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SNUTP03-002\\
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\begin{center}
\baselineskip=16pt {\Large\bf On the Almost-zero-energy
Eigenvalues of \\Some Broken Supersymmetric Systems }
%
\vskip 10mm {Min-Young Choi $^{1}$ and Choonkyu Lee$^{2}$}
%
\vskip 8mm {\small\it
School of Physics and Center for Theoretical Physics\\
Seoul National University, Seoul, 151-747, Korea \\}
\vskip 10mm
\par
{\bf ABSTRACT}
\begin{quote}
For a quantum mechanical system with broken supersymmetry, it is
demonstrated that reformulation of the problem as that of a
$(1+1)$-D Dirac equation allows an easy determination of the
ground state if the corresponding energy eigenvalue is
sufficiently small. A simple expression is derived for the
approximate ground state energy in an associated, well-separated,
asymmetric double-well-type potential. Our discussion is also
relevant for the analysis of the fermion bound state in the
kink-antikink scalar background.
\end{quote}
\end{center}

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\begin{quote}
{\small
\noindent
$^1$ E-mail: witt@phya.snu.ac.kr\\
$^2$ E-mail: cklee@phya.snu.ac.kr\\
 }
\end{quote}
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Supersymmetry(SUSY) and its breaking are fundamental issues in
theoretical particle physics. There have also been numerous
applications of SUSY to quantum-mechanical potential problems
\cite{Witt,Junk}, based on the observation that the spectrum of
the Hamiltonian
%
\be H_+ = -\frac{d^2}{dx^2} + V_+ (x)~,~~~~~V_+ (x)=W^2 (x) +
W'(x)~~\ee
%
($W(x)$ is the superpotential, and we set $\hbar = 2m=1$) is
related through SUSY to that of the partner Hamiltonian
%
\be H_- = -\frac{d^2}{dx^2} + V_- (x)~,~~~~~V_- (x)=W^2 (x) -
W'(x)~. \ee
%
This formalism has provided us with a number of exactly solvable
quantum mechanical systems for which energy eigenvalues and
eigenfunctions can be found in closed forms. The key properties
that made such feat possible are unbroken SUSY, manifested by the
vanishing energy for the ground state of $H_-$ (or $H_+$), and
shape invariance under the change of parameters for the given
potentials \cite{Gend}-\cite{CKS}. This approach can sometimes be
extended to parameter ranges corresponding to (spontaneously)
broken SUSY, and authors of Refs. \cite{Dutt2,Gang} have found
several additional exactly solvable systems by such consideration.
But, with SUSY broken, the ground state energy is no longer equal
to zero and this jeopardizes the possibility of obtaining exact
analytic results by the SUSY-based method in a crucial way.

In this work we will show that, in some broken SUSY case for which
the lowest energy $\bar E(>0)$ for the Hamiltonian $H_+$ or $H_-$
is sufficiently small, reformulating the system into that for a
Dirac Hamiltonian (defined on a line) makes possible an easy
evaluation of $\bar E$. The conspicuous role assumed by the Dirac
operator as regards the zero eigenmodes is well-known \cite{Schw},
and we here extend this role to the case of {\it
almost-zero-energy eigenmodes} in a restricted sense at least. Our
method finds useful application in studying the almost-zero-energy
fermion modes in the background of a soliton-antisoliton pair.

The superpotential relevant for our discussion is given as
follows. Let $\sigma_R (x)$ be a generic function with the
properties
%
\be \begin{split}  \sigma_R (x)>0~~~~~&,~~{\rm for}~x>0~{\rm
and}~|x|~{\rm not}~{\rm very}~{\rm small}~, \\  \sigma_R (x)
\rightarrow -v~~~~~&,~~{\rm for}~x<0~{\rm and}~|x|~{\rm not}~{\rm
very}~{\rm small} \end{split} \ee
%
and $\sigma_L (x)$ the one with the properties
%
\be \begin{split}  \sigma_L (x)>0~~~~~&,~~{\rm for}~x<0~{\rm
and}~|x|~{\rm not}~{\rm very}~{\rm small}~, \\  \sigma_L (x)
\rightarrow -v~~~~~&,~~{\rm for}~x>0~{\rm and}~|x|~{\rm not}~{\rm
very}~{\rm small} \end{split} \ee
%
so that the related potentials $V_{R\pm}(x)\equiv \sigma_R^2 \pm
\sigma_R' (x)$ and $V_{L\pm}(x)\equiv \sigma_L^2 \pm \sigma_L'
(x)$ may have the general shapes shown in Figs.1 and 2,
respectively. [Note that for any Hamiltonian having $V_{R-}(x)$ or
$V_{L+}(x)$ as its entire potential, we have a ground state of
strictly zero energy]. Then the superpotential appropriate to our
case is obtained by combining these two types of functions as
%
\be\label{sp} W(x)=\sigma_R (x-l_1)+\sigma_L (x-l_2) +v \ee
%
with $L\equiv|l_1 -l_2|$ taken to be reasonably large (so that
$W(x)$ may have a {\it flat} basin between the points $x=l_2$ and
$x=l_1$).\footnote{At the later stage of our discussion we will
use the fact if $l^*$ denotes a certain point in the flat basin,
the approximation $\sigma_R (x-l_1)+v \simeq 0$ for $x<l^*$ or
$\sigma_L(x-l_2)+v \simeq 0$ for $x>l^*$ is valid.} See Fig.3 for
the schematic plots of $W(x)$ and the related potentials $V_{\pm}
(x) \equiv W^2 (x) \pm W'(x)$. Both $W(\infty)$ and $W(-\infty)$
being positive, this corresponds to the case of broken SUSY
\cite{Witt,Junk}; but, for the posited superpotential (with $L$
large), the ground state energy $\bar E$ is expected to be rather
small. [Our superpotential will be an even function of $x$ if
$\sigma_L (x)$ happens to be the mirror image of $\sigma_R (x)$,
i.e., $\sigma_L (x)=\sigma_R (-x)$, and $(l_1 ,l_2)$ are equal to
$(\frac{L}{2},-\frac{L}{2})$].

