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\section*{Figure Captions}
\small
\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.

\item[{\sc Fig.} {\rm 2 a,b.}]
The energy levels $E_{\balp}^\iKn=E^\iKn(\alp,|F_3|,\cE\evd;\nv)$ (\ref{eq2.85}) for a ``hydrogen'' atom with $I=0$ in an external electric field $\cE\evd$ and $\nv=(n_1,n_2,n_3)$. The real parts for $\alp=2S_{1/2}$ and $2P_{1/2}$ and $n_2=n_3=1$ are shown in {\bf a}, the decay widths (twice the negative of the imaginary parts in units of s$^{-1}$) for $\alp=2S_{1/2}$, $2P_{1/2}$ and $2P_{3/2}$ in {\bf b.} In our calculation the imaginary parts are independent of $\nv$.

\item[{\sc Fig.} {\rm 3.}]
The real and imaginary parts of the P-violating energy-difference $\Delta E_{\balp}^\iKd\cdot\cE^3/(\eta\cE^{'3}\dele h)$ (cf.\ (\ref{eq2.133})) as a function of $\cE$ for the state $\balp=(\alp,|F_3|;\nv)=(2S_{1/2},1/2;(1,1,1))$ within a range of $\cE$, where extrema due to a level crossing occur (cf.\ Fig.\ 2a).  

\item[{\sc Fig.} {\rm 4.}]
The field strength value $\cE'_{max}$ characterizing the limit of applicability of our perturbative treatment (cf.\ (\ref{eq2.134})) as a function of the field strength $\cE$ for the same level $\balp$ as in Fig.\ 3.

\item[{\sc Fig.} {\rm 5 a,b.}]
The P-violating energy shift $\Delta E_{\balp}^\iKd$ (\ref{eq2.133}) for $\cE'=\cE'_{max}$. The real part is plotted in {\bf a}, twice the imaginary part in {\bf b.}

\item[{\sc Fig.} {\rm 6 a,b.}]
The decadic logarithms of the moduli of ({\bf a}) the real and ({\bf b}) twice the imaginary parts of the P-violating energy-difference $E_{\balp\bbet,+}(\{\cEv^{'\iKsig}\})
-E_{\balp\bbet,+}(\{\cEvR^{'\iKsig}\})$ (\ref{eq2.168.13})
for our ``hydrogen'' vs.\ the deviation $\eta_3^\iKz-\eta_{3,res,-}^\iKz$ of $\eta_3^\iKz$ from the resonance value $\eta_{3,res,-}^\iKz=574.9$ for which the P-even splitting of levels is removed, revealing a P-violating splitting of the order of $\sqrt{\dele}$. 

\item[{\sc Fig.} {\rm 7.}]
The lowest energy levels of the c.m.\ motion of a $\vzHep$ ion moving in one dimension in a linearily rising electric field potential of a field strength $\cE$ within an interval of length $A=1\,$nm. These are the eigenenergies of the Airy states (cf.\ (\ref{eq2.63})). For $\cE=0$ they show the quadratic behaviour of the eigenenergies of sinus states (cf.\ (\ref{eq2.9})).

\item[{\sc Fig.} {\rm 8.}]
The field strength value $\cE'_{max}$ characterizing the limit of applicability of our perturbative treatment of $\vzHep$ as a function of the field strength $\cE$ for the level $\balp=(2S_{1/2},1/2;(1,1,1))$.

\item[{\sc Fig.} {\rm 9 a,b.}]
The P-violating energy shift $\Delta E_{\balp}^\iKd$ in $\vzHep$ for $\cE'=\cE'_{max}$ for the same level $\balp$ as in Fig.\ 8. The real part is plotted in {\bf a.}, the imaginary part in {\bf b.}

\item[{\sc Fig.} {\rm 10 a,b.}]
The decadic logarithms of the moduli of ({\bf a}) the real and ({\bf b}) twice the imaginary parts of the P-violating energy-difference  $E_{\balp\bbet,+}(\{\cEv^{'\iKsig}\})
-E_{\balp\bbet,+}(\{\cEvR^{'\iKsig}\})$ (\ref{eq2.178.7})
for $\vzHep$ vs.\ the deviation $\eta_3^\iKd-\eta_{3,res,-}$ of $\eta_3^\iKd$ from the resonance value $\eta_{3,res,-}^\iKz=525.4$ for which the P-even splitting of levels is removed in 2nd order perturbation theory, revealing a P-violating splitting of the order of $\sqrt{\dele}$. 

