LaTeX 2.09 
BODY

%&latex209
\documentstyle[a4,%11pt,%
twoside]{article}
\sloppy
% \textheight=240mm
% \voffset=-10mm
% \textwidth=160mm
% \hoffset=-10mm

%%%%%%%%%%%%
\makeatletter
\renewcommand{\@oddhead}{The Ground, a String, Two Elastic Springs
 \hfill \thepage}
\renewcommand{\@evenhead}{\thepage \hfill S. A. Choro\v{s}avin }
\renewcommand{\@oddfoot}{}
\renewcommand{\@evenfoot}{}
\makeatother
%%%%%%%%%%%%%%%%%%%




\author{S.A.~Choro\v{s}avin}
\title{ 
 The Ground, a String, Two Elastic Springs.
 \bigskip\\
  Simple Exactly Solvable Models
  of One-Dimensional Scalar Fields 
  with Concentrated Factors. 
  }
\date{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%\hfill S. A. Skarst Choro\v{s}avin. \hfill \\
\maketitle 
\begin{abstract}
 This paper is an electronic application to my set of lectures, 
 subject:`Formal methods in solving differential equations and 
 constructing models of physical phenomena'. Addressed, mainly: 
 postgraduates and related readers. 
 Content: 
 a discussion of the simple models
 ( I would rather say, toy models )
 of the interaction based 
 on equation arrays of the kind: 
% d^2 u(t,x)/dt^2=c^{2}d^2 u(t,x)/dx^2
%                 -4{\gamma_a}c\delta(x-x_a) Q_a(t)
%                 -4{\gamma_b}c\delta(x-x_b) Q_b(t), 
% Q_a(t) = u(t,x_a),
% Q_b(t) = u(t,x_b),
\begin{eqnarray*}
 \frac{\partial^2 u(t,x)}{\partial t^2}
 &=&
 c^2\frac{\partial^2 u(t,x)}{\partial x^2}
                 -4{\gamma_a}c\delta(x-x_a) Q_a(t)
                 -4{\gamma_b}c\delta(x-x_b) Q_b(t), 
\\
 Q_a(t)
 &=& u(t,x_a)
\\
 Q_b(t)
 &=& u(t,x_b)
\\
\end{eqnarray*}
% Besides, less detailed discussion of related models.
 Central mathematical points: d'Alembert-Kirchhoff-like Formulae,
 Finite Rank Perturbations.
\end{abstract}


\newpage
\section*%
{ Introduction }

 As long as the researcher takes an interest in resolvent formulae 
 of finite rank perturbed operators, in other words, 
 as long as he or she, the researcher,
 prefers to remain in the space-FREQUENCE framework,
 so long he has an infinite series of good papers, articles, books,
 manuals...

 But as soon as
 the researcher is turning his attention to the problems of the "space-TIME", 
 in other words, as soon as
 he has need to solve the associated wave equation,
 as soon as he has need of a suitable d'Alembert-like formula, or so, - 
 in that moment the situation is changing dramatically. 
  
 I cannot say "there is no paper on the subject at all",
 I cannot say "I have not seen it",
 but if anyone asks me "where have you seen it?, where is this `there'?",
 I will become very "pensativo",
 and I am in doubt that I will be "solitario" in this state.

 So, I tried, try and (I hope) will try to collect 
 the suitable examples of simple exactly solvable models of wave'n'particle.

 Whether my collection is worth to discuss it, you solve.
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%\newpage
\section%
{ Models of Two-Point Interaction with 
  an only one-dimensional Scalar Field }

\subsection%
{ Preliminaries }
 In this paper we fix measure units 
% so that 
%$c >0$ 
 and let
$x$ 
 be dimensionless position parameter, i.e., 
%
$$
 \mbox{ physical position coordinate }
  =  [\mbox{ length unit }]\times x +const \,.
$$
%
 Otherwise a confusion can ocurr, in relating to the definition 
%
$$
 \int_{-\infty}^{\infty}\delta(x-x_0)f(x)dx=f(x_0) \,.
$$
%
 We assume the standard foramalism, where 
%
$$ 
 \delta(x-x_0) = \frac{\partial 1_{+}(x-x_0)}{\partial x}
$$
%
 and where 
$1_{+}$
 stands for a unit step function (Heaviside function):
%
$$ 
 1_{+}(\xi) := 1(\xi \geq 0) := \Bigl\{
 \begin{array}{ccc}
 1&,&\mbox{ if } \xi \geq 0 \,,
 \\
 0&,&\mbox{ if } \xi < 0 \,.
 \\
 \end{array}
$$

\bigskip

 Another feature of the notations is this. 
 We will handle the functions which have a special variable,
$t$,
 that means no doubt "time", and we will be interested in the case where 
$t \geq 0 $.
 So, we could consider the restrictions of these functions, 
 onto positive
 $t$-half-line. 
 But it will be {\bf technically } more convenient to
 {\bf redefine } the functions,
 putting them zero on negative
 $t$-half-line.
 For more details of this feature of the notations see 
 {\bf The 
$ \cdot 1_{+}()$
 Convention },
 in the next subsection.

\bigskip

 A few words about the models:
 Recently I have already presented some models of 
 ONE particle of FINITE mass, 
 interacting with scalar field,
 and the interaction has been concentrated at one, only ONE, point. 
 
 Now I discuss models of TWO-point concentrated interaction, but the mass 
 of the particle (or particles) I assume to be INFINITE,
 or, more precisely, I assume the particle(s)
 to be of infinite mass and immobile: motionless, fixed. 

 A naive formulation of the situation is: 
 let us look at the Ground (the infinite immobile mass),
 the Earth or the Moon say,
 connected with 
 a String (very-very long thin tensed cord) by means of two Ideal
 (we never approve Imperfection, don't we?) Springs.
 What will we tell then, ideally?

\newpage
\subsection%
{ D'Alembert-Kirchhoff-like formulae }

 Recall that a standard D'Alembert-Kirchhoff-like formula reads: 
 if 
%
$$
 \frac{\partial^2 u}{\partial t^2}
  = c^2\frac{\partial^2 u}{\partial x^2} + f
 \,,\quad
 u=u(t,x)
 \,,\quad
 f=f(t,x)
 \,,\quad
 (t\geq 0)
           \eqno(*)
$$
%
$$
  f=-4{\gamma_a}c\delta(x-x_a)\Big(F_{src\ a}(t)\Big)
    -4{\gamma_b}c\delta(x-x_b)\Big(F_{src\ b}(t)\Big)
 \qquad x_a \leq x_b
$$
%
 and given initial data, 
%
$u(0,\cdot )$ 
 and 
$\frac{\partial u(t,\xi)}{\partial t})\Big|_{t=0}$,
 then, for 
$t\geq 0$, 
%

\medskip

\begin{eqnarray*}
 u(t,x)
 &=&
 \displaystyle
 -{2\gamma_a}
 \int_0^{t-|x-x_a|/c}
 \Big(F_{src\ a}(\tau)\Big)d\tau \cdot 1_{+}(t-|x-x_a|/c)
 \\&&{}
 -{2\gamma_b}
 \int_0^{t-|x-x_b|/c}
 \Big(F_{src\ b}(\tau)\Big)d\tau \cdot 1_{+}(t-|x-x_b|/c)
 \\[\medskipamount]\\
 &&{}\qquad + u_{0}(t,x)
 \\[\smallskipamount] \\
\end{eqnarray*}
%
 where 
%%%
\begin{eqnarray*}
 u_{0}(t,x)
 &:=& 
 c_{+}(x+ct) + c_{-}(x-ct)
 \\&=&
 \frac12 \Big(u(0,x+ct) + u(0,x-ct)\Big)
 \\&&{}
   + \frac{1}{2c}\Big(
  \widetilde{\dot u}(0,x+ct) - \widetilde{\dot u}(0,x-ct)\Big)
 \\&&{}
\end{eqnarray*}
 and where 
$\widetilde{\dot u}$ 
 stands for any function defined by 
$$
 \frac{\partial\widetilde{\dot u}(0,\xi)}{\partial \xi}
   =\Bigl(\frac{\partial u(t,\xi)}{\partial t}\Bigr)\Big|_{t=0}
\,.
$$
\footnote{
 Note that
$$
\widetilde{\dot u}(0,x+ct) - \widetilde{\dot u}(0,x-ct)
$$
 does not depend on what the primitive is which one has chosen!!! 
 Moreover, we need only 
$ \widetilde{\dot u}|_{t=0} $
 and not 
$ \dot u|_{t=0}$
 itself!
} % end footnote


 If we put 
%
\begin{eqnarray*}
 F_a(t)
 &:=&
 \displaystyle
 -{2\gamma_a}
 \int_0^{t}
 \Big(F_{src\ a}(\tau)\Big)d\tau \cdot 1_{+}(t) \,,
\\
 F_b(t)
 &:=&
 -{2\gamma_b}
 \int_0^{t}
 \Big(F_{src\ b}(\tau)\Big)d\tau \cdot 1_{+}(t) \,,
\end{eqnarray*}
%
 for short, then 
%
\newpage\noindent
%
%

\medskip\noindent
%
\fbox{
\parbox{\textwidth}{
%
\begin{eqnarray*}
 u(t,x)
 &=&
 F_{a}(t-|x-x_a|/c)
 + F_{b}(t-|x-x_b|/c)
 + u_{0}(t,x)
\qquad (t \geq 0)
\\
\end{eqnarray*}
%
} % end parbox
} % end fbox

\medskip\noindent
%
 Another and a little more correct form of this expression is:
% Eine andere und ein wenig korrektere Form dieses Ausdrucks ist:

\medskip\noindent
%
\fbox{
\parbox{\textwidth}{
%
\begin{eqnarray*}
 u(t,x) \cdot 1_{+}(t)
 &=&
 F_{a}(t-|x-x_a|/c)
 + F_{b}(t-|x-x_b|/c)
 + u_{0}(t,x) \cdot 1_{+}(t)
\\
\end{eqnarray*}
%
} % end parbox
} % end fbox

\medskip\noindent
%
 Now we set 
% Nun setzten wir 
$$
 T := |x_a-x_b|/c
$$
 and concentrate on 
% und konzentrieren wir auf 
$$
 x = x_a , x_b
$$ 
%
 by writing 
% auf solche Weise:
%
\begin{eqnarray*}
 u(t,x_a)
 &=&
 F_{a}(t)
 + F_{b}(t-T)
 + u_{0}(t,x_a)
\qquad (t \geq 0)
\\
\end{eqnarray*}
%
%
\begin{eqnarray*}
 u(t,x_b)
 &=&
 F_{a}(t-T)
 + F_{b}(t)
 + u_{0}(t,x_b)
\qquad (t \geq 0)
\\
\end{eqnarray*}
%
 We transform these expressions into 
% Diese Ausdr\"ucke stelle (transformiere) ich so dar:
%
%