For $W(x)$ specified as above, the corresponding Hamiltonians
$H_{\pm}$ involve the potentials which can approximately be
described by the sum of two well-separated potentials (aside from
a constant term), i.e.,
%
\be \label{v} V_{\pm}(x) \sim V_{R\pm}(x-l_1) + V_{L\pm}(x-l_2) -
v^2 ~.\ee
%
These correspond to asymmetric double wells (even when $W(x)$ is
an even function) and hence the instanton method for instance
would not be useful.\footnote{See also Ref. \cite{Bern} for
previous applications of the (unbroken) SUSY in treating certain
double-well-type potentials.} In principle, one can study the
energy eigenstates of this system by using perturbation theory
that takes full energy eigenstates of the two local Hamiltonian
(involving the potentials $V_{R\pm}$, $V_{L\pm}$ separately) as
inputs \cite{Seok}. But, beyond the level of the tight-binding
approximation which may be applied with respect to degenerate
states of the local Hamiltonians, it is not a simple matter to use
this kind of perturbation theory since one needs to perform a
complicated state sum in general. Moreover, as for the ground
state of our Hamiltonian with the potential $V_+ \sim V_{R+} +
V_{L+} -v^2$ (or $V_- \sim V_{R-} + V_{L-} -v^2$), the
tight-binding approximation is plainly not available --- while the
local Hamiltonian involving $V_{L+}$ (or $V_{R-}$) allows a
zero-energy bound state, no zero-energy state exists for the local
Hamiltonian involving $V_{R+}$ (or $V_{L-}$). But the situation
changes if the problem is reformulated using the first-order Dirac
operator, and for the system under consideration we obtain,
through the analysis of the latter, the remarkably simple formula
for the lowest eigenvalue $\bar E$. It is simply the {\it square}
of the product of the two zero-energy eigenfunctions (allowed with
the potentials $V_{L+} (x-l_2)$ and $V_{R-} (x-l_1)$
separately)\footnote{This may be contrasted with the case of a
symmetric double well for which the ground state energy splitting
can approximately be given by a formula involving a single factor
of the state (or wave function) overlap term \cite{landau}. For an
asymmetric double well, however, having {\it two} factors of the
state overlap term is natural \cite{Seok}.}, evaluated at an
arbitrary chosen point $l^*$ in the flat middle region of the
superpotential $W(x)$. See the expression (\ref{18}) below.

The $(1+1)$-D Dirac equation in an external scalar field $W(x)$
reads
%
\be \{i\gamma^0 \partial_t + i \gamma^1 \partial_x -W(x)\}
\psi(x,t)=0~~,\ee
%
where the $\gamma$-matrices may be chosen as $\gamma^0 =\sigma^1
=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ and $\gamma^1 =
i\sigma^3 = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}$. Here
we have a first order Hamiltonian
%
\be \label{hd}H_D \equiv -\gamma^0 \gamma^1 \partial_x + \gamma^0
W(x) =\begin{pmatrix} 0 & A^\dag \\ A & 0 \end{pmatrix} \ee
%
with
%
\be A= \partial_x + W(x)~~~,~~~A^\dag = -\partial_x +W(x)~,\ee
%
and the corresponding eigenvalue equation, $H_D \Psi (x)=\omega
\Psi (x)$ with $\Psi (x)=\begin{pmatrix} \Psi_1 (x) \\ \Psi_2 (x)
\end{pmatrix}$, reduces to a pair of differential equations
%
\begin{subeqnarray} \slabel{d1} A~\Psi_1 (x)= \omega \Psi_2 (x)~, \\
\slabel{d2} A^\dag ~\Psi_2 (x)= \omega \Psi_1 (x)~.
