%\documentstyle[12pt,fleqn,a4wide,german,twoside,epsf,titlepage]{article}
%\input{defTG}
%\begin{document}
\setlength{\unitlength}{1.4mm}
\begin{picture}(100,80)
\put(35,78){\bf Figure 1}
\put( 5,15){\line(3,1){60}}
\put(35, 5){\line(3,1){60}}
\put( 5,45){\line(3,1){60}}
\put(35,35){\line(3,1){60}}
\put( 5,15){\line(3,-1){30}}
\put(65,35){\line(3,-1){30}}
\put( 5,45){\line(3,-1){30}}
\put(65,65){\line(3,-1){30}}
\put( 5,15){\line(0, 1){30}}
\put(35, 5){\line(0, 1){30}}
\put(65,35){\line(0, 1){30}}
\put(95,25){\line(0, 1){30}}
\multiput(55,11.66)(0,3){10}{\circle*{0.2}}
\multiput(75,18.33)(0,3){10}{\circle*{0.2}}
\multiput(25,21.66)(0,3){10}{\circle*{0.2}}
\multiput(45,28.33)(0,3){10}{\circle*{0.2}}
\multiput(25,21.66)(3,-1){10}{\circle*{0.2}}
\multiput(45,28.33)(3,-1){10}{\circle*{0.2}}
\multiput(25,51.66)(3,-1){10}{\circle*{0.2}}
\multiput(45,58.33)(3,-1){10}{\circle*{0.2}}
\put(12,10.66){\vector(-3, 1){10}}
\put(22, 7.33){\vector( 3,-1){10}}
\put(48, 7.33){\vector(-3,-1){10}}
\put(88,20.66){\vector( 3, 1){10}}
\put( 2,24   ){\vector( 0,-1){10}}
\put( 2,34   ){\vector( 0, 1){10}}
\put(47.5, 4.16){\vector(-3,-1){5}}
\put(57.5, 7.50){\vector( 3, 1){5}}
\put(57, 4.33){\vector(-3,-1){10}}
\put(77,11.00){\vector( 3, 1){10}}
\put(15,45){(1)}
\put(35,51.66){(2)}
\put(55,58.33){(3)}
\put(68,13){$A_1$}
\put(14, 7){$A_2$}
\put( 0,29){$A_3$}
\put(51.0, 5.33){$R_1^\iKe$}
\put(65.5, 7.16){$R_1^\iKz$}
\put(39, 8){$\eve$}
\put(29, 8){$\evz$}
\put(36,11){$\evd$}

\thicklines
\put(24,15.33){\vector(1,2){12}}
\put(50,20   ){\vector(0,1){30}}
\put(64,24.66){\vector(1,4){8}}
\put(35, 5){\vector( 3, 1){6}}
\put(35, 5){\vector(-3, 1){6}}
\put(35, 5){\vector( 0, 1){8}}
\thinlines
\put(53,49){$\vec{\cal E}$}
\end{picture}

\ \vspace{1cm}\\
\begin{description}
\item[{\sc Fig.} {\rm 1.}]
Sketch of the electric field configuration in a box divided into three segments (1)--(3). The electric field is homogeneous within every segment and varies suddenly along the 1--direction at the interfaces.
\end{description}


%\end{document}
%\documentstyle[rotating,12pt,fleqn,a4wide,german,twoside,epsfig,titlepage] {article}
%\input{defTG}
%\begin{document}
%\thispagestyle{empty}
\setlength{\unitlength}{0.8mm}

%\ \vspace{1cm}\newline
\begin{center}
\begin{picture}(200,95)

\put(0,0){\epsfig{file=figure2a.eps,width=180\unitlength,height=100\unitlength}}
  

%\multiput(0,0)(0,10){11}{\line(1,0){180}}
%\multiput(0,0)(10,0){20}{\line(0,1){100}}

\put(75,105){\bf Figure 2a}
%\put( 38, 8){$\balp=(2S\eh,|F_3|=\eh;\nv=(1,1,1))$}
\put( 75,-10){$\cE_2^{'\iKd}-\cE_{2,res}^{'\iKd}$ [V/cm]}

\begin{rotate}{90}
\put(10,0){$\lg|\Re\lk(E_{\pm}(\cEv^{'\iKsig})-E_{\pm}(\cEvR^{'\iKsig})\rk)/(h\,{\mathrm Hz})|$}
\end{rotate}

\end{picture}
\ \vspace{2cm}\nopagebreak\newline
%\ \newline
\begin{picture}(200,95)

\put(0,0){\epsfig{file=figure2b.eps,width=180\unitlength,height=100\unitlength}}
  

%\multiput(0,0)(0,10){11}{\line(1,0){180}}
%\multiput(0,0)(10,0){20}{\line(0,1){100}}

\put(75,105){\bf Figure 2b}
%\put( 38, 8){$\balp=(2S\eh,|F_3|=\eh;\nv=(1,1,1))$}
\put( 75,-10){$\cE_2^{'\iKd}-\cE_{2,res}^{'\iKd}$ [V/cm]}

\begin{rotate}{90}
\put(10,0){$\lg|2\Im\lk(E_{\pm}(\cEv^{'\iKsig})-E_{\pm}(\cEvR^{'\iKsig})\rk)/(\hbar\,{\mathrm s^{-1}})|$}
\end{rotate}

\end{picture}
\end{center}

\small
%
\ \vspace{0.5cm}\\
\begin{description}
\item[{\sc Fig.} {\rm 2.}]
The decadic logarithm of the absolute values of the real (a) and imaginary (b) parts of the P--violating energy--difference $E_{\pm}(\cEv^{'\iKsig})-E_{\pm}(\cEvR^{'\iKsig})$ (\ref{eq14}) vs.\ the deviation $\cE_2^{'\iKd}-\cE_{2,res}^{'\iKd}$ of $\cE_2^{'\iKd}$ from the resonance value (\ref{eq16}) for which the P--even splitting of levels is removed, revealing a P--violating splitting of the order of $\sqrt{\delez}$. 
%
\end{description}

%\end{document}


\newpage
\input{Fig1}
\newpage
\input{Fig2a-b}
% --------------------------------------------------------------------------
%
%   Letter: PARITY VIOLATING ENERGY SHIFTS AND BERRY PHASES IN ATOMS
%
%              1997 by D. Bruss, T. Gasenzer and O. Nachtmann
%
% --------------------------------------------------------------------------


\documentstyle[rotating,a4wide,epsfig,twoside,12pt]{article}
\textwidth15.5cm
\textheight22.0cm
\setlength{\topmargin}{-1cm}
\addtolength{\textheight}{1cm}
\oddsidemargin+1.2cm