\medskip\noindent
%
\fbox{
\parbox{\textwidth}{
%
\begin{eqnarray*}
 F_{a}(t)
 &=&
 \Bigl(
 u(t,x_a)
 - u_{0}(t,x_a)
 - F_{b}(t-T)
 \Bigr) \cdot 1_{+}(t)
\\
\end{eqnarray*}
%
\begin{eqnarray*}
 F_{b}(t)
 &=&
 \Bigl(
 u(t,x_b)
 - u_{0}(t,x_b)
 - F_{a}(t-T)
 \Bigr) \cdot 1_{+}(t)
\\
\end{eqnarray*}
%
} % end parbox
} % end fbox

\medskip\noindent
%
 and then 
% und dann 
%
\begin{eqnarray*}
 F_{a}(t)
 &=&
 \Bigl( u(t,x_a) - u_{0}(t,x_a) \Bigr) \cdot 1_{+}(t)
\\&&{}
 -
 \Bigl(
 u(t-T,x_b) - u_{0}(t-T,x_b)
 \Bigr) \cdot 1_{+}(t-T)
\\&&{}
 + F_{a}(t-2T) \cdot 1_{+}(t-T)
\\[\smallskipamount]\\
 F_{b}(t)
 &=&
 \Bigl( u(t,x_b) - u_{0}(t,x_b) \Bigr) \cdot 1_{+}(t)
\\&&{}
 -
 \Bigl(
 u(t-T,x_a) - u_{0}(t-T,x_a)
 \Bigr) \cdot 1_{+}(t-T)
\\&&{}
 + F_{b}(t-2T) \cdot 1_{+}(t-T)
\\
\end{eqnarray*}
%
 \newpage\noindent
%
\begin{eqnarray*}
 F_{a}(t)
 &=&
 \Bigl( u(t,x_a) - u_{0}(t,x_a) \Bigr) \cdot 1_{+}(t)
\\&&{}
 -
 \Bigl(
 u(t-T,x_b) - u_{0}(t-T,x_b)
 \Bigr) \cdot 1_{+}(t-T)
\\&&{}
 +
 \Bigl(
 u(t-2T,x_a) - u_{0}(t-2T,x_a)
 \Bigr) \cdot 1_{+}(t-2T)
\\&&{}
 -
 \Bigl(
 u(t-3T,x_b) - u_{0}(t-3T,x_b)
 \Bigr) \cdot 1_{+}(t-3T)
\\&&{}
 + F_{a}(t-4T) \cdot 1_{+}(t-3T)
\\[\smallskipamount]\\
 F_{b}(t)
 &=&
 \Bigl( u(t,x_b) - u_{0}(t,x_b) \Bigr) \cdot 1_{+}(t)
\\&&{}
 -
 \Bigl(
 u(t-T,x_a) - u_{0}(t-T,x_a)
 \Bigr) \cdot 1_{+}(t-T)
\\&&{}
 +
 \Bigl(
 u(t-2T,x_b) - u_{0}(t-2T,x_b)
 \Bigr) \cdot 1_{+}(t-2T)
\\&&{}
 -
 \Bigl(
 u(t-3T,x_a) - u_{0}(t-3T,x_a)
 \Bigr) \cdot 1_{+}(t-3T)
\\&&{}
 + F_{b}(t-4T) \cdot 1_{+}(t-3T)
\\
\end{eqnarray*}
%
$$
 \mbox{\bf and so on. }
% \mbox{\bf und so weiter. }
$$
%
 Take here in account that 
% Man beachte hier, dass 
$$
 F_{a}(t) = 0 \,,\, F_{b}(t) = 0 \,,\, 1_{+}(t) = 0 \mbox{ falls } t < 0 \,;
$$
 in any case we see that
% Jedenfalls
%
\begin{eqnarray*}
 1_{+}(t-NT) = 0 \,,\,
\\
 \mbox{ if } t < NT \,.
% \mbox{ falls } t < NT \,.
\end{eqnarray*}
%
 It follows that 
% Folglich, 
%
$$
  \mbox{ if } T \not= 0 \,, \mbox{ then }
%  \mbox{ falls } T \not= 0 \,, \mbox{ dann }
$$
%
\begin{eqnarray*}
 F_{a}(t)
 &=&
 \Bigl( u(t,x_a) - u_{0}(t,x_a) \Bigr) \cdot 1_{+}(t)
\\&&{}
 -
 \Bigl(
 u(t-T,x_b) - u_{0}(t-T,x_b)
 \Bigr) \cdot 1_{+}(t-T)
\\&&{}
 +
 \Bigl(
 u(t-2T,x_a) - u_{0}(t-2T,x_a)
 \Bigr) \cdot 1_{+}(t-2T)
\\&&{}
 -
 \Bigl(
 u(t-3T,x_b) - u_{0}(t-3T,x_b)
 \Bigr) \cdot 1_{+}(t-3T)
\\&&{}
 + \cdots
\\[\smallskipamount]\\
 F_{b}(t)
 &=&
 \Bigl( u(t,x_b) - u_{0}(t,x_b) \Bigr) \cdot 1_{+}(t)
\\&&{}
 -
 \Bigl(
 u(t-T,x_a) - u_{0}(t-T,x_a)
 \Bigr) \cdot 1_{+}(t-T)
\\&&{}
 +
 \Bigl(
 u(t-2T,x_b) - u_{0}(t-2T,x_b)
 \Bigr) \cdot 1_{+}(t-2T)
\\&&{}
 -
 \Bigl(
 u(t-3T,x_a) - u_{0}(t-3T,x_a)
 \Bigr) \cdot 1_{+}(t-3T)
\\&&{}
 + \cdots
\\
\end{eqnarray*}
%
\newpage 
\noindent 
%
\fbox{
\parbox{\textwidth}{
\medskip\noindent
\begin{center}
{\large \bf
 The 
$ \cdot 1_{+}()$
 Convention. }
\end{center}
 In the following text, we will often redefine functions of 
$t$ 
 by multiplying them by 
$1_{+}(t)$, 
 i.e., by resetting, e.g.,
% Im folgenden setzen wir oft ( um sich kurz zu fassen )
%
\begin{eqnarray*}
 u_{0}(t,\cdots)
 &:=&
 u_{0}(t,\cdots) \cdot 1_{+}(t)
\\
 u(t,\cdots)
 &:=&
 u(t,\cdots) \cdot 1_{+}(t)
\\
 u_{0}(t-T,\cdots)
 &:=&
 u_{0}(t-T,\cdots) \cdot 1_{+}(t-T)
\\
 &\makebox[0ex][c]{ and so on. }&
% &\makebox[0ex][c]{ und dergleichen. }&
\\
\end{eqnarray*}
 Nevertheless we will sometimes write the multiplier 
$1_{+}()$,
 although we will mostly do it for emphasis,
 or there, where the formula does not become too long. 
} % end parbox
} % end fbox
%

\bigskip

\noindent
 To take an example of this convention implementation,
 we say that 
 the recent relations concerning 
$F_{a},  F_{b}$,  
 we may write them as 
% Zum Beispiel: 
%
%
\begin{eqnarray*}
 F_{a}(t)
 &=&
 u(t,x_a) - u_{0}(t,x_a)
 - F_{b}(t-T)
\\
 F_{b}(t)
 &=&
 u(t,x_b) - u_{0}(t,x_b)
 - F_{a}(t-T)
\\
\end{eqnarray*}
%
 In addition to The 
$ \cdot 1_{+}()$
 Convention, we will

\medskip\noindent 
%
\fbox{
\parbox{\textwidth}{
\medskip\noindent
\begin{center}
{\large \bf
 assume that 
$ T \not= 0 $ .
 }
\end{center}
} % end parbox
} % end fbox
%

\bigskip

\noindent

\newpage
\section%
{ Particular Cases. Recurrence Relations }
\subsection%
{The Case of %%Der Fall von 
 $ \gamma_b = 0 $}
%
 We begin the analysis of this case turning to the following relations, 
 which we have seen in the previous section:
% In diesem Falle,
% wir richten unsere Aufmerksamkeit auf solche Formeln der vorigen Sektion
%

\medskip\noindent
%
%
\begin{eqnarray*}
 u(t,x)
 &=&
 F_{a}(t-|x-x_a|/c)
 + F_{b}(t-|x-x_b|/c)
 + u_{0}(t,x)
\qquad (t \geq 0)
\\
 F_{a}(t)
 &=&
 \Bigl(
 u(t,x_a)
 - u_{0}(t,x_a)
 - F_{b}(t-T)
 \Bigr) \cdot 1_{+}(t)
\\
 F_{b}(t)
 &=&
 \Bigl(
 u(t,x_b)
 - u_{0}(t,x_b)
 - F_{a}(t-T)
 \Bigr) \cdot 1_{+}(t)
\end{eqnarray*}
%
%
  Thus, we see, that in the case of
% Nun sehen wir, dass, im Falle von
$\gamma_b = 0$, 
 these relations become 
% diese Formeln werden 

\medskip\noindent
\fbox{
\parbox{\textwidth}{
%
\begin{eqnarray*}
 u(t,x)
 &=&
 F_{a}(t-|x-x_a|/c)
 + u_{0}(t,x)
\qquad (t \geq 0)
\\
\end{eqnarray*}
%
\begin{eqnarray*}
 u(t,x)
 &=&
 \Bigr( u(t-|x-x_a|/c,x_a)
 - u_{0}(t-|x-x_a|/c,x_a) \Bigr) \cdot 1_{+}(t-|x-x_a|/c)
\\&&{}
 + u_{0}(t,x)
\qquad (t \geq 0)
\\
\end{eqnarray*}
%
 Take here into account that 
% Man beachte hier, dass 
%
\begin{eqnarray*}
 u(t,x_a)
 &=&
 F_{a}(t)
 + u_{0}(t,x_a)
\qquad (t \geq 0)
\\
\end{eqnarray*}
%
} % end parbox
} % end fbox

\medskip\noindent
%
 In particular, if 
% Insbesondere, falls 
%
$$
 u(t,x_a) \equiv 0 \qquad ( t \geq 0 ) \,,
$$
%
 then 
% dann
%
$$
 F_{a}(t)
 = - u_{0}(t,x_a)
$$
%
 and, hence, 
% und, folglich, 
%
\medskip\noindent
\medskip

\noindent
\fbox{
\parbox{\textwidth}{

\medskip\noindent
%
 if 
% wenn 
%
$$
 u(t,x_a) \equiv 0 \qquad ( t \geq 0 ) \,,
$$
%
 then 
% dann
%
\begin{eqnarray*}
 u(t,x)
 &=&
 - u_{0}(t-|x-x_a|/c,x_a) \cdot 1_{+}(t-|x-x_a|/c)
 + u_{0}(t,x)
\qquad (t \geq 0)
\\
\end{eqnarray*}
%
} % end parbox
} % end fbox