\end{subeqnarray}
%
An immediate consequence of these is that the functions
$\Psi_{1,2} (x)$ here are eigenfunctions (with energy
$E=\omega^2$) of the Schr\"odinger Hamiltonians we started with
\cite{Coop}, i.e.,
%
\begin{subeqnarray} H_- \Psi_1 (x)= \omega^2 \Psi_1 (x)~, \\
H_+ \Psi_2 (x)= \omega^2 \Psi_2 (x)~,\end{subeqnarray}
%
since
%
\be A^\dag A = -\partial^2_x + W^2 (x) -W'(x)\equiv H_- ~~~,~~~
AA^\dag = -\partial^2_x + W^2 (x) +W'(x)\equiv H_+ ~.\ee

The above observation may be used to establish the precise
connection between the bound-state eigenfunctions of $H_\pm$ and
those of the Dirac Hamiltonian $H_D$. Here it is not difficult to
show that
\begin{itemize}
\item[(i)] if $\Psi (x) = \begin{pmatrix} \Psi_1 (x) \\ \Psi_2 (x)
\end{pmatrix}$ is an eigenspinor of $H_D$ with eigenvalue
$\omega$, $\tilde\Psi (x) = \begin{pmatrix} \Psi_1 (x) \\ -\Psi_2
(x) \end{pmatrix}$ corresponds to an eigenspinor of $H_D$ with
eigenvalue $-\omega$, and
%
\item[(ii)] if $\Psi_1 (x) = \varphi^{(\omega^2)} (x)$ represents
a real, normalized, eigenfunction of $H_-$ with energy $\omega^2
(\neq 0)$, we have in $\Psi_2 (x)=\frac{1}{\omega} A
\varphi^{(\omega^2)} (x)$ a real, normalized, eigenfunction of
$H_+$ with the same energy.
\end{itemize}
%
Also, bound states of a one-dimensional Schr\"odinger Hamiltonian
are always nondegenerate. Hence, if $\varphi_1(x)$
($\varphi_2(x)$) is a real, normalized, eigenfunction of $H_-$
($H_+$) with energy $E=\omega^2(>0)$, we conclude that either the
relation $\varphi_2(x)=\frac{1}{\omega}A\varphi_1(x)$ or
$\varphi_2(x)=-\frac{1}{\omega}A\varphi_1(x)$ holds and the
normalized (real) eigenspinor of $H_D$ with eigenvalue $+\omega$
or $-\omega$ can be identified with
%
\be\label{13} \varphi(x)=\frac{1}{\sqrt2} \begin{pmatrix}
\varphi_1 (x)
\\ \varphi_2 (x) \end{pmatrix}~~~~~{\rm or}~~~~~\frac{1}{\sqrt2}
\begin{pmatrix} \varphi_1 (x) \\ -\varphi_2 (x) \end{pmatrix}~.\ee
%
Moreover, if $W(x)$ happens to be an even function (and hence $V_+
(x)=V_-(-x)$), one can immediately identify $\varphi_2(x)$ with
$\pm \varphi_1(-x)$; in this case, the eigenfunctions of $H_-$ may
be found by solving the single equation
%
\be A\varphi_1(x)= \pm\omega\varphi_1(-x)~.\ee
%

We will now turn to the case with the superpotential (scalar
field) given by the form (\ref{sp}) with $L$ large. Let
$\varphi_1(x)$ denote the ground state of the Hamiltonian $H_-$
with a small energy $\bar E\equiv \bar\omega^2$, and $\varphi_2
(x)$ that of its isospectral partner $H_+$ (with the same energy
eigenvalue). With the Dirac equation in Eqs. (\ref{d1}) and
(\ref{d2}), the very fact that $|\bar \omega|$ becomes tiny as
$L\rightarrow \infty$ makes the related eigenspinor be amenable to
a simple perturbative analysis, regarding the terms on the right
(i.e., the pieces proportional to $\omega$) as small
perturbation.\footnote{But this is of different nature from the
usual Rayleigh-Schr\"odinger perturbation theory in that we have
no small perturbing potential --- what is small in our case is the
energy eigenvalue itself.} This almost-zero-energy eigenmode of
the Dirac Hamiltonian with the eigenvalue $\pm \bar\omega$ is
still given by the form (\ref{13}). In particular, the
corresponding zeroth-order expression (i.e., the expression for
infinite $L$), $\varphi^0_\pm (x) = \frac{1}{\sqrt 2}
\begin{pmatrix} \varphi^0_1 (x) \\ \pm \varphi^0_2 (x)
\end{pmatrix}$, can be written down using the appropriate wave
functions of the relevant Schr\"odinger Hamiltonian: in view of
Eq.$~$(\ref{v}) and considering the general shapes of the
potentials $V_\pm$ shown in Fig.3, $\varphi^0_1(x)$
($\varphi^0_2(x)$) can be taken as the normalized zero-energy
eigenfunction of $-\partial_x^2 + V_{R-}(x-l_1)$ ($-\partial_x^2 +
V_{L+}(x-l_2)$). According to this, $\varphi^0_1(x)$ and
$\varphi^0_2(x)$ will have to satisfy the first order differential
equations
%
\be\label{15} [\partial_x +
\sigma_R(x-l_1)]\varphi^0_1(x)=0~~~~~,~~~~~ [-\partial_x +
\sigma_L(x-l_2)]\varphi^0_2(x)=0~,\ee
%
and hence acquire the explicit expressions
%
\be \label{16} \varphi^0_1(x)= C_1 e^{-\int^x_{l_1} \sigma_R (y-
l_1)dy}~~~~~,~~~~~\varphi^0_2(x)= C_2 e^{\int^x_{l_2} \sigma_L (y-
l_2)dy}~\ee
%
with appropriate (positive) normalization constants $C_1$ and
$C_2$.

