%Paper: hep-th/9402110
%From: Grosche Christian <I02GRO@DSYIBM.DESY.DE>
%Date: FRI, 18 FEB 94 13:54:37 +0100

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\edef\BADUb{\the\Refno}\add
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{\nopagenumbers
\pageno=0
\centerline{DESY 94 - 019 \hfill ISSN 0418 - 9833}
\centerline{February 1994\hfill}
%\centerline{\hfill hep-th/9402110}
\vskip1cm
\centerline{\fourteenpoint $\delta'$-Function Perturbations and
                           Neumann}
\bigskip
\centerline{\fourteenpoint Boundary-Conditions by Path Integration}
\bigskip
\bigskip
\centerline{\twelverm CHRISTIAN GROSCHE$^*$}
\bigskip
\centerline{\it II.\ Institut f\"ur Theoretische Physik}
\centerline{\it Universit\"at Hamburg, Luruper Chaussee 149}
\centerline{\it 22761 Hamburg, Germany}
\vfill
\midinsert
\narrower
\noindent
{\bf Abstract.}
$\delta'$-function perturbations and Neumann boundary conditions are
incorporated into the path integral formalism. The starting point is
the consideration of the path integral representation for the one
dimensional Dirac particle together with a relativistic point
interaction. The non-relativistic limit yields either a usual
$\delta$-function or a $\delta'$-function perturbation; making their
strengths infinitely repulsive one obtains Dirichlet, respectively
Neumann boundary conditions in the path integral.
\endinsert

\bigskip\noindent
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\noindent
{\eightpoint\eightrm
 $^*$ Supported by Deutsche Forschungsgemeinschaft under contract
 number GR 1031/2--1.}
\eject}
\pageno=1

%----------------------------------------------------------------------
%                          END OF FILE0
%----------------------------------------------------------------------

\glno=0                      %I
Attempts to incorporate Dirichlet and Neumann boundary conditions into
the path integral formalism are e.g.~due to Barut and Duru$^{\BADUb}$,
Clark et al.$^{\CMS}$ and Carreau et al.$^{\CFG}$. Barut and Duru used a
canonical transformation to Hamilton-Jacobi coordinates in a phase space
path integral to perform the path integration as explicitly as possible
yielding an integral representation of the Feynman kernel; they also
could discuss step potentials within their formalism. In Refs.~[\CMS,
\CFG] general boundary conditions were addressed, but only for the free
particle case.

In a previous paper I have discussed how to implement Dirichlet
boundary conditions into the path integral$^{\GROx}$. This was achieved
by considering a one dimensional $\delta$-function perturbation in the
path integral. This problem can be solved in a straightforward manner by
means of a perturbation expansion$^{\FH,\GBD}$ which can be explicitly
summed yielding the corresponding (energy-dependent) Green function
$G^{(\delta)}(E)$ in terms of the unperturbed one $G^{(V)}(E)$ where $V$
refers to an arbitrary potential which can be included$^\GROh$. Making
the strength of the $\delta$-function infinitely repulsive yields
Dirichlet boundary conditions at the location of the  $\delta$-function
perturbation$^{\AGHH, \GROw}$. It is desireable to have also an
analogous representation for a $\delta'$-function perturbation. Making
in this case the strength of the coupling infinitely repulsive, produces
Neumann boundary conditions at the location of the $\delta'$-function.
However, things turn out to be awkward if one tries to consider by a
similar reasoning as for the usual $\delta$-, a $\delta'$-function
perturbation in the path integral. An expansion into a perturbation
expansion yields interrelated complicated terms with no obvious
resolution of the summation problem. Alternatively, an approximation of
the $\delta'$-function in terms of two usual $\delta$-functions with
distance $\epsilon$ does not make sense in an obvious way. Having this
in mind and that the existing literature concerning $\delta'$-function
perturbations and Neumann boundary conditions in the path integral does
not look satisfactorially, something new is needed and one has to look
for an appropriate regularization procedure to fill the gap.

In this paper this problem is resolved by means of the path integral
representation of the one dimensional Dirac particle$^{\FH}$. The
incorporation of a point-interaction yields a two parameter family for
the corresponding self-adjoint extension$^{\AGHH}$: One can choose
either the up (or electron component) or the down (or positron)
component where the point-interaction is acting on. Considering a
perturbation expansion for both problems, it is found that they can be
explicitly summed in terms of the corresponding Green functions (which
are a $2\times2$ matrices). In the non-relativistic limit the former
case yields the usual $\delta$-function perturbation, whereas in the
latter we obtain the equivalent of a $\delta'$-function perturbation
in the path integral. We will concentrate on this case.

