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\begin{center}
{\Large \textbf{QUASI-RELATIVISM, NARROW-GAP PROPERTY AND FORCED DYNAMICS OF
ELECTRONS IN SOLIDS: FEW SOLVABLE MODELS. }}
\end{center}

\vskip30pt

\begin{center}
\textbf{{\large B.S.Pavlov$^{1,2}$, A.A.Pokrovski$^1$, A.V.Strepetov$^3$ }}
\end{center}

\vskip27pt

\noindent \textit{${}^1$ Laboratory of Complex Systems Theory, Institute for
Physics, St.-Petersburg State University, Ulyanovskaya~1,
St.Petersburg~198904, Russia \\${}^2$ Department of Mathematics, University
of Auckland, Private Bag 92019, Auckland, New Zealand\\$^3$ Department of
the Applied Mathematics, State University for Airspace Instrument-making,
St.-Petersburg, 190000, Russia }
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\pagebreak

\section{Introduction}

Based on Landauer formula \cite{Land.} for quantum conductance one may
reduce the description of the quantum current \cite{BH} in quasi --
one-dimensional nano-wires to solution of relevant one-dimensional
scattering problem and calculation of the corresponding transmission
coefficient $\overleftarrow{\mathcal{T}}(E)$ which connect the amplitudes in
front of the asymptotically plain waves constituting the scattered waves $%
\psi (x,E)$ at $\pm \infty $
\[
\psi (x,E)\approx \left\{
\begin{array}{rl}
e^{-i\sqrt{E}x}+\overrightarrow{\mathcal{R}}e^{i\sqrt{E}x} &
,\,x\to +\infty ,
\\
\overleftarrow{\mathcal{T}}e^{-i\sqrt{E}x} & ,\,x\to -\infty .
\end{array}
\right.
\]
The averaging of the conductance over the Fermi distribution, \cite{Gir.}
\[
F(E,E_F)=\frac{CV\sqrt{E}}{\exp (\frac{E-E_F}{kT})+1}\quad
\]
gives another version of the Landauer formula - the current-voltage
dependence $j(U)$ - for higher temperature and nonzero voltage applied to
the wire :
\begin{equation}
j(U)=\frac{e^2}h\int \frac{|\overleftarrow{\mathcal{T}}(E)|^2}{1-|%
\overleftarrow{\mathcal{T}}(E)|^2}[F(E,E_F+eU)-F(E,E_F-eU)]dE\quad ,
\label{LA1}
\end{equation}
here $E_F$ is the Fermi level, $U$ - applied voltage. Assuming that
electrons in reservoirs on both ends of the quantum wire are almost free, we
may express the density of states in (\ref{LA1}) in terms of momentum as $%
dE=2kdk$. For sufficiently small values of the transmission coefficient $%
\mathcal{T}(E)$ the formula (\ref{LA1}) may be reduced to the formula:

\begin{equation}
j(U)=\frac{e^2}h\int |{\mathcal{T}(E)}|^2[F(E,E_F+eU)-F(E,E_F-eU)]dE\qquad{\
},  \label{LA2}
\end{equation}

We discuss below \textit{ballistic} electron's transmission problem in the
nanowire neglecting at this stage the electron-electron and electron-phonon
interaction and considering only the one-body problem. But we pay a special
attention to the calculation of the effective mass \cite{Titch.} of electron
in the lattice. In fact the magnitude of the effective mass defines the
mobility of electrons : the free path and the De-Broghlie wavelength. The
assumption about the small effective mass serves a base of application of
methods of the scattering theory to the description of ballistic electrons
in nano-electronic networks, see \cite{P_S}, \cite{Mir.}, \cite{Pav.}, \cite
{Ant.}. We analyze here two possible mechanisms of appearing of a small
effective mass in super-lattices: the mechanism based on a strong
interaction of blocks in the superlattice and the mechanism based on a weak
interaction between sub-lattices. Respectively two examples are considered
which show that electrons in strongly connected and weakly-connected
superlattices may exhibit a quasi-relativistic behaviour and may be
described by an analog of a one-dimensional Dirac equation, which \textit{is
derived from the first principles } under assumption of low temperature and
weak external electric field.

Basing on the derived Dirac equation we consider the time-periodic
perturbations in form of running wave and a localized singular potential. In
both cases the transmission coefficients exhibit spectral band structure.
This phenomenon may be used for manipulation of quantum current in quantum
networks constructed of the narrow-gap superlattices.

\section{ A superlattice with a small effective mass.}

In this section we construct a solvable model of a quantum waveguide with a
small effective mass of charge-carriers caused by splitting between energy
levels of electrons in the basic block of the corresponding lattice and
strong interaction between blocks.