In what follows, we will derive the following simple formula for
the very small eigenvalue of the Dirac Hamiltonian (in the leading
approximation):
%
\be \label{w}\begin{split} \bar\omega &=
\varphi^0_1(l^*)\varphi^0_2(l^*)
\\ &= C_1 C_2 e^{-\int^{l^*}_{l_1} \sigma_R (y-l_1)dy +
\int^{l^*}_{l_2} \sigma_L (y-l_2)dy}~,\end{split} \ee
%
where $l^*$ can be any point in the flat middle region of the
superpotential (see Fig.3). [Note that we do not have to be more
specific on the value of $l^*$, the expression in Eq. (\ref{w})
being independent of such choice]. Then, from $\bar E=
\bar\omega^2$, the lowest eigenvalue of the Schr\"odinger
Hamiltonian $H_\pm$ may immediately be identified with
%
\be\label{18}\bar E=[\varphi^0_1(l^*)\varphi^0_2(l^*)]^2 ~.\ee
%
As one recalls that the potential in our case has the structure
given in Eq.$~$(\ref{v}), this formula has a somewhat surprising
appearance: while one of the wave function entering the formula
(\ref{18}) is that of the zero-energy state allowed in one of the
local potential actually present, the other corresponds to the
zero-energy state allowed in the {\it superpartner} of the
remaining local potential (or, one might say, the zero-energy
state allowed in the `ghost' potential related to the actual local
potential by supersymmetry). The approximate ground state wave
functions will also be found below.

Note that the functions $\varphi_1(x)$ and $\varphi_2(x)$, as
solutions of Eqs. (\ref{d1}) and (\ref{d2}) with $\omega=\bar
\omega$, may be written as
%
\begin{subeqnarray} \slabel{19a}\varphi_1 (x) = e^{-\int^{x}_{l_1} W(y)dy}
\left\{D_1 + \bar\omega \int^{x}_{\alpha_1} dy
\left[e^{\int^y_{l_1} W(z)dz} \varphi_2(y)\right] \right\} ~, \\
\slabel{19b} \varphi_2(x)=e^{\int^{x}_{l_2} W(y)dy} \left\{D_2 -
\bar\omega \int^{x}_{\alpha_2} dy \left[e^{-\int^y_{l_2} W(z)dz}
\varphi_1(y)\right]\right\} ~,
\end{subeqnarray}
%
where $\alpha_1$, $\alpha_2$, $D_1$ and $D_2$ are suitable
constants. Our wave functions $\varphi_1(x)$ and $\varphi_2(x)$
should reduce for large $L$ to $\varphi^0_1(x)$ and
$\varphi^0_2(x)$, respectively, and then, for large but finite $L$
(and hence small $\bar\omega$), we may use Eqs. (\ref{19a}) and
(\ref{19b}) to express them by
%
\begin{subeqnarray} \slabel{20a}\varphi_1 (x) \simeq e^{-\int^{x}_{l_1} W(y)dy}
\left\{C_1 + \bar\omega \int^{x}_{l^*} dy
\left[e^{\int^y_{l_1} W(z)dz} \varphi_2^0(y)\right] \right\} ~, \\
\slabel{20b} \varphi_2(x) \simeq e^{\int^{x}_{l_2} W(y)dy}
\left\{C_2 - \bar\omega \int^{x}_{l^*} dy \left[e^{-\int^y_{l_2}
W(z)dz} \varphi_1^0(y)\right]\right\}~~
\end{subeqnarray}
%
with the value $l^*$ chosen conveniently at some point in the flat
middle region of the superpotential so that both $|l_1 - l^*|$ and
$|l^* - l_2|$ may become ${\cal O}(L)$. But the expression
(\ref{20a}) as $x\rightarrow -\infty$ (and similarly that in Eq.