%\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}


\input{defTG}

%
%
%
\date{}
\begin{document}
{\sloppy
\input{Titlepage}
%
\input{Text}
%
\input{Acknowl}
%
\input{Lit} 
%
\newpage
\input{Figures}
}
\end{document}
\newpage
\small
\begin{thebibliography}{MM}
%
\bibitem{Bouchiat74} 
{M.A.\ Bouchiat and C.\ Bouchiat}, {Phys.\ Lett.\ B }{48} (1974) 111; {J.\ Phys.\ (Paris) }{35} (1974) 899;\\
{C.\ Bouchiat}, {Phys.\ Lett.\ B }{57} (1975) 284.
%
\bibitem{Moskalev76} 
{A.N.\ Moskalev, R.M.\ Ryndin, and I.B.\ Khriplovich}, {Sov.\ Phys.\ Usp.\ }{19} (1976) 220.
%
\bibitem{Khriplovich91} 
{I.B.\ Khriplovich}, Parity Nonconservation in Atomic Phenomena (Gordon \& Breach, Philadelphia, 1991).
%
\bibitem{PVandHERA}
{L. Giusti, and A. Strumia}, Atomic Parity Violation and the HERA Anomaly, IFUP--TH 23/97, {eprint archive}  June 1997.\\
{G. Altarelli}, HERA Data and Leptoquarks in Supersymmetry, CERN--TH--97--195, {eprint archive}  May 1997.\\
{V. Barger, Kingman Cheung, D.P. Roy, D. Zeppenfeld}, Relaxing Atomic Parity Violation Constraints on New Physics, MADPH--97--1015, {eprint archive}  Oct.\ 1997.
%
\bibitem{Wood97}
{C.S.\ Wood, S.C.\ Bennett, D.\ Cho, B.P.\ Masterson, J.L.\ Roberts,
C.E.\ Tanner, and C.E.\ Wieman}, {Science} {275} (1997) 1759.
%
\bibitem{Erickson77}
{G.W. Erickson}, {J.\ Phys.\ Chem.\ Ref.\ Data} {6} (1977) 831.
%
\bibitem{Sapirstein90} 
{J.R.\ Sapirstein and D.\ R.\ Yennie}, Theory of Hydrogenic Bound States,
in Quantum Electrodynamics (T.\ Kinoshita, Ed., World Scientific, Singapore, 1990).
%
\bibitem{BBN95}
{G.W.\ Botz, D.\ Bru\ss, and O.\ Nachtmann}, {Ann.\ Phys.\ (N.Y.) }{240} (1995) 107.
%
\bibitem{Ashman89} 
{J.\ Ashman} {et al.\ }(EMC), {Phys.\ Lett.\ B }{206} (1988) 364;
%\newline\phantom{{J.\ Ashman} {et al.\ }(EMC)}
{Nucl.\ Phys.\ B }{328} (1989) 1.
%
\bibitem{Adams94} 
{B.\ Adeva} {et al.} (SMC), {Phys.\ Lett.\ B }{302} (1993) 533;\\
{P.L.\ Anthony} {et al.\ }(E142), {Phys.\ Rev.\ Lett.\ }{71} (1993) 959;\\
{D.\ Adams} {et al.\ }(SMC), {Phys.\ Lett.\ B }{329} (1994) 399; {336} (1994) 125;\\
{T.\ Pussieux} (SMC), Measurement of the Proton Spin Structure Function $g_1^p$, {in:} Proceedings, 6th Renc.\ de Blois, The Heart of the Matter, Blois, 1994, (J.--F.\ Mathiot {et al.}, Eds., Editions Frontieres, Gif--sur--Yvette, 1995);\\
{K. Abe} {et al.\ }(E143), {Phys.\ Lett.\ B }{364} (1995) 61; {Phys.\ Rev.\ Lett.\ }{74} (1995) 346; {76} (1996) 587; {78} (1997) 815;\\
{D.\ Adams} {et al.\ }(SMC), Spin Structure of the Proton from Polarized Inclusive
Deep--Inelastic Muon--Proton Scattering, {eprint archive}  Feb.\ 1997.
%
\bibitem{Devenish97}
{R.\ Devenish}, Structure Functions, talk given at the Int.\ Europhys.\ Conf.\ on High Energy Physics, Jerusalem, 1997, {http://www.cern.ch/hep97/pl3.htm}.
%
\bibitem{BGNI97}
{D.\ Bru\ss, T.\ Gasenzer, and O.\ Nachtmann}, Parity violating energy shifts and Berry Phases in Atoms I, to be submitted to {Ann.\ Phys.\ (N.Y.)}.
%
\bibitem{BGNII98}
{D.\ Bru\ss, T.\ Gasenzer, and O.\ Nachtmann}, Parity violating energy shifts and Berry Phases in Atoms II, in preparation.
%
\bibitem{Messiah} 
{A.\ Messiah}, M\'ecanique Quantique, Vol.\ 2 (Dunod, Paris, 1959).
%
\bibitem{Bloch58}
{C.\ Bloch}, {Nucl.\ Phys.\ }{6} (1958) 329.
%
\end{thebibliography}

% --------------------------------------------------------------------------
%
%    Titlepage.tex: Titelseite des Letters PARITY VIOLATING...
%
% --------------------------------------------------------------------------
%
\begin{titlepage}

\title{
{\normalsize 
\hfill HD--THEP--97--52
}
\vspace{1cm}
\\
%{\bf\Large PARITY VIOLATING\\
%ENERGY SHIFTS AND BERRY PHASES\\
%IN ATOMS\thanks{
%Work supported by Deutsche Forschungsgemeinschaft, Project No.\ ???}
%}
{\LARGE\bf\sf New Observables\\
for Parity Violation in Atoms:\\
Energy Shifts in External Electric Fields\thanks{
Work supported by Deutsche Forschungsgemeinschaft, Project No.\ Na 296/1--1} \stepcounter{footnote}}
}

\author{
{\sc 
D. Bru\ss,\thanks{Present address: ISI, Villa Gualino, Viale Settimio Severo 65, 10133 Torino, Italy.}\ \
T. Gasenzer,\thanks{Supported by Cusanuswerk}\ \ and 
O. Nachtmann}
}

\date{\small\sl 
Institut  f\"ur Theoretische Physik, Universit\"at Heidelberg\\
Philosophenweg 16, D-69120 Heidelberg, Germany
}

\maketitle

% --------------------------------------------------------------------------
%
%  Abstract
% 
% --------------------------------------------------------------------------
%
\begin{center}
%{\bf Abstract:}\\
\parbox[t]{\textwidth}{\small
We consider hydrogen--like atoms in unstable levels of principal quantum number $n=2$, confined to a finite size region in a non--homogeneous electric field carrying handedness. The interplay between the internal degrees of freedom of the atoms and the external ones of their c.m.\ motion can produce P--odd contributions to the eigenenergies.
The nominal order of such shifts is $10^{-8}\,$Hz. Typically such energy shifts depend linearly on the small P--violation parameters $\deli\simeq10^{-12}$ ($i=1,2$), essentially the ratios of the P--violating mixing matrix elements of the $2S$ and $2P$ states over the Lamb shift, with $i=1$ ($i=2$) corresponding to the nuclear spin independent (dependent) term.
We show how such energy shifts can be enhanced by a factor of $\simeq 10^6$ in a resonance like way for special field configurations where a crossing of unstable levels occurs, leading to P--violating effects proportional to $\sqrt{\deli}$. Measurements of such effects can give information concerning the ``spin crisis'' of the nucleons.\vspace{1cm}\\
PACS. 11.30.Er, 21.10.Hw, 31.15.Md, 31.50.+w, 31.90.+s, 32.80.Ys.\\
Submitted to Phys.\ Lett.\ A
} 
\end{center}