Consider an hermitian $2\times 2$ matrix $Q:E\to E$ with real eigenvalues $%
q_1,\,q_2$. Introducing the mean value $\displaystyle q_0:=\frac{q_1+q_2}2$
and splitting $\displaystyle \delta :=\frac{q_1-q_2}2>0$, we may present $Q$
as
\begin{equation}
q_0\left(
\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}
\right) +\frac \delta {\sqrt{2}}\left(
\begin{array}{cc}
1 & 1 \\
1 & -1
\end{array}
\right) .  \label{block}
\end{equation}
We use $Q$ as a block for construction of a periodic lattice operator $%
\mathcal{Q}$ in the Hilbert space $\mathcal{E}:=l_2\times E=l_2(E)$ of
infinite sequences $\vec {\mathcal{F}}=\left\{ \vec f_l\right\} _{{l=-\infty
}}^\infty ,\,\quad \vec f_l\in E$.

Denote in this section by $\mathcal{T}$ the shift operator in $\mathcal{E}$
\[
\mathcal{T}\left\{ \vec f_l\right\} _{{l=-\infty }}^\infty =\left\{ \vec
f_l^{\prime }\right\} _{{l=-\infty }}^\infty ,\quad \,\vec f_l^{\prime
}=\vec f_{l-1}
\]
and by $\mathcal{I}$ the unity in $l_2$. Introduce the operators $P_1,\,P_2$
projecting respectively onto the first and second components of each vector $%
{\vec f}_l$:
\[
P_1\vec {\mathcal{F}}=\left\{
\begin{array}{c}
f_{l,1} \\
0
\end{array}
\right\} _{l=-\infty }^\infty ,
\]
\[
P_2\vec {\mathcal{F}}=\left\{
\begin{array}{c}
0 \\
f_{l,2}
\end{array}
\right\} _{l=-\infty }^\infty ,
\]
and by $\pi _{12}$ the operators swoping the components of each vector $\vec
f_l$ ,for instance
\[
\pi _{12}\left\{
\begin{array}{c}
f_{l,1} \\
f_{l,2}
\end{array}
\right\} _{l=-\infty }^\infty =\left\{
\begin{array}{c}
f_{l,2} \\
f_{l,1}
\end{array}
\right\} _{l=-\infty }^\infty .
\]
The required one-body lattice Hamiltonian is defined as
\begin{equation}
\mathcal{Q}=Q\times \mathcal{I}_{L_2}-i\alpha \pi _{12}P_2\mathcal{T}%
+i\alpha \pi _{12}P_1\mathcal{T}^{+}.  \label{operator}
\end{equation}
It is presented by an infinite periodic Jacobian matrix with $2\times 2$
blocks $Q$ on the main diagonal and the interaction between blocks
introduced by the shifts:
\[
\left(
\begin{array}{ccccc}
\dots & \dots & \dots & \dots & \dots \\
\dots & -i\alpha & 0 & \dots & \dots \\
i\alpha & q_0+\frac \delta {\sqrt{2}} & \frac \delta {\sqrt{2}} & \dots &
\dots \\
0 & \frac \delta {\sqrt{2}} & q_0-\frac \delta {\sqrt{2}} & -i\alpha & \dots
\\
\dots & \dots & i\alpha & \dots & \dots \\
\dots & \dots & \dots & \dots & \ldots
\end{array}
\right)
\]
Fourier transform
\[
\mathbf{F}\vec {\mathcal{F}}\longrightarrow \sum_{l=-\infty }^\infty
e^{il\varphi }\vec f_l:=F(\varphi ),
\]
\[
\mathbf{F}\mathcal{T}\left\{ \vec f_l\right\} _{l=-\infty }^\infty
=e^{i\varphi }F(\varphi )
\]
defines the similarity of the operator $\mathcal{Q}$ to the multiplication
in $L_2((-\pi ,\pi ),E)$ by the matrix- function
\begin{equation}
Q(\varphi )=\left(
\begin{array}{cc}
q_0+\frac \delta {\sqrt{2}} & \frac \delta {\sqrt{2}}+i\alpha e^{-i\varphi }
\\
\frac \delta {\sqrt{2}}-i\alpha e^{i\varphi } & q_0-\frac \delta {\sqrt{2}}
\end{array}
\right) .  \label{Operator}
\end{equation}
The spectrum $\sigma _Q$ of the operator $\mathcal{Q}$ is defined by
dispersion equation
\begin{equation}
{\det }\left(Q(\varphi )-\lambda E \right)
=(\lambda -q_0)^2-\delta ^2-\alpha ^2+\alpha \delta \sqrt{2%
}-\alpha \delta \sqrt{2}(1-\sin \varphi )=  \label{dispersion}
\end{equation}
\[
(\lambda -q_0)^2-\delta ^2-\alpha ^2+\alpha \delta \sqrt{2}-\alpha \delta
\sqrt{2}\left( \sin \varphi /2-\cos \varphi /2\right) ^2=0,\,\,-\pi <\varphi
<\pi ,
\]
and consists of two intervals - upper and lower \textit{spectral bands} with
the spectral gap between them :
\[
\left( q_0-[\delta ^2+\alpha ^2-\alpha \delta \sqrt{2}]^{1/2},\,q_0+[\delta
^2+\alpha ^2-\alpha \delta \sqrt{2}]^{1/2}\right)
\]

The effective mass
\[
m_\delta =\left[ \frac{d^2\lambda }{d\varphi ^2}\right] ^{-1}
\]
is calculated from the dispersion function $\lambda (\varphi )$ defined by
the dispersion equation (\ref{dispersion}). At the bottom of the upper
spectral band $\varphi =\pi /2,\lambda =q_0+[\delta ^2+\alpha ^2-\alpha
\delta \sqrt{2}]^{1/2}$ it is positive and equal to
\[
\frac{2[\delta ^2+\alpha ^2-\alpha \delta \sqrt{2}]^{1/2}}{\alpha \delta
\sqrt{2}}\;.
\]
It is as small as an inverse of the minimal of two (large) parameters $%
\alpha ,\,\delta $ is.