(\ref{20b}) as $x\rightarrow \infty$) would blow up unless the
value of $\bar\omega$ were chosen such that
%
\be \label{eig}C_1 + \bar\omega \int^{-\infty}_{l^*}
dy\left[e^{\int_{l_1}^y W(z)dz} \varphi_2^0 (y) \right] = 0 ~.\ee
%
Then, with the explicit expression for $W(x)$ and $\varphi^0_2(x)$
given in Eqs. (\ref{sp}) and (\ref{16}), we observe that
%
\be \begin{split} e^{\int^y_{l_1} W(z)dz} & \simeq
e^{\int^{l^*}_{l_1} \sigma_R (z-l_1)dz} e^{\int^{l_2}_{l^*}
\sigma_L (z-l_2)dz} e^{\int^{y}_{l_2} \sigma_L (z-l_2)dz} \\
&= C_1 [\varphi^0_1(l^*)\varphi^0_2(l^*)]^{-1}
\varphi^0_2(y)~~,~~~({\rm for}~y\in [-\infty,l^*])~,\end{split}
\ee
%
\be \int^{-\infty}_{l^*}dy[\varphi^0_2(y)]^2 \simeq
-\int^{\infty}_{-\infty}dy[\varphi^0_2(y)]^2
=-1~,~~~~~~~~~~~~~~~~~~~~~\ee
%
and hence the `eigenvalue condition' (\ref{eig}) reduces to the
form
%
\be C_1 -\bar\omega C_1 [\varphi^0_1(l^*)\varphi^0_2(l^*)]^{-1}
=0~.\ee
%
This is our equation (\ref{w}), and the same follows from the
consideration based on Eq. (\ref{20b}). With the eigenvalue
$\bar\omega$ determined in this manner, the corresponding wave
functions $\varphi_1 (x)$ and $\varphi_2 (x)$ (up to
normalization) are now expressed as
%
\be\label{25} \varphi_1(x)\simeq \begin{cases}
\varphi_1^0(x)\left\{ 1+ \bar\omega \int^{x}_{l^*}dy
[\varphi^0_1(y)]^{-1}\varphi^0_2(y)\right\}~,& x>l^* \\
\bar\omega [\varphi^0_2(x)]^{-1}
\int^{x}_{-\infty}dy[\varphi^0_2(y)]^2~,& x<l^* ~~\end{cases} \ee
%
\be \label{26}\varphi_2(x) \simeq \begin{cases}
\bar\omega[\varphi^0_1(x)]^{-1}\int^{\infty}_{x}dy[\varphi^0_1(x)]^2~,&x>l^*\\
\varphi^0_2(x)\left\{1+\bar\omega\int^{l^*}_x
dy[\varphi^0_2(y)]^{-1} \varphi^0_1(y)
\right\}~,&x<l^*~.\end{cases}
\ee
%

We expect that a judicious use of the tight-binding approximation
with the {\it Dirac Hamiltonian} (\ref{hd}), taking
$\begin{pmatrix}\varphi^0_1(x) \\ 0 \end{pmatrix}$ and
$\begin{pmatrix} 0 \\ \varphi^0_2(x) \end{pmatrix}$ as the
degenerate (i.e., zero energy) eigenstates of the corresponding
local Hamiltonians, lead to the same conclusion as
above.\footnote{But, since the Dirac Hamiltonian is unbounded from
below, some care must be exercised.} This is supported by the
observation that, for the eigenvalue $\bar\omega$, the same result
(i.e., Eq. (\ref{w})) follows from the calculation based on the
formula
%
\be \label{27}\pm\bar\omega = \int^{\infty}_{-\infty}dx
\varphi^0_\pm (x) H_D \varphi^0_\pm (x) = \pm
\int^{\infty}_{-\infty}dx [\sigma_R(x-l_1) +v]\varphi_1^0(x)
\varphi^0_2(x)~,\ee
%
where $\varphi^0_\pm
(x)=\frac{1}{\sqrt2}\begin{pmatrix}\varphi^0_1(x) \\
\pm \varphi_2^0(x)\end{pmatrix}$ is the zeroth-order state
mentioned above. Indeed,
%
\be\begin{split}\int^{\infty}_{-\infty}dx [\sigma_R(x-l_1)
+v]\varphi_1^0(x) \varphi^0_2(x) &\simeq \int^{\infty}_{l^*}dx
[\sigma_R(x-l_1) +v]\varphi_1^0(x) \varphi^0_2(x) \\ &\simeq
\varphi^0_2(l^*)\int^{\infty}_{l^*}dx [\sigma_R(x-l_1)
+v]\varphi_1^0(x)e^{-v(x-l^*)}\\&=-\varphi^0_2(l^*)\int^{\infty}_{l^*}dx
\frac{d}{dx}\left[\varphi^0_1(x)e^{-v(x-l^*)}\right]\\&=\varphi^0_1(l^*)
\varphi^0_2(l^*)~, \end{split}\ee
%
since, for $x>l^*$, $\varphi^0_2(x)$ may well be replaced by
$\varphi^0_2(l^*)e^{-v(x-l^*)}$.