\end{titlepage}
% Preprint Energy Shifts/Doktorarbeit Thomas Gasenzer
%
% Definitionsfile    defTG.tex
%
% Dieses LATEX File wird mittels \include{defTG} ins Dokument eingebunden
% und enth"alt die in den Formeln und im Text verwendeten benutzerdefinierten
% Befehle.
%

%
%  Umdefinition des \vec Befehls fuer Boldface Vektoren
%
\def\vec#1{\mathchoice{\mbox{\boldmath$\mathrm\displaystyle#1$}}
{\mbox{\boldmath$\mathrm\textstyle#1$}}
{\mbox{\boldmath$\mathrm\scriptstyle#1$}}
{\mbox{\boldmath$\mathrm\scriptscriptstyle#1$}}}

%
%      Abk. von gebr. LaTeX-Befehlen
%      -----------------------------
%

\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\mtext}[1]{\mbox{\rm #1}}
%
\newsavebox{\TRS}
\sbox{\TRS}{\hspace{.5em} = \hspace{-1.8em}
                 \raisebox{1ex}{\mbox{\scriptsize TRS}} }
\newcommand{\eqtrs}{\usebox{\TRS}}
%
\newsavebox{\defgleich}
\sbox{\defgleich}{\ :=\ }
\newcommand{\eqdef}{\usebox{\defgleich}}
%
\newsavebox{\LSIM}
\sbox{\LSIM}{\raisebox{-1ex}{$\ \stackrel{\textstyle<}{\sim}\ $}}
\newcommand{\lsim}{\usebox{\LSIM}}
%
\newsavebox{\GSIM}
\sbox{\GSIM}{\raisebox{-1ex}{$\ \stackrel{\textstyle>}{\sim}\ $}}
\newcommand{\gsim}{\usebox{\GSIM}}
%
\newcommand{\lrar}{\longrightarrow}
\newcommand{\rar}{\rightarrow}
\newcommand{\Rar}{\Rightarrow}

\newcommand{\lk}{\left}
\newcommand{\rk}{\right}
%
\newcommand{\itdot}{\hspace*{2cm}$\bullet$\ \ }

%
%   Aendern und Ruecksetzen der Formelnumerierung
%
\newcounter{saveeqn}
\newcommand{\alphaeqos}{\setcounter{saveeqn}{\value{equation}}
                        \setcounter{equation}{0}
                        \renewcommand{\theequation}{\mbox
                        {\arabic{section}.\arabic{saveeqn}\alph{equation}}}}
\newcommand{\alphaeq}{\stepcounter{equation}
                      \setcounter{saveeqn}{\value{equation}}
                      \setcounter{equation}{0}
                      \renewcommand{\theequation}{\mbox
                       {\arabic{section}.\arabic{saveeqn}\alph{equation}}}}
\newcommand{\reseteq}{\setcounter{equation}{\value{saveeqn}}
                      \renewcommand{\theequation}{\mbox
                       {\arabic{section}.\arabic{equation}}}}



%
%   Schriftstile, -groessen
%
\newcommand{\tsty}{\textstyle}
\newcommand{\ssty}{\scriptstyle}
\newcommand{\sssty}{\scriptscriptstyle}
%
\newcommand{\fns}{\footnotesize}

%
% mathematische Zeichen und Operatoren
%
\newcommand{\ra}{\,\rangle}
\newcommand{\la}{\langle\,}
%
\newcommand{\lDk}{(\!\!(}
\newcommand{\rDk}{)\!\!)}
\newcommand{\lkDk}{\lk(\!\!\lk(}
\newcommand{\rkDk}{\rk)\!\!\rk)}
%
\newcommand{\ptd}{\partial}
\newcommand{\ptdnd}[1]{\frac{\partial}{\partial #1}}
\newcommand{\ptdxndy}[2]{\frac{\partial #1}{\partial #2}}
%
\newcommand{\nab}{\vec{\nabla}}
\newcommand{\nabq}{\vec{\nabla}^2}

\newcommand{\after}{\mbox{\raisebox{0.35ex}{$\ssty\circ$}}}
\newcommand{\cross}{\times}

\newcommand{\Tr}{\mbox{Tr}}
%
\newcommand{\eins}{\mbox{$1 \hspace{-1.0mm}  {\rm l}$}}
%
\newcommand{\NatZ}{{\mbox{\,N$\!\!\!\!\!\!\!\:${\rm I}$\,\,\,\,$}}}

%
%   Brueche
%
\newcommand{\eh}{\mbox{$\frac{1}{2}$}}
\newcommand{\dh}{\mbox{$\frac{3}{2}$}}
\newcommand{\ev}{\mbox{$\frac{1}{4}$}}
\newcommand{\ed}{\mbox{$\frac{1}{3}$}}
\newcommand{\es}{\mbox{$\frac{1}{6}$}}
\newcommand{\dv}{\mbox{$\frac{3}{4}$}}
\newcommand{\da}{\mbox{$\frac{3}{8}$}}
\newcommand{\ih}{\mbox{$\frac{i}{2}$}}
\newcommand{\pih}{\mbox{$\frac{\pi}{2}$}}
\newcommand{\edwz}{\mbox{$\frac{1}{\sqrt{2}}$}}

\newcommand{\neh}{\frac{1}{2}}

%
% Integralzeichen, Delta-, Thetafunktion
%
\newcommand{\deld}{\delta^{\sssty (3)}}
%
\newcommand{\The}{\Theta}
\newcommand{\Thed}{\Theta^{\sssty (3)}}
\newcommand{\Thes}{\Theta^{\sssty (\sigma)}}
\newcommand{\Thesi}{\Theta^{\sssty (\sigma_i)}}
\newcommand{\Thesj}{\Theta^{\sssty (\sigma_j)}}
\newcommand{\Thesk}{\Theta^{\sssty (\sigma_k)}}
\newcommand{\Thesl}{\Theta^{\sssty (\sigma_l)}}
\newcommand{\Ther}{\Theta^{\sssty (\rho)}}
\newcommand{\Thet}{\Theta^{\sssty (\tau)}}
%

