e^{(\ell)0}_{\alpha,k}\mathrm{e}^{\mathrm{i}k(x-t)}$
$\displaystyle\hskip
58.0pt+\big{(}(\lambda^{0}_{\alpha}e^{(\ell)0}_{\alpha,k}+\mathrm{i}(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})e^{(\ell)0}_{\overline{\alpha},k})(t\\!-\\!x)-e^{(\ell)0}_{\alpha,k}\big{)}\mathrm{e}^{\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma})(x-t)}\Big{]}\tilde{\theta}(t\\!-\\!x)\theta(x),$
(2.3floa) $\displaystyle
e^{(\ell)}_{\alpha,k}(t)=e^{(\ell)0}_{\alpha,k}\mathrm{e}^{-\mathrm{i}kt}+\Big{[}\big{(}\lambda^{0}_{\alpha}e^{(\ell)0}_{\alpha,k}+\mathrm{i}(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})e^{(\ell)0}_{\overline{\alpha},k}\big{)}t-e^{(\ell)0}_{\alpha,k}\Big{]}\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma})t},$ (2.3flob)

where $\lambda^{0}_{\alpha}=\lambda_{\alpha,s}|_{\eta=0}$ and
$e^{(\ell)0}_{\alpha,k}=e^{(\ell)}_{\alpha,k}|_{\eta=0}$.

Proof. By following the three steps above, we show a proof in the case
$\eta\neq 0$.

Step 1) We obtain the wave function $g_{m}(x,t)$ in the region of $x<0$ for
$t>0$ through the translation invariance in Eq. (2.3fa) as

$\displaystyle
g_{m}(x,t)=g_{m}(x-t,0)=\frac{1}{\sqrt{2\pi}}\delta_{m,\ell}\mathrm{e}^{\mathrm{i}k(x-t)}$
(2.3fp)

due to the relation $x(x-t)>0$.

Step 2) Process (i)$\to$(ii): By setting $x=0-$ in the above equation, we have

$\displaystyle
g_{m}(0-,t)=\frac{1}{\sqrt{2\pi}}\delta_{m,\ell}\mathrm{e}^{-\mathrm{i}kt}.$
(2.3fq)

Then the inhomogeneous term $G_{\alpha}(t)$ in Eq. (2.3fg) is calculated as

$\displaystyle
G_{\alpha}(t)=\frac{-1}{\sqrt{2\pi}}\mathrm{e}^{\mathrm{i}(-k+\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha})t}\bigg{[}v_{\ell\alpha}^{\ast}\big{(}k-\epsilon_{{\rm
d}\overline{\alpha}}+\mathrm{i}\Gamma_{\overline{\alpha}\,\overline{\alpha}})\\!+\\!v_{\ell\overline{\alpha}}^{\ast}(v^{\prime}_{\alpha}\\!-\\!\mathrm{i}\Gamma_{\alpha\overline{\alpha}})\bigg{]}.$
(2.3fr)

By inserting this into the general solution in Eq. (2.3fj), we obtain

$\displaystyle\tilde{e}_{\alpha}(t)=\sum_{s}C_{\alpha,s}\mathrm{e}^{\lambda_{\alpha,s}t}+e^{(\ell)}_{\alpha,k}\mathrm{e}^{\mathrm{i}(-k+\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha})t},$ $\displaystyle\therefore\quad$
$\displaystyle e_{\alpha}(t)=\mathrm{e}^{-\mathrm{i}(\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha})t}\tilde{e}_{\alpha}(t)=\sum_{s}C_{\alpha,s}\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)t}+e^{(\ell)}_{\alpha,k}\mathrm{e}^{-\mathrm{i}kt}.$
(2.3fs)

Here we used the relation $\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha}+\mathrm{i}\lambda_{\alpha,s}=\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2$. By applying the
solution (2.3fs) to Eq. (2.3fd), we obtain

$\displaystyle\tilde{e}_{\overline{\alpha}}(t)=\sum_{s}\frac{\mathrm{i}\lambda_{\alpha,s}}{v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}C_{\alpha,s}\mathrm{e}^{\mathrm{i}(\Delta\epsilon_{{\rm
d}\overline{\alpha}}-\mathrm{i}\Delta\Gamma_{\overline{\alpha}}-\mathrm{i}\lambda_{\alpha,s})t}+e^{(\ell)}_{\overline{\alpha},k}\mathrm{e}^{\mathrm{i}(-k+\epsilon_{{\rm
d}\overline{\alpha}}-\mathrm{i}\Gamma_{\overline{\alpha}\,\overline{\alpha}})t},$
$\displaystyle\therefore\quad$ $\displaystyle
e_{\overline{\alpha}}(t)=\sum_{s}\frac{\mathrm{i}\lambda_{\alpha,s}}{v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}C_{\alpha,s}\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)t}+e^{(\ell)}_{\overline{\alpha},k}\mathrm{e}^{-\mathrm{i}kt}.$
(2.3ft)

The initial conditions $e_{\alpha}(0)=e_{\overline{\alpha}}(0)=0$ determine
the integration constants $C_{\alpha,+}$ and $C_{\alpha,-}$ as

$\displaystyle
e_{\alpha}(0+)=C_{\alpha,+}+C_{\alpha,-}+e^{(\ell)}_{\alpha,k}=0,$
$\displaystyle
e_{\overline{\alpha}}(0+)=\frac{\mathrm{i}\lambda_{\alpha,+}}{v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}C_{\alpha,+}+\frac{\mathrm{i}\lambda_{\alpha,-}}{v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}C_{\alpha,-}+e^{(\ell)}_{\overline{\alpha},k}=0,$
$\displaystyle\therefore\quad$ $\displaystyle
C_{\alpha,\pm}=\frac{1}{\eta_{\pm}}\bigg{[}\lambda_{\alpha,\mp}e^{(\ell)}_{\alpha,k}+\mathrm{i}(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})e^{(\ell)}_{\overline{\alpha},k}\bigg{]}.$
(2.3fu)

Then, by inserting these into Eqs. (2.3fs) and (2.3ft), we obtain

$\displaystyle e_{\alpha}(t)$
$\displaystyle=e^{(\ell)}_{\alpha,k}\mathrm{e}^{-\mathrm{i}kt}\\!+\\!\sum_{s}\frac{1}{\eta_{s}}\bigg{[}\lambda_{\alpha,\overline{s}}e^{(\ell)}_{\alpha,k}\\!+\\!\mathrm{i}(v^{\prime}_{\alpha}\\!-\\!\mathrm{i}\Gamma_{\alpha\overline{\alpha}})e^{(\ell)}_{\overline{\alpha},k}\bigg{]}\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)t}.$ (2.3fv)

Step 3) Prosess (i), (ii)$\to$(iii): By using the matching condition of
$g_{m}(x,t)$ in Eq. (2.3fb), we have

$\displaystyle g_{m}(0+,t)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\delta_{m,\ell}\mathrm{e}^{-\mathrm{i}kt}-\mathrm{i}\tilde{\theta}(t)\sum_{\alpha}v_{m\alpha}e_{\alpha}(t).$
(2.3fw)

Here, in order to extend the relation to the region $t\leq 0$, we used the
step function $\tilde{\theta}(t)$ defined in Eq. (2.3fln). Through the
translation invariance in Eq. (2.3fa), we obtain the wave function
$g_{m}(x,t)$ for $x>0$ and $t>0$ as

$\displaystyle g_{m}(x,t)$ $\displaystyle=g_{m}(0+,t-x)$ (2.3fx)
$\displaystyle=\frac{1}{\sqrt{2\pi}}\delta_{m,\ell}\mathrm{e}^{\mathrm{i}k(x-t)}-\mathrm{i}\tilde{\theta}(t-x)\sum_{\alpha}v_{m\alpha}e_{\alpha}(t-x).$

The wave functions for $\eta=0$ are obtained in a similar way by using the
general solution for $\eta=0$ in Eqs. (2.3fj).

$\square$

We find that the second terms of the wave functions $g^{(\ell)}_{m,k}(x,t)$
and $e^{(\ell)}_{\alpha,k}(t)$ in Eqs. (2.3fl), (2.3flb), (2.3flo) and
(2.3flob) contain the terms with the time-dependent factor
$\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)t}$, which decays
exponentially with the inverse relaxation time $\overline{\Gamma}-{\rm
Re}(\eta_{s}/2)$. The purely exponential decay is due to the unbounded linear
dispersion relation [44]; if there were the lower limit of the energy
dispersion, we would have found deviations from the exponential decay. We also
find that the time-dependent factor corresponds to the one-body resonant
state, which is explicitly given in A.1. The wave functions for $\eta=0$ in
Eqs. (2.3flo) and (2.3flob) include an exponential function multiplied by a
term linear in time $t$. We remark that they are reproduced by taking the
limit $\eta\to 0$ of Eqs. (2.3fl) and (2.3flb).

In the long-time limit $t\to\infty$, the wave functions in Eqs. (2.3fl) and
(2.3flb) converge to the stationary scattering eigenstates [20],

$\displaystyle\lim_{t\to\infty}\mathrm{e}^{\mathrm{i}kt}g^{(\ell)}_{m,k}(x,t)=\frac{1}{\sqrt{2\pi}}\bigg{[}\delta_{m,\ell}-\mathrm{i}\sqrt{2\pi}\theta(x)\sum_{\alpha}v_{m\alpha}e^{(\ell)}_{\alpha,k}\bigg{]}\mathrm{e}^{\mathrm{i}kx}=:g^{(\ell)}_{m,k}(x),$
(2.3fya)
$\displaystyle\lim_{t\to\infty}\mathrm{e}^{\mathrm{i}kt}e^{(\ell)}_{\alpha,k}(t)=e^{(\ell)}_{\alpha,k},$
(2.3fyb)

where $e^{(\ell)}_{\alpha,k}$ is defined in Eq. (3.1). We remark that the wave
functions $g^{(\ell)}_{m,k}(x)$ and $e^{(\ell)}_{\alpha,k}$ have resonance
poles at $k=\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}\pm\mathrm{i}\eta/2$ on the complex $k$ plane.
The imaginary parts of the resonance poles correspond to the relaxation time
of the time-evolving scattering states.

###  3.2 Two-electron scattering states

Next we consider the two-electron case. In a way similar to the one-electron
case, we derive several relations among the wave functions from the set of
Schrödinger equations (2.3f), (2.3f) and (2.3fc). We consider the case of
$\eta\neq 0$ since the wave functions for $\eta=0$ are obtained by taking the
limit $\eta\to 0$. First, we find that the wave function
$g_{m_{1}m_{2}}(x_{1},x_{2},t)$ is discontinuous both at $x_{1}=0$ and
$x_{2}=0$. In each quadrant of the $(x_{1},x_{2})$-plane, the general solution
$g_{m_{1}m_{2}}(x_{1},x_{2},t)$ is given by an arbitrary function
$F(x_{1}-t,x_{2}-t)$ of the two variables $x_{1}-t$ and $x_{2}-t$. In other
words, the wave function $g_{m_{1}m_{2}}(x_{1},x_{2},t)$ has the following
translation invariance:

$\displaystyle g_{m_{1}m_{2}}(x_{1}+a,x_{2}+a,t+a)$
$\displaystyle=g_{m_{1}m_{2}}(x_{1},x_{2},t)\quad\mbox{ if
}x_{i}(x_{i}+a)>0\mbox{ for }i=1,2.$ (2.3fyz)

The matching conditions of $g_{m_{1}m_{2}}(x_{1},x_{2},t)$ at $x_{1}=0$ and
$x_{2}=0$ are respectively given by

$\displaystyle
g_{m_{1}m_{2}}(x,0+,t)-g_{m_{1}m_{2}}(x,0-,t)+\mathrm{i}\sum_{\alpha}v_{m_{2}\alpha}e_{m_{1},\alpha}(x,t)=0,$
$\displaystyle
g_{m_{1}m_{2}}(0+,x,t)-g_{m_{1}m_{2}}(0-,x,t)-\mathrm{i}\sum_{\alpha}v_{m_{1}\alpha}e_{m_{2},\alpha}(x,t)=0,$
(2.3fyaa)

as was in Eq. (2.3fb). We note that the two relations are consistent with the
anti-symmetries in Eqs. (2.3e). Similarly to Eq. (2.3fc), we assume the values
of $g_{m_{1}m_{2}}(x_{1},x_{2},t)$ at $x_{1}=0$ and $x_{2}=0$ as follows:

$\displaystyle
g_{m_{1}m_{2}}(0,x,t)=\frac{1}{2}\big{(}g_{m_{1}m_{2}}(0+,x,t)+g_{m_{1}m_{2}}(0-,x,t)\big{)},$
$\displaystyle
g_{m_{1}m_{2}}(x,0,t)=\frac{1}{2}\big{(}g_{m_{1}m_{2}}(x,0+,t)+g_{m_{1}m_{2}}(x,0-,t)\big{)}.$
(2.3fyab)

Next, we find that the wave function $e_{m,\alpha}(x,t)$ is also discontinuous
at $x=0$, and its matching condition at $x=0$ is given by

$\displaystyle
e_{m,\alpha}(0+,t)-e_{m,\alpha}(0-,t)-\mathrm{i}v_{m\overline{\alpha}}f_{\alpha\overline{\alpha}}(t)=0.$
(2.3fyac)

We assume the value of $e_{m,\alpha}(x,t)$ at $x=0$ as follows:

$\displaystyle
e_{m,\alpha}(0,t)=\frac{1}{2}\big{(}e_{m,\alpha}(0+,t)+e_{m,\alpha}(0-,t)\big{)}.$
(2.3fyad)

By applying Eqs. (3.2) and (3.2) to Eq. (2.3f) for $x\neq 0$, we have the
coupled partial-differential equations for $e_{m,\alpha}(x,t)$ and
$e_{m,\overline{\alpha}}(x,t)$ as

$\displaystyle\big{(}\mathrm{i}(\partial_{t}+\partial_{x})-\epsilon_{{\rm
d}\alpha}+\mathrm{i}\Gamma_{\alpha\alpha}\big{)}e_{m,\alpha}(x,t)-\big{(}v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}}\big{)}e_{m,\overline{\alpha}}(x,t)=\sum_{n}v^{\ast}_{n\alpha}g_{mn}(x,0-,t).$
(2.3fyae)