In the following I will outline how to implement point-interactions
in the one dimensional Dirac particle path integral. We obtain in
the case of the $\delta'$-function perturbation automatically a correct
regularization prescription in terms of the unperturbed Green function
$G^{(V)}(E)$. The general method for the time-ordered perturbation
expansion is quite simple. We assume that we have a potential
$W(x)=V(x)+\widetilde V(x)$ in the path integral and we suppose that
$W$ is so complicated that a direct path integration is not possible.
However, the path integral $K^{(V)}$ corresponding to $V(x)$ is assumed
to be known. We expand the path integral containing $\widetilde V(x)$
in a perturbation expansion about $V(x)$ in the following way. The
initial kernel corresponding to $V$ propagates in $\Delta t$-time
unperturbed, then it interacts with $\widetilde V$, propagates again in
another $\Delta t$-time unperturbed, a.s.o, up to the final state. This
gives the series expansion$^{\FH,\GBD}$ ($x\in\bbbr$)
\plus
$$\myalign
  K&(x'',x';T)
  =K^{(V)}(x'',x';T)+\sum_{n=1}^\infty\bigg(-\ih\bigg)^n
  \left(\prod_{j=1}^n
  \int_{t'}^{t_{j+1}} dt_j\int_{-\infty}^\infty dx_j\right)
  \\   &\times K^{(V)}(x_1,x';t_1-t')
  \widetilde V(x_1)K^{(V)}(x_2,x_1;t_2-t_1)
  \times\dotsc
  \\   &\dots\times
  \widetilde V(x_{n-1})K^{(V)}(x_n,x_{n-1};t_n-t_{n-1})
  \widetilde V(x_n)K^{(V)}(x'',x_n;t''-t_n)\enspace.
  \tag\num\endalign$$
I have ordered time as $t'=t_0<t_1<t_2<\dots<t_{n+1}=t''$ and paid
attention to the fact that $K(t_j-t_{j-1})$ is different from zero only
if $t_j>t_{j-1}$. We consider the path integral representation for the
one dimensional Dirac equation$^{\FH, \JASCH, \ICHTA}$
($p_x=-\i\hbar\partial_x$)
\plus$$\myalign
  \vKV(x'',x';T)
  &=\bigg<x''\bigg\vert\exp\bigg[-\ih T\Big(c\sigma_xp_x+mc^2\sigma_z
           +\vec V(x)\Big)\bigg]\bigg\vert x'\bigg>
        \\   &
  =\int\limits_{x(t')=x'}^{x(t'')=x''}\CD\nu(t)
    \exp\Bigg(-\ih\int_{t'}^{t''}\vec V(x)dt\Bigg)\enspace.
  \tag\num\endalign$$
\edef\numf{\num}%
$\vec V$ may be a matrix-valued potential. The support property of the
measure $\CD\nu$ is defined in such a way that the motion it is
describing selects paths of $N$ steps each of length $c\epsilon$
($\epsilon=T/N$ in the lattice representation) that start at $x'$ in
the direction $\alpha$, and end at $x''$ in the direction $\beta$, where
 $\alpha$ and $\beta$ take the values ``right'' and ``left''. The path
integration then is a summation over all reversings of directions$^\FH$.
$\sigma_x,\sigma_y,\sigma_z$ are the Pauli matrices. We introduce the
Green function $\vGV(E)$ with its matrix representation
\plus$$
   \vGV(x'',x';E)=\pmatrix
    \GV_{11}(x'',x';E)  &\GV_{12}(x'',x';E)   \\
    \GV_{21}(x'',x';E)  &\GV_{22}(x'',x';E)\endpmatrix\enspace.
  \tag\num$$
\edef\numa{\num}%
We first consider a $\delta$-function perturbation in the electron
($=\hbox{``$+$''}$-) component, i.e.\ $\tilde{\vec V}=-\alpha\pmatrix1&0
\\0&0\endpmatrix\delta(x-a)$. We obtain by inserting it into the path
integral and summing the perturbation expansion
\plus$$\myalign
  &{\vec G}^{(\delta_+)}(x'',x';E)