As an explicit example, consider the superpotential of the form
%
\be W(x)=v ~{\rm sgn}(x-\frac{L}{2}) - v ~{\rm sgn}(x+\frac{L}{2})
+v~,\ee
%
i.e., in our notation, $(l_l,l_2)=(\frac{L}{2},-\frac{L}{2})$,
$\sigma_R(x-l_1)=v ~{\rm sgn}(x-\frac{L}{2})$, and
$\sigma_L(x-l_2)=-v ~{\rm sgn}(x+\frac{L}{2})$. Given this, the
potentials of the Schr\"odinger Hamiltonians $H_\pm$ will be (see
Fig.4)
%
\be V_\pm(x)=v^2 \pm 2v~\delta (x-\frac{L}{2}) \mp 2v ~\delta
(x+\frac{L}{2})~.\ee
%
For these systems, one can of course find the exact ground state
energy $\bar E$ by solving the appropriate Schr\"odinger
equations. This exercise shows that $\bar E$ is the root of the
equation $\bar E= v^2 e^{-2L{\sqrt{v^2-\bar E}}}$, and hence, for
large $L$, we have
%
\be \label{31}\bar E \simeq v^2 e^{-2vL}~.\ee
%
Let us see whether our formula (\ref{18}) yields the same. The
normalized solutions of Eq. (\ref{15}) are trivially found here:
%
\be \varphi^0_1(x)={\sqrt v}~e^{-v|x-\frac{L}{2}|}~~~~~,~~~~~
\varphi^0_2(x)={\sqrt v}~e^{-v|x+\frac{L}{2}|}~.\ee
%
Then, from Eq. (\ref{18}), we have that $\bar E$ (for large $L$)
should equal $[\varphi^0_1(0)\varphi^0_2(0)]^2 = v^2 e^{-2vL}$.
Hence a complete agreement.

More physically relevant example is provided by the scalar field
(superpotential) appropriate to the kink-antikink pair,
%
\be \label{33}W(x)= v\tanh\left[\frac{\mu}{2}
(x-\frac{L}{2})\right] - v\tanh\left[\frac{\mu}{2}
(x+\frac{L}{2})\right] +v~.\ee
%
Here the scalar field $\sigma_R(x-\frac{L}{2}) = v\tanh
\left[\frac{\mu}{2} (x-\frac{L}{2})\right]$ represents a kink
located at $x=\frac{L}{2}$, and $\sigma_L(x+\frac{L}{2}) = -v\tanh
\left[\frac{\mu}{2} (x+\frac{L}{2})\right]$ an antikink at
$x=-\frac{L}{2}$ \cite{Soli}. The widely-separated kink-antikink
configuration, described by the form (\ref{33}), has received
attention in Refs. \cite{Jack,Schon,Alt}. Especially interesting
is the almost zero-energy mode of the Dirac Hamiltonian
(\ref{hd}), in connection with the role of the so-called
Jackiw-Rebbi mode \cite{Jack} (which refers to the zero-energy
fermion mode \cite{Choon} in the kink or antikink background) when
a kink and an antikink are simultaneously present. In the kink or
antikink background the Jackiw-Rebbi mode is represented by
$\begin{pmatrix}\varphi^0_1(x) \\ 0
\end{pmatrix}$ or $\begin{pmatrix} 0 \\ \varphi^0_2(x)
\end{pmatrix}$, if $\varphi^0_1(x)$ and $\varphi^0_2(x)$ denote
the normalized solutions of Eq. (\ref{15}):
%
\be
\varphi^0_1(x)=C\left[\cosh\frac{\mu}{2}(x-\frac{L}{2})\right]^{-\frac{2v}{\mu}}
~~~~~,~~~~~
\varphi^0_2(x)=C\left[\cosh\frac{\mu}{2}(x+\frac{L}{2})\right]^{-\frac{2v}{\mu}}
~,\ee
%
where $C=\left\{
\frac{\mu}{2}\frac{\Gamma(\frac{2v}{\mu}+\frac{1}{2})}
{\Gamma(\frac{2v}{\mu})\Gamma(\frac{1}{2})}\right\}^{\frac{1}{2}}$.
Then, in the above kink-antikink scalar background, one can
immediately find the energy of the almost-zero-energy fermion
eigenmode by using our formula (\ref{w}) --- it is equal to $\pm
\bar\omega$, with\footnote{Based on some heuristic arguments the
author of Ref.\cite{Alt} also identified $\bar\omega$ with the
expectation value in Eq. (\ref{27}). But his final expression for
$\bar\omega$ is apparently not consistent with our result in Eq.