%
%  griech. Schriftzeichen
%
\newcommand{\alp}{\alpha}
\newcommand{\bet}{\beta}
\newcommand{\gam}{\gamma}
\newcommand{\Gam}{\Gamma}
\newcommand{\del}{\delta}
\newcommand{\Del}{\Delta}
\newcommand{\eps}{\epsilon}
\newcommand{\kap}{\kappa}
\newcommand{\lam}{\lambda}
\newcommand{\ome}{\omega}
\newcommand{\sig}{\sigma}
\newcommand{\Tet}{\Theta}
\newcommand{\tet}{\theta}
\newcommand{\vph}{\varphi}

\newcommand{\balp}{\bar\alpha}
\newcommand{\bbet}{\bar\beta}
\newcommand{\bgam}{\bar\gamma}

%
%  Paulimatrizen
%
\newcommand{\paue}{\sigma_1}
\newcommand{\pauz}{\sigma_2}
\newcommand{\paud}{\sigma_3}
\newcommand{\pauv}{\vec{\sigma}}



%-------------------------------------------------------------------
%
%   Spezielle Zeichen fuer P-odd Energy Shifts
%

%
%   Hamiltonoperator
%
\newcommand{\hpv}{\mbox{$H_{\sssty\mathrm PV}$}}
\newcommand{\hpve}{\mbox{$H_{\sssty\mathrm PV}^{\sssty (1)}$}}
\newcommand{\hpvz}{\mbox{$H_{\sssty\mathrm PV}^{\sssty (2)}$}}
\newcommand{\hpvez}{\mbox{$H_{\sssty\mathrm PV}^{\sssty (1,2)}$}}

\newcommand{\hcm}{\mbox{$H_{c.m.}$}}
\newcommand{\hext}{\mbox{$H_{ext}$}}
\newcommand{\heff}{\mbox{$H_{e\!f\!\!f}$}}
\newcommand{\heffn}{\mbox{$H_{e\!f\!\!f, 0}$}}
\newcommand{\heffe}{\mbox{$H_{e\!f\!\!f, 1}$}}

\newcommand{\HINT}{H^{\sssty INT}}
\newcommand{\HEXT}{H^{\sssty EXT}}


%
%   Massen
%
\newcommand{\mA}{m_{\sssty \!A}}
\newcommand{\ml}{m_{\ell}}


%
%   schwache Ladungen, Weinberg-Winkel, Bohrradius
%
\newcommand{\qwe}{\mbox{$Q_{\sssty W}^{\sssty (1)}$}}
\newcommand{\qwz}{\mbox{$Q_{\sssty W}^{\sssty (2)}$}}
\newcommand{\qwez}{\mbox{$Q_{\sssty W}^{\sssty (1,2)}$}}
\newcommand{\qwi}{\mbox{$Q_{\sssty W}^{\sssty (i)}$}}

\newcommand{\sint}{\mbox{$\sin^2\theta_{\sssty W}$}}
\newcommand{\theW}{\theta_{\sssty W}}

\newcommand{\rbz}{\mbox{$r_{\sssty B}(Z)$}}

%
%   kalligr. Buchstaben
%
\newcommand{\cB}{{\cal B}}
\newcommand{\cE}{{\cal E}}
\newcommand{\cF}{{\cal F}}
\newcommand{\cH}{{\cal H}}
\newcommand{\cN}{{\cal N}}
\newcommand{\cO}{{\cal O}}
\newcommand{\cP}{{\cal P}}
\newcommand{\cR}{{\cal R}}
\newcommand{\cS}{{\cal S}}
\newcommand{\cT}{{\cal T}}


%
%   delta - Parameter
%
\newcommand{\deli}{\delta_i}
\newcommand{\delj}{\delta_j}
\newcommand{\dele}{\delta_1}
\newcommand{\delz}{\delta_2}
\newcommand{\delez}{\delta_{1,2}}

\newcommand{\dlin}{$\delta$--linear\ }
\newcommand{\delin}{$\delta_1$--linear\ }
\newcommand{\dilin}{$\delta_i$--linear\ }

%
%   n=2 Unterraum Matrizen
%
\newcommand{\unC}{\mbox{$\underline{C\!}\,$}}
\newcommand{\unCt}{\mbox{$\underline{C\!\!\:t\!}\,$}}

\newcommand{\unD}{\mbox{$\underline{D\!}\,$}}
\newcommand{\unDn}{\mbox{$\underline{D\!}\,$}_0}
\newcommand{\unDe}{\mbox{$\underline{D\!}\,$}_1}
\newcommand{\unDz}{\mbox{$\underline{D\!}\,$}_2}
\newcommand{\unDd}{\mbox{$\underline{D\!}\,$}_3}
\newcommand{\unDv}{\mbox{$\underline{\vec{D}}$}}
\newcommand{\uncD}{\mbox{$\underline{\cal D\!}\,$}}

\newcommand{\uneins}{\mbox{$\underline{1\hspace{-1.0mm} {\rm l}}$}}
\newcommand{\unE}{\mbox{$\underline{E}$}}

\newcommand{\unF}{{\underline{F}}}
\newcommand{\unFe}{{\underline{F\!}\,}_1}
\newcommand{\unFz}{{\underline{F\!}\,}_2}
\newcommand{\unFd}{{\underline{F\!}\,}_3}
\newcommand{\unFv}{\mbox{$\underline{\vec{F}}$}}

\newcommand{\unGa}{\mbox{$\underline{\Gamma}$}}

\newcommand{\unH}{\mbox{$\underline{\cal H}$}}
\newcommand{\unhext}{\mbox{$\underline{H}_{ext}$}}
%
\newcommand{\unmuv}{\mbox{$\underline{\muv}$}}

\newcommand{\unM}{\mbox{$\underline{\cal M}$}}
\newcommand{\uncM}{\mbox{$\underline{\cal M}$}}
\newcommand{\unMn}{\mbox{$\underline{\cal M\!}\,_{\sssty 0}$}}
\newcommand{\unMpc}{\mbox{$\underline{\cal M\!}\,_{\sssty\mathrm PC}$}}
\newcommand{\unMpv}{\mbox{$\underline{\cal M\!}\,_{\sssty\mathrm PV}$}}

\newcommand{\unMc}{\mbox{$\underline{M\!}\,_{c}$}}
\newcommand{\unMcL}{\mbox{$\underline{M\!}\,_{L}$}}
\newcommand{\unMcR}{\mbox{$\underline{M\!}\,_{R}$}}

\newcommand{\unN}{\mbox{$\underline{\cal N}$}}

\newcommand{\unphi}{\mbox{$\underline{\phi}$}}
\newcommand{\unpsi}{\mbox{$\underline{\psi}$}}

\newcommand{\Pro}{{ \mbox{\,P$\!\!\!\!\!${\rm I}$\,\,\,$} }}
\newcommand{\Protil}{{ {\widetilde{\mbox{\,P$\!\!\!\!\!${\rm I}$\,\,\,$}}} }}