\item[{\sc Fig.} {\rm 11 a,b.}]
The decadic logarithm of the moduli squared of ({\bf a}) the real and ({\bf b}) twice the imaginary parts of the P-violating energy-difference $E_{\balp,\pm}(\{\cEv^{'\iKsig}\})-E_{\balp,\pm}(\{\cEvR^{'\iKsig}\})$ (\ref{eq2.213}) for $\eeH$ vs.\ the deviation $\cE_2^{'\iKd}-\cE_{2,res}^{'\iKd}$ of $\cE_2^{'\iKd}$ from the resonance value $\cE_{2,res}^{'\iKd}=-1.64\,$V/cm for which the P-even splitting of levels is removed in 2nd order perturbation theory, revealing a P-violating splitting of the order of $\sqrt{\delz}$. 


\end{description}
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\small
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{\sc A.N.\ Moskalev, R.M.\ Ryndin, and I.B.\ Khriplovich}, {\it Sov.\ Phys.\ Usp.\ }{\bf 19} (1976), 220.
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\bibitem{Khriplovich91} 
{\sc I.B.\ Khriplovich}, ``Parity Nonconservation in Atomic Phenomena'', Gordon \& Breach, Philadelphia, 1991.
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\end{thebibliography}

% --------------------------------------------------------------------------
%
%   Preprint: PARITY VIOLATING ENERGY SHIFTS AND BERRY PHASES IN ATOMS
%
%           1998 by D. Bruss, T. Gasenzer and O. Nachtmann
%
%				Part I, ver 2
%
% --------------------------------------------------------------------------
%
% Guide to changes from version 1:
% Some calculations from sections 2 and 3 where simply shifted to
% appendices A,B,C and the new app. D.
% Changes in content: Table I and eqn. (3.134). 
% Legends of Figs. 2b and 6b
% Some remarks to chiral 
% molecules were added in Section 4 
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%
TABLE I
\vspace*{.5cm}
  
\begin{tabular}{ccc|cccc|cccc}
\hline 
\hline 
 $\alp,|F_3|$ & 
 $\cE$ & 
 $\cE'_{max}$ &
 \multicolumn{2}{c}{$E_{\balp}^\iKvar{2,0}/h$}   	& 
 \multicolumn{2}{c|}{$\Del E_{\balp}^\iKvar{2,0}/h$}   	& 
 \multicolumn{2}{c}{$\Del E_{\balp}^\iKvar{2,1}/h$}	&
 \multicolumn{2}{c}{$\Del E_{\balp}^\iKvar{2,2}/h$}      
\\
 & 
 [V/cm] &
 [V/cm] &
 \multicolumn{2}{c}{[MHz]} &
 \multicolumn{2}{c|}{[MHz]} &
 \multicolumn{2}{c}{[$10^{4}\,$Hz]} &
 \multicolumn{2}{c}{[$10^{4}\,$Hz]} 
\\
 & & &
 Re & Im &
 Re & Im &
 Re & Im &
 Re & Im 
\\
\hline
%
%
$2S\eh 1,1$ & 
$340	 	$ & $ 84	$ &
$-13 		$ & $- 0,\!89	$ &
$  0,\!091 	$ & $  0,\!082	$ &
$  0,\!12 	$ & $- 0,\!085	$ &
$- 2,\!0 	$ & $  1,\!3	$ 
\\
%\hline
%
$2P\eh1,1	$ & 
$380	 	$ & $ 228	$ &
$-44	 	$ & $   0,\!57	$ &
$  0,\!34	$ & $   0,\!0026$ &
$- 0,\!027 	$ & $   0,\!0045$ &
$- 0,\!70	$ & $-  0,\!041	$ 
\\
%\hline
%
$2P\dh1,1	$ & 
$ 370	 	$ & $ 284	$ &
$- 15	 	$ & $   3,\!5	$ &
$   6,\!5	$ & $-  1.5	$ &
$  25 	 	$ & $   1,\!0	$ &
$-450	 	$ & $- 14	 $ 
\\
%\hline
%
$2P\dh2,1 	$ & 
$ 350	 	$ & $ 206	$ &
$   3,\!9	$ & $   1,\!9 	$ &
$   0,\!20 	$ & $-  0,\!064	$ &
$  80	 	$ & $- 12	$ &
$-1250	 	$ & $ 240	$ 
\\
\hline
\hline 
\end{tabular}
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\section*{Table Captions}
\small
\begin{description}
%
\item[{\sc Table} {\rm I.}]
The P-conserving as well as the P-violating perturbative contributions to the eigenenergies $E_{\balp,\pm}$ (\ref{eq2.201})\,ff.\ up to 2nd order perturbation theory for different levels $\balp=(\alp,|F_3|;\nv=(1,1,1))$ of $\eeH$ at different electric fields $\cE$ for the corresponding maximally allowed perturbation field strengths $\cE'_{max}$.