  =\vGV(x'',x';E)+{1\over1/\alpha-G^{(V)}_{11}(a,a;E)}
        \\   &\qquad\times
   \pmatrix
   G^{(V)}_{11}(a,x';E)G^{(V)}_{11}(x'',a;E)
  &G^{(V)}_{11}(a,x';E)G^{(V)}_{12}(x'',a;E)  \\
   G^{(V)}_{21}(a,x';E)G^{(V)}_{11}(x'',a;E)
  &G^{(V)}_{21}(a,x';E)G^{(V)}_{12}(x'',a;E)  \endpmatrix\enspace.
  \tag\num\endalign$$
Similarly for the positron ($=\hbox{``$-$''}$-) component, i.e.\
$\tilde{\vec V}=(4m^2\beta c^2/\hbar^2)\pmatrix0&0\\0&1\endpmatrix
\delta (x-a)$ (the constants have been chosen for convenience)
\plus$$\myalign
  &{\vec G}^{(\delta_-)}(x'',x';E)
  =\vGV(x'',x';E)-{1\over \hbar^2/4m^2c^2\beta+G^{(V)}_{22}(a,a;E)}
        \\   &\qquad\times
   \pmatrix
   G^{(V)}_{12}(a,x';E)G^{(V)}_{21}(x'',a;E)
  &G^{(V)}_{12}(a,x';E)G^{(V)}_{22}(x'',a;E)  \\
   G^{(V)}_{22}(a,x';E)G^{(V)}_{21}(x'',a;E)
  &G^{(V)}_{22}(a,x';E)G^{(V)}_{22}(x'',a;E)  \endpmatrix\enspace.
  \tag\num\endalign$$
\edef\numc{\num}%
We consider the unperturbed free particle; the explicit expression for
$\vGz(E)$ has the form$^{\AGHH}$:
\plus$$\vGz(x'',x';E)
  ={\i\over2c\hbar}\pmatrix
   \zeta          &\sign(x''-x')   \\
   \sign(x''-x')  &1/\zeta         \endpmatrix
   \e^{\i k\vert x''-x'\vert }\enspace,
  \tag\num$$
where $\zeta=(E+mc^2)/ck\hbar$, $ck\hbar=\sqrt{E^2-m^2c^4}$. This
yields for a $\delta$-function perturbation in the electron component:
\plus$$\myalign
  &{\vec G}^{(\delta_+)}(x'',x';E)
  ={\i\over2c\hbar}\pmatrix
   \zeta          &\sign(x''-x')   \\
   \sign(x''-x')  &1/\zeta  \endpmatrix \e^{\i k\vert x''-x'\vert }
        \\   &\qquad
  -{\alpha\e^{\i k(\vert x''-a\vert +\vert a-x'\vert )}\over
    4c\hbar(c\hbar-\i\alpha\zeta/2)}
   \pmatrix
   \zeta^2           &\zeta\sign(x''-a)       \\
   \zeta\sign(a-x')  &\sign(x''-a)\sign(a-x') \endpmatrix\enspace.
  \tag\num\endalign$$
For $[\alpha]>0$ there is one bound state with energy $E=mc^2(1-\lambda^
2)/(1+\lambda^2)$ ($\lambda=\alpha/2c\hbar$). Similarly for a
$\delta$-function perturbation in the positron component
\plus$$\myalign
  &{\vec G}^{(\delta_-)}(x'',x';E)
  ={\i\over2c\hbar}\pmatrix
   \zeta          &\sign(x''-x')   \\
   \sign(x''-x')  &1/\zeta  \endpmatrix \e^{\i k\vert x''-x'\vert }
        \\   &\qquad
  +{2m^2\beta\e^{\i k(\vert x''-a\vert +\vert a-x'\vert )}\over
    \hbar(2\hbar^3+4\i m^2c\beta/\zeta)}
   \pmatrix
   \sign(x''-a)\sign(a-x')  &\sign(a-x')/\zeta  \\
   \sign(x''-a)/\zeta       &1/\zeta^2          \endpmatrix\enspace.
  \tag\num\endalign$$
For $[\beta]>0$ there is one bound state with energy $E=-mc^2(1-\lambda^
2)/(1+\lambda^2)$ ($\lambda=2m^2c\beta/\hbar^3$). Let us assume for
simplicity that the component $\GV_{11}(E)$ in (\numa) is known and
$\vec V$ is a scalar, we then can derive
$$\myalign
  \GV_{12}(x,y;E)
  &={c\over mc^2+V+E}p_x \GV_{11}(x,y;E)\enspace,
  \tag\num\\   \global\plus
  \GV_{22}(x,y;E)
  &={-1\over mc^2+V+E}\Bigg(
    {c^2\over mc^2+V+E}p_xp_y\GV_{11}(x,y;E)+\delta(x-y)\Bigg)\enspace.
  \tag\num\endalign$$
\edef\numb{\num}%
{}From these representations it is easily seen that if $\GV_{11}(E)$
is of $O(1)$ for $c\to\infty$, $\GV_{12}(E)$ and $\GV_{22}(E)$
vanish according to $\propto1/c$ and $\propto1/c^2$ for $c\to\infty$,
respectively.