\medskip\noindent
%
\newpage\noindent
%

 The restriction 
$ u(t,x_a) \equiv 0 \quad ( t \geq 0 ) $
 states that the string is absolutely motionless, fixed at the point 
$x=x_a$.
 If we replace this restriction by 
% Falls aber 
%
\begin{eqnarray*}
 F_a(t)
 &:=&
 -{2\gamma_a}
 \int_0^{t}
 \Big(F_{src\ a}(\tau)\Big)d\tau \cdot 1_{+}(t) \,,
\\
 &:=&
 -{2\gamma_a}
 \int_0^{t}
 u(\tau,x_a) d\tau \cdot 1_{+}(t) \,,
\\
\end{eqnarray*}
%
 then we obtain: 
% erhalten wir, dass 
%
\begin{eqnarray*}
 u(t,x_a)
 &=&
 -{2\gamma_a}
 \int_0^{t}
 u(\tau,x_a) d\tau \cdot 1_{+}(t)
 + u_{0}(t,x_a)
\qquad (t \geq 0)
\\
\end{eqnarray*}
%
 and, finally, 
% und, schliesslich,
%
\medskip

\noindent
\fbox{
\parbox{\textwidth}{
%
\begin{eqnarray*}
 u(t,x_a)
 &=&
 e^{-{2\gamma_a} t}u_{0}(0,x_a)
 +
 \int_0^{t}
  e^{-{2\gamma_a} (t-\tau)} d_{\tau}u_{0}(\tau,x_a) \cdot 1_{+}(t) \,,
\qquad (t \geq 0)
\\
\end{eqnarray*}
%
%
\begin{eqnarray*}
 u(t,x)
 &=&
 \Bigr( u(t-|x-x_a|/c,x_a)
 - u_{0}(t-|x-x_a|/c,x_a) \Bigr) \cdot 1_{+}(t-|x-x_a|/c)
\\&&{}
 + u_{0}(t,x)
\qquad (t \geq 0)
\\
\end{eqnarray*}
%
} % end parbox
} % end fbox

\medskip\noindent
%


\newpage
\subsection%
{The Case of %%Der Fall von 
 $\quad u(t,x_a) = 0\,,\quad u(t,x_b) = 0 $}
%
 We have yet observed that in any case 
% Wir haben gesehen, dass jedenfalls 
%
\begin{eqnarray*}
 F_{a}(t)
 &=&
 \Bigl( u(t,x_a) - u_{0}(t,x_a) \Bigr) \cdot 1_{+}(t)
\\&&{}
 -
 \Bigl(
 u(t-T,x_b) - u_{0}(t-T,x_b)
 \Bigr) \cdot 1_{+}(t-T)
\\&&{}
 +
 \Bigl(
 u(t-2T,x_a) - u_{0}(t-2T,x_a)
 \Bigr) \cdot 1_{+}(t-2T)
\\&&{}
 -
 \Bigl(
 u(t-3T,x_b) - u_{0}(t-3T,x_b)
 \Bigr) \cdot 1_{+}(t-3T)
\\&&{}
 + \cdots 
\\[\smallskipamount]\\
 F_{b}(t)
 &=&
 \Bigl( u(t,x_b) - u_{0}(t,x_b) \Bigr) \cdot 1_{+}(t)
\\&&{}
 -
 \Bigl(
 u(t-T,x_a) - u_{0}(t-T,x_a)
 \Bigr) \cdot 1_{+}(t-T)
\\&&{}
 +
 \Bigl(
 u(t-2T,x_b) - u_{0}(t-2T,x_b)
 \Bigr) \cdot 1_{+}(t-2T)
\\&&{}
 -
 \Bigl(
 u(t-3T,x_a) - u_{0}(t-3T,x_a)
 \Bigr) \cdot 1_{+}(t-3T)
\\&&{}
 + \cdots 
\\
\end{eqnarray*}
%
 Thus we infer that, if 
% Somit ergibt sich, dass, falls 
%
$$
 u(t,x_a) = 0\,,\quad u(t,x_b) = 0 \,,
$$
 then exactly 
% dann ist genau 
%
\begin{eqnarray*}
\makebox[0ex][l]{$ u(t,x) $}
\\
 &=&
 u_{0}(t,x)
\\&&{}
 -u_{0}(t-|x-x_a|/c,x_a)
 -u_{0}(t-|x-x_b|/c,x_b)
\\&&{}
 +u_{0}(t-|x-x_a|/c-T,x_b)
 +u_{0}(t-|x-x_b|/c-T,x_a)
\\&&{}
 -u_{0}(t-|x-x_a|/c-2T,x_a)
 -u_{0}(t-|x-x_b|/c-2T,x_b)
\\&&{}
 +u_{0}(t-|x-x_a|/c-3T,x_b)
 +u_{0}(t-|x-x_b|/c-3T,x_a)
\\&&{}
 \cdots
\qquad (t \geq 0)
\\
\end{eqnarray*}

\medskip

 Needs there a comment?

\newpage
\subsection%
{The Case of %%Der Fall von 
 $\quad u(t,x_a) = 0\,,\quad \gamma_b \not= \infty $ }
%
 It this case, we observe
\footnote{
 exploiting the  
{\bf The 
$ \cdot 1_{+}()$
 Convention },
 }
 that 
% In diesem Fall, nach der Konvention \"uber die kurzen Bezeichnungen, 
%
%
%
\begin{eqnarray*}
 F_{a}(t)
 &=&
 0 - u_{0}(t,x_a)
 - F_{b}(t-T)
\\
 F_{b}(t)
 &=&
 u(t,x_b) - u_{0}(t,x_b)
 - F_{a}(t-T)
\\
\end{eqnarray*}
%
%
\begin{eqnarray*}
 F_{a}(t)
 &=&
 0 - u_{0}(t,x_a)
\\&&{}
 -
 u(t-T,x_b) + u_{0}(t-T,x_b)
\\&&{}
 + F_{a}(t-2T)
\\[\smallskipamount]\\
 F_{b}(t)
 &=&
 u(t,x_b) - u_{0}(t,x_b)
\\&&{}
 - 0 + u_{0}(t-T,x_a)
\\&&{}
 + F_{b}(t-2T)
\\
\end{eqnarray*}
%
%
\noindent 
 Thus we infer that, if 
% Somit ergibt sich, dass, falls 
%
\begin{eqnarray*}
 F_b(t)
 &:=&
 -{2\gamma_b}
 \int_0^{t}
 \Big(F_{src\ b}(\tau)\Big)d\tau \cdot 1_{+}(t) \,,
\\
 &:=&
 -{2\gamma_b}
 \int_0^{t}
 u(\tau,x_b) d\tau \cdot 1_{+}(t) \,,
\\
\end{eqnarray*}
%
 then 
% dann 
%
\begin{eqnarray*}
 F_{a}(t)
 &=&
 - u_{0}(t,x_a)
\\&&{}
 -
 u(t-T,x_b) + u_{0}(t-T,x_b)
\\&&{}
 + F_{a}(t-2T)
\\[\smallskipamount]\\
 -{2\gamma_b}
 \int_0^{t}
 u(\tau,x_b) d\tau \cdot 1_{+}(t)
 &=&
 u(t,x_b) - u_{0}(t,x_b)
\\&&{}
 + u_{0}(t-T,x_a)
\\&&{}
 -
 {2\gamma_b}
 \int_0^{t-2T}
 u(\tau,x_b) d\tau \cdot 1_{+}(t-2T)
\\
\end{eqnarray*}
%
%
 We now focus on the latter relation, which we write in the following terms: 

% Ich richte meinen Blick nach
% die letztere, j\"ungste Relation
% die ich so schreiben will: 
%
%\footnote{
% Ich richte meine Aufmerksamkeit auf
% die letzte (letztere), j\"ungste Relation
% ( Beziehung, das letzte Verh\"altnis ),
% die ich so beschreiben will: 
% }
%
%
\begin{eqnarray*}
 u(t,x_b)
 &=&
 -{2\gamma_b}
 \int_0^{t}
 u(\tau,x_b) d\tau
 +
 {2\gamma_b}
 \int_0^{t-2T}
 u(\tau,x_b) d\tau
\\&&{}
 + u_{0}(t,x_b) - u_{0}(t-T,x_a)
\\
\end{eqnarray*}
%
\newpage 
\noindent
%
 Next, let us define 
% Ferner seien 
%
$$
 Q_{b}(t) := u(t,x_b) \,;\,
 Q_{0b}(t) := u_{0}(t,x_b) \,;\,
 Q_{0a}(t) := u_{0}(t,x_a) \,;\,
$$
%
 so that the former relation becomes 
% so dass die fr\"uhere Beziehung wird so aussehen:
%
\begin{eqnarray*}
 Q_{b}(t)
 &=&
 -{2\gamma_b}
 \int_0^{t}
 Q_{b}(\tau) d\tau
 +
 {2\gamma_b}
 \int_0^{t-2T}
 Q_{b}(\tau) d\tau
\\&&{}
 + Q_{0b}(t) - Q_{0a}(t-T)
\\
\end{eqnarray*}
%
%
\begin{eqnarray*}
 Q_{b}(t)
 &=&
 -{2\gamma_b}
 \int_{t-2T}^{t}
 Q_{b}(\tau) d\tau
\\&&{}
 + Q_{0b}(t) - Q_{0a}(t-T)
\\
\end{eqnarray*}
%
%
\begin{eqnarray*}
 \frac{\partial Q_{b}(t)}{\partial t}
 &=&
 -{2\gamma_b}
 Q_{b}(t)
 +
 {2\gamma_b}
 Q_{b}(t-2T)
\\&&{}
 + \frac{\partial ( Q_{0b}(t) - Q_{0a}(t-T) )}{\partial t}
\\
\end{eqnarray*}
%
%
 Let
% Es sei 
%
\footnote{ Method of Variation of Constants }
%\footnote{ Methode von Variation der Konstanten }
%
$$
 Q_{b}(t) = e^{-{2\gamma_b}t} C(t) \,.
$$
%
 Then we obtain:
% Dann haben wir: 
%
\begin{eqnarray*}
 Q_{b}(t-2T) &=& e^{-{2\gamma_b}t} e^{{2\gamma_b}T}C(t-2T) \,,
\\
 \frac{\partial Q_{b}(t)}{\partial t}
 &=&
 -{2\gamma_b} e^{-{2\gamma_b}t} C(t)
 +
 e^{-{2\gamma_b}t} \frac{\partial C(t)}{\partial t}
\\
 &=&
 -{2\gamma_b}Q_{b}(t)
 +
 e^{-{2\gamma_b}t} \frac{\partial C(t)}{\partial t}
\\
\end{eqnarray*}
%
%
\begin{eqnarray*}
 e^{-{2\gamma_b}t} \frac{\partial C(t)}{\partial t}
 &=&
 {2\gamma_b}
 Q_{b}(t-2T)
\\&&{}
 + \frac{\partial ( Q_{0b}(t) - Q_{0a}(t-T) )}{\partial t}
\\
\end{eqnarray*}
%
%
\newpage 
\noindent 
%
\begin{eqnarray*}
 e^{-{2\gamma_b}t} \frac{\partial C(t)}{\partial t}
 &=&
 {2\gamma_b}
 e^{-{2\gamma_b}t} e^{{2\gamma_b}T}C(t-2T)
\\&&{}
 + \frac{\partial ( Q_{0b}(t) - Q_{0a}(t-T) )}{\partial t}
\\
\end{eqnarray*}
%
%
\begin{eqnarray*}
 \frac{\partial C(t)}{\partial t}
 &=&
 {2\gamma_b} e^{{2\gamma_b}T}
 C(t-2T)
\\&&{}
 + e^{{2\gamma_b}t}
 \frac{\partial ( Q_{0b}(t) - Q_{0a}(t-T) )}{\partial t}
\\
\end{eqnarray*}
%
%
\begin{eqnarray*}
 \frac{\partial C(t)}{\partial t}
 &=&
 {2\gamma_b}
 e^{{2\gamma_b}t} Q_{b}(t-2T)
\\&&{}
 +e^{{2\gamma_b}t}
 \frac{\partial ( Q_{0b}(t) - Q_{0a}(t-T) )}{\partial t}
\\
\end{eqnarray*}
%
%
\begin{eqnarray*}
 C(t)
 &=&
 {2\gamma_b}
 \int_0^{t}
 e^{{2\gamma_b}\tau } Q_{b}(\tau-2T)
 d\tau
\\&&{}
 +
 \int_0^{t}
 e^{{2\gamma_b}\tau}
 \frac{\partial ( Q_{0b}(\tau) - Q_{0a}(\tau-T) )}{\partial \tau}
 d\tau
 +
 C(0)
\\
\end{eqnarray*}
%
%
\begin{eqnarray*}
 Q_{b}(t)
 &=&
 e^{-{2\gamma_b}t}
 {2\gamma_b}
 \int_0^{t}
 e^{{2\gamma_b}\tau } Q_{b}(\tau-2T)
 d\tau
\\&&{}
 +
 e^{-{2\gamma_b}t}
 \int_0^{t}
 e^{{2\gamma_b}\tau}
 \frac{\partial ( Q_{0b}(\tau) - Q_{0a}(\tau-T))}{\partial \tau}
 d\tau
 +
 e^{-{2\gamma_b}t} Q_{b}(0)
\\
\end{eqnarray*}
%
%
\begin{eqnarray*}
 Q_{b}(t)
 &=&
 {2\gamma_b}
 \int_0^{t}
 e^{-{2\gamma_b}(t-\tau )} Q_{b}(\tau-2T)
 d\tau
\\&&{}
 +
 \int_0^{t}
 e^{-{2\gamma_b}(t-\tau )}
 d_{\tau} 
 ( Q_{0b}(\tau) - Q_{0a}(\tau-T) )
 +
 e^{-{2\gamma_b}t} Q_{b}(0)
\\
\end{eqnarray*}
%
%
\begin{eqnarray*}
 Q_{b}(t)
 &=&
 {2\gamma_b}
 \int_0^{t}
 e^{-{2\gamma_b}(t-\tau )} Q_{b}(\tau-2T)
 d\tau
\\&&{}
 +
  I_{0b}(t) \,,
\\
\makebox[0ex][l]{ where }
%\makebox[0ex][l]{ wobei }
\\
  I_{0b}(t)
&:=&
 \int_0^{t}
 e^{-{2\gamma_b}(t-\tau )}
 d_{\tau} 
 ( Q_{0b}(\tau) - Q_{0a}(\tau-T) )
 +
 e^{-{2\gamma_b}t} Q_{b}(0)
\\
\end{eqnarray*}
%