\newcommand{\unPro}{{\mbox{\underline{\,P$\!\!\!\!\!${\rm I}$\,\,$}$\,$}}}
\newcommand{\unProann}{{\mbox{\underline{\,P$\!\!\!\!\!${\rm  
I}$\,\,$}$\,$}}_{\alpha,\vec{n}}^{\sssty (0)}}
\newcommand{\unProanFFsn}{{\mbox{\underline{\,P$\!\!\!\!\!${\rm  
I}$\,\,$}$\,$}}_{\alpha,\vec{n};F_3,F'_3}^{\sssty (0)}}

\newcommand{\QPro}{\mbox{Q$\!\!\!\!\!\:${\small \sf  
l\normalsize}$\,\,\,$}}

\newcommand{\unQ}{\mbox{\underline{Q$\!\!\!\!\!\:${\small \sf  
l\normalsize}$\,\,$}$\,$}}
\newcommand{\unQann}{\mbox{\underline{Q$\!\!\!\!\!\:${\small \sf  
l\normalsize}$\,\,$}$\,$}_{\alpha,\vec{n}}^{\sssty (0)}}
\newcommand{\unQadnn}{\mbox{\underline{Q$\!\!\!\!\!\:${\small \sf  
l\normalsize}$\,\,$}$\,$}_{\alpha,\dele,\vec{n}}^{\sssty (0)}}
\newcommand{\unQamdnn}{\mbox{\underline{Q$\!\!\!\!\!\:${\small \sf  
l\normalsize}$\,\,$}$\,$}_{\alpha,-\dele,\vec{n}}^{\sssty (0)}}

\newcommand{\unR}{\mbox{$\underline{\cal R\!\!\:}\,$}}
\newcommand{\uncR}{\mbox{$\underline{\cal R\!\!\:}\,$}}

\newcommand{\unS}{\mbox{$\underline{S\!}\,$}}

\newcommand{\uncT}{\mbox{$\underline{\cal T\!}\,$}}
\newcommand{\unT}{\mbox{$\underline{\cal T\!}\,$}}

\newcommand{\unTg}{\mbox{$\underline{T\!g\!}\,$}}

\newcommand{\unU}{\mbox{$\underline{U\!}\,$}}

\newcommand{\unX}{\mbox{$\underline{X\!}\,$}}

%
%   behutete Buchstaben
%
\newcommand{\Hh}{\hat H}
\newcommand{\Kh}{\hat K}

%\newcommand{\doublehat}[1]{
%  \hat{\phantom{\mbox{\raisebox{-1pt}{$\hat{\mbox{#1}}$}}}}
%  \!\!\!\!\!\!\: \hat{\phantom{\mbox{#1}}}
%  \!\!\!\!\!\!\: #1  }
%\newcommand{\doublewidehat}[1]{
%  \widehat{\phantom{\mbox{\raisebox{-1pt}{$\widehat{\mbox{#1}}$}}}}
%  \!\!\!\!\!\!\: \widehat{\phantom{\mbox{#1}}}
%  \!\!\!\!\!\!\: #1  }
\newcommand{\doublehat}[1]{
  \hat{\hat{\mbox{\phantom{#1}}}}
  \!\!\!\!\!\!\!\: #1
  }
\newcommand{\doublewidehat}[1]{
  \widehat{\widehat{\mbox{\phantom{#1}}}}
  \!\!\!\!\!\!\!\: #1
  }

\newcommand{\Hhh}{\doublehat{H}}
\newcommand{\Khh}{\doublehat{K}}
\newcommand{\Hwhh}{\doublewidehat{H}}
\newcommand{\Kwhh}{\doublewidehat{K}}





%
%   Bras und Kets
%
\newcommand{\bra}[1]{\mbox{$\la#1\,$}}
\newcommand{\brak}[1]{\mbox{$\la#1\,|$}}
\newcommand{\ket}[1]{\mbox{$ |\,#1\ra$}}


%
%   n=2 rechte und linke Eigenvektoren
%
\newcommand{\lki}[1]{\mbox{$\left.^{#1}(\right.$}}

\newcommand{\lbra}[1]{\mbox{$ ( \widetilde{#1}$}}
\newcommand{\lbrak}[1]{\mbox{$ ( \widetilde{#1}|$}}
\newcommand{\rket}[1]{\mbox{$ | #1 )$}}


\newcommand{\evrza}{\mbox{$ | {a})$}}
\newcommand{\evlza}{\mbox{$ ( {a}|$}}
\newcommand{\evrzb}{\mbox{$ | {b})$}}
\newcommand{\evlzb}{\mbox{$ ( {b}|$}}
\newcommand{\levlzb}{\mbox{$ ( \widetilde{b}|$}}

\newcommand{\evrzza}{\mbox{$ | {2,a})$}}
\newcommand{\evrzzas}{\mbox{$ | {2,a'})$}}

\newcommand{\evrbet}{\mbox{$ | {\beta})$}}
\newcommand{\evlbet}{\mbox{$ ( {\beta}|$}}
\newcommand{\levlbet}{\mbox{$ ( \widetilde{\beta}|$}}

\newcommand{\revrFd}{\mbox{$ | {F_3})$}}
\newcommand{\revrFds}{\mbox{$ | {F'_3})$}}
\newcommand{\revrFdss}{\mbox{$ | {F''_3})$}}
\newcommand{\revrbFdss}{\mbox{$ | {\beta,F''_3})$}}
\newcommand{\levlFd}{\mbox{$ ( \widetilde{F_3}|$}}
\newcommand{\levlFds}{\mbox{$ ( \widetilde{F'_3}|$}}
\newcommand{\levlFdss}{\mbox{$ ( \widetilde{F''_3}|$}}
\newcommand{\levlbFdss}{\mbox{$ ( \widetilde{\beta,F''_3}|$}}

\newcommand{\revrzaE}{\mbox{$ | {a,{\cal E}\evd})$}}
\newcommand{\levlzaE}{\mbox{$ ( \widetilde{a,{\cal E}\evd}|$}}
\newcommand{\revrzcE}{\mbox{$ | {c,{\cal E}\evd})$}}
\newcommand{\levlzcE}{\mbox{$ ( \widetilde{c,{\cal E}\evd}|$}}

\newcommand{\revrzaEsig}{\mbox{$ | {a,{\cal E}^\iKsig\evd})$}}
\newcommand{\revrzcEsig}{\mbox{$ | {c,{\cal E}^\iKsig\evd})$}}

\newcommand{\revraEsig}{\mbox{$ | {a,\cEvs})$}}
\newcommand{\levlaEsig}{\mbox{$ ( \widetilde{a,\cEvs}|$}}
\newcommand{\revrcEsig}{\mbox{$ | {c,\cEvs})$}}

\newcommand{\levr}{\mbox{$ | \widetilde{\alpha,F_3 })$}}
\newcommand{\levl}{\mbox{$ ( \widetilde{\alpha,F_3 }|$}}
\newcommand{\revr}{\mbox{$ | \alpha,F_3 )$}}
\newcommand{\revl}{\mbox{$ ( \alpha,F_3 |$}}