Through the change of variables $z=(x+t)/2$ and $z_{1}=x-t$, the partial-
differential equation is rewritten as

$\displaystyle\big{(}\mathrm{i}\partial_{z}-\epsilon_{{\rm
d}\alpha}+\mathrm{i}\Gamma_{\alpha\alpha}\big{)}e_{m,\alpha}(x,t)-\big{(}v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}}\big{)}e_{m,\overline{\alpha}}(x,t)$
$\displaystyle=\sum_{n}v^{\ast}_{n\alpha}g_{mn}\Big{(}z+\frac{z_{1}}{2},0-,z-\frac{z_{1}}{2}\Big{)}.$
(2.3fyaf)

By introducing an auxiliary function
$e_{m,\alpha}(x,t)=\mathrm{e}^{-\mathrm{i}(\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha})z}\tilde{e}_{m,\alpha}(x,t)$, we
obtain

$\displaystyle\mathrm{i}\partial_{z}\tilde{e}_{m,\alpha}(x,t)=\big{(}v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}}\big{)}\mathrm{e}^{\mathrm{i}(\Delta\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Delta\Gamma_{\alpha})z}\tilde{e}_{m,\overline{\alpha}}(x,t)$
$\displaystyle\hskip
75.0pt+\sum_{n}v^{\ast}_{n\alpha}\mathrm{e}^{\mathrm{i}(\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha})z}g_{mn}\Big{(}z+\frac{z_{1}}{2},0-,z-\frac{z_{1}}{2}\Big{)},$
(2.3fyag)

which is transformed to a second-order differential equation for
$\tilde{e}_{m,\alpha}(x,t)$ for each $\alpha$ as

$\displaystyle\partial^{2}_{z}\tilde{e}_{m,\alpha}(x,t)-\mathrm{i}(\Delta\epsilon_{{\rm
d}\alpha}\\!-\\!\mathrm{i}\Delta\Gamma_{\alpha})\partial_{z}\tilde{e}_{m,\alpha}(x,t)$
$\displaystyle+(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})(v^{\prime}_{\overline{\alpha}}-\mathrm{i}\Gamma_{\overline{\alpha}\alpha})\tilde{e}_{m,\alpha}(x,t)=G_{m,\alpha}(z)$
(2.3fyah)

with the inhomogeneous term

$\displaystyle G_{m,\alpha}(z)$
$\displaystyle=\\!-\\!\sum_{n}\mathrm{e}^{\mathrm{i}(\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha})z}\bigg{[}v^{\ast}_{n\alpha}\big{(}\mathrm{i}\partial_{z}\\!-\\!\epsilon_{{\rm
d}\overline{\alpha}}\\!+\\!\mathrm{i}\Gamma_{\overline{\alpha}\,\overline{\alpha}})+v^{\ast}_{n\overline{\alpha}}(v^{\prime}_{\alpha}\\!-\\!\mathrm{i}\Gamma_{\alpha\overline{\alpha}})\bigg{]}$
(2.3fyai) $\displaystyle\quad\times
g_{mn}\Big{(}z\\!+\\!\frac{z_{1}}{2},0-,z\\!-\\!\frac{z_{1}}{2}\Big{)}.$

The general solution of Eq. (2.3fyah) is given by

$\displaystyle\tilde{e}_{m,\alpha}(x,t)=\sum_{s}C_{m,\alpha,s}(x\\!-\\!t)\mathrm{e}^{\lambda_{\alpha,s}(x+t)/2}$
$\displaystyle\hskip
57.0pt+\sum_{s}\frac{1}{\lambda_{\alpha,s}\\!-\\!\lambda_{\alpha,\overline{s}}}\int^{(x+t)/2}_{-\infty}dz^{\prime}\,\mathrm{e}^{\lambda_{\alpha,s}((x+t)/2-z^{\prime})}G_{m,\alpha}(z^{\prime}),$
$\displaystyle\therefore\quad$ $\displaystyle
e_{m,\alpha}(x,t)=\sum_{s}C_{m,\alpha,s}(x\\!-\\!t)\mathrm{e}^{-\mathrm{i}(\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha}+\mathrm{i}\lambda_{\alpha,s})(x+t)/2}$
(2.3fyaj) $\displaystyle\hskip
57.0pt+\sum_{s}\frac{1}{\lambda_{\alpha,s}\\!-\\!\lambda_{\alpha,\overline{s}}}\int_{-\infty}^{(x+t)/2}dz^{\prime}\,\mathrm{e}^{-\mathrm{i}(\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha})(x+t)/2+\lambda_{\alpha,s}((x+t)/2-z^{\prime})}G_{m,\alpha}(z^{\prime}),$

where $C_{m,\alpha,s}(x-t)$ for $s=\pm$ are arbitrary functions of the
variable $x-t$.

Finally, by using Eqs. (2.3fyac) and (2.3fyad), the differential equation
(2.3fc) for the double-occupancy wave function
$f_{\alpha\overline{\alpha}}(t)$ is rewritten as

$\displaystyle(\mathrm{i}\partial_{t}-2\overline{\epsilon}_{\rm
d}-U^{\prime}+2\mathrm{i}\overline{\Gamma})f_{\alpha\overline{\alpha}}(t)=\sum_{m,\beta}(-)^{\alpha+\beta}v^{\ast}_{m\beta}e_{m,\overline{\beta}}(0-,t).$
(2.3fyak)

By setting
$f_{\alpha\overline{\alpha}}(t)=\mathrm{e}^{-\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})t}\tilde{f}_{\alpha\overline{\alpha}}(t)$,
we have

$\displaystyle\partial_{t}\tilde{f}_{\alpha\overline{\alpha}}(t)=-\mathrm{i}\mathrm{e}^{\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})t}\sum_{m,\beta}(-)^{\alpha+\beta}v^{\ast}_{m\beta}e_{m,\overline{\beta}}(0-,t),$
(2.3fyal)

which is easily integrated in the form

$\displaystyle\tilde{f}_{\alpha\overline{\alpha}}(t)=C_{\alpha}-\mathrm{i}\int^{t}_{0}dt^{\prime}\,\mathrm{e}^{\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})t^{\prime}}\sum_{m,\beta}(-)^{\alpha+\beta}v^{\ast}_{m\beta}e_{m,\overline{\beta}}(0-,t^{\prime}),$
$\displaystyle\therefore\quad$ $\displaystyle
f_{\alpha\overline{\alpha}}(t)=C_{\alpha}\mathrm{e}^{-\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})t}$ (2.3fyam) $\displaystyle\hskip
42.0pt-\mathrm{i}\int^{t}_{0}dt^{\prime}\,\mathrm{e}^{\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})(t^{\prime}-t)}\sum_{m,\beta}(-)^{\alpha+\beta}v^{\ast}_{m\beta}e_{m,\overline{\beta}}(0-,t^{\prime}).$

Here $C_{\alpha}$ for $\alpha=1,2$ are integration constants.

Now we propose a systematic construction of the time-evolving two-electron
states under arbitrary initial conditions. The construction is given by the
flow chart in Fig. 3:

Step 1)

The wave function $g_{m_{1}m_{2}}(x_{1},x_{2},t)$ for $x_{1}<x_{2}<0$ in (i)
of Fig. 3 is obtained by the translation invariance in Eq. (3.2) with the wave
function $g_{m_{1}m_{2}}(x_{1},x_{2},0)$ of given initial states.

Step 2)

In the process (i)$\to$(ii) in Fig. 3, we use the general solution
$e_{m,\alpha}(x,t)$ in Eq. (2.3fyaj). The arbitrary function
$C_{m,\alpha,s}(x-t)$ is determined by the initial condition for
$e_{m,\alpha}(x,t)$.

Step 3)

In the process (ii)$\to$(iii) in Fig. 3, we use the general solution
$f_{\alpha\overline{\alpha}}(t)$ in Eq. (2.3fyam). The integration constant
$C_{\alpha}$ is determined by the initial condition for
$f_{\alpha\overline{\alpha}}(t)$.

Step 4)

In the process (i), (ii)$\to$(iv) in Fig. 3, we use the matching conditions
$g_{m_{1}m_{2}}(x_{1},x_{2},t)$ in Eqs. (3.2).

Step 5)

In the process (iii), (iv)$\to$(v) in Fig. 3, we use the general solution
$e_{m,\alpha}(x,t)$ in Eq. (2.3fyaj) again. The arbitrary function
$C_{m,\alpha,s}(x-t)$ is determined by the initial condition for
$e_{m,\alpha}(x,t)$ and the matching condition for $e_{m,\alpha}(x,t)$ in Eq.
(2.3fyac).

Step 6)

In the process (iv), (v)$\to$(vi) in Fig. 3, we use the matching conditions
for $g_{m_{1}m_{2}}(x_{1},x_{2},t)$ in Eqs. (3.2) again.

Step 7)

Through the anti-symmetrization in the variables $x_{1}$ and $x_{2}$, we
obtain the wave function $g_{m_{1}m_{2}}(x_{1},x_{2},t)$ in the case
$x_{2}<x_{1}$.

Figure 3: The flow chart of the construction of time-evolving two-electron
states.

Let us demonstrate the construction of the time-evolving two-electron
scattering states under the initial conditions

$\displaystyle
g_{m_{1}m_{2}}(x_{1},x_{2},0)=\frac{1}{2\pi}\sum_{P}\mathrm{sgn}(P)\delta_{m_{1},\ell_{P_{1}}}\delta_{m_{2},\ell_{P_{2}}}\mathrm{e}^{\mathrm{i}(k_{P_{1}}x_{1}+k_{P_{2}}x_{2})}\,\mbox{
for }m_{1},m_{2}=1,2,$ $\displaystyle e_{m,\alpha}(x,0)=0,\quad
f_{\alpha\overline{\alpha}}(0)=0\,\mbox{ for }m=1,2\mbox{ and }\alpha=1,2,$
(2.3fyan)

where $P=(P_{1},P_{2})$ is a permutation of $(1,2)$. The initial state stands
for the two-electron plane-wave state with wave numbers $k_{1}$ and $k_{2}$ on
the entire leads $\ell_{1}$ and $\ell_{2}$, respectively, and the absence of
electrons on the quantum dots.

###### Proposition 3.2

For $\eta\neq 0$, the solution of the set of Schrödinger equations (2.3f),
(2.3f) and (2.3fc) for the initial conditions (2.3fyan) is given by

$\displaystyle
g^{(\ell_{1}\ell_{2})}_{m_{1}m_{2},k_{1}k_{2}}(x_{1},x_{2},t)=\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m_{1},k_{P_{1}}}(x_{1},t)g^{(\ell_{P_{2}})}_{m_{2},k_{P_{2}}}(x_{2},t)$
$\displaystyle-\mathrm{i}\\!\sum_{Q,\alpha,s}\\!\mathrm{sgn}(Q)v_{m_{Q_{1}}\alpha}y_{m_{Q_{2}}\alpha,s}X^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}(x_{Q_{2}}\\!\\!-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)x_{Q_{2}Q_{1}}}\theta(x_{Q_{2}Q_{1}})\theta(x_{Q_{1}}),$
(2.3fyaoa) $\displaystyle e^{(\ell_{1}\ell_{2})}_{m\alpha,k_{1}k_{2}}(x,t)$
$\displaystyle=\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m,k_{P_{1}}}(x,t)e^{(\ell_{P_{2}})}_{\alpha,k_{P_{2}}}(t)-\sum_{s}y_{m\alpha,s}X^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}(x-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)x}\theta(x),$ (2.3fyaob)
$\displaystyle
f^{(\ell_{1}\ell_{2})}_{\alpha\overline{\alpha},k_{1}k_{2}}(t)=\sum_{P}\mathrm{sgn}(P)e^{(\ell_{P_{1}})}_{\alpha,k_{P_{1}}}(t)e^{(\ell_{P_{2}})}_{\overline{\alpha},k_{P_{2}}}(t)+X^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}(-t),$
(2.3fyaoc)

where $P$ and $Q$ are permutations of $(1,2)$, $x_{12}=x_{1}-x_{2}$ and we
introduced

$\displaystyle
X^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}(-t)=\sum_{P}\mathrm{sgn}(P)\Big{[}\frac{1}{2}Z^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}\big{(}\mathrm{e}^{-\mathrm{i}(k_{1}+k_{2})t}-\mathrm{e}^{-\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})t}\big{)}$ (2.3fyaoap)
$\displaystyle\hskip
75.0pt\quad-e^{(\ell_{P_{1}})}_{\alpha,k_{P_{1}}}e^{(\ell_{P_{2}})}_{\overline{\alpha},k_{P_{2}}}\big{(}\mathrm{e}^{-2\mathrm{i}(\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma})t}-\mathrm{e}^{-\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})t}\big{)}$ $\displaystyle\hskip
75.0pt+\frac{1}{\sqrt{2\pi}}\sum_{\beta,s}(-)^{\alpha+\beta}\frac{\mathrm{i}\tilde{y}_{\ell_{P_{1}}\beta,s}U^{\prime}}{(k_{P_{1}}\\!-\\!\overline{\epsilon}_{\rm
d}\\!+\\!\mathrm{i}\overline{\Gamma}\\!-\\!\mathrm{i}\eta_{s}/2)(k_{P_{1}}\\!-\\!\overline{\epsilon}_{\rm
d}\\!-\\!U^{\prime}\\!+\\!\mathrm{i}\overline{\Gamma}\\!-\\!\mathrm{i}\eta_{s}/2)}e^{(\ell_{P_{2}})}_{\overline{\beta},k_{P_{2}}}$
$\displaystyle\hskip
75.0pt\quad\times\big{(}\mathrm{e}^{-\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})t}-\mathrm{e}^{-\mathrm{i}(k_{P_{1}}+\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}-\mathrm{i}\eta_{s}/2)t}\big{)}\Big{]}\tilde{\theta}(t),$
$\displaystyle
Z^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}=\frac{U^{\prime}}{k_{1}\\!+\\!k_{2}\\!-\\!2\overline{\epsilon}_{\rm
d}\\!-\\!U^{\prime}\\!+\\!2\mathrm{i}\overline{\Gamma}}\big{(}e^{(\ell_{1})}_{\alpha,k_{1}}e^{(\ell_{2})}_{\overline{\alpha},k_{2}}-e^{(\ell_{2})}_{\alpha,k_{2}}e^{(\ell_{1})}_{\overline{\alpha},k_{1}}\big{)}.$

Proof. We follow the seven steps above.