\medskip\noindent
\newpage\noindent
%
 and then, after returning to the terms of 
$ u \,,\, u_{0} $,
 we infer that 
%
%

\medskip\noindent
\fbox{
\parbox{\textwidth}{
\begin{eqnarray*}
 u(t,x_b)
 &=&
 {2\gamma_b}
 \int_{2T}^{t}
 e^{-{2\gamma_b}(t-\tau )} u(\tau-2T,x_b)
 d\tau
\\&&{}
 +
  I_{0b}(t) \,,
\\
\makebox[0ex][l]{ where }
\\
 I_{0b}(t)
&:=&
 \int_{0+0}^{t}
 e^{-{2\gamma_b}(t-\tau )}
 d_{\tau} 
 ( u_{0}(\tau,x_b) - u_{0}(\tau-T,x_a)1_{+}(\tau-T) )
 +
 e^{-{2\gamma_b}t} u(0,x_b)
\\
\end{eqnarray*}
%
} % end parbox
} % end fbox
%

\medskip\noindent
 Finally, recall that 

\medskip\noindent
%
\fbox{
\parbox{\textwidth}{
%
\begin{eqnarray*}
 u(t,x)
 &=&
 F_{a}(t-|x-x_a|/c)
 - F_{b}(t-|x-x_b|/c)
 + u_{0}(t,x)
\qquad (t \geq 0)
\\
\end{eqnarray*}
%
} % end parbox
} % end fbox

\medskip\noindent
%
 where
%
\begin{eqnarray*}
 F_{a}(t)
 &=&
 \Bigl( 0 - u_{0}(t,x_a) \Bigr) \cdot 1_{+}(t)
\\&&{}
 -
 \Bigl(
 u(t-T,x_b) - u_{0}(t-T,x_b)
 \Bigr) \cdot 1_{+}(t-T)
\\&&{}
 +
 \Bigl(
 0 - u_{0}(t-2T,x_a)
 \Bigr) \cdot 1_{+}(t-2T)
\\&&{}
 -
 \Bigl(
 u(t-3T,x_b) - u_{0}(t-3T,x_b)
 \Bigr) \cdot 1_{+}(t-3T)
\\&&{}
 + \cdots 
\\[\smallskipamount]\\
 F_{b}(t)
 &=&
 \Bigl( u(t,x_b) - u_{0}(t,x_b) \Bigr) \cdot 1_{+}(t)
\\&&{}
 -
 \Bigl(
  0 - u_{0}(t-T,x_a)
 \Bigr) \cdot 1_{+}(t-T)
\\&&{}
 +
 \Bigl(
 u(t-2T,x_b) - u_{0}(t-2T,x_b)
 \Bigr) \cdot 1_{+}(t-2T)
\\&&{}
 -
 \Bigl(
 0 - u_{0}(t-3T,x_a)
 \Bigr) \cdot 1_{+}(t-3T)
\\&&{}
 + \cdots 
\\
\end{eqnarray*}
%
%
\newpage
\subsection%
{The Case of %%Der Fall von 
 $\gamma_a \not= \infty \,,\quad \gamma_b \not= \infty $ }
%
 In this case 
% In diesem Fall 
\footnote{
 Recall and take into account
 ``The 
$ \cdot 1_{+}()$
 Convention''
} % end footnote
%
\begin{eqnarray*}
 u(t,x_a)
 &=&
 \displaystyle
 -{2\gamma_a}
 \int_0^{t} 
 \Big(F_{src\ a}(\tau)\Big)d\tau
 -{2\gamma_b}
 \int_0^{t-T} 
 \Big(F_{src\ b}(\tau)\Big)d\tau \cdot 1_{+}(t-T)
 \\&&{}
 + u_{0}(t,x_a)
\end{eqnarray*}
%
\begin{eqnarray*}
 u(t,x_b)
 &=&
 \displaystyle
 -{2\gamma_a}
 \int_0^{t-T} 
 \Big(F_{src\ a}(\tau)\Big)d\tau \cdot 1_{+}(t-T)
 -{2\gamma_b}
 \int_0^{t} 
 \Big(F_{src\ b}(\tau)\Big)d\tau 
 \\&&{}
  + u_{0}(t,x_b)
\end{eqnarray*}
%
 We put, for short, 
% Ferner seien 
%
$$
 Q_{a}(t) := u(t,x_a) \,;\,
 Q_{0a}(t) := u_{0}(t,x_a) \,;\,
 Q_{b}(t) := u(t,x_b) \,;\,
 Q_{0b}(t) := u_{0}(t,x_b) \,;\,
$$
%
 so that the relationships become 
% so dass die fr\"uhere Beziehungen werden so aussehen:
%
\begin{eqnarray*}
 Q_{a}(t)
 &=&
 \displaystyle
 -{2\gamma_a}
 \int_0^{t} 
 \Big(F_{src\ a}(\tau)\Big)d\tau
 -{2\gamma_b}
 \int_0^{t-T} 
 \Big(F_{src\ b}(\tau)\Big)d\tau \cdot 1_{+}(t-T)
 \\&&{}
 + Q_{0a}(t)
\end{eqnarray*}
%
\begin{eqnarray*}
 Q_{b}(t)
 &=&
 \displaystyle
 -{2\gamma_a}
 \int_0^{t-T} 
 \Big(F_{src\ a}(\tau)\Big)d\tau \cdot 1_{+}(t-T)
 -{2\gamma_b}
 \int_0^{t} 
 \Big(F_{src\ b}(\tau)\Big)d\tau 
 \\&&{}
  + Q_{0b}(t)
\end{eqnarray*}
%
 and, hence, 
% und, folglich,
%
\begin{eqnarray*}
 \frac{\partial Q_{a}(t)}{\partial t} 
 &=&
 \displaystyle
 -{2\gamma_a}
 F_{src\ a}(t)
 -{2\gamma_b}
 F_{src\ b}(t-T) \cdot 1_{+}(t-T)
 \\&&{}
 + Q_{0a}(t)
\end{eqnarray*}
%
\begin{eqnarray*}
 \frac{\partial Q_{b}(t)}{\partial t} 
 &=&
 \displaystyle
 -{2\gamma_a}
 F_{src\ a}(t-T) \cdot 1_{+}(t-T)
 -{2\gamma_b}
 F_{src\ b}(t)
 \\&&{}
  + Q_{0b}(t)
\,.
\end{eqnarray*}
%
  We now assume that 
% Hier nehmen wir an, dass 
$$
 F_{src\ a}(t) := u(t, x_a) \,,\,
 F_{src\ b}(t) := u(t, x_b) \,.
$$
%
 Then 
% Dann 
%
\begin{eqnarray*}
 \frac{\partial Q_{a}(t)}{\partial t} 
 &=&
 \displaystyle
 -{2\gamma_a}
 Q_{a}(t)
 -{2\gamma_b}
 Q_{b}(t-T) \cdot 1_{+}(t-T)
 \\&&{}
 +
 \frac{\partial Q_{0a}(t)}{\partial t}
\end{eqnarray*}
%
\begin{eqnarray*}
 \frac{\partial Q_{b}(t)}{\partial t} 
 &=&
 \displaystyle
 -{2\gamma_a}
 Q_{a}(t-T) \cdot 1_{+}(t-T)
 -{2\gamma_b}
 Q_{b}(t)
 \\&&{}
  +
 \frac{\partial Q_{0b}(t)}{\partial t} 
\end{eqnarray*}
%
%
 Let 
% Es seien 
\footnote{ Method of Variation of Constants }
%\footnote{ Metode von Variation der Konstanten }
%
$$
 Q_{a}(t) = e^{-{2\gamma_a}t} C_{a}(t) \,,
 Q_{b}(t) = e^{-{2\gamma_b}t} C_{b}(t) \,.
$$
%
\newpage
\noindent
%
 Then we infer that:
% Dann bekommen wir, dass: 
%
\begin{eqnarray*}
% Q_{a}(t-T) &=& e^{-{2\gamma_a}t} e^{{\gamma_a}T}C_{a}(t-2T) \,,
%\\
 \frac{\partial Q_{a}(t)}{\partial t}
 &=&
 -{2\gamma_a} e^{-{2\gamma_a}t} C_{a}(t)
 +
 e^{-{2\gamma_a}t} \frac{\partial C_{a}(t)}{\partial t}
\\
 &=&
 -{2\gamma_a}Q_{a}(t)
 +
 e^{-{2\gamma_a}t} \frac{\partial C_{a}(t)}{\partial t}
\\
\end{eqnarray*}
%
%
\begin{eqnarray*}
% Q_{b}(t-T) &=& e^{-{2\gamma_b}t} e^{{\gamma_b}T}C_{b}(t-2T) \,,
%\\
 \frac{\partial Q_{b}(t)}{\partial t}
 &=&
 -{2\gamma_b} e^{-{2\gamma_b}t} C_{b}(t)
 +
 e^{-{2\gamma_b}t} \frac{\partial C_{b}(t)}{\partial t}
\\
 &=&
 -{2\gamma_b}Q_{b}(t)
 +
 e^{-{2\gamma_b}t} \frac{\partial C_{b}(t)}{\partial t}
\\
\end{eqnarray*}
%
%
\begin{eqnarray*}
 e^{-{2\gamma_a}t} \frac{\partial C_{a}(t)}{\partial t} 
 &=&
 -{2\gamma_b}
 Q_{b}(t-T) \cdot 1_{+}(t-T)
 \\&&{}
 +
 \frac{\partial Q_{0a}(t)}{\partial t}
\end{eqnarray*}
%
\begin{eqnarray*}
 e^{-{2\gamma_b}t} \frac{\partial C_{b}(t)}{\partial t} 
 &=&
 \displaystyle
 -{2\gamma_a}
 Q_{a}(t-T) \cdot 1_{+}(t-T)
 \\&&{}
  +
 \frac{\partial Q_{0b}(t)}{\partial t} 
\end{eqnarray*}
%
%
\begin{eqnarray*}
 \frac{\partial C_{a}(t)}{\partial t} 
 &=&
 -{2\gamma_b}
 e^{{2\gamma_a}t}Q_{b}(t-T) \cdot 1_{+}(t-T)
 \\&&{}
 +
 e^{{2\gamma_a}t} \frac{\partial Q_{0a}(t)}{\partial t}
\end{eqnarray*}
%
\begin{eqnarray*}
 \frac{\partial C_{b}(t)}{\partial t} 
 &=&
 \displaystyle
 -{2\gamma_a}
 e^{{2\gamma_b}t}Q_{a}(t-T) \cdot 1_{+}(t-T)
 \\&&{}
  +
 e^{{2\gamma_b}t}\frac{\partial Q_{0b}(t)}{\partial t} 
\end{eqnarray*}
%
%
\begin{eqnarray*}
 C_{a}(t)
 &=&
 -{2\gamma_b}
 \int_0^{t}
 e^{{2\gamma_a}\tau}Q_{b}(\tau-T)
 d\tau \cdot 1_{+}(t-T)
 \\&&{}
 +
 \int_0^{t}
 e^{{2\gamma_a}\tau} \frac{\partial Q_{0a}(\tau)}{\partial \tau} d\tau
 + C_{a}(0)
\end{eqnarray*}
%
\begin{eqnarray*}
 C_{b}(t)
 &=&
 -{2\gamma_a}
 \int_0^{t}
 e^{{2\gamma_b}\tau}Q_{a}(\tau-T)
 d\tau \cdot 1_{+}(t-T)
 \\&&{}
 +
 \int_0^{t}
 e^{{2\gamma_b}\tau} \frac{\partial Q_{0b}(\tau)}{\partial \tau} d\tau
 + C_{b}(0)
\end{eqnarray*}
%
\newpage\noindent 
%
%
\begin{eqnarray*}
 Q_{a}(t)
 &=&
 -{2\gamma_b}
 \int_0^{t}
 e^{-{2\gamma_a}(t-\tau)}Q_{b}(\tau-T)
 d\tau \cdot 1_{+}(t-T)
 \\&&{}
 +
 \int_{0+0}^{t}
 e^{-{2\gamma_a}(t-\tau)} d_{\tau} Q_{0a}(\tau)
 + e^{-{2\gamma_a}t} Q_{a}(0)
\end{eqnarray*}
%
\begin{eqnarray*}
 Q_{b}(t)
 &=&
 -{2\gamma_a}
 \int_0^{t}
 e^{-{2\gamma_b}(t-\tau)}Q_{a}(\tau-T)
 d\tau \cdot 1_{+}(t-T)
 \\&&{}
 +
 \int_{0+0}^{t}
 e^{{-2\gamma_b}(t-\tau)} d_{\tau} Q_{0b}(\tau)
 + e^{-{2\gamma_b}t} Q_{b}(0)
\end{eqnarray*}
%
 Thus, after returning to the terms of 
$ u \,,\, u_{0} $,
 we conclude that 
%