We consider the limit $c\to\infty$ in ${\vec G}^{(\delta_\pm)}(E)$. On
the one hand we know that the path integral representation (\numf)
gives the usual one dimensional path integral in non-relativistic
quantum mechanics$^{\FH}$. On the other we find that only the $(1,1)$
component in the Green functions remains finite, all others vanish.
Furthermore we have $G_{11}^{(\delta_+)}(E)\to G^{(\delta)}(E)$ and
$G_{11}^{(\delta_-)}(E)\to G^{(\delta')}(E)$, where $G^{(\delta)}(E)
$ is the Green function for a potential problem $V$ with a usual
$\delta$-function perturbation in non-relativistic quantum mechanics,
and $G^{(\delta')}(E)$ is the Green function for a potential problem
$V$ together with a $\delta'$-function perturbation, respectively.
Putting everything together we obtain for the latter an explicit path
integral representation yielding
\minus$$\align
  &G^{(\delta')}(x'',x';E)
         \\   &
  =\ih        \int_0^\infty dT\,\e^{\i ET/\hbar}
  \int\limits_{x(t')=x'}^{x(t'')=x''}\CD x(t)
  \exp\left\{\ih\int_{t'}^{t''}\bigg[{m\over2}\dot x^2
       -V(x)+\beta\delta'(x-a)\bigg]dt\right\}\qquad
  \tag\num\\   \global\plus
              &
  =G^{(V)}(x'',x';E)
   -\dsize\thickfrac{G^{(V)}_{,x'}(x'',a;E)G^{(V)}_{,x''}(a,x';E)}
   {\hGV_{,x'x''}(a, a;E)+1/\beta}\enspace,
  \tag\num\\   \global\plus
  &\hGV_{,xy}(a, a;E)
  =\Big[\partial_x\partial_y G^{(V)}(x,y;E)
      -2m\delta(x-y)/\hbar^2\Big]\bigg\vert_{x=y=a}\enspace.
  \tag\num\endalign$$
\minus\minus
\edef\numd{\num}\plus\plus%
The path integral (\numd) has been {\it derived\/} in a unique way
through the regularization (\numc) in the limit $c\to\infty$. Note that
(\numb) yields automatically the correct regularization of the formal
expression ``$G^{(V)}_{,xy}(a,a;E) $''. For $V\equiv0$, i.e.\ the free
particle, we obtain the explicit representation (together with an
inverse Laplace-Fourier transformation)
\plus$$\myalign
  &\int\limits_{x(t')=x'}^{x(t'')=x''}\CD x(t)
  \exp\Bigg\{\ih\int_{t'}^{t''}
  \bigg[{m\over2}\dot x^2+\beta\delta'(x-a)\bigg]dt\Bigg\}
         \\   &
  =\sqrt{m\over2\pi\i\hbar T}
  \exp\bigg({\i m\over2\hbar T}\vert x''-x'\vert ^2\bigg)
         \\   &\qquad
  +\sqrt{m\over2\pi\i\hbar T}
   \exp\bigg[{\i m\over2\hbar T}(\vert x''-a\vert+\vert x'-a\vert)^2
  \bigg]\sign(x''-a)\sign(x'-a)
         \\   &\qquad
  +{\hbar^2\over2m\beta}\exp\Bigg[-{\hbar^2\over m\beta}
        \Big(\vert x''-a\vert +\vert x'-a\vert \Big)
        +\ih{\hbar^6\over2m^3\beta^2}T\Bigg]
         \\   &\qquad\qquad\times
  \erfc\Bigg\{\sqrt{m\over2\i\hbar T}\,
       \bigg[\Big(\vert x''-a\vert +\vert x'-a\vert \Big)
       -{\i\hbar^3 T\over m^2\beta}\bigg]\Bigg\}
  \sign(x''-a)\sign(x'-a)\enspace.
  \tag\num\endalign$$
The one bound state has energy $E=-\hbar^6/2m^3\beta^2$.
Repeating the procedure for N-fold $\delta'$-function perturbations
gives similarly (compare also Ref.~[\AGHH], $a_i\not= a_j$ ($i\not=j$)
\goodbreak\noindent
\plus$$\myalign
  & \ih \int_0^\infty  dT\,\e^{\i TE/\hbar}
  \int\limits_{x(t')=x'}^{x(t'')=x''}\CD x(t)\exp\left\{\ih
   \int_{t'}^{t''}\left[{m\over2}\dot x^2-V(x)
   +\sum_{j=1}^N\beta_j\delta'(x-a_j)\right]dt\right\}
         \\   &
  =\dsize\thickfrac{\left\vert\matrix
  \GV(x'',x';E)  &\GV_{,x'}(x'',a_1;E) &\hdots
                                 &\GV_{,x'}(x'',a_N;E)       \\
  \GV_{,x''}(a_1,x';E)
                 &\hGV_{,x'x''}(a_1,a_1;E)+1/\beta_1
                 &\hdots         &\GV_{,x'x''}(a_1,a_N;E)    \\
  \vdots         &\vdots         &\ddots            &\vdots  \\
  \GV_{,x''}(a_N,x';E)  &\GV_{,x'x''}(a_N,a_1)   &\hdots
                 &\hGV(a_N,a_N;E)+1/\beta_N
  \endmatrix\right\vert}{\left\vert\matrix
  \hGV_{,x'x''}(a_1,a_1;E)+1/\beta_1
                 &\hdots         &\GV_{,x'x''}(a_1,a_N;E)    \\
  \vdots         &\ddots         &\vdots                     \\
  \GV_{,x'x''}(a_N,a_1;E)        &\hdots
                 &\hGV_{,x'x''}(a_N,a_N;E)+1/\beta_N
  \endmatrix\right\vert}\enspace.
  \tag\num\endalign$$
Of course, any combination of N-fold $\delta$-functions and M-fold
$\delta'$-functions perturbations is possible yielding a closed
expression in terms of the corresponding Green functions.