Note that the multiplication operator (\ref{Operator}) has a uniformly
convergent asymptotic expansion for small deviations of quasimomentum $%
\varphi $ from the lowest point of the upper spectral band $\varphi =\pi /2$%
. This expansion is defined by the Taylor series of exponentials after the
change of variables $\alpha (\varphi -\pi /2)\to p$%
\begin{equation}
Q(\varphi )=\left(
\begin{array}{cc}
q_0+\frac \delta {\sqrt{2}} & \frac \delta {\sqrt{2}}+i\alpha \\
\frac \delta {\sqrt{2}}-i\alpha & q_0-\frac \delta {\sqrt{2}}
\end{array}
\right) +\left(
\begin{array}{cc}
0 & -ip \\
ip & 0
\end{array}
\right) +\left(
\begin{array}{cc}
0 & \alpha \sum_{l=2}^\infty \frac{(-ip)^l}{l!\alpha ^l} \\
\alpha \sum_{l=2}^\infty \frac{(ip)^l}{l!\alpha ^l} & 0.
\end{array}
\right)  \label{asymptotics}
\end{equation}
After the corresponding transformation of the spectral variable $%
\displaystyle{\lambda \to \mu :=\lambda -q_0}$ we obtain a new operator
\[
\begin{array}{rl}
\mathcal{D}=Q-q_0 E = & \sigma _yp+\frac \delta {\sqrt{2}}\sigma
_z+(\frac \delta {\sqrt{2}}+\alpha )\sigma _x+ \\ &  \\ & +\left(
\begin{array}{cc}
0 & \alpha \left( e^{-ip/\alpha }-1+ip/\alpha \right) \\ \alpha
\left( e^{ip/\alpha }-1-ip/\alpha \right) & 0
\end{array}
\right) = \\ & \\ = & \mathcal{D}_0 + \left(
\begin{array}{cc}
0 & \alpha \left( e^{-ip/\alpha }-1+ip/\alpha \right) \\ \alpha
\left( e^{ip/\alpha }-1-ip/\alpha \right) & 0
\end{array}
\right).
\end{array}
\]
The main part of it $\sigma _yp$ has the spectrum $\sigma ^D$ centered at
the origin :
\[
\sigma _D=\left\{ \mu :q_0+\mu \in \sigma _Q\right\} .
\]
One can see that the last term in the expression for the operator $D$ may be
estimated near the origin as $p^2$. In fact small values of the $p$ play an
important role in conductance processes for low temperature and weak
potential $U$. Really, assuming that the potential $U$ is small and the
temperature $T$ is low we may introduce a new norm in the space of the
spectral representation of the operator $D$ :
\[
\langle u,\,v\rangle =\int u\overline{v}\left[ F(E,E_F+U)-F(E,E_F-U)\right]
dE.
\]
The weight function in this norm which is combined of Fermi distributions is
almost a step- function $\Pi _\Delta $ under the above assumptions. It is
localized on a small interval $\Delta ,|\Delta |\approx \kappa T+2eU$ near
the Fermi -level. In particular assuming that the Fermi-level coincides with
the bottom of the upper spectral band
\[
E_F=Q_0+[\delta ^2+\alpha ^2-\alpha \delta \sqrt{2}]^{1/2},
\]
and introducing the quasimomentum $\varphi $ instead of the energy
$\lambda =E$ we see that the restriction of the operator $D$ onto
the spectral subspace corresponding to the step-function $\Pi
_\Delta $ differs from the $\mathcal{D}_0$ -- Fourier image of the
Dirac operator -- only by the small addend
\[
\left(
\begin{array}{cc}
0 & \alpha \left( e^{-ip/\alpha }-1+ip/\alpha \right) \\
\alpha \left( e^{ip/\alpha }-1-ip/\alpha \right) & 0
\end{array}
\right)
\]
with the norm not greater than $|\alpha |^{-1}\Delta ^2$. Summarizing the
above observations we obtain the following important assertion

\begin{theorem}
\label{relativism} . The dynamics of electrons in the quasi-one-dimensional
lattice combined of strongly interacting two-dimensional blocks with large
splitting of energy levels is described for low temperatures and weak fields
by the quasi-relativistic equation, if the Fermi -- level sits at the bottom
of the upper spectral band.
\end{theorem}

This fact is in natural agreement with physical observation of
quasi-relativistic behavior of electrons in some semiconductors.
But the same effect for the effective mass may arise in totally
different situation, see the next section. \vskip0.5cm