\newcommand{\levrE}{\mbox{$ | \widetilde{\alpha,F_3,{\cal E}\evd })$}}
\newcommand{\levlE}{\mbox{$ ( \widetilde{\alpha,F_3 ,{\cal E}\evd}|$}}
\newcommand{\revrE}{\mbox{$ | \alpha,F_3,{\cal E}\evd )$}}
\newcommand{\revlE}{\mbox{$ ( \alpha,F_3 ,{\cal E}\evd|$}}
\newcommand{\revrsE}{\mbox{$ | \alpha,F'_3,{\cal E}\evd )$}}

\newcommand{\levlaE}{\mbox{$ ( \widetilde{\alpha,a,F_3,{\cal E}\evd}|$}}
\newcommand{\revrbE}{\mbox{$ | \alpha,b,F_3,{\cal E}\evd )$}}

\newcommand{\levlvar}[3]{\mbox{$ \lk.^{{\sssty (#1)}}
                      ( \widetilde{#2,#3,{\cal E}\evd}|\rk.$}}
\newcommand{\revrvar}[3]{\mbox{$ | #2,#3,{\cal E}\evd )^{\sssty (#1)}$}}

\newcommand{\levlRsvar}[4]{\mbox{$
                       ( \widetilde{#1,#2,{\cal E}^{\sssty (#3)}\evd},#4 |$}}
\newcommand{\revrRsvar}[4]{\mbox{$ | #1,#2,{\cal E}^{\sssty (#3)}\evd ,#4 )$}}
\newcommand{\levlRvar}[3]{\mbox{$
                       ( \widetilde{#1,#2,{\cal E}\evd},#3 |$}}
\newcommand{\revrRvar}[3]{\mbox{$ | #1,#2,{\cal E}\evd ,#3 )$}}

\newcommand{\levlEd}{\mbox{$(\widetilde{\alpha,F_3,{\cal E}\evd},\dele|$}}
\newcommand{\revrEd}{\mbox{$ | \alpha,F_3,{\cal E}\evd,\dele)$}}
\newcommand{\levlEmd}{\mbox{$(\widetilde{\alpha,F_3,{\cal E}\evd},-\dele|$}}
\newcommand{\revrEmd}{\mbox{$ | \alpha,F_3,{\cal E}\evd,-\dele)$}}
\newcommand{\levlEdi}{\mbox{$(\widetilde{\alpha,F_3,{\cal E}\evd},\deli|$}}
\newcommand{\revrEdi}{\mbox{$ | \alpha,F_3,{\cal E}\evd,\deli)$}}
\newcommand{\levlEmdi}{\mbox{$(\widetilde{\alpha,F_3,{\cal E}\evd},-\deli|$}}
\newcommand{\revrEmdi}{\mbox{$ | \alpha,F_3,{\cal E}\evd,-\deli)$}}

\newcommand{\levlnE}{\mbox{$(\widetilde{\alpha,F_3,{\cal E}\evd},\vec{n}|$}}
\newcommand{\levlmE}{\mbox{$(\widetilde{\alpha,F_3,{\cal E}\evd},\vec{m}|$}}
\newcommand{\levlsnE}{\mbox{$(\widetilde{\alpha,F'_3,{\cal E}\evd},\vec{n}|$}}
\newcommand{\revrnE}{\mbox{$ | \alpha,F_3,{\cal E}\evd,\vec{n})$}}
\newcommand{\revrmE}{\mbox{$ | \alpha,F_3,{\cal E}\evd,\vec{m})$}}

\newcommand{\levlnEd}{\mbox{$(\widetilde{\alpha,F_3,{\cal E}\evd},\dele,\vec{n}|$}}
\newcommand{\revrnEd}{\mbox{$ | \alpha,F_3,{\cal E}\evd,\dele,\vec{n})$}}
\newcommand{\levlnEmd}{\mbox{$(\widetilde{\alpha,F_3,{\cal  
E}\evd},-\dele,\vec{n}|$}}
\newcommand{\revrnEmd}{\mbox{$ | \alpha,F_3,{\cal E}\evd,-\dele,\vec{n})$}}

\newcommand{\levlEv}{\mbox{$ ( \widetilde{\alpha,F_3 ,\vec{\cal E}}|$}}
\newcommand{\revrEv}{\mbox{$ | \alpha,F_3,\vec{\cal E} )$}}
\newcommand{\revrsEv}{\mbox{$ | \alpha,F'_3,\vec{\cal E} )$}}

\newcommand{\levlER}{\mbox{$ ( \widetilde{\alpha,F_3 ,\vec{\cal E}(R)}|$}}
\newcommand{\revrER}{\mbox{$ | \alpha,F_3,\vec{\cal E}(R) )$}}

\newcommand{\levlR}{\mbox{$ ( \widetilde{\alpha,F_3,R }|$}}
\newcommand{\revrR}{\mbox{$ | \alpha,F_3,R )$}}

\newcommand{\levlRs}{\mbox{$ ( \widetilde{\alpha,F_3,R' }|$}}
\newcommand{\revrRs}{\mbox{$ | \alpha,F_3,R' )$}}
\newcommand{\revrsRs}{\mbox{$ | \alpha,F'_3,R' )$}}

\newcommand{\levlzaER}{\mbox{$ ( \widetilde{a,\vec{\cal E}(R)}|$}}
\newcommand{\revrzaER}{\mbox{$ | a,\vec{\cal E}(R) )$}}

\newcommand{\levlzaR}{\mbox{$ ( \widetilde{a,R }|$}}
\newcommand{\revrzaR}{\mbox{$ | a,R )$}}

\newcommand{\levlzaRs}{\mbox{$ ( \widetilde{a,R' }|$}}
\newcommand{\revrzaRs}{\mbox{$ | a,R' )$}}
\newcommand{\revrzbRs}{\mbox{$ | b,R' )$}}

\newcommand{\levley}[1]{\mbox{$ ( \widetilde{#1}|$}}
\newcommand{\revrey}[1]{\mbox{$ | #1 )$}}


\newcommand{\rPsir}{|\,\Psi\,)}
\newcommand{\rPsisr}{|\,\Psi^{\sssty (\sigma)}\,)}

%
%   Wellenfunktionen
%
\newcommand{\PsiaRv}{\,\Psi_a(\vec{R}\,)\,}
\newcommand{\PsiasRv}{\,\Psi_a^{\iKsig}(\vec{R}\,)\,}
\newcommand{\phicsRv}{\,\phi_c^{\iKsig}(\vec{R}\,)\,}


%
%   Koeffizienten
%
\newcommand{\zcmemzs}{\zeta}


%
%   Normierungen
%
\newcommand{\Na}{{\cal N}_{\alpha}}
\newcommand{\NaI}{{\cal N}_{\alpha}^{-1}}
\newcommand{\NaFd}{{\cal N}_{\alpha,F_3}}
\newcommand{\Nan}{{\cal N}_{\alpha,\vec{n}}}
\newcommand{\Nn}{{\cal N}_{\vec{n}}}
\newcommand{\Nvar}[1]{{\cal N}^{\sssty(#1)}}
\newcommand{\NIvar}[1]{{\cal N}^{{\sssty(#1)}-1}}