Step 1) Since we have $x_{1}-t<x_{2}-t<0$ for $t>0$ and $x_{1}<x_{2}<0$ in
this step, the wave function $g_{m_{1}m_{2}}(x_{1},x_{2},t)$ is obtained
through the invariance of the translation
$(x_{1},x_{2},t)\mapsto(x_{1}-t,x_{2}-t,0)$ from the initial state
$g_{m_{1}m_{2}}(x_{1}-t,x_{2}-t,0)$ as

$\displaystyle g_{m_{1}m_{2}}(x_{1},x_{2},t)$
$\displaystyle=g_{m_{1}m_{2}}(x_{1}-t,x_{2}-t,0)$ (2.3fyaq)
$\displaystyle=\frac{1}{2\pi}\sum_{P}\mathrm{sgn}(P)\delta_{m_{1},\ell_{P_{1}}}\delta_{m_{2},\ell_{P_{2}}}\mathrm{e}^{\mathrm{i}k_{P_{1}}(x_{1}-t)+\mathrm{i}k_{P_{2}}(x_{2}-t)}.$

Step 2) Process (i)$\to$(ii): By setting $x_{2}=0-$ in the above equation, we
have

$\displaystyle
g_{m_{1}m_{2}}(x_{1},0-,t)=\frac{1}{2\pi}\sum_{P}\mathrm{sgn}(P)\delta_{m_{1},\ell_{P_{1}}}\delta_{m_{2},\ell_{P_{2}}}\mathrm{e}^{\mathrm{i}k_{P_{1}}(x_{1}-t)-\mathrm{i}k_{P_{2}}t},$
(2.3fyar)

which gives the inhomogeneous term in Eq. (2.3fyai) in the form

$\displaystyle G_{m,\alpha}(z)$
$\displaystyle=-\sum_{n}\bigg{[}v^{\ast}_{n\alpha}\big{(}k_{P_{2}}-\epsilon_{{\rm
d}\overline{\alpha}}+\mathrm{i}\Gamma_{\overline{\alpha}\,\overline{\alpha}})+v^{\ast}_{n\overline{\alpha}}(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})\bigg{]}$
(2.3fyas)
$\displaystyle\quad\times\frac{1}{2\pi}\sum_{P}\mathrm{sgn}(P)\delta_{m,\ell_{P_{1}}}\delta_{n,\ell_{P_{2}}}\mathrm{e}^{\mathrm{i}(k_{P_{1}}+k_{P_{2}}/2)z_{1}+\mathrm{i}(-k_{P_{2}}+\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha})z}.$

Then, for $x<0$, we obtain the general solution in Eq. (2.3fyaj) as

$\displaystyle e_{m,\alpha}(x,t)$
$\displaystyle=\sum_{s}C_{m,\alpha,s}(x-t)\mathrm{e}^{\mathrm{i}(-\epsilon_{{\rm
d}\alpha}+\mathrm{i}\Gamma_{\alpha\alpha}-\mathrm{i}\lambda_{\alpha,s})(x+t)/2}$
(2.3fyata)
$\displaystyle\quad+\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m,k_{P_{1}}}(x,t)e^{(\ell_{P_{2}})}_{\alpha,k_{P_{2}}}\mathrm{e}^{-\mathrm{i}k_{P_{2}}t},$
$\displaystyle e_{m,\overline{\alpha}}(x,t)$
$\displaystyle=\sum_{s}\frac{\mathrm{i}\lambda_{\alpha,s}}{v^{\prime}_{\alpha}\\!-\\!\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}C_{m,\alpha,s}(x-t)\mathrm{e}^{\mathrm{i}(-\epsilon_{{\rm
d}\overline{\alpha}}+\mathrm{i}\Gamma_{\overline{\alpha}\overline{\alpha}}-\mathrm{i}\lambda_{\overline{\alpha},s})(x+t)/2}$
(2.3fyatb)
$\displaystyle\quad+\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m,k_{P_{1}}}(x,t)e^{(\ell_{P_{2}})}_{\overline{\alpha},k_{P_{2}}}\mathrm{e}^{-\mathrm{i}k_{P_{2}}t},$

where $e_{m,\overline{\alpha}}(x,t)$ is obtained by using Eq. (2.3fyaf) with
the above expression for $e_{m,\alpha}(x,t)$. Here we used the wave function
$g^{(\ell)}_{m,k}(x,t)$ of the one-electron scattering states in Eq. (2.3fl).
The initial conditions in Eqs. (2.3fyan) for $e_{m,\alpha}(x,t)$ and
$e_{m,\overline{\alpha}}(x,t)$ determine the arbitrary functions
$C_{m,\alpha,s}(x)$ and $C_{m,\alpha,\overline{s}}(x)$ as

$\displaystyle
e_{m,\alpha}(x,0+)=\sum_{s}C_{m,\alpha,s}(x)\mathrm{e}^{\mathrm{i}(-\epsilon_{{\rm
d}\alpha}+\mathrm{i}\Gamma_{\alpha\alpha}-\mathrm{i}\lambda_{\alpha,s})x/2}+\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m,k_{P_{1}}}(x,0)e^{(\ell_{P_{2}})}_{\alpha,k_{P_{2}}}=0,$
$\displaystyle
e_{m,\overline{\alpha}}(x,0+)=\sum_{s}\frac{\mathrm{i}\lambda_{\alpha,s}}{v^{\prime}_{\alpha}\\!-\\!\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}C_{m,\alpha,s}(x)\mathrm{e}^{\mathrm{i}(-\epsilon_{{\rm
d}\overline{\alpha}}+\mathrm{i}\Gamma_{\overline{\alpha}\,\overline{\alpha}}-\mathrm{i}\lambda_{\overline{\alpha},s})x/2}$
$\displaystyle\hskip
70.0pt+\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m,k_{P_{1}}}(x,0)e^{(\ell_{P_{2}})}_{\overline{\alpha},k_{P_{2}}}=0,$
$\displaystyle\therefore\quad$ $\displaystyle
C_{m,\alpha,s}(x)=\frac{1}{\eta_{s}}\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m,k_{P_{1}}}(x,0)\bigg{[}\lambda_{\alpha,\overline{s}}e^{(\ell_{P_{2}})}_{\alpha,k_{P_{2}}}+\mathrm{i}(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})e^{(\ell_{P_{2}})}_{\overline{\alpha},k_{P_{2}}}\bigg{]}$
(2.3fyatau) $\displaystyle\hskip
60.0pt\times\mathrm{e}^{\mathrm{i}(\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha}+\mathrm{i}\lambda_{\alpha,s})x/2}.$

By using the translation invariance
$g^{(\ell_{P_{1}})}_{m,k_{P_{1}}}(x,t)=g^{(\ell_{P_{1}})}_{m,k_{P_{1}}}(x-t,0)$
for $t>0$ and $x<0$, we have

$\displaystyle C_{m,\alpha,s}(x-t)$
$\displaystyle=\frac{1}{\eta_{s}}\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m,k_{P_{1}}}(x,t)\bigg{[}\lambda_{\alpha,\overline{s}}e^{(\ell_{P_{2}})}_{\alpha,k_{P_{2}}}+\mathrm{i}(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})e^{(\ell_{P_{2}})}_{\overline{\alpha},k_{P_{2}}}\bigg{]}$
(2.3fyatav) $\displaystyle\quad\times\mathrm{e}^{\mathrm{i}(\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha}+\mathrm{i}\lambda_{\alpha,s})(x-t)/2}.$

By inserting this into Eqs. (2.3fyata) and (2.3fyatb), we obtain
$e_{m,\alpha}(x,t)$ and $e_{m,\overline{\alpha}}(x,t)$ in the same form

$\displaystyle e_{m,\alpha}(x,t)$
$\displaystyle=\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m,k_{P_{1}}}(x,t)e^{(\ell_{P_{2}})}_{\alpha,k_{P_{2}}}(t).$
(2.3fyataw)

Here $e^{(\ell)}_{\alpha,k}(t)$ is the wave function of the one-electron
scattering state in Eq. (2.3flb).

Step 3) Process (ii)$\to$(iii): We insert Eq. (2.3fyataw) into the solution in
Eq. (2.3fyam) for $f_{\alpha\overline{\alpha}}(t)$. Because of the initial
condition $f_{\alpha\overline{\alpha}}(0)=0$, we find $C=0$, obtaining

$\displaystyle
f_{\alpha\overline{\alpha}}(t)=\sum_{P}\mathrm{sgn}(P)\Big{[}\frac{k_{P_{1}}\\!+\\!k_{P_{2}}\\!-\\!2\overline{\epsilon}_{\rm
d}\\!+\\!2\mathrm{i}\overline{\Gamma}}{k_{P_{1}}\\!+\\!k_{P_{2}}\\!-\\!2\overline{\epsilon}_{\rm
d}\\!-\\!U^{\prime}\\!+\\!2\mathrm{i}\overline{\Gamma}}e^{(\ell_{P_{1}})}_{\alpha,k_{P_{1}}}e^{(\ell_{P_{2}})}_{\overline{\alpha},k_{P_{2}}}\big{(}\mathrm{e}^{-\mathrm{i}(k_{P_{1}}+k_{P_{2}})t}\\!-\\!\mathrm{e}^{-\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})t}\big{)}$ $\displaystyle\hskip
50.0pt+\frac{1}{\sqrt{2\pi}}\sum_{\beta}(-)^{\alpha+\beta}e^{(\ell_{P_{2}})}_{\beta,k_{P_{2}}}\sum_{s}\frac{\mathrm{i}\tilde{y}_{\ell_{P_{1}}\overline{\beta},\overline{s}}}{k_{P_{1}}\\!-\\!\overline{\epsilon}_{\rm
d}\\!-\\!U^{\prime}\\!+\\!\mathrm{i}\overline{\Gamma}\\!-\\!\mathrm{i}\eta_{\overline{s}}/2}$
$\displaystyle\hskip
50.0pt\quad\times\big{(}\mathrm{e}^{-\mathrm{i}(k_{P_{1}}+\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)t}-\mathrm{e}^{-\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})t}\big{)}\Big{]}.$ (2.3fyatax)

After some calculations, we arrive at the expression

$\displaystyle
f_{\alpha\overline{\alpha}}(t)=\sum_{P}\mathrm{sgn}(P)e^{(\ell_{P_{1}})}_{\alpha,k_{P_{1}}}(t)e^{(\ell_{P_{2}})}_{\overline{\alpha},k_{P_{2}}}(t)+X^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}(-t),$
(2.3fyatay)

with $X^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}(-t)$ defined in Eqs.
(2.3fyaoap).

Step 4) Process (i), (ii)$\to$(iv): The first matching condition in Eqs. (3.2)
gives

$\displaystyle g_{m_{1}m_{2}}(x,0+,t)$
$\displaystyle=\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m_{1},k_{P_{1}}}(x,t)g^{(\ell_{P_{2}})}_{m_{2},k_{P_{1}}}(0+,t).$
(2.3fyataz)

Since $x_{1}-x_{2}<0$ for $x_{1}<0<x_{2}$ in this step, we have

$\displaystyle g_{m_{1}m_{2}}(x_{1},x_{2},t)$
$\displaystyle=g_{m_{1}m_{2}}(x_{1}-x_{2},0+,t-x_{2})$ (2.3fyatba)
$\displaystyle=\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m_{1},k_{P_{1}}}(x_{1},t)g^{(\ell_{P_{2}})}_{m_{2},k_{P_{1}}}(x_{2},t)$

from the translation invariance of the wave function
$g_{m_{1}m_{2}}(x_{1},x_{2},t)$ in Eq. (3.2).

Step 5) Process (iii), (iv)$\to$(v): By setteing $x_{1}=0-$ in Eq.
(2.3fyatba), we have

$\displaystyle
g_{m_{1}m_{2}}(0-,x,t)=\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m_{1},k_{P_{1}}}(0-,t)g^{(\ell_{P_{2}})}_{m_{2},k_{P_{1}}}(x,t),$
(2.3fyatbb)

which gives the inhomogeneous term in Eq. (2.3fyai) as

$\displaystyle
G_{n,\alpha}(z)=\sum_{m}\bigg{[}v^{\ast}_{m\alpha}\big{(}k_{P_{1}}\\!-\\!\epsilon_{{\rm
d}\overline{\alpha}}\\!+\\!\mathrm{i}\Gamma_{\overline{\alpha}\,\overline{\alpha}})+v^{\ast}_{m\overline{\alpha}}(v^{\prime}_{\alpha}\\!-\\!\mathrm{i}\Gamma_{\alpha\overline{\alpha}})\bigg{]}$
$\displaystyle\hskip
50.0pt\times\frac{1}{\sqrt{2\pi}}\sum_{P}\mathrm{sgn}(P)\delta_{m\ell_{P_{1}}}\mathrm{e}^{\mathrm{i}k_{P_{1}}z_{1}/2}\mathrm{e}^{\mathrm{i}(-k_{P_{1}}+\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha})z}$ $\displaystyle\hskip
50.0pt\times\Big{[}\frac{1}{\sqrt{2\pi}}\delta_{n\ell_{P_{2}}}\mathrm{e}^{\mathrm{i}k_{P_{2}}z_{1}}+\sum_{\beta,s}e^{(\ell_{P_{2}})}_{\beta,k_{P_{2}}}\,y_{n\overline{\beta},\overline{s}}\big{(}\mathrm{e}^{\mathrm{i}k_{P_{2}}z_{1}}\\!-\\!\mathrm{e}^{\mathrm{i}(\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)z_{1}}\big{)}\theta(-z_{1})\Big{]}.$
(2.3fyatbc)