\subsection{ Narrow-gap superlattice.}

Consider a double superlattice combined of two parallel weakly interacting
branches with equal periods. Each branch is constituted by equivalent
quantum dots (atoms or quasi-molecular complexes) with vacant normalized
orbitals ${\psi }_l=\psi (x-le),\,l=0,\pm 1,\,\pm 2\dots ;\varphi _m=\phi
(x-me),\,m=0,\,\pm 1,\,\pm 2\dots $. We use the following assumption about
the overlapping integrals:
\[
\langle \psi _l,\,\psi _{l^{\prime }}\rangle =\langle \varphi _l,\,\varphi
_{l^{\prime }}\rangle =0,\,\,\mbox{if}\;|l-l^{\prime }|\geq 2,
\]
\[
\langle \varphi _l,\,\psi _{l^{\prime }}\rangle =0,\;\mbox{if}\;l\neq
l^{\prime },
\]
\begin{eqnarray}
\langle \varphi _l,\,\varphi _{l+j}\rangle &=&a_j,\;\mbox {if}\;j=0,\pm
1,\,a_0=1,\;a_1=a_{-1}>0;\;\langle \psi _l,\,\psi _{l+j}\rangle =b_j,
\nonumber  \label{cond} \\
\;\mbox {if}\;j &=&0,\pm 1,\;\,b_0=1,\;b_1=b_{-1}<0;\;\langle \psi
_l,\,\varphi _l\rangle =\alpha =\bar \alpha .
\end{eqnarray}
We assume, that the one-electron Hamiltonian of the superlattice is
presented as a sum of the two weakly interacting chains of quantum dots:
\begin{equation}\label{Hamilt}
\mathcal{H}u=A\sum_{l=-\infty }^{+\infty }\varphi _l\langle u,\,\varphi
_l\rangle +A\sum_{l^{\prime }=-\infty }^{+\infty }\psi _{l^{\prime }}\langle
u,\,\psi _{l^{\prime }}\rangle .
\end{equation}
The formal eigenvalue $A$ which corresponds to the vacant orbitals is
assumed to be equal to the Fermi level of the total superlattice. We may
assume that $A=1$ since it gives just an additional factor in the
description of the spectrum. The interaction $\alpha $ between the branches
of the superlattice is assumed to be weak, $|\alpha |<<1$ compared with the
Fermi level and the overlapping integrals $a_l,b_m$.

We find the spectrum of the superlattice and calculate the corresponding
Bloch waves and effective mass. We shall use the ansatz for the Bloch waves
in form :
\[
u=\sum_{l=-\infty }^\infty u_l^\varphi \varphi _l+u_l^\psi \psi _l.
\]
Substitution of the above ansatz into the constructed Hamiltonian gives the
following infinite system of linear equations :
\begin{equation}
\left\{
\begin{array}{ccccc}
a_1u_{l-1}^\varphi + & a_1u_{l+1}^\varphi + & a_0u_l^\varphi + & \alpha
u_l^\psi = & \lambda u_l^\varphi \\
b_1u_{l-1}^\psi + & b_1u_{l+1}^\psi + & b_0u_l^\psi + & \alpha u_l^\varphi =
& \lambda u_l^\psi .
\end{array}
\right.  \label{eq}
\end{equation}
The Bloch-type solution of it is defined by the property
\[
\left(
\begin{array}{c}
u_l^\varphi \\
u_l^\psi
\end{array}
\right) :=\vec u_l=\Theta \vec u_{l-1},
\]
which gives the following equation for the basic vector $\vec u_0$:
\begin{equation}
\left\{ \left(
\begin{array}{cc}
a_1 & 0 \\
0 & b_1
\end{array}
\right) \left[ \Theta +\Theta ^{-1}\right] +\left(
\begin{array}{cc}
a_0 & \alpha \\
\alpha & b_0
\end{array}
\right) +\lambda I\right\} \vec u_0=0.  \label{bloch}
\end{equation}
The Bloch exponent $\Theta $ may be found from the condition
\[
\det \left\{ \left(
\begin{array}{cc}
a_1 & 0 \\
0 & b_1
\end{array}
\right) \left[ \Theta +\Theta ^{-1}\right] +\left(
\begin{array}{cc}
a_0 & \alpha \\
\alpha & b_0
\end{array}
\right) +\lambda I\right\} =0.
\]
We accomplish the calculation assuming that $a_1=1=-b_1$ and $a_0=b_0$. Then
we obtain for quasimomentum $p,\,\Theta =e^{ip}$ the simple algebraic
equation
\[
(a_0-\lambda )^2-4(\cos p)^2=\alpha ^2.
\]
The spectrum of the Hamiltonian  (\ref{Hamilt}) consists of two
spectral bands separated by the gap $(-\alpha ,\alpha )$. The gap
is centered at the value of the quasimomentum $p_0=\pi
/2,\,\lambda _0^{\pm }=a_0\pm \alpha $, so the gap is
\textit{narrow} if the interaction between the branches of the
superlattice is \textit{weak}. On the other hand the direct
calculation of the effective mass gives the following result :
\begin{equation}
2(\lambda _0^{\pm }-a_0)\lambda _{pp}=8(\sin p_0)^2,
\end{equation}
hence $m_{eff}=(\lambda _{pp})^{-1}=\frac \alpha 4$ is small. This result is
in good agreement with observation of the narrow-gap property for
weakly-connected superlattices, in particular for non-stable semiconductors
like $HgTe$, \cite{Ant.}. Analysis of the operator in the invariant subspace
corresponding to the small neighbourhood of the Fermi level at the bottom of
the upper spectral band reveals the quasi-relativistic properties of the
dynamics of electrons subject to the condition $\kappa T+2eU<<1$, similarly
to the case discussed in the previous section.