\medskip\noindent
\fbox{
\parbox{\textwidth}{
\medskip
\begin{eqnarray*}
 u(t,x_a)
 &=&
 -{2\gamma_b}
 \int_T^{t}
 e^{-{2\gamma_a}(t-\tau)}u(\tau-T,x_b)
 d\tau \cdot 1_{+}(t-T)
 \\&&{}
 +
 \int_{0+0}^{t}
 e^{-{2\gamma_a}(t-\tau)} d_{\tau} u_{0}(\tau,x_a)
 + e^{-{2\gamma_a}t} u_{0}(0,x_a)
\end{eqnarray*}
%
\begin{eqnarray*}
 u(t,x_b)
 &=&
 -{2\gamma_a}
 \int_T^{t}
 e^{-{2\gamma_b}(t-\tau)}u(\tau-T,x_a)
 d\tau \cdot 1_{+}(t-T)
 \\&&{}
 +
 \int_{0+0}^{t}
 e^{{-2\gamma_b}(t-\tau)} d_{\tau} u_{0}(\tau,x_b)
 + e^{-{2\gamma_b}t} u_{0}(0,x_b)
\end{eqnarray*}
} % end parbox
} % end fbox
%

\medskip\noindent
 Finally, recall that 

\medskip\noindent
%
\fbox{
\parbox{\textwidth}{
%
\begin{eqnarray*}
 u(t,x)
 &=&
 F_{a}(t-|x-x_a|/c)
 + F_{b}(t-|x-x_b|/c)
 + u_{0}(t,x)
\qquad (t \geq 0)
\\
\end{eqnarray*}
%
} % end parbox
} % end fbox

\medskip\noindent
%
 where
%
\begin{eqnarray*}
 F_{a}(t)
 &=&
 \Bigl( u(t,x_a) - u_{0}(t,x_a) \Bigr) \cdot 1_{+}(t)
\\&&{}
 -
 \Bigl(
 u(t-T,x_b) - u_{0}(t-T,x_b)
 \Bigr) \cdot 1_{+}(t-T)
\\&&{}
 +
 \Bigl(
 u(t-2T,x_a) - u_{0}(t-2T,x_a)
 \Bigr) \cdot 1_{+}(t-2T)
\\&&{}
 -
 \Bigl(
 u(t-3T,x_b) - u_{0}(t-3T,x_b)
 \Bigr) \cdot 1_{+}(t-3T)
\\&&{}
 + \cdots 
\\[\smallskipamount]\\
 F_{b}(t)
 &=&
 \Bigl( u(t,x_b) - u_{0}(t,x_b) \Bigr) \cdot 1_{+}(t)
\\&&{}
 -
 \Bigl(
 u(t-T,x_a) - u_{0}(t-T,x_a)
 \Bigr) \cdot 1_{+}(t-T)
\\&&{}
 +
 \Bigl(
 u(t-2T,x_b) - u_{0}(t-2T,x_b)
 \Bigr) \cdot 1_{+}(t-2T)
\\&&{}
 -
 \Bigl(
 u(t-3T,x_a) - u_{0}(t-3T,x_a)
 \Bigr) \cdot 1_{+}(t-3T)
\\&&{}
 + \cdots 
\\
\end{eqnarray*}
%

\newpage\noindent 
%
\section%
{ Explicit Relations }
\subsection%
{ The Algebra of
 ${\bf I}_{\gamma, T_0}$
 }
%
 In the previous section we have seen the recurrence relations, 
 which normally contain "retarded integral operations".
%
 When solving such relations, 
 a usual machinery involves a treatment of the various compositions
 of the operations that build the relations. 
%
 So,
 we try to find an effective representation
 of the referred compositions.