%
%   Saekularmatrizen
%
\newcommand{\cSane}{{\cal S}_{\alpha,\vec{n}}^{\sssty (1)}}
\newcommand{\cSanz}{{\cal S}_{\alpha,\vec{n}}^{\sssty (2)}}
\newcommand{\cSand}{{\cal S}_{\alpha,\vec{n}}^{\sssty (3)}}
\newcommand{\cSanv}{{\cal S}_{\alpha,\vec{n}}^{\sssty (4)}}

\newcommand{\cSanId}{{\cal S}_{\alpha,\vec{n}}^{\sssty (3)\,I}}


%
%   Eigenwerte
%
\newcommand{\EINT}{E^{\sssty INT}}
\newcommand{\EEXT}{E^{\sssty EXT}}

\newcommand{\EKcEvdedz}{E(\cEv,\dele,\delz)}

\newcommand{\Ezcs}{E_{2,c}^\iKsig}

\newcommand{\EKaFd}{E(\alpha,F_3)}

\newcommand{\EKaFdE}{E(\alpha,F_3,\cE)}
\newcommand{\EiKaFdEevd}{E^{\sssty INT}(\alpha,F_3,\cE\evd)}

\newcommand{\Eal}{E_\alpha}

\newcommand{\EKalE}{E(\alpha,{\cal E})}

\newcommand{\EanId}{E_{\alpha,\vec{n}}^{\sssty (3)\,I}}
\newcommand{\EanIv}{E_{\alpha,\vec{n}}^{\sssty (4)\,I}}
\newcommand{\Eann}{E_{\alpha,\vec{n}}^{\sssty (0)}}
\newcommand{\Eane}{E_{\alpha,\vec{n}}^{\sssty (1)}}
\newcommand{\Eanz}{E_{\alpha,\vec{n}}^{\sssty (2)}}
\newcommand{\Eand}{E_{\alpha,\vec{n}}^{\sssty (3)}}
\newcommand{\Eanv}{E_{\alpha,\vec{n}}^{\sssty (4)}}
\newcommand{\Eandn}{E_{\alpha,\vec{n}}^{\sssty (3,0)}}
\newcommand{\Eanvn}{E_{\alpha,\vec{n}}^{\sssty (4,0)}}
\newcommand{\Eande}{E_{\alpha,\vec{n}}^{\sssty (3,1)}}
\newcommand{\Eanve}{E_{\alpha,\vec{n}}^{\sssty (4,1)}}

\newcommand{\Etila}{\tilde{E}_{a}}
\newcommand{\Etilb}{\tilde{E}_{b}}
\newcommand{\Etilal}{\tilde{E}_{\alpha}}

\newcommand{\candeeds}{c_{\alpha,\vec{n}}^{\sssty (3,1)1\,3\,\sigma}}
\newcommand{\canveedsr}{c_{\alpha,\vec{n}}^{\sssty (4,1)1\,3\,\sigma\,\rho}}
\newcommand{\canie}[1]{c_{\alpha,\vec{n}}^{\sssty
                                              (#1,1)}}
\newcommand{\cande}[3]{c_{\alpha,\vec{n}}^{\sssty
                                              (3,1)#1#2#3}}
\newcommand{\canve}[4]{c_{\alpha,\vec{n}}^{\sssty
                                              (4,1)#1#2#3#4}}

%
% Tilde-Zeichen
%
\newcommand{\ctil}{\tilde c}
\newcommand{\Htil}{\widetilde H}
\newcommand{\cHtil}{\widetilde \cH}
\newcommand{\ntil}{\tilde n}
\newcommand{\cTtil}{\widetilde \cT}
\newcommand{\cRtil}{\widetilde \cR}

%
% kleine Grossbuchstaben-Subskripte
%
\newcommand{\AkT}{A_{\sssty T}}
\newcommand{\AkR}{A_{\sssty R}}
\newcommand{\AvT}{\vec{A}_{\sssty T}}
\newcommand{\AvR}{\vec{A}_{\sssty R}}



%
% Orts-, Impuls-, etc. -vektoren
%
\newcommand{\Av}{\vec{A}}
\newcommand{\Bv}{\vec{B}}
\newcommand{\Dv}{\vec{D}}

\newcommand{\eve}{\vec{e}_1}
\newcommand{\evz}{\vec{e}_2}
\newcommand{\evd}{\vec{e}_3}
\newcommand{\evi}{\vec{e}_i}

\newcommand{\cEv}{\vec{\cal E}}
\newcommand{\cEvP}{\vec{\cal E}_{\mbox{\tiny P}}}
\newcommand{\cEvR}{\vec{\cal E}_{\mbox{\tiny R}}}
\newcommand{\cEvs}{\vec{\cal E}^{\sssty (\sigma)}}
\newcommand{\cEsvs}{\vec{\cal E}'^{\sssty (\sigma)}}
\newcommand{\cEsvz}{\vec{\cal E}'^{\sssty (2)}}
\newcommand{\cEsvd}{\vec{\cal E}'^{\sssty (2)}}
\newcommand{\cEvr}{\vec{\cal E}^{\sssty (\rho)}}
\newcommand{\cEvt}{\vec{\cal E}^{\sssty (\tau)}}
\newcommand{\cEvse}{\vec{\cal E}^{\sssty (\sigma_1)}}
\newcommand{\cEvsz}{\vec{\cal E}^{\sssty (\sigma_2)}}
\newcommand{\cEvsd}{\vec{\cal E}^{\sssty (\sigma_3)}}
\newcommand{\cEvsv}{\vec{\cal E}^{\sssty (\sigma_4)}}
\newcommand{\cEve}{\vec{\cal E}^{\sssty (1)}}
\newcommand{\cEvz}{\vec{\cal E}^{\sssty (2)}}
\newcommand{\cEvd}{\vec{\cal E}^{\sssty (3)}}

\newcommand{\cEP}{{\cal E}_{\mbox{\tiny P}}}
\newcommand{\cER}{{\cal E}_{\mbox{\tiny R}}}
\newcommand{\cEs}{{\cal E}^{\sssty (\sigma)}}
\newcommand{\cEr}{{\cal E}^{\sssty (\rho)}}
\newcommand{\cEt}{{\cal E}^{\sssty (\tau)}}
\newcommand{\cEk}{{\cal E}^{\sssty (\kap)}}
\newcommand{\cEse}{{\cal E}^{\sssty (\sigma_1)}}
\newcommand{\cEsz}{{\cal E}^{\sssty (\sigma_2)}}
\newcommand{\cEsd}{{\cal E}^{\sssty (\sigma_3)}}
\newcommand{\cEsv}{{\cal E}^{\sssty (\sigma_4)}}