\section{Dirac equation with running potential}

The quasi-relativistic property of the dynamics of electrons observed in the
previous sections permits to suggest a method of manipulation of the quantum
current through the wire with use of the running wave, similar to the method
suggested in \cite{Prok1} and \cite{Prok2}.

Consider a non-stationary Dirac operator with running potential $\epsilon
_1\cos \epsilon _2(x-vt)$.
\begin{equation}
\frac 1i\frac{\partial \psi }{\partial t}=\left(
\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}
\right) i\frac{\partial \psi }{\partial x}+\delta \left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right) \psi +\left( q_0+\epsilon _1\cos \epsilon _2(x-vt)\right) \left(
\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}
\right) \psi .  \label{running}
\end{equation}
Following \cite{Prok2} we apply the Lorentz transform
\begin{equation}
\frac{x-vt}{\sqrt{1-v^2}}=s,\,\,\frac{t-vx}{\sqrt{1-v^2}}=\tau
\label{Lorentz}
\end{equation}
and obtain a new equation
\[
\left(
\begin{array}{cc}
\frac 1{\sqrt{1-v^2}} & \frac{iv}{\sqrt{1-v^2}} \\
-\frac{iv}{\sqrt{1-v^2}} & \frac 1{\sqrt{1-v^2}}
\end{array}
\right) i\frac{\partial \psi }{\partial \tau }+\left(
\begin{array}{cc}
\frac v{\sqrt{1-v^2}} & \frac{-i}{\sqrt{1-v^2}} \\
-\frac i{\sqrt{1-v^2}} & \frac v{\sqrt{1-v^2}}
\end{array}
\right) i\frac{\partial \psi }{\partial s}+
\]
\begin{equation}
+\delta \left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right) \psi +\left( q_0+\epsilon _1\cos \epsilon _2\sqrt{1-v^2}s\right)
\left(
\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}
\right) \psi .  \label{Running}
\end{equation}
which admits separation of variables. Introducing the spectral variable $%
\Omega $ representation $\psi$ as $\psi =e^{i\Omega \tau
}\varphi_{\Omega}$, we obtain the spectral problem :

\[
\left(
\begin{array}{cc}
\frac{1}{\sqrt{1-v^2}} & \frac{iv}{\sqrt{1-v^2}} \\
-\frac{iv}{\sqrt{1-v^2}} & \frac{1}{\sqrt{1-v^2}}
\end{array}
\right) \Omega \varphi_{\Omega} = \left(
\begin{array}{cc}
\frac{v}{\sqrt{1-v^2}} & \frac{-i}{\sqrt{1-v^2}} \\
-\frac{i}{\sqrt{1-v^2}} & \frac{v}{\sqrt{1-v^2}}
\end{array}
\right) i\frac{\partial \varphi_{\Omega}}{\partial s} +
\]
\begin{equation}  \label{stationary}
+ \delta \left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right) \varphi_{\Omega} + \left( q_0 + \epsilon_1 \cos \epsilon_2
\sqrt{1-v^2} s \right) \left(
\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}
\right) \varphi_{\Omega} := L \varphi_{\Omega}
\end{equation}

The differential operator $L$ which corresponds to this spectral problem in
the corresponding weighted space of all square integrable function has
periodic coefficients and band spectrum $\sigma _L$. Hence the equation (\ref
{running}) has also running solutions of the form $\displaystyle{\
e^{i\Omega \frac{t-vx}{\sqrt{1-v^2}}}\varphi_{\Omega} (\frac{x-vt}{\sqrt{1-v^2}})%
}$ for $\Omega \in \sigma _L$, parametrized by the spectral points $\Omega $
.\vskip0.5cm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Time-periodic Dirac equation}