Then, for $x>0$, we obtain the general solution in Eq. (2.3fyaj) as

$\displaystyle e_{n,\alpha}(x,t)$
$\displaystyle=\sum_{s}C_{n,\alpha,s}(x-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\frac{\mathrm{i}}{2}\eta_{s})(x+t)/2}$
(2.3fyatbda)
$\displaystyle\quad-\sum_{P}\mathrm{sgn}(P)e^{(\ell_{P_{1}})}_{\alpha,k_{P_{1}}}\mathrm{e}^{-\mathrm{i}k_{P_{1}}t}g^{(\ell_{P_{2}})}_{n,k_{P_{2}}}(x,t),$
$\displaystyle e_{n,\overline{\alpha}}(x,t)$
$\displaystyle=\sum_{s}\frac{\mathrm{i}\lambda_{\alpha,s}}{v^{\prime}_{\alpha}\\!-\\!\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}C_{n,\alpha,s}(x-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\frac{\mathrm{i}}{2}\eta_{s})(x+t)/2}$
(2.3fyatbdb)
$\displaystyle\quad-\sum_{P}\mathrm{sgn}(P)e^{(\ell_{P_{1}})}_{\overline{\alpha},k_{P_{1}}}\mathrm{e}^{-\mathrm{i}k_{P_{1}}t}g^{(\ell_{P_{2}})}_{n,k_{P_{2}}}(x,t).$

The initial conditions in Eqs. (2.3fyan) for $e_{n,\alpha}(x,t)$ and
$e_{n,\overline{\alpha}}(x,t)$ determine the functions $C_{n,\alpha,s}(x)$ and
$C_{n,\alpha,\overline{s}}(x)$ as

$\displaystyle
e_{n,\alpha}(x,0+)=\sum_{s}C_{n,\alpha,s}(x)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)x/2}-\sum_{P}\mathrm{sgn}(P)e^{(\ell_{P_{1}})}_{\alpha,k_{P_{1}}}g^{(\ell_{P_{2}})}_{n,k_{P_{2}}}(x,0)=0,$
$\displaystyle
e_{n,\overline{\alpha}}(x,0+)=\sum_{s}\frac{\mathrm{i}\lambda_{\alpha,s}}{v^{\prime}_{\alpha}\\!-\\!\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}C_{n,\alpha,s}(x)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)x/2}\\!-\\!\sum_{P}\mathrm{sgn}(P)e^{(\ell_{P_{1}})}_{\overline{\alpha},k_{P_{1}}}g^{(\ell_{P_{2}})}_{n,k_{P_{2}}}(x,0)=0,$
$\displaystyle\therefore\quad
C_{n,\alpha,s}(x)=\frac{1}{\eta_{s}}\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{n,k_{P_{1}}}(x,0)\Big{[}\lambda_{\alpha,\overline{s}}e^{(\ell_{P_{2}})}_{\alpha,k_{P_{2}}}+\mathrm{i}(v^{\prime}_{\alpha}\\!-\\!\mathrm{i}\Gamma_{\alpha\overline{\alpha}})e^{(\ell_{P_{2}})}_{\overline{\alpha},k_{P_{2}}}\Big{]}\mathrm{e}^{\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)x/2}.$ (2.3fyatbdbe)

It should be noted that, in order to obtain $C_{n,\alpha,s}(x-t)$ for $t>0$
and $x>0$, we consider both the cases $x-t>0$ and $x-t<0$. Since Eq.
(2.3fyatbdbe) determines the function $C_{n,\alpha,s}(x-t)$ only in the case
$x-t>0$, we need another function $D_{n,\alpha,s}(x-t)$ to express
$C_{n,\alpha,s}(x-t)$ in the case $x-t<0$ as

$\displaystyle C_{n,\alpha,s}(x-t)=$ $\displaystyle
D_{n,\alpha,s}(x-t)\theta(t-x)$ (2.3fyatbdbf)
$\displaystyle+\frac{1}{\eta_{s}}\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{n,k_{P_{1}}}(x,t)\Big{[}\lambda_{\alpha,\overline{s}}e^{(\ell_{P_{2}})}_{\alpha,k_{P_{2}}}\\!+\\!\mathrm{i}(v^{\prime}_{\alpha}\\!-\\!\mathrm{i}\Gamma_{\alpha\overline{\alpha}})e^{(\ell_{P_{2}})}_{\overline{\alpha},k_{P_{2}}}\Big{]}$
$\displaystyle\quad\times\mathrm{e}^{\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)(x-t)/2}\theta(x-t).$

By inserting this into Eqs. (2.3fyatbda) and (2.3fyatbdb), we obtain

$\displaystyle e_{m,\alpha}(x,t)$
$\displaystyle=\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m,k_{P_{1}}}(x,t)e^{(\ell_{P_{2}})}_{\alpha,k_{P_{2}}}(t)$
(2.3fyatbdbga)
$\displaystyle\quad+\sum_{s}D_{m,\alpha,s}(x-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)(x+t)/2}\theta(t-x),$
$\displaystyle e_{m,\overline{\alpha}}(x,t)$
$\displaystyle=\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m,k_{P_{1}}}(x,t)e^{(\ell_{P_{2}})}_{\overline{\alpha},k_{P_{2}}}(t)$
(2.3fyatbdbgb)
$\displaystyle\quad+\sum_{s}\frac{\mathrm{i}\lambda_{\alpha,s}}{v^{\prime}_{\alpha}\\!-\\!\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}D_{m,\alpha,s}(x-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)(x+t)/2}\theta(t-x).$

By using the matching condition for $e_{m,\alpha}(x,t)$ in Eq. (2.3fyac), we
have

$\displaystyle\sum_{s}D_{m,\alpha,s}(-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)t/2}\theta(t)=\mathrm{i}v_{m\overline{\alpha}}X^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}(-t).$
(2.3fyatbdbgbh)

In a similar way, we use the matching condition for
$e_{m,\overline{\alpha}}(x,t)$ in Eq. (2.3fyac), obtaining

$\displaystyle\sum_{s}\frac{\mathrm{i}\lambda_{\alpha,s}}{v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}D_{m,\alpha,s}(-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)t/2}=-\mathrm{i}v_{m\alpha}X^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}(-t).$
(2.3fyatbdbgbi)

By solving the coupled equations for $D_{m,\alpha,+}(-t)$ and
$D_{m,\alpha,-}(-t)$, we obtain

$\displaystyle
D_{m,\alpha,s}(-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)t/2}\theta(t)=-y_{m\alpha,s}X^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}(-t).$
(2.3fyatbdbgbj)

By inserting this into Eqs. (2.3fyatbdbga) and (2.3fyatbdbgb), we obtain
$e_{m,\alpha}(x,t)$ and $e_{m,\overline{\alpha}}(x,t)$ in the same form

$\displaystyle e_{m,\alpha}(x,t)$
$\displaystyle=\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m,k_{P_{1}}}(x,t)e^{(\ell_{P_{2}})}_{\alpha,k_{P_{2}}}(t)$
(2.3fyatbdbgbk)
$\displaystyle\quad-\sum_{s}y_{m\alpha,s}X^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}(x-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/s)x}.$

Step 6) Process (iv), (v)$\to$(vi): The second matching condition in Eqs.
(3.2) gives

$\displaystyle g_{m_{1}m_{2}}(0+,x,t)$
$\displaystyle=\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m_{1},k_{P_{1}}}(0+,t)g^{(\ell_{P_{2}})}_{m_{2},k_{P_{2}}}(x,t)$
(2.3fyatbdbgbl)
$\displaystyle\quad-\mathrm{i}\sum_{\alpha,s}v_{m_{1}\alpha}y_{m_{2}\alpha,s}X^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}(x-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)x}\theta(x).$

Since $x_{2}-x_{1}>0$ for $0<x_{1}<x_{2}$ in this step, we have

$\displaystyle g_{m_{1}m_{2}}(x_{1},x_{2},t)$
$\displaystyle=g_{m_{1}m_{2}}(0+,x_{2}-x_{1},t-x_{1})$ (2.3fyatbdbgbm)
$\displaystyle=\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m_{1},k_{P_{1}}}(x_{1},t)g^{(\ell_{P_{2}})}_{m_{2},k_{P_{2}}}(x_{2},t)$
$\displaystyle\quad-\mathrm{i}\sum_{\alpha,s}v_{m_{1}\alpha}y_{m_{2}\alpha,s}X^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}(x_{2}\\!-\\!t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)x_{21}}\theta(x_{21})\theta(x_{1}).$

Step 7) After the anti-symmetrization in the variables $x_{1}$ and $x_{2}$, we
obtain the wave function $g_{m_{1}m_{2}}(x_{1},x_{2},t)$ in Eqs. (2.3fyao).

$\square$

We find that the first term of each of the wave functions in Eqs. (2.3fyao),
(2.3fyao) and (2.3fyaoc) is a noninteracting part given by the Slater
determinant of the one-electron wave functions $g^{(\ell)}_{m,k}(x,t)$ and
$e^{(\ell)}_{\alpha,k}(t)$ in Eqs. (2.3fl) and (2.3flb). The effect of the
interdot interaction appears in the second term of each of Eqs. (2.3fyao),
(2.3fyao) and (2.3fyaoc); the factor
$\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)x_{21}}\theta(x_{21})$ in
the second term in Eq. (2.3fl) decays exponentially as the two electrons
separate from each other, which indicates that the two electrons form two-body
bound states after the scattering at the quantum dots. We also find that the
binding strength $\overline{\Gamma}-\mathrm{Re}(\eta_{s}/2)$ of the two-body
bound states corresponds to the resonance pole of the one-electron wave
functions $g^{(\ell)}_{m,k}(x,t)$ and $e^{(\ell)}_{\alpha,k}(t)$ in Eqs.
(2.3fl) and (2.3flb).

Let us now consider the long-time limit $t\to\infty$ of the time-evolving two-
electron scattering states. The second term of each of the three wave
functions in Eqs. (2.3fyao), (2.3fyao) and (2.3fyaoc) contains the function
$X^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}(x-t)$ or
$X^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}(-t)$, which includes several terms
that decay exponentially in time $t$. By using the limit relation

$\displaystyle\lim_{t\to\infty}\mathrm{e}^{\mathrm{i}(k_{1}+k_{2})t}X^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}(x-t)$
$\displaystyle=\sum_{P}\mathrm{sgn}(P)\frac{U^{\prime}}{k_{P_{1}}\\!+\\!k_{P_{2}}\\!-\\!2\overline{\epsilon}_{\rm
d}\\!-\\!U^{\prime}\\!+\\!2\mathrm{i}\overline{\Gamma}}e^{(\ell_{1})}_{\alpha,k_{1}}e^{(\ell_{2})}_{\overline{\alpha},k_{P_{2}}}\mathrm{e}^{\mathrm{i}(k_{P_{1}}+k_{P_{2}})x}=Z^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}\mathrm{e}^{\mathrm{i}(k_{1}+k_{2})x},$
(2.3fyatbdbgbn)

we find

$\displaystyle\lim_{t\to\infty}\mathrm{e}^{\mathrm{i}(k_{1}+k_{2})t}g^{(\ell_{1}\ell_{2})}_{m_{1}m_{2},k_{1}k_{2}}(x_{1},x_{2},t)$
$\displaystyle=\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m_{1},k_{P_{1}}}(x_{1})g^{(\ell_{P_{2}})}_{m_{2},k_{P_{2}}}(x_{2})$
$\displaystyle\quad-\\!\mathrm{i}\sum_{Q\atop\alpha,s}\mathrm{sgn}(Q)v_{m_{Q_{1}}\alpha}y_{m_{Q_{2}}\alpha,s}Z^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)x_{Q_{2}Q_{1}}}\mathrm{e}^{\mathrm{i}Ex_{Q_{2}}}\theta(x_{Q_{2}Q_{1}})\theta(x_{Q_{1}})$
$\displaystyle=:g^{(\ell_{1}\ell_{2})}_{m_{1}m_{2},k_{1}k_{2}}(x_{1},x_{2}),$
(2.3fyatbdbgboa)
$\displaystyle\lim_{t\to\infty}\mathrm{e}^{\mathrm{i}(k_{1}+k_{2})t}e^{(\ell_{1}\ell_{2})}_{m\alpha,k_{1}k_{2}}(x,t)$
$\displaystyle=\sum_{P}\mathrm{sgn}(P)g^{(\ell_{P_{1}})}_{m,k_{P_{1}}}(x)e^{(\ell_{P_{2}})}_{\alpha,k_{P_{2}}}-\sum_{s}y_{m\alpha,s}Z^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)x}\mathrm{e}^{\mathrm{i}Ex}\theta(x)$
$\displaystyle=:e^{(\ell_{1}\ell_{2})}_{m\alpha,k_{1}k_{2}}(x),$
(2.3fyatbdbgbob)
$\displaystyle\lim_{t\to\infty}\mathrm{e}^{\mathrm{i}(k_{1}+k_{2})t}f^{(\ell_{1}\ell_{2})}_{\alpha\overline{\alpha},k_{1}k_{2}}(t)=\sum_{P}\mathrm{sgn}(P)e^{(\ell_{P_{1}})}_{\alpha,k_{P_{1}}}e^{(\ell_{P_{2}})}_{\overline{\alpha},k_{P_{2}}}+Z^{(\ell_{1}\ell_{2})}_{\alpha,k_{1}k_{2}}=:f^{(\ell_{1}\ell_{2})}_{\alpha\overline{\alpha},k_{1}k_{2}},$
(2.3fyatbdbgboc)

which are the stationary two-electron scattering eigenstates obtained in our
previous work [20]. The wave functions of the stationary two-electron
scattering eigenstates have a resonance pole at
$k_{1}+k_{2}=2\overline{\epsilon}_{{\rm
d}}+U^{\prime}-2\mathrm{i}\overline{\Gamma}$ on the complex energy plane.