 Now, let 
${\bf I}_{\gamma, T_0}$
 denote the operation defined by
\footnote
{
 recall that, in the context, 
$\int_a^b$
 means 
$\int_a^b \cdot 1_{+}(b-a) $
} % end footnote 
%
\begin{eqnarray*}
 \Bigl( {\bf I}_{\gamma, T_0} f \Bigr)(t)
 &:=&
 \int_{T_0}^{t}
 e^{-{2\gamma}(t-\tau)} f(\tau-T_0)
 d\tau 
\\&&{} 
 \equiv
 \int_{T_0}^{t}
 e^{-{2\gamma}(t-\tau)} f(\tau-T_0)
 d\tau \cdot 1_{+}(t-T_0)
\\&&{} 
 =
 \int_{0}^{t-T_0}
 e^{-{2\gamma}(t-T_0-\tau)} f(\tau)
 d\tau \cdot 1_{+}(t-T_0)
\\&&{}
 \equiv
 \int_{0}^{t-T_0}
 e^{-{2\gamma}(t-T_0-\tau)} f(\tau)
 d\tau \cdot
\end{eqnarray*}
%
 Firstly, notice that
%
\begin{eqnarray*}
 \Bigl( {\bf I}_{\gamma, T_0} f \Bigr)(t)
 &=&
 \int_{0}^{t}
 e^{-{2\gamma}(t-T_0-\tau)}
 f(\tau) \cdot 1_{+}(t-T_0-\tau)
 d\tau \cdot 1_{+}(t)
\end{eqnarray*}
%
 In particular, the operations under consideration
 are usual convolution operations, and as such they are commutative:
%
\begin{eqnarray*}
  {\bf I}_{\gamma_1, T_1}{\bf I}_{\gamma_2, T_2}
 &=&
  {\bf I}_{\gamma_2, T_2}{\bf I}_{\gamma_1, T_1} \,. 
\end{eqnarray*}
%
 In addition notice that 
%
\begin{eqnarray*}
 \Bigl( {\bf I}_{\gamma, T_0} f \Bigr)(t+T_0)
 &=&
 \int_{0}^{t}
 e^{-{2\gamma}(t-\tau)} f(\tau)
 d\tau \cdot 1_{+}(t)
\\
 &=&
 \Bigl( {\bf I}_{\gamma,0} f \Bigr)(t)
\end{eqnarray*}
%
 Next, let us calculate
%
$$
  {\bf I}_{\gamma, T_1}{\bf I}_{\gamma, T_2}
$$
%
 The usual way is:
%
\begin{eqnarray*}
\makebox[5ex][l]{$\displaystyle
 \Bigl( {\bf I}_{\gamma, T_1}{\bf I}_{\gamma, T_2} f \Bigr)(t)
$} % end makebox
\\
&&{} =
 \int_{0}^{t-T_1}
 e^{-{2\gamma}(t-T_1-\tau)}
 \Bigl( {\bf I}_{\gamma, T_2} f \Bigr)(\tau)
 d\tau \cdot 1_{+}(t-T_1)
\\&&{} =
 \int_{0}^{t-T_1}
 e^{-{2\gamma}(t-T_1-\tau)}
 \Bigl( {\bf I}_{\gamma, T_2} f \Bigr)(\tau)
 \cdot 1_{+}(\tau-T_2)
 d\tau \cdot 1_{+}(t-T_1)
\\&&{} =
 \int_{0}^{t-T_1}
 e^{-{2\gamma}(t-T_1-\tau)}
 \Bigl( {\bf I}_{\gamma, T_2} f \Bigr)(\tau)
 \cdot 1_{+}(\tau-T_2)
 d\tau \cdot 1_{+}(t-T_1-T_2)
\\&&{} =
 \int_{T_2}^{t-T_1}
 e^{-{2\gamma}(t-T_1-\tau)}
 \Bigl( {\bf I}_{\gamma, T_2} f \Bigr)(\tau)
 d\tau \cdot 1_{+}(t-T_1-T_2)
\\&&{} =
 \int_{0}^{t-T_1-T_2}
 e^{-{2\gamma}(t-T_1-T_2-\tau)}
 \Bigl( {\bf I}_{\gamma, T_2} f \Bigr)(\tau+T_2)
 d\tau \cdot 1_{+}(t-T_1-T_2)
\\&&{} =
 \int_{0}^{t-T_1-T_2}
 e^{-{2\gamma}(t-T_1-T_2-\tau)}
 \Bigl( {\bf I}_{\gamma, 0} f \Bigr)(\tau)
 d\tau \cdot 1_{+}(t-T_1-T_2)
\\&&{} =
 \Bigl( {\bf I}_{\gamma, T_1+T_2}{\bf I}_{\gamma, 0} f \Bigr)(t)
\end{eqnarray*}
%
 Thus, we have seen that 
%
\begin{eqnarray*}
 {\bf I}_{\gamma, T_1}{\bf I}_{\gamma, T_2}
 &=&
 {\bf I}_{\gamma, T_1+T_2}{\bf I}_{\gamma, 0} \,.
\end{eqnarray*}
%
  Consequences which we need:
%
\begin{eqnarray*}
 {\bf I}_{\gamma_b, 2T}^N
 &=&
 {\bf I}_{\gamma_b, 2NT}{\bf I}_{\gamma_b, 0}^{N-1} \,,
\end{eqnarray*}
%
%
\begin{eqnarray*}
 ({\bf I}_{\gamma_b, T})^{2N}
 &=&
 {\bf I}_{\gamma_b, 2NT}{\bf I}_{\gamma_b, 0}^{2N-1} \,,
\end{eqnarray*}
%
%
\begin{eqnarray*}
 ({\bf I}_{\gamma_b, T})^{2N+1}
 &=&
 {\bf I}_{\gamma_b, 2NT}{\bf I}_{\gamma_b, 0}^{2N} \,,
\end{eqnarray*}
%
%
\begin{eqnarray*}
 ({\bf I}_{\gamma_a, T}{\bf I}_{\gamma_b, T})^N
 &=&
 ({\bf I}_{\gamma_b, T}{\bf I}_{\gamma_a, T})^N
\\
 &=&
 {\bf I}_{\gamma_b, T}{\bf I}_{\gamma_a, T}
 {\bf I}_{\gamma_b, 0}^{N-1}{\bf I}_{\gamma_a, 0}^{N-1} \,.
\end{eqnarray*}
%
 In addition, let 
$E_{\gamma}$
 denote the operation defined by 
%
\begin{eqnarray*}
 \Bigl( E_{\gamma} f \Bigr)(t)
 &:=&
 e^{\gamma t}f(t) \,.
\end{eqnarray*}
%
 Then we can write
%
\begin{eqnarray*}
 {\bf I}_{\gamma, 0} 
 &=&
 E_{\gamma}^{-1}{\bf I}_{0,0}E_{\gamma} \,,
\end{eqnarray*}
%
 and then 
%
\begin{eqnarray*}
 {\bf I}_{\gamma, 0}^2 
 &=&
 E_{\gamma}^{-1}{\bf I}_{0,0}E_{\gamma}E_{\gamma}^{-1}{\bf I}_{0,0}E_{\gamma}
\\
&&{} =
E_{\gamma}^{-1}{\bf I}_{0,0}^2E_{\gamma} \,.
\end{eqnarray*}
%
 Iterating this kind of arguments we infer that 
%
\begin{eqnarray*}
 {\bf I}_{\gamma, T_0}^N
 &=&
 {\bf I}_{\gamma, T_0}E_{\gamma}^{-1}{\bf I}_{0,0}^{N-1}E_{\gamma}
\end{eqnarray*}
%
 and then 
%
\begin{eqnarray*}
 ( {\bf I}_{\gamma, T_0}^N f )(t)
 &=&
 \int_{0}^{t-NT_0}
 \int_{0}^{\tau}
 \int_{0}^{\tau_1}
 \cdots
 \int_{0}^{\tau_{N-2}}
 e^{-{2\gamma}(t-NT_0-\tau_{N-2})} f(\tau_{N-2})
 d\tau_{N-2}
 \cdots
 d\tau_1
 d\tau
\\
 &=&
 \int_{0}^{t-NT_0}
 \frac{(t-NT_0-\tau)^{N-1}}{(N-1)!}
 e^{-{2\gamma}(t-NT_0-\tau)} f(\tau)
 d\tau
\\
 &\equiv&
 \int_{0}^{t}
 \frac{(t-NT_0-\tau)^{N-1}}{(N-1)!}
 e^{-{2\gamma}(t-NT_0-\tau)} f(\tau) \cdot 1_{+}(t-NT_0-\tau)
 d\tau \,.
\end{eqnarray*}
%
 Now, we go on to see consequences of these consequences.
%
\newpage\noindent 
%
\subsection%
{ Explicit Relations:
 The Case of %%Der Fall von 
 $\quad u(t,x_a) = 0\,,\quad \gamma_b \not= \infty $ }
%
 We have seen that
%

\begin{eqnarray*}
 u(t,x_b)
 &=&
 {2\gamma_b}
 \int_{2T}^{t}
 e^{-{2\gamma_b}(t-\tau )} u(\tau-2T,x_b)
 d\tau
\\&&{}
 +
  I_{0b}(t) \,,
\\
\makebox[0ex][l]{ where }
\\
 I_{0b}(t)
&:=&
 \int_{0+0}^{t}
 e^{-{2\gamma_b}(t-\tau )}
 d_{\tau} 
 ( u_{0}(\tau,x_b) - u_{0}(\tau-T,x_a)1_{+}(\tau-T) )
 +
 e^{-{2\gamma_b}t} u(0,x_b)
 \,.
\\
\end{eqnarray*}
%
 Recall, 
${\bf I}_{\gamma, T_0}$
 denote the operation defined by
%
\begin{eqnarray*}
 \Bigl( {\bf I}_{\gamma, T_0} f \Bigr)(t)
 &:=&
 \int_{T_0}^{t}
 e^{-{2\gamma}(t-\tau)} f(\tau-T_0)
 d\tau 
\\&&{} 
 =
 e^{2{\gamma}T_0}
 \int_{0}^{t-T_0}
 e^{-{2\gamma}(t-\tau)} f(\tau)
 d\tau 
\\&&{}
 \equiv
 e^{2{\gamma}T_0}
 \int_{-\infty}^{\infty}
 e^{-{2\gamma_b}(t-\tau)} f(\tau) \cdot 1_{+}(t-T_0-\tau) \cdot 1_{+}(\tau)
 d\tau \cdot 1_{+}(t-T_0) \,.
\end{eqnarray*}
%
 Thus, we can write 
%
\begin{eqnarray*}
 u(t,x_b)
 &=&
 I_{0b}(t)
\,,\quad ( \mbox{ if } 0 \leq t < 2T )
\end{eqnarray*}
%
%
\begin{eqnarray*}
 u(t,x_b)
 &=&
 \biggl(\Bigl(
 1 + ({2\gamma_b}) {\bf I}_{\gamma_b, 2T}
 \Bigr) I_{0b}\biggr) (t)
\,,\quad ( \mbox{ if } 0 \leq t < 4T )
\end{eqnarray*}
%
%
\begin{eqnarray*}
 u(t,x_b)
 &=&
 \biggl(\Bigl(
 1 + ({2\gamma_b}) {\bf I}_{\gamma_b, 2T}
 + ({2\gamma_b})^2 {\bf I}_{\gamma, 2T}^2
 + \cdots
 + ({2\gamma_b})^N {\bf I}_{\gamma_b, 2T}^N
 \Bigr) I_{0b}\biggr) (t)
\,,\quad ( \mbox{ if } 0 \leq t < (N+1)2T )
\end{eqnarray*}
%
 So, iterating, 
%
\begin{eqnarray*}
 u(t,x_b)
 &=&
 \biggl(\Bigl(
 1 + ({2\gamma_b}) {\bf I}_{\gamma_b, 2T}
 + ({2\gamma_b})^2 {\bf I}_{\gamma, 2T}^2
 + \cdots
 + ({2\gamma_b})^N {\bf I}_{\gamma_b, 2T}^N \cdots 
 \Bigr) I_{0b}\biggr) (t)
\end{eqnarray*}
%
%
 Finally, by recalling that 
%
\begin{eqnarray*}
 ( {\bf I}_{\gamma, T_0}^N f )(t)
 &=&
 \int_{0}^{t}
 \frac{(t-NT_0-\tau)^{N-1}}{(N-1)!}
 e^{-{2\gamma}(t-NT_0-\tau)} f(\tau) \cdot 1_{+}(t-NT_0-\tau)
 d\tau
\end{eqnarray*}
%
 and after introducing 
%
\begin{eqnarray*}
\makebox[3ex][l]{$ Exp(\lambda,T_0,t) $}
\\&:=&
 1 \cdot 1_{+}(t) + \lambda (t-T_0) \cdot 1_{+}(t-T_0)
 + \lambda^2 \frac{(t-2T_0)^2}{2!} \cdot 1_{+}(t-2T_0) 
\\&&{}
 + \cdots + \lambda^N \frac{(t-NT_0)^N}{N!} \cdot 1_{+}(t-NT_0) \cdots
\end{eqnarray*}
%
 we conclude that
%
%
\begin{eqnarray*}
 u(t,x_b)
 &=&
  I_{0b}(t)
\\&&{}
 +
 ({2\gamma_b})
 \int_{2T}^{t}
 e^{-{2\gamma_b}(t-\tau )}
  I_{0b}(\tau-2T)
 d\tau
\\&&{}
 +
 ({2\gamma_b})^2
 \int_{4T}^{t}
 \int_{4T}^{\tau}
 e^{-{2\gamma_b}(t-\tau_1 )} I_{0b}(\tau_1-4T)
 d\tau_1
 d\tau
\\&&{}
 + \cdots 
\\
 &=&
  I_{0b}(t)
\\&&{}
 +
 ({2\gamma_b})
 \int_{0}^{t-2T}
 e^{-{2\gamma_b}(t-2T-\tau )}
  I_{0b}(\tau)
 d\tau
\\&&{}
 +
 ({2\gamma_b})^2
 \int_{0}^{t-4T}
 \int_{0}^{\tau}
 e^{-{2\gamma_b}(t-4T-\tau_1 )} I_{0b}(\tau_1)
 d\tau_1
 d\tau
\\&&{}
 + \cdots 
\\
\\&=&
 I_{0b}(t)
 +2{\gamma_b}e^{4{\gamma_b}T}
 \int_{0}^{t}
 Exp(2{\gamma_b}e^{4{\gamma_b}T}, 2T, t-\tau-2T)
 e^{-{2\gamma_b}(t-\tau )}
  I_{0b}(\tau)
 d\tau
\\
\\
\makebox[0ex][l]{ where }
\\
 I_{0b}(t)
 &:=&
 \int_{0+0}^{t}
 e^{-{2\gamma_b}(t-\tau )}
 d_{\tau} 
 ( u_{0}(\tau,x_b) - u_{0}(\tau-T,x_a)1_{+}(\tau-T) )
 +
 e^{-{2\gamma_b}t} u(0,x_b) \,.
\\
\end{eqnarray*}
%