Making now the coupling of the $\delta'$-function perturbation
infinitely repulsive produces Neumann boundary conditions at $x=a$, i.e.
\plus$$\myalign
  &\ih        \int_0^\infty dT\,\e^{\i ET/\hbar}
  \int\limits_{x(t')=x'}^{x(t'')=x''}\CD_{x=a}^{(N)} x(t)
  \exp\left\{\ih\int_{t'}^{t''}\bigg[{m\over2}\dot x^2
       -V(x)\bigg]dt\right\}
         \\   &
  =G^{(V)}(x'',x';E)
   -G^{(V)}_{,x'}(x'',a;E)G^{(V)}_{,x''}(a,x';E)
   \Big/\hGV_{,x'x''}(a, a;E)\enspace.
  \tag\num\endalign$$
The notation $\CD_{x=a}^{(N)}$ stands for Neumann boundary
conditions at $x=a$, and for the corresponding Green function we write
shorthand $G^{(V,N)}_{x=a}(E)$. Note that $\lim_{\gamma\to-\infty}
G^{(\delta)}(E)=G^{(V,D)}_{x=a}(E)$, where $D$ stands for Dirichlet
boundary conditions$^{\GROw}$. Let us consider a particle moving under
the influence of a potential in the box $a<x<b$ with Neumann boundary
conditions for $x=a$ and $x=b$; we obtain
\plus$$\myalign
   \ih                    \int_0^\infty dT\,&\e^{\i ET/\hbar}
  \int\limits_{x(t')=x'}^{x(t'')=x''}\CD_{(a<x<b)}^{(NN)}x(t)
  \exp\left\{\ih\int_{t'}^{t''}\bigg[{m\over2}\dot x^2
       -V(x)\bigg]dt\right\}
         \\   &
  =\dsize\thickfrac{\left\vert
  \matrix\GV(x'',x';E)      &\GV_{,x'}(x'',b;E)  &\GV_{,x'}(x'',a;E)  \\
         \GV_{,x''}(b,x';E) &\hGV_{,x'x''}(b,b;E)&\GV_{,x'x''}(b,a;E)\\
         \GV_{,x''}(a,x';E) &\GV_{,x'x''}(a,b;E)&\hGV_{,x'x''}(a,a;E)
  \endmatrix\right\vert}{\left\vert\matrix
  \hGV_{,x'x''}(b,b;E) &\GV_{,x'x''}(b,a;E)\\
  \GV_{,x'x''}(a,b;E) &\hGV_{,x'x''}(a,a;E)\endmatrix\right\vert}
  \enspace.
  \tag\num\endalign$$
Of course, any combination of boundary conditions of a particle moving
in the box $a<x<b$ is allowed yielding closed expression in terms of
the corresponding Green functions.