##  4 Time-evolving resonant states

###  4.1 Time-evolving one-body resonant states

In Section 3, we presented exact solutions for the initial conditions in which
electrons are absent on the two quantum dots. We now move onto the solution
for the initial condition in which electrons are present only on the dots. As
a result, we discover new time-evolving states whose wave functions are given
in the form

$\displaystyle\Psi(x,t)=\psi_{R}(x)\mathrm{e}^{-\mathrm{i}E_{R}t}\theta(t-x)\quad\mbox{
for }x>0,$ (2.3fyatbdbgboa)

where $\psi_{R}(x)$ is the wave function of a stationary resonant state and
$E_{R}$ is the resonance energy with an imaginary part. We refer to the time-
evolving states described by Eq. (2.3fyatbdbgboa) as time-evolving resonant
states. Here the step function $\theta(t-x)$ in Eq. (2.3fyatbdbgboa), which is
due to the linear dispersion relation, restricts the wave function $\Psi(x,t)$
to the space interval $0<x<t$; see Fig. 4 below. Hence the time-evolving
resonant states are normalizable even if the corresponding stationary resonant
states diverge in space [29]. We expect that decaying states are observed in
experiments not as the stationary resonant state but as the time-evolving
resonant state. In the previous work [29], one of the authors showed that the
electron number of resonant states is conserved if the integration interval in
the calculation of the electron number is extended with the electron velocity.
The present time-evolving resonant states clarify the meaning of the extension
of the integration interval.

First, we consider the time-evolving one-electron state for the initial
conditions

$\displaystyle g_{m}(x,0)=0,\quad e_{\alpha}(0)=\psi_{\alpha},$
(2.3fyatbdbgbob)

where the constants $\psi_{\alpha}$ for $\alpha=1,2$ satisfy the normalization
condition $|\psi_{1}|^{2}+|\psi_{2}|^{2}=1$.

###### Proposition 4.1

The set of Schrödinger equations (2.3a) and (2.3b) under the initial
conditions (2.3fyatbdbgbob) is solved as follows: i) For $\eta\neq 0$, the
solution is given by

$\displaystyle
g_{m}(x,t)=\sum_{\alpha,s}y_{m\alpha,\overline{s}}\,\psi_{\overline{\alpha}}\,\mathrm{e}^{\mathrm{i}E^{(1)s}_{R}(x-t)}\tilde{\theta}(t-x)\theta(x),$
(2.3fyatbdbgboca) $\displaystyle
e_{\alpha}(t)=\sum_{s}\frac{1}{\eta_{\overline{s}}}\bigg{[}\lambda_{\alpha,\overline{s}}\psi_{\alpha}+\mathrm{i}(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})\psi_{\overline{\alpha}}\bigg{]}\mathrm{e}^{-\mathrm{i}E^{(1)s}_{R}t},$
(2.3fyatbdbgbocb)

where $E^{(1)\pm}_{R}=\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}\pm\mathrm{i}\eta/2$ is the one-body resonance
energy.
ii) For $\eta=0$, the solution is given by

$\displaystyle
g_{m}(x,t)=\mathrm{i}\sum_{\alpha}v_{m\alpha}((\lambda^{0}_{\alpha}\psi_{\alpha}\\!+\\!\mathrm{i}(v^{\prime}_{\alpha}\\!-\\!\mathrm{i}\Gamma_{\alpha\overline{\alpha}})\psi_{\overline{\alpha}})(t\\!-\\!x)\\!-\\!\psi_{\alpha})\mathrm{e}^{\mathrm{i}E^{(1)0}_{R}(x-t)}\tilde{\theta}(t-x)\theta(x),$
(2.3fyatbdbgbocda) $\displaystyle
e_{\alpha}(t)=-((\lambda^{0}_{\alpha}\psi_{\alpha}\\!+\\!\mathrm{i}(v^{\prime}_{\alpha}\\!-\\!\mathrm{i}\Gamma_{\alpha\overline{\alpha}})\psi_{\overline{\alpha}})t\\!-\\!\psi_{\alpha})\mathrm{e}^{-\mathrm{i}E^{(1)0}_{R}t},$
(2.3fyatbdbgbocdb)

where $E^{(1)0}_{R}=\overline{\epsilon}_{\rm d}-\mathrm{i}\overline{\Gamma}$.

Proof. We employ the systematic construction of three steps proposed in
Section 3.1 for $\eta\neq 0$. The wave functions for $\eta=0$ in Eqs.
(2.3fyatbdbgbocda) and (2.3fyatbdbgbocdb) are obtained by taking the limit
$\eta\to 0$ of Eqs. (2.3fyatbdbgboca) and (2.3fyatbdbgboca).

Step 1) In the region of $x<0$ for $t>0$, we have $g_{m}(x,t)=g_{m}(x-t,0)=0$
through the translation invariance in Eq. (2.3fa).

Step 2) Process (i)$\to$(ii): Since the inhomogeneous term in Eq. (2.3fg) also
vanishes, that is, $G_{\alpha}(t)=0$, the general solution in Eq. (2.3fj) is
given by

$\displaystyle\tilde{e}_{\alpha}(t)=\sum_{s}C_{\alpha,s}\mathrm{e}^{\lambda_{\alpha,s}t},$
$\displaystyle\therefore\quad$ $\displaystyle
e_{\alpha}(t)=\mathrm{e}^{-\mathrm{i}(\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha})t}\tilde{e}_{\alpha}(t)=\sum_{s}C_{\alpha,s}\mathrm{e}^{\mathrm{i}(-\epsilon_{{\rm
d}\alpha}+\mathrm{i}\Gamma_{\alpha\alpha}-\mathrm{i}\lambda_{\alpha,s})t}.$
(2.3fyatbdbgboe)

By inserting this and $g_{m}(0-,t)=0$ into Eq. (2.3fd), we obtain

$\displaystyle\tilde{e}_{\overline{\alpha}}(t)=\sum_{s}\frac{\mathrm{i}\lambda_{\alpha,s}}{v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}C_{\alpha,s}\mathrm{e}^{\mathrm{i}(\Delta\epsilon_{{\rm
d}\overline{\alpha}}-\mathrm{i}\Delta\Gamma_{\overline{\alpha}}-\mathrm{i}\lambda_{\alpha,s})t},$
$\displaystyle\therefore\quad$ $\displaystyle
e_{\overline{\alpha}}(t)=\mathrm{e}^{-\mathrm{i}(\epsilon_{{\rm
d}\overline{\alpha}}-\mathrm{i}\Gamma_{\overline{\alpha}\,\overline{\alpha}})t}\tilde{e}_{\overline{\alpha}}(t)=\sum_{s}\frac{\mathrm{i}\lambda_{\alpha,s}}{v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}C_{\alpha,s}\mathrm{e}^{\mathrm{i}(-\epsilon_{{\rm
d}\alpha}+\mathrm{i}\Gamma_{\alpha\alpha}-\mathrm{i}\lambda_{\alpha,s})t}.$
(2.3fyatbdbgbof)

The initial condition $e_{\alpha}(0)=\psi_{\alpha}$ in Eqs. (2.3fyatbdbgbob)
determines the constants $C_{\alpha,+}$ and $C_{\alpha,-}$ in Eqs.
(2.3fyatbdbgboe) and (2.3fyatbdbgbof) as

$\displaystyle\tilde{e}_{\alpha}(0+)=C_{\alpha,+}+C_{\alpha,-}=\psi_{\alpha},$
$\displaystyle\tilde{e}_{\overline{\alpha}}(0+)=\frac{\mathrm{i}\lambda_{\alpha,+}}{v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}\tilde{C}_{\alpha,+}+\frac{\mathrm{i}\lambda_{\alpha,-}}{v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}\tilde{C}_{\alpha,-}=\psi_{\overline{\alpha}},$
$\displaystyle\therefore\quad$ $\displaystyle
C_{\alpha,s}=-\frac{1}{\eta_{s}}\bigg{[}\lambda_{\alpha,\overline{s}}\psi_{\alpha}+\mathrm{i}(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})\psi_{\overline{\alpha}}\bigg{]}.$
(2.3fyatbdbgbog)

By inserting this into Eqs. (2.3fyatbdbgboe) and (2.3fyatbdbgbof), we obtain

$\displaystyle e_{\alpha}(t)$
$\displaystyle=-\sum_{s}\frac{1}{\eta_{s}}\bigg{[}\lambda_{\alpha,\overline{s}}\psi_{\alpha}+\mathrm{i}(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})\psi_{\overline{\alpha}}\bigg{]}\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)t}.$ (2.3fyatbdbgboh)

Step 3) Process (i), (ii)$\to$(iii): Through the matching condition for
$g_{m}(x,t)$ in Eq. (2.3fb), we have

$\displaystyle g_{m}(0+,t)$
$\displaystyle=-\mathrm{i}\tilde{\theta}(t)\sum_{\alpha}v_{m\alpha}e_{\alpha}(t),$
(2.3fyatbdbgboi)

where, in order to extend the relation to the region $t\leq 0$, we used the
step function $\tilde{\theta}(t)$ which is defined in Eq. (2.3fln). Hence,
through the translation invariance in Eq. (2.3fa), we obtain the wave function
$g_{m}(x,t)$ for $x>0$ as

$\displaystyle
g_{m}(x,t)=g_{m}(0+,t-x)=-\mathrm{i}\tilde{\theta}(t-x)\sum_{\alpha,s}v_{m\alpha}e_{\alpha}(t-x).$
(2.3fyatbdbgboj)

$\square$

The wave functions $g_{m}(x,t)$ in Eqs. (2.3fyatbdbgboca) and
(2.3fyatbdbgbocda) include the step function $\tilde{\theta}(t-x)$, which
indicates that they increase exponentially only inside the space interval
$0<x<t$. This is a result of causality since the electron on the leads travels
with the Fermi velocity $v_{\rm F}=1$ without reflection. A similar
restriction to the time-evolving wave functions due to causality was observed
in the Friedrichs model [39]. It should be noted that the strict step function
is due to the precisely linear dispersion of electrons on the leads; the lower
limit of the dispersion would be a branch point and result in deviations from
the purely exponential behavior [36, 37, 38, 39, 40, 41, 42].

Except for the step function $\tilde{\theta}(t-x)$, the time-evolving state
for $\eta\neq 0$ in Eqs. (2.3fyatbdbgboca) and (2.3fyatbdbgbocb) is a
superposition of the one-body stationary resonant states with the resonance
energies $E^{(1)\pm}_{R}$. Indeed, by using the wave functions
$g^{\pm}_{m,R}(x)$ and $e^{\pm}_{\alpha,R}$ of the one-body resonant state,
which are explicitly given in A.1, the wave functions $g_{m}(x,t)$ and
$e_{\alpha}(t)$ in Eqs. (2.3fyatbdbgboca) and (2.3fyatbdbgbocb) are rewritten
as

$\displaystyle
g_{m}(x,t)=\sum_{s}g^{s}_{m,R}(x)\mathrm{e}^{-\mathrm{i}E^{(1)s}_{R}t}\tilde{\theta}(t-x),$
(2.3fyatbdbgboka) $\displaystyle
e_{\alpha}(t)=\sum_{s}e^{s}_{\alpha,R}\mathrm{e}^{-\mathrm{i}E^{(1)s}_{R}t},$
(2.3fyatbdbgbokb)

where the constants $\psi_{\alpha}$, which characterize the initial state in
Eqs. (2.3fyatbdbgbob), are absorbed into the coefficients $g^{s}_{m,R}(x)$ and
$e^{s}_{\alpha,R}$. Since the wave function $g_{m}(x,t)$ is finite in the
space interval $0<x<t$, the time-evolving resonant state is normalizable; see
Fig. 4 below. Indeed, the time-evolving resonant state in Eqs.
(2.3fyatbdbgboca) and (2.3fyatbdbgbocb) is normalized because the initial
state in Eqs. (2.3fyatbdbgbob) is normalized and the particle number is
conserved.

The wave functions for $\eta=0$ in Eqs. (2.3fyatbdbgbocda) and
(2.3fyatbdbgbocdb) are not in the form of Eqs. (2.3fyatbdbgboka) and
(2.3fyatbdbgbokb) but include an exponential function multiplied by a term
linear in $t$. This is because the two resonance energies $E^{(1)\pm}_{R}$
merge into one at $\eta=0$ and the corresponding one-body resonant states
become parallel to each other, which shall be described in A.1.

###  4.2 Time-evolving two-body resonant states

Next, we consider the time-evolving two-body resonant state by solving the
time-dependent Schrödinger equations (2.3f), (2.3f) and (2.3fc) under the
initial conditions

$\displaystyle g_{m_{1}m_{2}}(x_{1},x_{2},0)=0,\quad e_{m,\alpha}(x,0)=0,\quad
f_{\alpha\overline{\alpha}}(0)=(-1)^{\overline{\alpha}},$ (2.3fyatbdbgbokl)

which stand for two electrons localized on the quantum dots. Recall the anti-
symmetry $f_{12}(t)=-f_{21}(t)$ of the wave function of double occupancy.

###### Proposition 4.2

For $\eta\neq 0$, the solution of the set of Schrödinger equations (2.3f),
(2.3f) and (2.3fc) for the initial conditions (2.3fyatbdbgbokl) is given by

$\displaystyle
g_{m_{1}m_{2}}(x_{1},x_{2},t)=\mathrm{i}\sum_{Q,\alpha,s}(-1)^{\alpha}\mathrm{sgn}(Q)v_{m_{Q_{1}}\alpha}y_{m_{Q_{2}}\alpha,s}\mathrm{e}^{\mathrm{i}E^{(2)}_{R}(x_{Q_{2}}-t)-\mathrm{i}E^{(1)s}_{R}x_{Q_{2}Q_{1}}}$
$\displaystyle\hskip
93.0pt\times\tilde{\theta}(t-x_{Q_{2}})\theta(x_{Q_{2}Q_{1}})\theta(x_{Q_{1}}),$
(2.3fyatbdbgbokma) $\displaystyle
e_{m,\alpha}(x,t)=\sum_{s}(-1)^{\alpha}y_{m\alpha,s}\mathrm{e}^{\mathrm{i}E^{(2)}_{R}(x-t)-\mathrm{i}E^{(1)s}_{R}x}\tilde{\theta}(t-x)\theta(x),$
(2.3fyatbdbgbokmb) $\displaystyle
f_{\alpha\overline{\alpha}}(t)=(-1)^{\overline{\alpha}}\mathrm{e}^{-\mathrm{i}E^{(2)}_{R}t},$
(2.3fyatbdbgbokmc)

where $Q=(Q_{1},Q_{2})$ is a permutation of $(1,2)$,
$E^{(1)\pm}_{R}=\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2$ is the one-body resonance
energy and $E^{(2)}_{R}=2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma}$ is the two-body resonance energy.