\newcommand{\etav}{\vec{\eta}}

\newcommand{\Fv}{\vec{F}}

\newcommand{\Iv}{\vec{I}}

\newcommand{\kv}{\vec{k}}

\newcommand{\mv}{\vec{m}}
\newcommand{\muv}{\vec{\mu}}
\newcommand{\nv}{\vec{n}}

\newcommand{\pv}{\vec{p}}

\newcommand{\Pv}{\vec{P}}
\newcommand{\Pvcs}{\vec{P}_c^\iKsig}
\newcommand{\Pvcsq}{\vec{P}_c^{\iKsig 2}}
\newcommand{\Pics}{P_{i,c}^\iKsig}

\newcommand{\Rv}{\vec{R}}
\newcommand{\Rvdot}{\dot{\vec{R}}}
\newcommand{\Rvsk}{\vec{R}^{\sssty (\sigma,1)}}
\newcommand{\Rvsg}{\vec{R}^{\sssty (\sigma,2)}}
\newcommand{\Risk}{R^{\sssty (\sigma,1)}_{\sssty i}}
\newcommand{\Risg}{R^{\sssty (\sigma,2)}_{\sssty i}}

\newcommand{\sigv}{\vec{\sigma}}

\newcommand{\xv}{\vec{x}}
\newcommand{\xvA}{\xv_{\sssty \!A}}
\newcommand{\xvl}{\xv_{\ell}}

%
%  elektronische Uebergangsmatrix U_i^(sig,tau)+-
%
\newcommand{\Unzepp}{U_{\sssty 0}^{\sssty (2,1)++}}
\newcommand{\Unzemm}{U_{\sssty 0}^{\sssty (2,1)--}}
\newcommand{\Uezepm}{U_{\sssty 1}^{\sssty (2,1)+-}}
\newcommand{\Uezemp}{U_{\sssty 1}^{\sssty (2,1)-+}}
\newcommand{\Unezpp}{U_{\sssty 0}^{\sssty (1,2)++}}
\newcommand{\Unezmm}{U_{\sssty 0}^{\sssty (1,2)--}}
\newcommand{\Ueezpm}{U_{\sssty 1}^{\sssty (1,2)+-}}
\newcommand{\Ueezmp}{U_{\sssty 1}^{\sssty (1,2)-+}}

\newcommand{\Unstpmpm}{U_{\sssty 0}^{\sssty (\sig,\tau)\pm\pm}}
\newcommand{\Untspmpm}{U_{\sssty 0}^{\sssty (\tau,\sig)\pm\pm}}
\newcommand{\Untsmpmp}{U_{\sssty 0}^{\sssty (\tau,\sig)\mp\mp}}
\newcommand{\Uestpmmp}{U_{\sssty 1}^{\sssty (\sig,\tau)\pm\mp}}
\newcommand{\Uetspmmp}{U_{\sssty 1}^{\sssty (\tau,\sig)\pm\mp}}
\newcommand{\Uetsmppm}{U_{\sssty 1}^{\sssty (\tau,\sig)\mp\pm}}

\newcommand{\Uzepp}{\unU^{\sssty (2,1)++}}
\newcommand{\Uzemm}{\unU^{\sssty (2,1)--}}
\newcommand{\Uzepm}{\unU^{\sssty (2,1)+-}}
\newcommand{\Uzemp}{\unU^{\sssty (2,1)-+}}
\newcommand{\Uezpp}{\unU^{\sssty (1,2)++}}
\newcommand{\Uezmm}{\unU^{\sssty (1,2)--}}
\newcommand{\Uezpm}{\unU^{\sssty (1,2)+-}}
\newcommand{\Uezmp}{\unU^{\sssty (1,2)-+}}

\newcommand{\Ustpmpm}{\unU^{\sssty (\sig,\tau)\pm\pm}}
\newcommand{\Utspmpm}{\unU^{\sssty (\tau,\sig)\pm\pm}}
\newcommand{\Utsmpmp}{\unU^{\sssty (\tau,\sig)\mp\mp}}
\newcommand{\Ustpmmp}{\unU^{\sssty (\sig,\tau)\pm\mp}}
\newcommand{\Utspmmp}{\unU^{\sssty (\tau,\sig)\pm\mp}}
\newcommand{\Utsmppm}{\unU^{\sssty (\tau,\sig)\mp\pm}}

\newcommand{\Ustrs}{\unU^{\sssty (\sig,\tau)r\,s}}
\newcommand{\Utsmsmr}{\unU^{\sssty (\tau,\sig)(-s)(-r)}}
\newcommand{\Utssr}{\unU^{\sssty (\tau,\sig)s\,r}}

%
%  Entwicklungsoperator
%
\newcommand{\Util}{\tilde{U}}
\newcommand{\UtRRn}{\tilde{U}(R,R_0)}
\newcommand{\UtgRRn}{\tilde{U}_g(R,R_0)}
%\newcommand{}{}


%
%  Spezielle Zeichenfolgen
%
\newcommand{\isig}{{\sssty \sigma }}
\newcommand{\iKkap}{{\sssty (\kappa )}}
\newcommand{\iKrho}{{\sssty (\rho )}}
\newcommand{\iKsig}{{\sssty (\sigma )}}
\newcommand{\iKtau}{{\sssty (\tau )}}
\newcommand{\iKi}{{\sssty (i)}}
\newcommand{\iKen}{{\sssty (n)}}
\newcommand{\iKn}{{\sssty (0)}}
\newcommand{\iKe}{{\sssty (1)}}
\newcommand{\iKz}{{\sssty (2)}}
\newcommand{\iKd}{{\sssty (3)}}
\newcommand{\iKvar}[1]{{\sssty (#1)}}
\newcommand{\isKvar}[1]{{'\sssty (#1)}}
\newcommand{\ikvar}[1]{\mbox{\raisebox{0.2ex}{$\ssty (#1)$}}}

\newcommand{\ivar}[1]{{\sssty #1}}

\newcommand{\phm}{\varphi_{max}}
\newcommand{\thm}{\theta_{max}}


\newcommand{\pianh}{\frac{\pi a_0}{2}}
\newcommand{\piFdh}{\frac{\pi |F_3|}{2}}
\newcommand{\aean}{\frac{a_1}{a_0}}

\newcommand{\DelE}{\Delta{\cal E}}
\newcommand{\DelEoE}{\frac{\Delta{\cal E}}{{\cal E}}}

\newcommand{\dndE}{\frac{\ptd}{\ptd\cE}}

\newcommand{\dkap}{\partial_\kappa}
\newcommand{\dlam}{\partial_\lambda}


%
%  Atombezeichnungen
%
\newcommand{\vzHep}{\left.^4_2\mbox{He}^+\right.}
\newcommand{\eeH}{\left.^1_1\mbox{H}\right.}


%
%  Woerter
%
\newcommand{\WWA}{Wigner--Weisskopf--approximation\ }


%\newcommand{}{}
%\newcommand{}{}
%\newcommand{}{}
%\newcommand{}{}
%\newcommand{}{}