(\ref{35})---this is due to the calculational error made in
Ref.\cite{Alt} (e.g., wrong fermion zero-energy wave functions
used).}
%
\be \label{35}\begin{split} \bar\omega &=
\varphi^0_1(0)\varphi^0_2(0)=
C^2[\cosh\frac{\mu L}{4}]^{-\frac{4v}{\mu}} \\
&\simeq \frac{\mu}{2}e^{\frac{4v}{\mu}}
\frac{\Gamma(\frac{2v}{\mu}+\frac{1}{2})}
{\Gamma(\frac{2v}{\mu})\Gamma(\frac{1}{2})} e^{-vL}~.
\end{split}\ee
%
[Note that, in the limit $\mu\rightarrow\infty$, this yields the
value $\bar\omega\simeq v~e^{-vL}$, which is equal to the square
root of the result in Eq. (\ref{31})]. The exponential dependence
of $\bar\omega$ on the distance $L$ was previously noted in
Ref.\cite{Jack}. This $L$-dependent fermion energy shift will
contribute to the effective potential between the kink and the
antikink. For instance, in the vacuum sector where all negative
energy fermion modes are to be occupied, the contribution from
this mode, i.e., that with energy $-\bar\omega$ will become more
negative as $L$ decreases, thus producing an attractive
interaction between the kink and the antikink (if only this mode
is taken into account). The related fermion eigenfunctions with
energy $\pm\bar\omega$ can also be obtained with the help of the
formulas (\ref{13}), (\ref{25}) and (\ref{26}).

In this work we investigated on some special properties pertaining
to the ground state of a quantum mechanical Hamiltonian with
broken supersymmetry, when the corresponding eigenvalue is small.
We showed that the reformulation of the system using a
one-dimensional Dirac Hamiltonian greatly facilitates the
analysis, providing a very simple (approximate) expression for the
ground state energy. Our formula (\ref{18}) should be useful in
finding the ground state energy of a Schr\"odinger Hamiltonian the
potential in which can be approximated by the form (\ref{v}).
Actually, for the validity of this formula, the separation
distance between the local potentials may not have to be very
large, for we believe that a good measure on the validity is
really the smallness of the state overlap (i.e., of the term
$\varphi^0_1(l^*)\varphi^0_2(l^*)$ in Eq. (\ref{18})). It would be
highly desirable if an analogous approach can be developed for
systems defined in more than one spatial dimensions (for instance
to study the role of Jackiw-Rebbi-type fermion modes in the
background of more general soliton-antisoliton pair configurations
\cite{Jack,Jack2}). This is left for future study.

\section*{Acknowledgment}
We would like to thank Seok Kim for very useful discussions. This
work was supported in part by the BK21 project of the Ministry of
Education, Korea, and also by Korea Research Foundation Grant
2001-015-DP0085(C.L.).

\begin{thebibliography}{99}
\bibitem{Witt} E.~Witten, Nucl.\ Phys.\ {\bf B188}, 513 (1981);
F.~Cooper and B.~Friedman, Ann.\ Phys.\ (NY) {\bf 146}, 262
(1983).
\bibitem{Junk} G.~Junker, Supersymmetric Methods in Quantum
and Statistical Physics,\ Springer, Berlin, 1996.
\bibitem{Gend} L.~Gendenshtein, JETP Lett. {\bf 38}, 356 (1983);
L.~Gendenshtein and I.~V.~Krive, Sov.\ Phys.\ Usp.\ {\bf 28}, 645
(1985).
\bibitem{Dutt} R.~Dutt, A.~Khare, and U.~Sukhatme, Phys.\ Lett.\ {\bf
B181}, 295 (1986); R.~Dutt, A.~Khare, and U.~Sukhatme, Am.\ J.\
Phys.\ {\bf 56}, 163 (1988).
\bibitem{Suku} C.~V.~Sukumar, J.\ Phys.\ {\bf A18}, 2917 (1985).
\bibitem{CKS} F.~Cooper, A.~Khare, and U.~Sukhatme, Phys.\ Rep.\ {\bf
251}, 268 (1995).
\bibitem{Dutt2} R.~Dutt, A.~Gangopadhyaya, A.~Khare,
A.~Pagnamenta, and U.~Sukhatme, Phys.\ Lett.\ {\bf A174}, 364
(1993).
\bibitem{Gang} A.~Gangopadhyaya, J.~V.~Mallow, and U.~P.~Sukhatme,
Phys.\ Lett.\ {\bf A283}, 279 (2001).
\bibitem{Schw} A.~Schwarz, Phys.\ Lett.\ {\bf B67}, 172 (1977);
L.~S.~Brown, R.~D.~Carlitz, and C.~Lee, Phys.\ Rev.\ {\bf D16},
417 (1977); J.~Kiskis, Phys.\ Rev.\ {\bf D15}, 2329 (1977);
C.~Callias, Comm.\ Math.\ Phys.\ {\bf 62}, 213 (1978); A.~J.~Niemi
and G.~W.~Semenoff, Phys.\ Rep.\ {\bf 135}, 99 (1986).