Proof. We employ the systematic construction of seven steps proposed in
Section 3.2.

Step 1) For $t>0$ and $x_{1}<x_{2}<0$, we find $x_{1}-t<x_{2}-t<0$ in this
step. Then, through the translation invariance in Eq. (3.2), we find that
there is no two-electron incident waves:

$\displaystyle
g_{m_{1}m_{2}}(x_{1},x_{2},t)=g_{m_{1}m_{2}}(x_{1}-t,x_{2}-t,0)=0.$
(2.3fyatbdbgbokn)

Step 2) Process (i)$\to$(ii): Because $g_{m_{1}m_{2}}(x,0-,t)=0$ for $x<0$,
the inhomogeneous term in Eq. (2.3fyai) vanishes: $G_{m,\alpha}(z)=0$ with
$z=(x+t)/2$. Then we obtain the general solution in Eq. (2.3fyaj) as

$\displaystyle
e_{m,\alpha}(x,t)=\sum_{s}C_{m,\alpha,s}(x-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)(x+t)/2},$ $\displaystyle
e_{m,\overline{\alpha}}(x,t)=\sum_{s}\frac{\mathrm{i}\lambda_{\alpha,s}}{v^{\prime}_{\alpha}\\!-\\!\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}C_{m,\alpha,s}(x-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)(x+t)/2},$
(2.3fyatbdbgboko)

where $C_{m,\alpha,s}(x-t)$ for $s=\pm$ are arbitrary functions of the
variable $x-t$. For the initial conditions in Eqs. (2.3fyatbdbgbokl) for
$e_{m,\alpha}(x,t)$ and $e_{m,\overline{\alpha}}(x,t)$, we obtain
$C_{m,\alpha,s}(x)=0$, that is, $e_{m,\alpha}(x,t)=0$ for $x<0$.

Step 3) Process (ii)$\to$(iii): By applying $e_{m,\alpha}(x,t)=0$ for $x<0$ to
the solution in Eq. (2.3fyam) for $f_{\alpha\overline{\alpha}}(t)$, we have

$\displaystyle f_{\alpha\overline{\alpha}}(t)$
$\displaystyle=C_{\alpha}\mathrm{e}^{-\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})t}.$ (2.3fyatbdbgbokp)

Under the initial condition
$f_{\alpha\overline{\alpha}}(0)=(-1)^{\overline{\alpha}}$ in Eqs.
(2.3fyatbdbgbokl), we have

$\displaystyle
f_{\alpha\overline{\alpha}}(0+)=C_{\alpha}=(-1)^{\overline{\alpha}},$
$\displaystyle\therefore\quad$ $\displaystyle
f_{\alpha\overline{\alpha}}(t)=(-1)^{\overline{\alpha}}\mathrm{e}^{-\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})t}.$ (2.3fyatbdbgbokq)

Step 4) Process (i), (ii)$\to$(iv): For $x_{1}<0<x_{2}$, we have
$x_{1}-x_{2}<0$ in this step. Then the first matching condition in Eqs. (3.2)
gives

$\displaystyle g_{m_{1}m_{2}}(x_{1},x_{2},t)$
$\displaystyle=g_{m_{1}m_{2}}(x_{1}-x_{2},0+,t-x_{2})=0.$ (2.3fyatbdbgbokr)

Step 5) Process (iii), (iv)$\to$(v): Because $g_{m_{1}m_{2}}(0-,x,t)=0$ for
$x>0$, the inhomogeneous term in Eq. (2.3fyai) also vanishes:
$G_{m,\alpha}(z)=0$ with $z=(x+t)/2$. Then, by using the solution in Eq.
(2.3fyaj), we have

$\displaystyle
e_{m,\alpha}(x,t)=\sum_{s}\tilde{C}_{m,\alpha,s}(x-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)(x+t)/2},$ $\displaystyle
e_{m,\overline{\alpha}}(x,t)=\sum_{s}\frac{\mathrm{i}\lambda_{\alpha,s}}{v^{\prime}_{\alpha}\\!-\\!\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}\tilde{C}_{m,\alpha,s}(x-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)(x+t)/2},$
(2.3fyatbdbgboks)

where $\tilde{C}_{m,\alpha,s}(x-t)$ for $s=\pm$ are arbitrary functions of the
variable $x-t$. From the initial conditions (2.3fyatbdbgbokl), we have

$\displaystyle
e_{m,\alpha}(x,0)=\sum_{s}\tilde{C}_{m,\alpha,s}(x)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)x/2}=0,$ $\displaystyle
e_{m,\overline{\alpha}}(x,0)=\sum_{s}\frac{\mathrm{i}\lambda_{\alpha,s}}{v^{\prime}_{\alpha}\\!-\\!\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}\tilde{C}_{m,\alpha,s}(x)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)x/2}=0,$
$\displaystyle\therefore\quad$
$\displaystyle\tilde{C}_{m,\alpha,s}(x)=0\quad\mbox{ for }x>0,$
(2.3fyatbdbgbokt)

which determines $\tilde{C}_{m,\alpha,s}(x-t)=0$ for $x-t>0$. In order to
express the function $\tilde{C}_{m,\alpha,s}(x-t)$ for $x-t<0$, we introduce
another function $D_{m,\alpha,s}(x-t)$ and put

$\displaystyle\tilde{C}_{m,\alpha,s}(x-t)=D_{m,\alpha,s}(x-t)\theta(t-x).$
(2.3fyatbdbgboku)

By inserting this into the solution in Eqs. (4.2), we have

$\displaystyle e_{m,\alpha}(x,t)$
$\displaystyle=\sum_{s}D_{m,\alpha,s}(x-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)(x+t)/2}\theta(t-x),$
(2.3fyatbdbgbokv) $\displaystyle e_{m,\overline{\alpha}}(x,t)$
$\displaystyle=\sum_{s}\frac{\mathrm{i}\lambda_{\alpha,s}}{v^{\prime}_{\alpha}\\!-\\!\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}D_{m,\alpha,s}(x-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)(x+t)/2}\theta(t-x).$
(2.3fyatbdbgbokw)

The matching condition in Eq. (2.3fyac) for $e_{m,\alpha}(x,t)$ gives

$\displaystyle\sum_{s}D_{m,\alpha,s}(-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)t/2}=(-1)^{\overline{\alpha}}\mathrm{i}v_{m\overline{\alpha}}\mathrm{e}^{-\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})t},$
$\displaystyle\sum_{s}\frac{\mathrm{i}\lambda_{\alpha,s}}{v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}D_{m,\alpha,s}(-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{{\rm
d}}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)t/2}=(-1)^{\alpha}\mathrm{i}v_{m\alpha}\mathrm{e}^{-\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})t}.$ (2.3fyatbdbgbokx)

By solving the coupled equations for $D_{m,\alpha,s}(-t)$, we obtain

$\displaystyle
D_{m,\alpha,s}(-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)t/2}=(-1)^{\alpha}y_{m\alpha,s}\psi_{\alpha\overline{\alpha}}\mathrm{e}^{-\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})t}.$ (2.3fyatbdbgboky)

By replacing $t(>0)$ with $t-x(>0)$ in the variable of $D_{m,\alpha,s}(-t)$,
we have

$\displaystyle
D_{m,\alpha,s}(x-t)\mathrm{e}^{-\mathrm{i}(\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)(x+t)/2}\tilde{\theta}(t-x)$
$\displaystyle=(-1)^{\alpha}\tilde{\theta}(t-x)y_{m\alpha,s}\mathrm{e}^{\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})(x-t)-\mathrm{i}(\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2)x},$ (2.3fyatbdbgbokz)

which gives

$\displaystyle
e_{m,\alpha}(x,t)=(-1)^{\alpha}\tilde{\theta}(t-x)\sum_{s}y_{m\alpha,s}\mathrm{e}^{\mathrm{i}(\overline{\epsilon}_{\rm
d}+U^{\prime}-\mathrm{i}\overline{\Gamma}-\mathrm{i}\eta_{s}/2)x-\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})t}\theta(x).$ (2.3fyatbdbgbokaa)

Step 6) Process (iv), (v)$\to$(vi): By using the second matching condition in
Eqs. (3.2), we have

$\displaystyle
g_{m_{1}m_{2}}(0+,x,t)=\mathrm{i}\tilde{\theta}(t-x)\sum_{\alpha,s}(-1)^{\alpha}v_{m_{1}\alpha}y_{m_{2}\alpha,s}\mathrm{e}^{\mathrm{i}(\overline{\epsilon}_{\rm
d}+U^{\prime}-\mathrm{i}\overline{\Gamma}-\mathrm{i}\eta_{s}/2)x-\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})t}\theta(x).$ (2.3fyatbdbgbokab)

Because $x_{2}-x_{1}>0$ for $0<x_{1}<x_{2}$, we have

$\displaystyle
g_{m_{1}m_{2}}(x_{1},x_{2},t)=g_{m_{1}m_{2}}(0+,x_{2}-x_{1},t-x_{1})$
$\displaystyle=\mathrm{i}\sum_{\alpha,s}(-1)^{\alpha}v_{m_{1}\alpha}y_{m_{2}\alpha,s}\mathrm{e}^{\mathrm{i}(\overline{\epsilon}_{\rm
d}+U^{\prime}-\mathrm{i}\overline{\Gamma}-\mathrm{i}\eta_{s}/2)x_{21}+\mathrm{i}(2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma})(x_{1}-t)}\tilde{\theta}(t-x_{2})\theta(x_{21})\theta(x_{1}).$
(2.3fyatbdbgbokac)

Step 7) After the anti-symmetrization with respect to the variables $x_{1}$
and $x_{2}$, we obtain the wave function $g_{m_{1}m_{2}}(x_{1},x_{2},t)$ in
the case $x_{2}<x_{1}$.

$\square$

The time-evolving two-body resonant state is normalizable since the wave
functions $g_{m_{1}m_{2}}(x_{1},x_{2},t)$ is finite in the space region
$0<x_{1},x_{2}<t$, and $e_{m,\alpha}(x,t)$ is also finite in the space
interval $0<x<t$, which is due to the step functions that imply causality. We
also find that the two electrons in the two-body resonant state form two-body
bound states with the binding strength
$\overline{\Gamma}-\mathrm{Re}(\eta_{s}/2)$, which is the imaginary part of
the resonance energy $E^{(1)s}_{R}$.

By using the wave functions $g_{m_{1}m_{2},R}(x_{1},x_{2})$,
$e_{m,\alpha,R}(x)$ and $f_{\alpha\overline{\alpha},R}$ of the two-body
stationary resonant state given in A.2, the wave functions for $\eta\neq 0$ in
Eqs. (2.3fyatbdbgbokma), (2.3fyatbdbgbokmb) and (2.3fyatbdbgbokmc) are
rewritten as

$\displaystyle
g_{m_{1}m_{2}}(x_{1},x_{2},t)=g_{m_{1}m_{2},R}(x_{1},x_{2})\mathrm{e}^{-\mathrm{i}E^{(2)}_{R}t}\tilde{\theta}(t-x_{1})\tilde{\theta}(t-x_{2}),$
(2.3fyatbdbgbokad) $\displaystyle
e_{m,\alpha}(x,t)=e_{m,\alpha,R}(x)\mathrm{e}^{-\mathrm{i}E^{(2)}_{R}t}\tilde{\theta}(t-x),$
(2.3fyatbdbgbokae) $\displaystyle
f_{\alpha\overline{\alpha}}(t)=f_{\alpha\overline{\alpha},R}\mathrm{e}^{-\mathrm{i}E^{(2)}_{R}t}.$
(2.3fyatbdbgbokaf)

Hence we refer to this time-evolving two-electron state as a time-evolving
two-body resonant state. The relation
$E^{(2)}_{R}=U^{\prime}+E^{(1)+}_{R}+E^{(1)-}_{R}$ among the resonance
energies shows that, at $U^{\prime}=0$, the time-evolving two-body resonant
state is reduced to a combination of the two time-evolving one-body resonant
states with different resonance energies $E^{(1)+}_{R}$ and $E^{(1)-}_{R}$. We
remark that, in a way similar to the one-electron case, the wave functions for
$\eta=0$, which are obtained by taking the limit $\eta\to 0$ of Eqs.
(2.3fyatbdbgbokma), (2.3fyatbdbgbokmb) and (2.3fyatbdbgbokmc), include an
exponential function multiplied by a term linear in $t$.

###  4.3 Existence and survival probabilities

The explicit wave functions of the time-evolving states enable us to calculate
the time-dependence of the existence probability of electrons on the leads and
the survival probability of electrons on the quantum dots. First, we
investigate the existence probability under the initial condition of electrons
localized on the quantum dots. By using the wave function $g_{m}(x,t)$ in Eq.
(2.3fyatbdbgboca) of the time-evolving one-body resonant state, we obtain the
existence probability distribution of an electron on the lead $m$ at time $t$
as

$\displaystyle
P^{(1)}_{m}(x,t)=|g_{m}(x,t)|^{2}=\Big{|}\sum_{\alpha,s}y_{m\alpha,s}\psi_{\overline{\alpha}}\,\mathrm{e}^{(\overline{\Gamma}+\eta_{s}/2)(x-t)}\Big{|}^{2}\tilde{\theta}(t-x)\theta(x).$
(2.3fyatbdbgbokag)

Similarly, by using the wave function $e_{m,\alpha}(x,t)$ in Eq.
(2.3fyatbdbgbokmb) of the time-evolving two-body resonant state, we obtain the
existence probability distribution of an electron on the lead $m$ in the
presence of another electron on the quantum dots at time $t$ as

$\displaystyle P^{(2)}_{m{\rm
d}}(x,t)=\sum_{\alpha}|e_{m,\alpha}(x,t)|^{2}=\sum_{\alpha}\Big{|}\sum_{s}y_{m\alpha,s}\mathrm{e}^{((\overline{\Gamma}+\eta_{s}/2)x-2\overline{\Gamma}t)}\Big{|}^{2}\tilde{\theta}(t-x)\theta(x).$
(2.3fyatbdbgbokah)

We note that the existence probability $P^{(2)}_{m{\rm d}}(x,t)$ in Eq.
(2.3fyatbdbgbokah) does not depend on the interaction $U^{\prime}$ since the
interaction $U^{\prime}$ appears only in the real part of the two-body
resonance energy $E^{(2)}_{R}=2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma}$ and hence in the phase factor of
the wave function $e_{m,\alpha}(x,t)$ in Eq. (2.3fyatbdbgbokae).