\newpage\noindent 
%
\subsection%
{ Explicit Relations:
 The Case of %%Der Fall von 
 $\gamma_a \not= \infty \,,\quad \gamma_b \not= \infty $ }
%
 Imitating the arguments of the previous subsections, we observe that
%
\begin{eqnarray*}
 u(t,x_a)
 &=&
 -{2\gamma_b}
 \int_T^{t}
 e^{-{2\gamma_a}(t-\tau)}u(\tau-T,x_b)
 d\tau \cdot 1_{+}(t-T)
 \\&&{}
 +
 \int_{0+0}^{t}
 e^{-{2\gamma_a}(t-\tau)} d_{\tau} u_{0}(\tau,x_a)
 + e^{-{2\gamma_a}t} u_{0}(0,x_a)
\end{eqnarray*}
%
\begin{eqnarray*}
 u(t,x_b)
 &=&
 -{2\gamma_a}
 \int_T^{t}
 e^{-{2\gamma_b}(t-\tau)}u(\tau-T,x_a)
 d\tau \cdot 1_{+}(t-T)
 \\&&{}
 +
 \int_{0+0}^{t}
 e^{{-2\gamma_b}(t-\tau)} d_{\tau} u_{0}(\tau,x_b)
 + e^{-{2\gamma_b}t} u_{0}(0,x_b)
\end{eqnarray*}
%
 and then after defining the operation
${\bf I}_{\gamma, T}$
 by
%
\begin{eqnarray*}
 \Bigl( {\bf I}_{\gamma, T} f \Bigr)(t)
 &:=&
 \int_{T}^{t}
 e^{-{2\gamma_b}(t-\tau)} f(\tau-T)
 d\tau 
\end{eqnarray*}
%
 we infer that 
%
\begin{eqnarray*}
\left(
 \begin{array}{c}
 Q_{a} \\ Q_{b}
 \end{array}
\right)
 &=&
\left(
 \begin{array}{cc}
 0 & -2{\gamma_b}{\bf I}_{\gamma, T}
 \\
  -2{\gamma_a}{\bf I}_{\gamma, T} & 0
 \end{array}
\right)
\left(
 \begin{array}{c}
 Q_{a} \\ Q_{b}
 \end{array}
\right)
 +
\left(
 \begin{array}{c}
 u_{0,a,\gamma_a} \\ u_{0,b,\gamma_b}
 \end{array}
\right)
\end{eqnarray*}
%
 where
%
\begin{eqnarray*}
\left(
 \begin{array}{c}
 Q_{a}(t) \\ Q_{b}(t)
 \end{array}
\right)
 &:=&
\left(
 \begin{array}{c}
 u(t,x_a) \\ u(t,x_b)
 \end{array}
\right) \,,
\end{eqnarray*}
%
\begin{eqnarray*}
\left(
 \begin{array}{c}
 u_{0,a,\gamma_a}(t) \\ u_{0,b,\gamma_b}(t)
 \end{array}
\right)
 &:=&
\left(
 \begin{array}{c}
 \int_{0+0}^{t}
 e^{-{2\gamma_a}(t-\tau)} d_{\tau} u_{0}(\tau,x_a)
 + e^{-{2\gamma_a}t} u_{0}(0,x_a)
\\
 \int_{0+0}^{t}
 e^{{-2\gamma_b}(t-\tau)} d_{\tau} u_{0}(\tau,x_b)
 + e^{-{2\gamma_b}t} u_{0}(0,x_b)
 \end{array}
\right)
 \quad .
\end{eqnarray*}
%
 So,
%
\begin{eqnarray*}
\left(
 \begin{array}{c}
 Q_{a} \\ Q_{b}
 \end{array}
\right)
 &=&
\sum_{N=0}^{\infty}
\left(
 \begin{array}{cc}
 0 & -2{\gamma_b}{\bf I}_{\gamma_b, T}
 \\
  -2{\gamma_a}{\bf I}_{\gamma_a, T} & 0
 \end{array}
\right)^N
\left(
 \begin{array}{c}
 u_{0,a,\gamma_a} \\ u_{0,b,\gamma_b}
 \end{array}
\right)
\end{eqnarray*}
%
 Since 
%
\begin{eqnarray*}
 {\bf I}_{\gamma_b, T}{\bf I}_{\gamma_a, T}
&=&
 {\bf I}_{\gamma_a, T}{\bf I}_{\gamma_b, T} \,,
\end{eqnarray*}
%
\begin{eqnarray*}
\left(
 \begin{array}{cc}
 0 & -2{\gamma_b}{\bf I}_{\gamma_b, T}
 \\
  -2{\gamma_a}{\bf I}_{\gamma_a, T} & 0
 \end{array}
\right)^{2n}
 &=&
 (4{\gamma_a}{\gamma_b})^n ({\bf I}_{\gamma_b, T}{\bf I}_{\gamma_a, T})^n
\left(
 \begin{array}{cc}
 1 & 0
 \\
 0 & 1
 \end{array}
\right) \,,
\end{eqnarray*}
%
%
\begin{eqnarray*}
\left(
 \begin{array}{cc}
 0 & -2{\gamma_b}{\bf I}_{\gamma_b, T}
 \\
  -2{\gamma_a}{\bf I}_{\gamma_a, T} & 0
 \end{array}
\right)^{2n+1}
 &=&
 -
 (4{\gamma_a}{\gamma_b})^n ({\bf I}_{\gamma_b, T}{\bf I}_{\gamma_a, T})^n
\left(
 \begin{array}{cc}
 0 & 2\gamma_b{\bf I}_{\gamma_b, T}
 \\
 2\gamma_a {\bf I}_{\gamma_a, T} & 0
 \end{array}
\right) \,,
\end{eqnarray*}
%
 we infer that 
%
\begin{eqnarray*}
\left(
 \begin{array}{c}
 Q_{a} \\ Q_{b}
 \end{array}
\right)
 &=&
\sum_{n=0}^{\infty}
 (4{\gamma_a}{\gamma_b})^n ({\bf I}_{\gamma_b, T}{\bf I}_{\gamma_a, T})^n
\left(
 \begin{array}{cc}
 1 & 0
 \\
 0 & 1
 \end{array}
\right)
\left(
 \begin{array}{c}
 u_{0,a,\gamma_a} \\ u_{0,b,\gamma_b}
 \end{array}
\right)
\\
&&{} -
\sum_{n=0}^{\infty}
 (4{\gamma_a}{\gamma_b})^n ({\bf I}_{\gamma_b, T}{\bf I}_{\gamma_a, T})^n
\left(
 \begin{array}{cc}
 0 & 2\gamma_b{\bf I}_{\gamma_b, T}
 \\
 2\gamma_a {\bf I}_{\gamma_a, T} & 0
 \end{array}
\right)
\left(
 \begin{array}{c}
 u_{0,a,\gamma_a} \\ u_{0,b,\gamma_b}
 \end{array}
\right)
\\[\bigskipamount]\\
 &=&
\sum_{n=0}^{\infty}
 (4{\gamma_a}{\gamma_b})^n ({\bf I}_{\gamma_b, T}{\bf I}_{\gamma_a, T})^n
\left(
 \begin{array}{c}
 u_{0,a,\gamma_a} \\ u_{0,b,\gamma_b}
 \end{array}
\right)
\\
&&{} -
\sum_{n=0}^{\infty}
(4{\gamma_a}{\gamma_b})^n ({\bf I}_{\gamma_b, T}{\bf I}_{\gamma_a, T})^n
\left(
 \begin{array}{c}
 2\gamma_b{\bf I}_{\gamma_b, T} u_{0,b,\gamma_b}
 \\
 2\gamma_a{\bf I}_{\gamma_a, T} u_{0,a,\gamma_a}
 \end{array}
\right)
\end{eqnarray*}
%
 Unfortunately, I do not know a good representation of this formala, 
 in the general case. Nevertheless, if
%
$$
  \gamma_a = \gamma_b =: \gamma_0  \,,
$$
 then the situation becomes very similar to that of the previous subsection:
 we infer that 
%
\begin{eqnarray*}
\left(
 \begin{array}{c}
 Q_{a} \\ Q_{b}
 \end{array}
\right)
 &=&
\sum_{n=0}^{\infty}
 (2{\gamma_0})^{2n}
\left(
 \begin{array}{cc}
 1 & 0
 \\
 0 & 1
 \end{array}
\right)
 {\bf I}_{\gamma_0, T}^{2n}
\left(
 \begin{array}{c}
 u_{0,a,\gamma_0} \\ u_{0,b,\gamma_0}
 \end{array}
\right)
\\
&&{} -
\sum_{n=0}^{\infty}
 (2{\gamma_0})^{2n+1}
\left(
 \begin{array}{cc}
 0 & 1
 \\
 1 & 0
 \end{array}
\right)
 {\bf I}_{\gamma_0, T}^{2n+1}
\left(
 \begin{array}{c}
 u_{0,a,\gamma_0} \\ u_{0,b,\gamma_0}
 \end{array}
\right)
\\[\bigskipamount]\\
 &=&
\sum_{n=0}^{\infty}
 (2{\gamma_0})^{2n}
 {\bf I}_{\gamma_0, T}^{2n}
\left(
 \begin{array}{c}
 u_{0,a,\gamma_0} \\ u_{0,b,\gamma_0}
 \end{array}
\right)
\\
&&{} -
\sum_{n=0}^{\infty}
 (2{\gamma_0})^{2n+1}
 {\bf I}_{\gamma_0, T}^{2n+1}
\left(
 \begin{array}{c}
 u_{0,b,\gamma_0} \\ u_{0,a,\gamma_0}
 \end{array}
\right)
\end{eqnarray*}
%
 Finally,
 as in the previous subsection,
 by recalling that 
%
\begin{eqnarray*}
 ( {\bf I}_{\gamma, T_0}^{2n} f )(t)
 &=&
 \int_{0}^{t}
 \frac{(t-2nT_0-\tau)^{2n-1}}{(2n-1)!}
 e^{-{2\gamma}(t-2nT_0-\tau)} f(\tau) \cdot 1_{+}(t-2nT_0-\tau)
 d\tau
\end{eqnarray*}
%
\begin{eqnarray*}
 ( {\bf I}_{\gamma, T_0}^{2n+1} f )(t)
 &=&
 \int_{0}^{t}
 \frac{(t-(2n+1)T_0-\tau)^{2n}}{(2n)!}
 e^{-{2\gamma}(t-(2n+1)T_0-\tau)} f(\tau) \cdot 1_{+}(t-(2n+1)T_0-\tau)
 d\tau
\end{eqnarray*}
%
 and after introducing 
%
\begin{eqnarray*}
\makebox[3ex][l]{$ Sinh(\lambda,T_0,t) $}
\\&:=&
 \lambda (t-T_0) \cdot 1_{+}(t-T_0)
 + \lambda^3 \frac{(t-3T_0)^3}{3!} \cdot 1_{+}(t-3T_0) 
\\&&{}
 + \cdots + \lambda^{2n+1} \frac{(t-(2n+1)T_0)^{2n+1}}{(2n+1)!}
 \cdot 1_{+}(t-(2n+1)T_0) \cdots
\\
\makebox[3ex][l]{$ Cosh(\lambda,T_0,t) $}
\\&:=&
 1 \cdot 1_{+}(t)
 + \lambda^2 \frac{(t-2T_0)^2}{2!} \cdot 1_{+}(t-2T_0) 
\\&&{}
 + \cdots + \lambda^{2n} \frac{(t-2nT_0)^{2n}}{(2n)!}
 \cdot 1_{+}(t-2nT_0) \cdots
\end{eqnarray*}
%
 we observe that
%
\begin{eqnarray*}
 \Bigl(
\sum_{n=0}^{\infty}
 (2{\gamma_0})^{2n}
 {\bf I}_{\gamma_0, T}^{2n} f
 \Bigr)(t)
&=&
 f(t) +
 2{\gamma_0}e^{2{\gamma_0}T}
 \int_{0}^{t}Sinh(2{\gamma_0}e^{2{\gamma_0}T} ,T ,t-T-\tau)
 e^{-{2\gamma}(t-\tau)} f(\tau) d\tau
\end{eqnarray*}
%
\begin{eqnarray*}
 \Bigl(
\sum_{n=0}^{\infty}
 (2{\gamma_0})^{2n+1}
 {\bf I}_{\gamma_0, T}^{2n+1} f
 \Bigr)(t)
&=&
 2{\gamma_0}e^{2{\gamma_0}T}
 \int_{0}^{t}Cosh(2{\gamma_0}e^{2{\gamma_0}T} ,T ,t-T-\tau)
  e^{-{2\gamma}(t-\tau)} f(\tau) d\tau
\end{eqnarray*}