Another possibility of manipulation quantum current may be achieved via
localized periodic excitations. Consider the Dirac operator with a singular
time- periodic potential defined as an extension of the non-perturbed Dirac
operator restricted (from $D$ to $D_0$) onto the class of all $W_2^1$-smooth
vector-functions vanishing near the origin $x=0$. The adjoint operator $%
D_0^{+}$ is defined on the piecewise continuous $W_2^1$-smooth vector -
functions $\vec \psi =\left( \psi _1,\,\psi _2\right) $ which may have a
jump at the origin, i.e. the left limit $\vec \psi (-0):=\left( \psi
_1^{-},\,\psi _2^{-}\right) $ is not equal to the right limit $\vec \psi
(+0):=\left( \psi _1^{+},\,\psi _2^{+}\right) $. The straightforward
calculation of the boundary form of the operator $D_0^{+}$ gives the
following result for any elements $\vec \psi ,\,\vec \phi $ from the domain
of the adjoint operator:
\begin{equation}
<D_0^{+}\vec \psi ,\,\vec \phi >-<\vec \psi ,\,D_0^{+}\vec \phi >=i\left(
\psi _1^{-}\bar \phi _2^{-}+\psi _2^{-}\bar \phi _1^{-}\right) -i\left( \psi
_1^{+}\bar \phi _2^{+}+\psi _2^{+}\bar \phi _1^{+}\right) .  \label{bform}
\end{equation}
This implies the following description of boundary conditions defining the
selfadjoint extensions $D_\Gamma $ of $D_0$ parametrized by $2\times 2$
hermitian matrices $\Gamma $,see\cite{BSP}:
\begin{equation}
\left(
\begin{array}{c}
i\psi _1^{-} \\
\psi _2^{+}
\end{array}
\right) =\Gamma \left(
\begin{array}{c}
\psi _2^{-} \\
-i\psi _1^{+}
\end{array}
\right) .  \label{bcond}
\end{equation}
In what follows we omit the vector notations $\vec \psi $, having in mind
that each time the lower indices as in $\psi _{1,2}$ are absent, we have a $%
2 $-vector, that is $\psi \equiv \vec \psi $.

One can easily calculate the spectral properties of operators $D_\Gamma $ in
terms of the matrix $\Gamma $. Really, the solutions of the non-perturbed
homogeneous equation may be presented in form
\begin{equation}
\psi ^{\pm }=\left(
\begin{array}{c}
{\pm }i\sqrt{\frac{\lambda -q_0+\delta }{\lambda -q_0-\delta }}e^{i\pm \sqrt{%
(\lambda -q_0)^2-\delta ^2}x} \\
e^{i\pm \sqrt{(\lambda -q_0)^2-\delta ^2}x}
\end{array}
\right) .  \label{waves}
\end{equation}
Solutions of the perturbed equation which satisfy the boundary conditions (%
\ref{bcond}) are combined of $\psi ^{\pm }$ as
\[
\overrightarrow{{\psi }}=\left\{
\begin{array}{cc}
\psi ^{+}+\overleftarrow{{R}}\psi ^{-} & \mbox{for}\,\,\,x<0 \\
\overrightarrow{{T}}\psi ^{+} & \mbox{for}\,\,\,x>0,
\end{array}
\right.
\]
\begin{equation}
\overleftarrow{{\psi }}=\left\{
\begin{array}{cc}
\psi ^{-}+\overrightarrow{{R}}\psi ^{+} & \mbox{for}\,\,\,x>0 \\
\overleftarrow{{T}}\psi ^{-} & \mbox{for}\,\,\,x<0
\end{array}
\right.  \label{swaves}
\end{equation}
Then the scattering matrix of the operator $D_\Gamma $ may be defined as
\begin{equation}
S_\Gamma =\left(
\begin{array}{cc}
\overleftarrow{{R}} & \overleftarrow{{T}} \\
\overrightarrow{{T}} & \overrightarrow{{R}}
\end{array}
\right) .  \label{smatrix}
\end{equation}

The values of transmission coefficients $\stackrel{\rightleftharpoons}{T}$
do not depend on the normalization constants, i.e. they remain invariant
with respect to replacement of $\psi ^{\pm }$ by $|\alpha |e^{\pm i\varphi
}\psi ^{\pm }$. For special choice of the normalizing constants $|\alpha
|e^{\pm i\varphi }$ the scattering matrix is unitary. But for our aims it is
sufficient to calculate the transmission coefficients only. The following
statement is true:

\begin{theorem}
The scattering matrix of the operator $D_\Gamma $ with respect to solutions $%
\psi ^{\pm }$ has the form
\[
S_\gamma =\left\{ \left(
\begin{array}{cc}
\sqrt{\frac{\lambda -q_0+\delta }{\lambda -q_0-\delta }} & 0 \\
0 & 1
\end{array}
\right) -\Gamma \left(
\begin{array}{cc}
1 & 0 \\
0 & \sqrt{\frac{\lambda -q_0+\delta }{\lambda -q_0-\delta }}
\end{array}
\right) \right\} ^{-1}\times
\]
\begin{equation}
\times \left\{ \left(
\begin{array}{cc}
\sqrt{\frac{\lambda -q_0+\delta }{\lambda -q_0-\delta }} & 0 \\
0 & 1
\end{array}
\right) +\Gamma \left(
\begin{array}{cc}
1 & 0 \\
0 & \sqrt{\frac{\lambda -q_0+\delta }{\lambda -q_0-\delta }}
\end{array}
\right) \right\} \left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right) .  \label{Smatrix}
\end{equation}
\end{theorem}