\bibitem{Bern} M.~Bernstein and L.~S.~Brown, Phys.\ Rev.\ Lett.\
{\bf 52}, 1933 (1984); P.~Kumar, M.~Ruiz-Altaba, and B.~S.~Thomas,
Phys.\ Rev.\ Lett.\ {\bf 57}, 2749 (1986); W.~-Y.~Keung,
E.~Kovacs, and U.~Sukhatme, Phys.\ Rev.\ Lett.\ {\bf 60}, 41
(1988).
\bibitem{Seok} S.~Kim and C.~Lee, quant-ph/0209046.
\bibitem{landau} L.~D.~Landau and E.~M.~Lifshitz, Quantum
Mechanics, 3rd edition, Pergamon, New York, 1977, Chap.VII,
Sec.50, Problem 3.
\bibitem{Coop} F.~Cooper, A.~Khare, R.~Musto, and A.~Wipf, Ann.\ Phys.\ {\bf
187}, 1 (1988).
\bibitem{Soli} R.~Rajaraman, Solitons and Instantons,
North-Holland, 1982.
\bibitem{Jack} R.~Jackiw and C.~Rebbi, Phys.\ Rev.\ {\bf D13}, 3398 (1976).
\bibitem{Schon} J.~F.~Schonfeld, Nucl.\ Phys.\ {\bf B161}, 125 (1979);
A.~S.~Goldhaber, A.~Litvintsev, and P.~van Nieuwenhuizen, Phys.\
Rev.\ {\bf D64}, 045013 (2001).
\bibitem{Alt} B.~Altschul, hep-th/0111042.
\bibitem{Choon} C.~Lee, unpublished;
R. Dashen, B.~Hasslacher, and A.~Neveu, Phys.\ Rev.\ {\bf D10},
4130 (1974); W.~Bardeen, M.~Chanowitz, S.~Drell, M.~Weinsrein, and
T.~-M.~Yan, Phys.\ Rev.\ {\bf D11}, 1094 (1975).
\bibitem{Jack2} R.~Jackiw and  P.~Rossi, Nucl.\ Phys.\ {\bf B190},
681 (1981); E.~J.~Weinberg, Phys.\ Rev.\ {\bf D24}, 2699 (1981).
\end{thebibliography}
\newpage
\begin{figure} %Fig.1
\centering\psfrag{sr}{$\sigma_R$}\psfrag{v2}{$v^2$}\psfrag{-v}{$-v$}
\psfrag{x}{$x$} \psfrag{V+}{$V_{R+}(x)$} \psfrag{V-}{$V_{R-}(x)$}
\psfrag{VR}{$V_R$}
\includegraphics[clip=,width=.8\textwidth]{fig1.eps}
\caption{\footnotesize{Schematic plots of $\sigma_R (x)$ and
$V_{R\pm}(x) \equiv \sigma_R^2 (x) \pm \sigma_R' (x)$}}
\end{figure}
%
\begin{figure} %Fig.2
\centering\psfrag{sl}{$\sigma_L$}\psfrag{v2}{$v^2$}\psfrag{-v}{$-v$}
\psfrag{x}{$x$} \psfrag{V+}{$V_{L+}(x)$} \psfrag{V-}{$V_{L-}(x)$}
\psfrag{VL}{$V_L$}
\includegraphics[clip=,width=.8\textwidth]{fig2.eps}
\caption{\footnotesize{Schematic plots of $\sigma_L (x)$ and
$V_{L\pm}(x) \equiv \sigma_L^2 (x) \pm \sigma_L' (x)$}}
\end{figure}
%

\newpage
\begin{figure} %Fig.3
\centering\psfrag{W}{$W$}\psfrag{v2}{$v^2$}
\psfrag{l1}{$l_1$}\psfrag{l2}{$l_2$}\psfrag{ls}{$l^*$}
\psfrag{x}{$x$} \psfrag{V+}{$V_{+}$} \psfrag{V-}{$V_{-}$}
\includegraphics[clip=,width=1.0\textwidth]{fig3.eps}
\caption{\footnotesize{Schematic plots of $W(x)$ and $V_\pm (x)
\equiv W^2 (x) \pm W'(x)$}}
\end{figure}
%
\begin{figure} %Fig.4
\centering\psfrag{V}{$V$}\psfrag{v2}{$v^2$}\psfrag{V+}{$V_+$}
\psfrag{V-}{$V_-$}\psfrag{x}{$x$}\psfrag{-L/2}{$-L/2$}\psfrag{L/2}{$L/2$}
\includegraphics[clip=,width=.8\textwidth]{fig4.eps}
\caption{\footnotesize{Potentials related to the form $W(x)=v~{\rm
sgn} (x-\frac{L}{2}) - v~{\rm sgn}(x+\frac{L}{2}) +v$}}
\end{figure}




\end{document}