We remind that the existence probability $P^{(1)}_{m}(x,t)$ in Eq.
(2.3fyatbdbgbokag) describes an electron on the lead $m$ in the one-electron
case, while $P^{(2)}_{m{\rm d}}(x,t)$ in Eq. (2.3fyatbdbgbokah) describes an
electron on the lead $m$ and another electron on the quantum dots in the two-
electron case. We find from Eqs. (2.3fyatbdbgbokag) and (2.3fyatbdbgbokah)
that, at fixed time $t$, both the existence probabilities $P^{(1)}_{m}(x,t)$
and $P^{(2)}_{m{\rm d}}(x,t)$ increase exponentially with the same strength
$2\overline{\Gamma}+|\mathrm{Re}(\eta)|$ in the space interval $0<x<t$ [39].
Thus, even in the two-electron case, the exponential increase in space is
dominated by the one-body resonance energy $E^{(1)s}_{R}$, which has been seen
in the two-body bound state of the two-electron scattering wave function in
Eq. (2.3fyao). On the other hand, at a fixed position $x$, the existence
probability $P^{(1)}_{m}(x,t)$ decays exponentially in time with the inverse
relaxation time $2\overline{\Gamma}-|\mathrm{Re}(\eta)|$, while
$P^{(2)}_{m{\rm d}}(x,t)$ decays exponentially in time with
$4\overline{\Gamma}$. The exponential decay in time in the two-electron case
is dominated by the two-body resonance energy $E^{(2)}_{R}$.

For numerical demonstration, we consider the parallel-coupled double quantum-
dot system by setting $v_{m\alpha}=v(\in\mathbb{R})$, $v^{\prime}_{\alpha}=0$
and $\epsilon_{{\rm d}1}=-\epsilon_{{\rm d}2}=\epsilon$, in which we have
$\eta=2(\Gamma^{2}-\epsilon^{2})^{1/2}$ with $\Gamma=v^{2}$. Here we find an
exceptional point $\eta=0$ at $\Gamma=|\epsilon|$. For $\Gamma>|\epsilon|$,
the one-body resonance energies
$E^{(1)\pm}_{R}=\epsilon-\mathrm{i}\Gamma\pm\mathrm{i}\eta/2$ have different
imaginary parts, while for $\Gamma<|\epsilon|$, they have different real
parts. We take the initial state in Eq. (2.3fyatbdbgbob) with
$\psi_{1}=\psi_{2}=1/\sqrt{2}$ for the existence probability
$P^{(1)}_{m}(x,t)$, while the initial state for $P^{(2)}_{m{\rm d}}(x,t)$ is
Eq. (2.3fyatbdbgbokl). Then the one- and two-electron existence probabilities
$P^{(1)}_{m}(x,t)$ and $P^{(2)}_{m{\rm d}}(x,t)$ in Eqs. (2.3fyatbdbgbokag)
and (2.3fyatbdbgbokah) inside the space interval $0<x<t$ are explicitly
calculated as

$\displaystyle
P^{(1)}_{m}(x,t)\\!=\\!\cases{2\Big{|}\frac{2\Gamma}{\eta}\sinh\Big{(}\frac{\eta}{2}(t-x)\Big{)}-\cosh\Big{(}\frac{\eta}{2}(t-x)\Big{)}\Big{|}^{2}\Gamma\mathrm{e}^{-2\Gamma(t-x)}&for
$\Gamma>|\epsilon|$,\\\
2\big{|}\Gamma(t-x)-1\big{|}^{2}\Gamma\mathrm{e}^{-2\Gamma(t-x)}&for
$\Gamma=|\epsilon|$,\\\
2\Big{|}\frac{2\Gamma}{\zeta}\sin\Big{(}\frac{\zeta}{2}(t-x)\Big{)}-\cos\Big{(}\frac{\zeta}{2}(t-x)\Big{)}\Big{|}^{2}\Gamma\mathrm{e}^{-2\Gamma(t-x)}&for
$\Gamma<|\epsilon|$,\\\ }$ (2.3fyatbdbgbokai) $\displaystyle P^{(2)}_{m{\rm
d}}(x,t)\\!=\\!\cases{\\!\sum_{\alpha}\Big{|}\frac{2(\mathrm{i}\epsilon_{{\rm
d}\alpha}-\Gamma)}{\eta}\sinh\Big{(}\frac{\eta}{2}x\Big{)}-\cosh\Big{(}\frac{\eta}{2}x\Big{)}\Big{|}^{2}\Gamma\mathrm{e}^{-2\Gamma(2t-x)}&for
$\Gamma>|\epsilon|$,\\\ \sum_{\alpha}\big{|}\big{(}\mathrm{i}\epsilon_{{\rm
d}\alpha}-\Gamma\big{)}x-1\big{|}^{2}\Gamma\mathrm{e}^{-2\Gamma(2t-x)}&for
$\Gamma=|\epsilon|$,\\\ \sum_{\alpha}\Big{|}\frac{2(\mathrm{i}\epsilon_{{\rm
d}\alpha}-\Gamma)}{\zeta}\sin\Big{(}\frac{\zeta}{2}x\Big{)}-\cos\Big{(}\frac{\zeta}{2}x\Big{)}\Big{|}^{2}\Gamma\mathrm{e}^{-2\Gamma(2t-x)}&for
$\Gamma<|\epsilon|$.\\\ }$ (2.3fyatbdbgbokaj)

Here we put $\zeta=2(\epsilon^{2}-\Gamma^{2})^{1/2}$ for $\Gamma<|\epsilon|$.
The existence probabilities for $\Gamma=|\epsilon|$ are obtained by taking the
limit $\eta\to 0$ of Eqs. (2.3fyatbdbgbokag) and (2.3fyatbdbgbokah). Figure 4
shows the existence probability distributions $P^{(1)}_{m}(x,t)$ and
$P^{(2)}_{m{\rm d}}(x,t)$ for (a) $\epsilon_{{\rm d}1}=-\epsilon_{{\rm
d}2}=0.8\Gamma$ and (b) $\epsilon_{{\rm d}1}=-\epsilon_{{\rm d}2}=1.2\Gamma$;
the solid lines indicate the rescaled existence probability
$P^{(1)}_{m}(x,t)/\Gamma$ and the dashed lines indicate $P^{(2)}_{m{\rm
d}}(x,t)/\Gamma$. The thick lines correspond to the existence probability at
time $t=\Gamma$ and the thin lines correspond to that at time $t=2\Gamma$. In
the case (b), the initial decrease of $P^{(2)}_{m{\rm d}}(x,2\Gamma)$ is a
part of oscillation of the existence probabilities before the exponential
increase, which is due to the interference of the two resonance energies
$E^{(1)\pm}_{R}$ with different real parts and is expressed by the
trigonometric functions in Eqs. (2.3fyatbdbgbokai) and (2.3fyatbdbgbokaj). As
Eqs. (2.3fyatbdbgbokai) and (2.3fyatbdbgbokaj) indicate, for fixed $x$, the
two-electron existence probability $P^{(2)}_{m{\rm d}}(x,t)$ with the
relaxation time $4\Gamma$ decays more rapidly than $P^{(1)}_{m}(x,t)$ with
$2\Gamma-|\mathrm{Re}(\eta)|$ as a function of $t$.

Figure 4:  Existence probability distributions of the time-evolving one- and
two-body resonant states in the case of the parallel-coupled DQD system with
$v_{\ell\alpha}=v\in\mathbb{R}$, $v^{\prime}_{\alpha}=0$ for (a)
$\epsilon_{{\rm d}1}=-\epsilon_{{\rm d}2}=0.8\Gamma$ and (b) $\epsilon_{{\rm
d}1}=-\epsilon_{{\rm d}2}=1.2\Gamma$. The solid lines indicate the existence
probability $P^{(1)}_{m}(x,t)/\Gamma$ of an electron on the lead $m$ in the
one-electron case, and the dashed lines indicate the existence probability
$P^{(2)}_{m{\rm d}}(x,t)/\Gamma$ of an electron on the lead $m$ and another
electron on the quantum dots in the two-electron case.

Next we investigate the survival probabilities of electrons on the quantum
dots. By using the wave function $e_{\alpha}(t)$ in Eq. (2.3fyatbdbgbocb), we
obtain the one-electron survival probability $Q^{(1)}(t)$ on the quantum dots
at time $t$ as

$\displaystyle
Q^{(1)}(t)=\sum_{\alpha}|e_{\alpha}(t)|^{2}=\sum_{\alpha}\Big{|}\sum_{s}\frac{1}{\eta_{\overline{s}}}\big{(}\lambda_{\alpha,\overline{s}}\psi_{\alpha}+\mathrm{i}(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})\psi_{\overline{\alpha}}\big{)}\mathrm{e}^{-(\overline{\Gamma}-\eta_{s}/2)t}\Big{|}^{2},$
(2.3fyatbdbgbokak)

which decays exponentially with the inverse lifetime
$2\overline{\Gamma}-|\mathrm{Re}(\eta)|$. On the other hand, the two-electron
survival probability $Q^{(2)}(t)$ on the quantum dots is obtained from the
wave function of double occupancy $f_{12}(t)$ in Eq. (2.3fyatbdbgbokmc) at
time $t$ as

$\displaystyle Q^{(2)}(t)=|f_{12}(t)|^{2}=\mathrm{e}^{-4\overline{\Gamma}t},$
(2.3fyatbdbgbokal)

which decays exponentially with the inverse lifetime $4\overline{\Gamma}$.
Thus the lifetime of the survival probability $Q^{(1)}(t)$ on the quantum dots
is equal to the relaxation time of the existence probability
$P^{(1)}_{m}(x,t)$ on the leads, while the lifetime of $Q^{(2)}(t)$ is equal
to the relaxation time of $P^{(2)}_{m{\rm d}}(x,t)$.

Let us analyze the case of the parallel-coupled double quantum-dot system with
$v_{m\alpha}=v(\in\mathbb{R})$, $v^{\prime}_{\alpha}=0$ and $\epsilon_{{\rm
d}1}=-\epsilon_{{\rm d}2}=\epsilon$ again. Recall that
$\eta=2(\Gamma^{2}-\epsilon^{2})^{1/2}$ for $\Gamma>|\epsilon|$ and
$\zeta=2(\epsilon^{2}-\Gamma^{2})^{1/2}$ for $\Gamma<|\epsilon|$ with
$\Gamma=v^{2}$. We take the initial state in Eq. (2.3fyatbdbgbob) with
$\psi_{1}=\psi_{2}=1/\sqrt{2}$ for the one-electron survival probability
$Q^{(1)}(t)$, which is the same as that of the existence probabilities in Eq.
(2.3fyatbdbgbokai). Then we have

$Q^{(1)}(t)\\!=\\!\cases{\\!\frac{1}{2}\sum_{\alpha}\Big{|}\frac{2(\mathrm{i}\epsilon_{{\rm
d}\alpha}+\Gamma)}{\eta}\sinh\Big{(}\frac{\eta}{2}t\Big{)}-\cosh\Big{(}\frac{\eta}{2}t\Big{)}\Big{|}^{2}\mathrm{e}^{-2\Gamma
t}&for $\Gamma>|\epsilon|$,\\\
\\!\frac{1}{2}\sum_{\alpha}\big{|}\big{(}\mathrm{i}\epsilon_{{\rm
d}\alpha}+\Gamma\big{)}t-1\big{|}^{2}\mathrm{e}^{-2\Gamma t}&for
$\Gamma=|\epsilon|$,\\\
\\!\frac{1}{2}\sum_{\alpha}\Big{|}\frac{2(\mathrm{i}\epsilon_{{\rm
d}\alpha}+\Gamma)}{\zeta}\sin\Big{(}\frac{\zeta}{2}t\Big{)}-\cos\Big{(}\frac{\zeta}{2}t\Big{)}\Big{|}^{2}\mathrm{e}^{-2\Gamma
t}&for $\Gamma<|\epsilon|$.\\\ }$ (2.3fyatbdbgbokam)

Figure 5 shows the one-electron survival probabilities $Q^{(1)}(t)$ for
$\epsilon=0.5\Gamma,\Gamma,1.5\Gamma,2\Gamma$ and the two-electron survival
probability $Q^{(2)}(t)$ in Eq. (2.3fyatbdbgbokal), which does not depend on
the energy level $\epsilon$. Due to the difference of their lifetime, the two-
electron survival probability $Q^{(2)}(t)$ decays more rapidly than
$Q^{(1)}(t)$ as a function of $t$. We also observe that, for
$\Gamma<|\epsilon|$, the survival probability $Q^{(1)}(t)$ oscillates in time,
which is expressed by the trigonometric functions in Eqs. (2.3fyatbdbgbokam).
The oscillation in time is a result of the interference of the two resonance
energies $E^{(1)\pm}_{R}$with different real parts.

Figure 5:  Survival probabilities of the time-evolving one- and two-body
resonant states in the case of the parallel-coupled double quantum-dot system
with $v_{\ell\alpha}=v\in\mathbb{R}$, $v^{\prime}_{\alpha}=0$ and
$\epsilon_{{\rm d}1}=-\epsilon_{{\rm d}2}$. The solid thin line indicates the
survival probabilities $Q^{(1)}(t)$ of the time-evolving one-body resonant
states at $\epsilon_{{\rm d}1}=0.5\Gamma$, the dotted line indicates that at
$\epsilon_{{\rm d}1}=1.0\Gamma$, the dashed line indicates that at
$\epsilon_{{\rm d}1}=1.5\Gamma$, the chain line indicates that at
$\epsilon_{{\rm d}1}=2.0\Gamma$ and the solid thick line indicates the
survival probability $Q^{(2)}(t)$ of the time-evolving two-body resonant
states.