It is now obvious how to describe potential problems with absolute
value dependence, i.e.~$V=V(\vert x\vert)$. Combing the results for
Dirichlet and Neumann boundary conditions we obtain the general formula
\plus$$\myalign
  &\ih                    \int_0^\infty dT\,\e^{\i ET/\hbar}
  \int\limits_{x(t')=x'}^{x(t'')=x''}\CD x(t)
  \exp\left\{\ih\int_{t'}^{t''}\bigg[{m\over2}\dot x^2
       -V(\vert x\vert )\bigg]dt\right\}
         \\   &
  =G^{(V)}(x'',x';E)-{1\over2}
  \Big[G^{(V,D)}_{x=0}(x'',x';E)+G^{(V,N)}_{x=0}(x'',x';E)\Big]\enspace.
  \tag\num\endalign$$
\edef\nume{\num}%
If the potential $V$ already contains only even powers in $x$, like the
harmonic oscillator, the last two term in (\nume) cancel. Simple
examples for the general case are, for instance, the double-oscillator
$V(x)={m\over2}\omega^2(\vert x\vert-a)^2$, the one dimensional
Coulomb-problem $V(x)=k\vert x\vert$, or the symmetric potential well.
For the one dimensional Coulomb problem one obtains e.g.\ the Green
function
\plus$$\myalign
  &\ih                    \int_0^\infty dT\,\e^{\i ET/\hbar}
   \int\limits_{x(t')=x'}^{x(t'')=x''}\CD x(t)
  \exp\left[\ih\int_{t'}^{t''}\bigg({m\over2}\dot x^2
       -k\vert x\vert\bigg)dt\right]
         \\   &
  ={4\over3}{m\over\hbar^2}\bigg[
   \bigg(x'-{E\over k}\bigg)\bigg(x''-{E\over k}\bigg)\bigg]^{1/2}
         \\   &\quad\times\Bigg\{
   K_{1/3}\left({2\over3}{\sqrt{2mk}\over\hbar}
   \bigg(x_>-{E\over k}\bigg)^{3/2}\right)
   I_{1/3}\left({2\over3}{\sqrt{2mk}\over\hbar}
   \bigg(x_<-{E\over k}\bigg)^{3/2}\right)
         \\   &\qquad-{1\over2}
   K_{1/3}\left({2\over3}{\sqrt{2mk}\over\hbar}
   \bigg(x'-{E\over k}\bigg)^{3/2}\right)
   K_{1/3}\left({2\over3}{\sqrt{2mk}\over\hbar}
   \bigg(x''-{E\over k}\bigg)^{3/2}\right)
         \\   &\qquad\quad\times\Bigg[
   I_{1/3}\left({2\over3}{\sqrt{2mk}\over\hbar}
   \bigg(-\dsize{E\over k}\bigg)^{3/2}\right)\Bigg/
   K_{1/3}\left({2\over3}{\sqrt{2mk}\over\hbar}
   \bigg(-\dsize{E\over k}\bigg)^{3/2}\right)
         \\   &\qquad\qquad-2\pi
   I_{2/3}\left({2\over3}{\sqrt{2mk}\over\hbar}
   \bigg(-\dsize{E\over k}\bigg)^{3/2}\right)\Bigg/
   K_{2/3}\left({2\over3}{\sqrt{2mk}\over\hbar}
   \bigg(-\dsize{E\over k}\bigg)^{3/2}\right)\Bigg]\Bigg\}\enspace,
  \tag\num\endalign$$
with the quantization condition
\plus$$
   K_\nu\left({2\over3}{\sqrt{2mk}\over\hbar}
   \bigg(-\dsize{E_n\over k}\bigg)^{3/2}\right)=0\enspace,
  \tag\num$$
with $n={1\over3},{2\over3}$ for the odd, respectively even
wave functions.