\textit{Proof} may be obtained by the straightforward calculation of the
transmission and reflection coefficients from the boundary conditions (\ref
{bcond}) for $\stackrel{\rightleftharpoons }{\psi }$ :
\[
\left(
\begin{array}{c}
\overleftarrow{{T}}\sqrt{\frac{\lambda -q_0+\delta }{\lambda -q_0-\delta }}
\\
1+\overrightarrow{{R}}
\end{array}
\right) =\Gamma \left(
\begin{array}{c}
\overleftarrow{{T}} \\
-\sqrt{\frac{\lambda -q_0+\delta }{\lambda -q_0-\delta }}+\overrightarrow{{R}%
}\sqrt{\frac{\lambda -q_0+\delta }{\lambda -q_0-\delta }}
\end{array}
\right) ,
\]
\[
\left(
\begin{array}{c}
-\sqrt{\frac{\lambda -q_0+\delta }{\lambda -q_0-\delta }}+\overleftarrow{{R}}%
\sqrt{\frac{\lambda -q_0+\delta }{\lambda -q_0-\delta }} \\
\overrightarrow{{T}}
\end{array}
\right) =\Gamma \left(
\begin{array}{c}
1+\overleftarrow{{R}} \\
\overrightarrow{{T}}\sqrt{\frac{\lambda -q_0+\delta }{\lambda -q_0-\delta }}
\end{array}
\right)
\]

In particular the last pair of equations may be solved as
\[
\begin{array}{c}
\left(
\begin{array}{c}
\overleftarrow{R} \\
\overrightarrow{T}
\end{array}
\right) =\left\{ \left(
\begin{array}{cc}
\sqrt{\frac{\lambda -q_0+\delta }{\lambda -q_0-\delta }} & 0 \\
0 & 1
\end{array}
\right) -\Gamma \left(
\begin{array}{cc}
1 & 0 \\
0 & \sqrt{\frac{\lambda -q_0+\delta }{\lambda -q_0-\delta }}
\end{array}
\right) \right\} ^{-1} \\
\\
\left\{ \left(
\begin{array}{cc}
\sqrt{\frac{\lambda -q_0+\delta }{\lambda -q_0-\delta }} & 0 \\
0 & 1
\end{array}
\right) +\Gamma \left(
\begin{array}{cc}
1 & 0 \\
0 & \sqrt{\frac{\lambda -q_0+\delta }{\lambda -q_0-\delta }}
\end{array}
\right) \right\} \left(
\begin{array}{c}
1 \\
0
\end{array}
\right) .
\end{array}
\]
The solution of the first pair of equations may be represented in a similar
way. The combination of them gives the announced result.

$\Box$

We consider now the Dirac equation with time-dependent singular potential as
a selfadjoint operator in space- time Hilbert space $L_2 (R) \times L_2 (R,
E)$ which is obtained as an extension of the ``non-perturbed" operator