\bigskip

 The conclusion is evident.









\newpage 

\bibliographystyle{unsrt}
\begin{thebibliography}{DalKre}
%\bibitem[]{}
\bibitem[AK]{AK} 
 {\sc S. Albeverio and P. Kurasov,} 
 {\it Singular Perturbations of Differential Operators. 
 Solvable Scr\"odinger Type Operators}, 
 London Mathematical Society: 
 Lecture Note Series. 271, 1999.
 {\sc Cambridge university press}


\bibitem[BF]{BF} 
 {\sc F.A. Berezin, L.D. Faddeev} , 
 {\it Remark on the Schr\"odinger equation with singular potential}, 
 Dokl. Akad. Nauk. SSSR, 137 (1961) 1011-1014 (in Russian).

\bibitem[BST56]{BST56} 
 {\sc Budak, B.M.; Samarskij, A.A.; Tikhonov, A.N. }
 {\it Aufgabensammlung zur mathematischen Physik.} (Russian)
 Moskau: Staatsverlag f\"ur technisch-theoretische Literatur 1956.
 684 S. (1956).
\\
 see especially `Chapter 2, \S 2, 2., Problems 69, 73, 74', and ibid., 
 `3., Problems 79, 80, 81, 82. '

\bibitem[BST80]{BST80} 
 {\sc
 Budak, B.M.; Samarskij, A.A.; Tikhonov, A.N.
 ( = Tichonov, Tichonow, Tychonoff)
 }
 {\it A collection of problems of mathematical physics } (English)
 Translated by A.R.M. Robson. Translation edited by D.M. Brink 
 (International Series of Monographs in Pure and Applied Mathematics Vo. 52) 
 Oxford-London-New York-Paris: Pergamon Press 1964, XII, 768 p. 80 s. (1964). 

\bibitem[Do]{Do} 
 {\sc W. Donoghue }, 
 {\it On the perturbation of spectra},
 Comm. Pure App. Math. 18 (1965) 559-579


\bibitem[Fog]{Fog} 
 {\sc S.R. Foguel}, 
 {\it Finite Dimensional Perturbations in Banach Spaces}, 
 American Journal of Mathematics, 
 Volume 82, Issue 2 ( Apr., 1960 ), 260-270


\bibitem[Jack]{Jack}
 {\sc J.D.~Jackson:} {\it Classical electrodynamics}.
\\ John Wiley \& Sons, Inc. New York-London, 1962.


\bibitem[RT79]{RT79}%
{\sc Razumov,A.V.; Taranov,A.Ju.} 

 Dipole interaction of an oscillator with a scalar field. 
 (Russian) 
\\{}
 {\it Teoret. Mat. Fiz.} {\bf 38} (1979), no. 3, 355--363. 
 {\bf MR 80b:81034} 

\bigskip 

 {\bf Electronic Print: }

 \mbox{ Mathematical Physics Preprint Archive \quad mp\_arc }

 (\verb" http://www.ma.utexas.edu/mp_arc/index.html ") :

\bigskip

\bibitem[BrDeB]{BrDeB}
  mp\_arc 01-275 

{\sc L. Bruneau, S. DeBievre }
A Hamiltonian model for linear friction in a homogeneous medium 
(548K,postscript) Jul 17, 01 

\begin{minipage}{\textwidth}
\begin{verbatim}
http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=01-275
http://mpej.unige.ch/mp_arc-bin/mpa?yn=01-275
http://www.maia.ub.es/mp_arc-bin/mpa?yn=01-275
\end{verbatim}
\end{minipage}


\bibitem[DerFr]{DerFr}
 mp\_arc 02-275

{\sc Derezinski J., Fruboes R. }
Renormalization of the Friedrichs Hamiltonian
(16K, LATeX 2e)

\begin{minipage}{\textwidth}
\begin{verbatim}
http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=02-275
http://mpej.unige.ch/mp_arc-bin/mpa?yn=02-275
http://www.maia.ub.es/mp_arc-bin/mpa?yn=02-275
\end{verbatim}
\end{minipage}


\bigskip

 LANL E-Print ( \verb"http://arXiv.org/" ):

\bibitem[AMN]{AMN}
 Paper: physics/0001009
\\ From: \verb"adolfo@lafexSu1.lafex.cbpf.br" (Adolfo Malbouisson)
\\ Date: Wed, 5 Jan 2000 19:20:08 GMT   (15kb)
\\[2ex]
 Title: 
 An Exact Approach to the Oscillator Radiation Process in an Arbitrarily
 Large Cavity
\\ Authors: N.P. Andion, A.P.C. Malbouisson and A. Mattos Neto
\\ Comments: 27 pages
\\ Subj-class: Atomic Physics; Mathematical Physics
\\

\bibitem[CGS]{CGS}
 Paper: quant-ph/0307232
\\From: Ricardo Moritz Cavalcanti \verb"rmoritz@if.ufrj.br"
\\Date: Wed, 30 Jul 2003 18:14:48 GMT   (20kb)
\\[2ex]
Title: Decay in a uniform field: An exactly solvable model
\\Authors: R. M. Cavalcanti, P. Giacconi and R. Soldati
\\Comments: 21 pages, 2 figures
\\Subj-class: Quantum Physics; Atomic Physics; Mathematical Physics
\\ ( \verb"http://arXiv.org/abs/quant-ph/0307232" ,  20kb)

\bibitem[Ch1]{Ch1}
 Paper: math.DS/0301167
\\ From: Sergej A. Choroszavin \verb"sergius@pve.vsu.ru"
\\ Date: Thu, 16 Jan 2003 04:34:16 GMT   (18kb)
\\[2ex]
 Title: An Interaction of An Oscillator with An One-Dimensional Scalar Field.
  Simple Exactly Solvable Models based on Finite Rank Perturbations Methods.
 I:
  D'Alembert-Kirchhoff-like formulae
\\ Author: Sergej A. Choroszavin
\\ Comments: Latex 2.09
\\ Subj-class: Dynamical Systems; Mathematical Physics
\\ ( \verb"http://arXiv.org/abs/math/0301167" ,  18kb)

\bibitem[Ch2]{Ch2}
 Paper: math-ph/0302038
\\ From: Sergej A. Choroszavin \verb"sergius@pve.vsu.ru"
\\ Date: Sat, 15 Feb 2003 02:14:09 GMT   (15kb)
\\[2ex]
Title: An Interaction of An Oscillator with An One-Dimensional Scalar Field.
  Simple Exactly Solvable Models based on Finite Rank Perturbations Methods.
  II: Resolvents formulae
\\ Author: Sergej A. Choroszavin
\\ Comments: Latex 2.09
\\ Subj-class: Mathematical Physics; Dynamical Systems
\\ ( \verb"http://arXiv.org/abs/math-ph/0302038" ,  15kb)

\bibitem[Ch3]{Ch3}
 Paper: math-ph/0307029
\\ From: Sergej Choroszavin \verb"sergius@pve.vsu.ru"
\\ Date: Sun, 13 Jul 2003 22:32:33 GMT   (14kb)
\\[2ex]
Title: 1D Particle, 1D Field, 1D Interaction. Simple Exactly Solvable Models
  based on Finite Rank Perturbations Methods. III. Linear Friction as Radiation
  Reaction
\\ Authors: Sergej A. Choroszavin
\\ Comments: Latex 2.09
\\ Subj-class: Mathematical Physics; Dynamical Systems
\\ ( \verb"http://arXiv.org/abs/math-ph/0307029" ,  14kb)


\bibitem[MPB]{MPB}
 Paper: hep-th/9207033
\\ From: \verb"physth@ulb.ac.be"
\\ Date: Fri, 10 Jul 92 16:52:54 +0200   (20kb)

 Title: 
 On the Problem of the Uniformly Accelerated Oscillator
\\ Authors: S. Massar, R. Parentani, R. Brout
\\ Comments: 14 pages (+postscript figures attached), ULB-TH-03/92


\end{thebibliography}  



\end{document}

ENDBODY