In this paper I have presented a perturbation expansion approach to the
problem of $\delta'$-function perturbations and Neumann boundary
conditions in the context of path integrals. This was achieved by
considering the path integral representation of the one dimensional
Dirac particle with a $\delta$-function perturbation in the positron
component. In the non-relativistic limit a $\delta'$-function
perturbation in the path integral emerges. I obtained closed formul\ae\
for both problems in terms of the corresponding energy-dependent Green
function. An analogous  discussion for the electron-component yields a
$\delta$-function perturbation in the path integral and Dirichlet
boundary condition, respectively. The formalism can be repeated in an
obvious way to incorporate multiple $\delta$- and $\delta'$-function
perturbations, and one can consider motion in a box $a<x<b$ with any
combination of Dirichlet and Neumann boundary conditions at the walls of
the box. Analogously to Ref.~[\GROw] one can also generalize our method
to higher dimensions to derive path integral formulations for
$\delta'$-function perturbations, Dirichlet and Neumann boundary
conditions along lines and hyperplanes, etc., respectively.

I could also derive a general expression for potentials with absolute
value dependence by combining the results from Dirichlet and Neumann
boundary conditions, c.f.~(\nume). In general, only the corresponding
Green function can be stated.

The definition of the path integral of the $\delta'$-function
perturbation and its (energy dependent) Green function via the path
integral representation of the one dimensional Dirac particle
looks at first sight circumstancial. However, specific regularization
prescriptions of singular potentials are familiar for path integrals:
For instance, the $1/r$ potential requires in a proper path integral
representation a regularization through the Kustaanheimo-Stiefel
transformation$^{\DKb}$, and the $1/r^2$ potential by means of the
Besselian functional weight$^{\GRSb}$. In the path integral formulation
the usual $\delta$-function perturbation is quite a simple
object$^{\GROh}$ in comparison to the $\delta'$-function perturbation
as shown in this paper. Actually both point interactions describe a
particular kind of boundary conditions of the wave functions in their
domains at the location of the interaction. The even more singular two
and three dimensional point interactions require also a regularization
prescription by means of their Green functions$^{\AGHH}$.

The achieved results for a proper approach to $\delta$-and
$\delta'$-function perturbations, and Dirichlet and Neumann boundary
conditions respectively, in the language of Feynman path integrals
cover properly combined a wide range of problems in path integral
technique. What remains is to develop a path integral formalism to
incorporate general boundary conditions, where Dirichlet and Neumann
boundary conditions are but special cases, respectively multiple
boundary conditions on the real line (e.g.\ combination of step
potentials). These open question will be subject to future
investigations.

\ack
I would like to thank C.\ Oldhoff for fruitful discussions.


\newpage\noindent
\numreferences
%-----------------------------------------------------------------------
\ref{$^{\BADUb}$}
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%-----------------------------------------------------------------------
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%-----------------------------------------------------------------------
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%-----------------------------------------------------------------------
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%-----------------------------------------------------------------------
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%-----------------------------------------------------------------------
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%-----------------------------------------------------------------------
\ref{$^{\ICHTA}$}
T.Ichinose and H.Tamura:
Path Integral Approach to Relativistic Quantum Mechanics;
{\it Prog.Theor.Phys.Supp.}\ {\bf 92} (1987) 144
%-----------------------------------------------------------------------
\ref{$^{\DKb}$}
I.H.Duru and H.Kleinert:
Quantum Mechanics of H-Atoms From Path Integrals;
{\it Fort\-schr.Phys.}\ {\bf 30} (1982) 401
%-----------------------------------------------------------------------
\ref{$^{\GRSb}$}
C.Grosche and F.Steiner:
Path Integrals on Curved Manifolds;
{\it Zeitschr.Phys.}\ {\bf C 36} (1987) 699
%-----------------------------------------------------------------------

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