\begin{equation}
i\frac{\partial \psi }{\partial t}+\left(
\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}
\right) i\frac{\partial \psi }{\partial x}+\delta \left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right) \psi :=\mathcal{D},  \label{time-space_operator}
\end{equation}
restricted $\mathcal{D}\to \mathcal{D}_0$ onto the linear set of all $W_2^1$
functions vanishing near the $t$- axis on the phase space $R_2=R_t\times R_x$%
. The corresponding adjoint operator is defined on the splitted space $%
W_2^1(R_2,t<0)\oplus W_2^1(R_2,t>0)$, and the boundary form of it may be
written as the integral of the boundary form (\ref{bform}) of the operator $%
D_0^{+}$ over $R_t$:
\[
<\mathcal{D}_0^{+}\vec \psi ,\,\vec \phi >-<\vec \psi ,\,\mathcal{D}%
_0^{+}\vec \phi >=
\]
\begin{equation}
\int_{-\infty }^\infty \left\{ i\left( \psi _1^{-}\bar \phi _2^{-}+\psi
_2^{-}\bar \phi _1^{-}\right) -i\left( \psi _1^{+}\bar \phi _2^{+}+\psi
_2^{+}\bar \phi _1^{+}\right) \right\} dt.  \label{Bform}
\end{equation}
Existence of the boundary values $\psi _{1,2}^{\pm }\in L_2(R_t)$ follows
from standard embedding theorems. To define the selfadjoint extension $%
\mathcal{D}_\Gamma $ we submit the boundary values $\psi _{1,2}^{\pm }$ to
the \textit{local} time-dependent boundary conditions:
\begin{equation}
\left(
\begin{array}{c}
i\psi _1^{-} \\
\psi _2^{+}
\end{array}
\right) =\Gamma \cos {\Omega t}\left(
\begin{array}{c}
\psi _2^{-} \\
-i\psi _1^{+}
\end{array}
\right)  \label{Bcond}
\end{equation}
with a constant Hermitian matrix $\Gamma $ . The solutions of the
homogeneous equation
\begin{equation}
i\frac{\partial \psi }{\partial t}+\left(
\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}
\right) i\frac{\partial \psi }{\partial x}+\delta \left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right) \psi =\Lambda \psi  \label{time-space_equation}
\end{equation}
serve the eigenfunctions of the operator $\mathcal{D}_\Gamma $. One can show
that the whole spectrum of this operator is absolutely continuous and
consists of two branches : the branch with infinite multiplicity, which
corresponds to the scattered waves combined of solutions of the
non-perturbed homogeneous equations
\[
i\frac \partial {\partial t}\psi _\Omega (t){\partial t}=\Omega \psi _\Omega
(t),
\]
\[
D\psi _\lambda ^{\pm }=\lambda \psi _\lambda ^{\pm }
\]
for $\Lambda =\Omega +\lambda $ as
\[
\overrightarrow{\Psi} _{\Lambda ,\Omega }=\psi _\Omega (t)\times \left\{
\begin{array}{cc}
\psi _\lambda ^{+}+\overleftarrow{\mathcal{R}}\psi ^{-}, & \mbox{for}\,x<0
\\
\overrightarrow{\mathcal{T}}\psi ^{-}, & \mbox{for}\,x>0,
\end{array}
\right. ,
\]
\begin{equation}
\overleftarrow{\Psi}_{\Lambda ,\Omega }=\psi _\Omega (t)\left\{
\begin{array}{cc}
\psi ^{-}+\overrightarrow{\mathcal{R}}\psi ^{+} & \mbox{for}\,\,\,x>0 \\
\overleftarrow{\mathcal{T}}\psi ^{-} & \mbox{for}\,\,\,x<0
\end{array}
\right. ,  \label{Swaves}
\end{equation}
and the waveguide branch which is constructed of outgoing solutions on $x$%
-axis:
\begin{equation}
\Psi _\Lambda (x,t)=\psi _\Omega (t)\left\{
\begin{array}{cc}
\psi ^{+}(x) & \mbox{for}\,x>0 \\
\mathcal{K}\psi ^{-}(x) & \mbox{for}\,x<0
\end{array}
\right. ,  \label{waveguide}
\end{equation}
The transmission and reflection coefficients $\stackrel{\rightleftharpoons}{%
\mathcal{T}},\,\stackrel{\rightleftharpoons}{\mathcal{R}}$ for given $%
\lambda $ still depend on $\Omega $ and may be defined from the boundary
conditions (\ref{Bcond}). One may construct of them the scattering matrix as
\begin{equation}
\mathcal{S}_\Gamma (\Lambda ,\Omega )=\left(
\begin{array}{cc}
\overleftarrow{\mathcal{R}} & \overleftarrow{\mathcal{T}} \\
\overrightarrow{\mathcal{T}} & \overrightarrow{\mathcal{R}}
\end{array}
\right) .  \label{SSmatrix}
\end{equation}
The amplitudes $K$ are defined uniquely for given $\lambda $ from the
additional condition of quasiperiodicity with respect to the time-variable: $%
\Psi _\Lambda (x,t+\frac{2\pi }\Omega )=\Theta \Psi _\Lambda (x,t)$ which
may be imposed just on the boundary values $\Psi _\Lambda (\pm 0,t)$. The
quasimomentum exponential $\Theta (\Lambda )$ may play the role of an
alternative spectral variable for the waveguide branch of an absolutely-
continuous spectrum. Similar spectral variable may be introduced also for
the scattered waves $\Psi _{\Lambda ,\Omega }^{\rightleftharpoons }$ to
substitute one of parameters $\Lambda ,\Omega $.

We concentrate now on the analysis of eigenfunctions of absolutely
continuous spectra which correspond to the spectral point $\Lambda =0$. The
corresponding eigenfunction satisfy the equation $\mathcal{D}_\gamma \Psi =0$%
. In particular the waveguide-type eigenfunction of this type (if exist) may
correspond to some sort of bound states which depend on time harmonically.
For investigation of the conductance of the quantum channel of the
narrow-gap semiconductor manipulated by the localized time-periodic
excitation an important role is played by the scattered waves $\stackrel{%
\rightleftharpoons}{\Psi }_{\Lambda ,\Omega }$. Precisely, the conductance
for given $\Omega $ is calculated from the corresponding transmission
coefficients via Landauer formula as $\displaystyle\frac{e^2}h\frac{|%
\mathcal{T}(0,\Omega )|^2}{1-|\mathcal{T}(0,\Omega )|^2}$. The investigation
of the behavior of the transmission coefficient $\overrightarrow{\mathcal{T}}%
(0,\Omega )$ with respect to frequency $\Omega $ of the excitation is an
important ongoing problem. %%%%%%%%%%%%%%%


Let us estimate the frequency necessary for effective control of
electrons dynamics in a narrow-gap semiconductor using this
method. It is clear that the required frequency must be of the
same order as the frequency of the ballistic electrons. When the
Fermi level coincides with the bottom of the upper spectral band
the frequency of electrons  $\nu_{el}$ at average thermal energy
$kT$ (for room temperature $T=300K$) is
\[
 \nu_{el}=2\pi\frac{kT}{h}\approx 3\cdot10^{13}Hz,
\]
which corresponds to IR wavelengths  $\lambda\approx 10\mu m$.
\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{document}
Eventually we may assume that $\epsilon _{1,2}$ act as small
parameters, but we still restrain of it at the moment.