##  5 Concluding remarks

We have constructed exact time-evolving states of the open double quantum-dot
system. Their purely exponential decay and strict step-function behavior are
consequences of the unbounded linear dispersion relation of electrons on the
leads. For noninteracting cases, it was shown [44] that the time-evolving
scattering states are characterized by a non-Markovian integro-differential
equation, because of which the behavior of the survival probability of
unstable states deviates from the purely exponential decay usually in an
extremely short time and in an extremely long time if the dispersion is
bounded below [36, 37, 38, 39, 40, 41, 42]. In the case of the linear
dispersion relation, the integro-differential equation is reduced to a
Markovian differential equation and hence produces exponential functions
without deviations. We have also shown that the exponential decay of the time-
evolving states leads to that of the existence probability of an electron on
the leads and the survival probability of localized electrons on the quantum
dots.

The systematic construction of time-evolving states that we have proposed can
be readily extended to the cases of three or more electrons. We have found
that the two-body bound states in the two-electron scattering states
correspond to the one-body resonance poles of the one-electron scattering
states. In a similar way, three-body bound states appear in three-electron
scattering states due to the two-body resonance poles of the two-electron
scattering states. Each term of the many-electron scattering states is
characterized by the arrangement of the two-body and the three-body bound
states, which is an essential difference from the interacting resonant-level
models with only two-body bound states [11, 12, 13, 33, 34, 35]. The time-
evolving many-electron scattering states shall enable us to analyze time-
dependent electric current across the quantum dots for the system under bias
voltages.

## Acknowledgments

The authors thank Prof. T. Y. Petrosky, Prof. G. Ordonez and Prof. S. Garmon
for helpful comments. The present study is partially supported by JSPS KAKENHI
Grants Numbers 19H00658, 21H01005, 22H01140.

##  Appendix A Resonant states and Siegert conditions

###  A.1 One-body resonant states

In this appendix, we study the stationary resonant states which are
eigenstates of the time-independent Schrödinger equation with complex energy
eigenvalues. First, we consider the one-electron case. By assuming the time-
dependent parts of the one-electron wave functions of the set of time-
dependent Schrödinger equations (2.3a) and (2.3b) as

$\displaystyle g_{m}(x,t)=\mathrm{e}^{-\mathrm{i}Et}g_{m}(x),\quad
e_{\alpha}(t)=\mathrm{e}^{-\mathrm{i}Et}e_{\alpha}$ (2.3fyatbdbgboka)

with an energy eigenvalue $E$ and inserting them into Eqs. (2.3fb) and
(2.3fd), we have

$\displaystyle
g_{m}(0+)-g_{m}(0-)=-\mathrm{i}\sum_{\alpha}v_{m\alpha}e_{\alpha},$
(2.3fyatbdbgbokb) $\displaystyle(E-\epsilon_{{\rm
d}\alpha}+\mathrm{i}\Gamma_{\alpha\alpha})e_{\alpha}-\big{(}v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}}\big{)}e_{\overline{\alpha}}=\sum_{m}v_{m\alpha}^{\ast}g_{m}(0-).$
(2.3fyatbdbgbokc)

To define the resonant states, we employ the Siegert boundary conditions that
impose no incident wave on the wave functions [22] as

$\displaystyle g_{m}(x)=0\mbox{ for }x<0.$ (2.3fyatbdbgbokd)

Hereafter we put the suffix $R$ on the wave functions as $g_{m,R}(x)$ and
$e_{\alpha,R}$ to denote the wave functions on which the Siegert boundary
conditions are imposed. Then Eq. (2.3fyatbdbgbokc) with $g_{m,R}(0-)=0$ is
reduced to the following eigenvalue problem of a non-Hermite matrix:

$\left(\begin{array}[]{cc}\epsilon_{{\rm
d}\alpha}-\mathrm{i}\Gamma_{\alpha\alpha}&v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}}\\\
v^{\prime}_{\overline{\alpha}}-\mathrm{i}\Gamma_{\overline{\alpha}\alpha}&\epsilon_{{\rm
d}\overline{\alpha}}-\mathrm{i}\Gamma_{\overline{\alpha}\,\overline{\alpha}}\end{array}\right)\left(\begin{array}[]{c}e_{\alpha,R}\\\
e_{\overline{\alpha},R}\end{array}\right)=E\left(\begin{array}[]{c}e_{\alpha,R}\\\
e_{\overline{\alpha},R}\end{array}\right).$ (2.3fyatbdbgboke)

The non-Hermite matrix on the left-hand side is referred to as an effective
Hamiltonian of the open system [29, 30]. It is characteristic to the systems
with linear dispersion relations that the imaginary term
$-\mathrm{i}\Gamma_{\alpha\beta}$, so-called the self-energy of the leads, in
each matrix element is independent of energy $E$. By diagonalizing the non-
Hermite matrix, we obtain resonance energies

$\displaystyle E^{(1)\pm}_{R}=\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\frac{\mathrm{i}}{2}\eta_{\pm},$
(2.3fyatbdbgbokf)

where $\eta_{\pm}$ has been defined in Eqs. (2.3fi). The corresponding
eigenvector $(e^{s}_{\alpha,R},e^{s}_{\overline{\alpha},R})$ for $s=\pm$ must
satisfy the relation

$\displaystyle\frac{e^{s}_{\alpha,R}}{e^{s}_{\overline{\alpha},R}}=\frac{v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}}}{\mathrm{i}\lambda_{\alpha,s}}=\frac{\mathrm{i}\lambda_{\overline{\alpha},s}}{v^{\prime}_{\overline{\alpha}}-\mathrm{i}\Gamma_{\overline{\alpha}\alpha}}.$
(2.3fyatbdbgbokg)

It is readily found that, at the exceptional points satisfying $\eta=0$, the
two eigenvalues $E^{(1)s}_{R}$ for $s=\pm$ merge into the one eigenvalue
$\overline{\epsilon}_{\rm d}-\mathrm{i}\overline{\Gamma}$ while the
eigenvectors $(e^{+}_{\alpha,R},e^{+}_{\overline{\alpha},R})$ and
$(e^{-}_{\alpha,R},e^{-}_{\overline{\alpha},R})$ become parallel to each
other.

In order to relate the eigenvectors to the time-evolving resonant state for
$\eta\neq 0$ in Eqs. (2.3fyatbdbgboca) and (2.3fyatbdbgbocb), we take the
expression

$\displaystyle e^{s}_{\alpha,R}$
$\displaystyle=\frac{1}{\eta_{\overline{s}}}\big{(}\psi_{\alpha}\lambda_{\alpha,\overline{s}}+\psi_{\overline{\alpha}}\,\mathrm{i}(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})\big{)}$
(2.3fyatbdbgbokh)

with the constants $\psi_{\alpha}$ and $\psi_{\overline{\alpha}}$. We note
that the ambiguity of the above expression of the eigenvectors is only a
constant multiple due to the relation
$\lambda_{\alpha,s}\lambda_{\overline{\alpha},s}=-(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})(v^{\prime}_{\overline{\alpha}}-\mathrm{i}\Gamma_{\overline{\alpha}\alpha})$.
By using the matching condition in Eq. (2.3fyatbdbgbokb), we obtain the wave
function $g_{m,R}(x)$ of the resonant state as

$\displaystyle
g^{s}_{m,R}(0+)=\sum_{\alpha}\psi_{\overline{\alpha}}\frac{1}{\eta_{\overline{s}}}\big{(}\mathrm{i}\lambda_{\alpha,s}v_{m\overline{\alpha}}+(v^{\prime}_{\alpha}-\mathrm{i}\Gamma_{\alpha\overline{\alpha}})v_{m\alpha}\big{)}=\sum_{\alpha}\psi_{\overline{\alpha}}y_{m\alpha,\overline{s}},$
$\displaystyle\therefore\quad$ $\displaystyle
g^{s}_{m,R}(x)=\sum_{\alpha}\psi_{\overline{\alpha}}y_{m\alpha,\overline{s}}\mathrm{e}^{\mathrm{i}E^{(1)s}_{R}x}\theta(x).$
(2.3fyatbdbgboki)

By inserting the wave function $g^{s}_{m,R}(x)$ in Eq. (2.3fyatbdbgboki) and
$e^{s}_{\alpha,R}$ in Eq. (2.3fyatbdbgbokh) into Eqs. (2.3fyatbdbgboka), we
obtain the one-body stationary resonant state. The imaginary part
$\overline{\Gamma}-\mathrm{Re}(\eta_{s}/2)$ of the resonance energy
$E^{(1)s}_{R}$ corresponds to the inverse lifetime of the resonant state.

It should be noted that the resonance energy $E^{(1)s}_{R}$ has appeared as a
resonance pole of the one-electron scattering eigenfunctions in Eqs. (2.3fya)
and (2.3fyb). We remark that the above wave functions $g^{s}_{m,R}(x)$ and
$e^{s}_{\alpha,R}$ of the one-body resonant state are also obtained by taking
a residue at the resonance pole $k=\overline{\epsilon}_{\rm
d}-\mathrm{i}\overline{\Gamma}+\mathrm{i}\eta_{s}/2$ of the one-electron
stationary scattering eigenfunctions $g^{(\ell)}_{m,k}(x)$ and
$e^{(\ell)}_{\alpha,k}$ in Eqs. (2.3fya) and (2.3fyb).

###  A.2 Two-body resonant states

Next, we consider the two-body stationary resonant states. In a way similar to
the one-body case, we assume the time-dependent parts of the two-electron wave
functions of the time-dependent Schrödinger equations (2.3f), (2.3f) and
(2.3fc) as

$\displaystyle
g_{m_{1}m_{2}}(x_{1},x_{2},t)=\mathrm{e}^{-\mathrm{i}Et}g_{m_{1}m_{2}}(x_{1},x_{2}),\quad$
$\displaystyle
e_{m,\alpha}(x,t)=\mathrm{e}^{-\mathrm{i}Et}e_{m,\alpha}(x),\quad
f_{\alpha\overline{\alpha}}(t)=\mathrm{e}^{-\mathrm{i}Et}f_{\alpha\overline{\alpha}}.$
(2.3fyatbdbgbokj)

By inserting these into Eq. (2.3fyak), we have

$\displaystyle(E-2\overline{\epsilon}_{\rm
d}-U^{\prime}+2\mathrm{i}\overline{\Gamma})f_{\alpha\overline{\alpha}}=\sum_{m,\beta}(-)^{\alpha+\beta}v^{\ast}_{m\beta}e_{m,\overline{\beta}}(0-).$
(2.3fyatbdbgbokk)

We recall that the wave function $e_{m,\alpha}(x)$ describes two electrons,
one of which is on the quantum dots and another is on the leads; see Eq.
(2.3d).

As an extension of the Siegert boundary conditions [22] to the two-electron
case, we impose the conditions that there is neither a one-electron nor a two-
electron incident wave:

$\displaystyle g_{m_{1}m_{2}}(x_{1},x_{2})=0\quad\mbox{ for }x_{1}<0\mbox{ or
}x_{2}<0,$ $\displaystyle e_{m,\alpha}(x)=0\quad\mbox{ for }x<0.$
(2.3fyatbdbgbokl)

Then, by applying $e_{m,\alpha}(0-)=0$ to Eq. (2.3fyatbdbgbokk), we obtain the
two-body resonance energy

$\displaystyle E^{(2)}_{R}=2\overline{\epsilon}_{\rm
d}+U^{\prime}-2\mathrm{i}\overline{\Gamma},$ (2.3fyatbdbgbokm)

which depends on the strength $U^{\prime}$ of the interdot interaction.

Since there is no restriction on the wave function
$f_{\alpha\overline{\alpha},R}$, we take
$f_{\alpha\overline{\alpha},R}=(-1)^{\overline{\alpha}}$. Through the
systematic construction of scattering eigenstates proposed in the previous
paper [20], we have

$\displaystyle
g_{m_{1}m_{2},R}(x_{1},x_{2})=\mathrm{i}\sum_{Q,\alpha,s}(-)^{\alpha}\mathrm{sgn}(Q)v_{m_{Q_{1}}\alpha}y_{m_{Q_{2}}\alpha,s}\mathrm{e}^{\mathrm{i}(U^{\prime}+E^{(1)\overline{s}}_{R})x_{Q_{2}Q_{1}}+\mathrm{i}E^{(2)}_{R}x_{Q_{1}}}\theta(x_{Q_{2}Q_{1}})\theta(x_{Q_{1}}),$
$\displaystyle
e_{m,\alpha,R}(x)=(-)^{\alpha}\sum_{s}y_{m\alpha,s}\mathrm{e}^{\mathrm{i}(U^{\prime}+E^{(1)\overline{s}}_{R})x}\theta(x).$
(2.3fyatbdbgbokn)

By inserting these into Eqs. (A.2), we obtain the two-body stationary resonant
state. The imaginary part $2\overline{\Gamma}$ of the resonance energy
$E^{(2)}_{R}$ gives the inverse lifetime of the two-body resonant state.

We note that the two-body resonance energy has appeared as the resonance pole
of the two-electron scattering eigenfunctions in Eqs. (2.3fyatbdbgboa),
(2.3fyatbdbgbob) and (2.3fyatbdbgboc) with Eq. (2.3fyatbdbgbn). We remark that
the above wave functions $g_{m_{1}m_{2},R}(x_{1},x_{2})$, $e_{m,\alpha,R}(x)$
and $f_{\alpha\overline{\alpha},R}$ of the two-body resonant state are also
obtained by taking a residue at the resonance pole
$E=2\overline{\epsilon}_{\rm d}+U^{\prime}-2\mathrm{i}\overline{\Gamma}$ of
the stationary two-electron scattering eigenfunctions
$g^{(\ell_{1}\ell_{2})}_{m_{1}m_{2},k_{1}k_{2}}(x_{1},x_{2})$,
$e^{(\ell_{1}\ell_{2})}_{m\alpha,k_{1}k_{2}}(x)$ and
$f^{(\ell_{1}\ell_{2})}_{\alpha\overline{\alpha},k_{1}k_{2}}$ in Eqs.
(2.3fyatbdbgboa), (2.3fyatbdbgbob) and (2.3fyatbdbgboc).

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