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\title{
{\normalsize 
\hfill HD--THEP--97--36
}
\vspace{1cm}
\\
%{\bf\Large PARITY VIOLATING\\
%ENERGY SHIFTS AND BERRY PHASES\\
%IN ATOMS\thanks{
%Work supported by Deutsche Forschungsgemeinschaft, Project No.\ ???}
%}
{\LARGE\bf\sf Parity Violating\\
Energy Shifts and Berry Phases in Atoms, I\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

\begin{center}
%{\bf Abstract:}\\
\parbox[t]{\textwidth}{\small
We present a study of parity (P) violating contributions to the eigenenergies of stationary systems containing atoms in spatially inhomogeneous external electric fields. In this context the subtle interplay of P-violation and time reversal (T) invariance plays an important role. If the entire field configuration is chosen to exhibit chirality the energies are in general shifted by pseudoscalar contributions which change sign under a planar reflection of the field. 
In part I we consider sudden variations of the fields and calculate P-violating energy shifts using perturbation theory.
In part II the adiabatic case will be treated and the connection to geometrical (Berry-) phases will be elucidated.
To calculate the effects we use the standard model of elementary particle physics where the P-odd interaction arises through the exchange of Z-bosons between the quarks in the nucleus and the atomic electrons. We consider in detail hydrogen-like systems in unstable levels of principal quantum number $n=2$. We study atoms with vanishing nuclear spin like $\vzHep$ and with nuclear spin $I=1/2$ like $\eeH$. The nominal order of P-violating effects is $10^{-5}\!\!\dots 10^{-9}\,$Hz which is determined by the mixing of the $2S_{1/2}$ and $2P_{1/2}$ states. However we point out that with certain configurations of the external fields, it is possible to enhance the P-violating energy shifts dramatically!
Instead of energy shifts linear in the P-violation parameters we get then shifts proportional to the square root of these parameters. Numerically we find such energy shifts which only appear for unstable states to be of order $10^{-5}...\,1\,\,$Hz. Under a reversal of the handedness of the external field configuration these P-violating shifts get multiplied by a phase factor $i$, i.e.\ the shifts in the real and imaginary part of the complex eigenenergies are exchanged. 
Application of our technique to hydrogen-like atoms with a nucleus of spin $I=1/2$ yields P-violating energy shifts which are very sensitive to the nuclear spin dependent P-odd force, which receives a rather large contribution from the polarized strange quark density in polarized nuclei. 
Thus, a measurement of these energy shifts could provide an important tool to elucidate nuclear properties connected to the so called ``spin crisis''.  
We also present a method for treating degenerate perturbation theory which combines advantages of both, Kato's and Bloch's methods.
} 
\end{center}

\end{titlepage}
% Preprint Energy Shifts/Doktorarbeit Thomas Gasenzer
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%
\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{\Ome}{\Omega}
\newcommand{\sig}{\sigma}
\newcommand{\Tet}{\Theta}
\newcommand{\tet}{\theta}
\newcommand{\vth}{\vartheta}
\newcommand{\vph}{\varphi}

\newcommand{\balp}{{\bar\alpha}}
\newcommand{\bbet}{{\bar\beta}}
\newcommand{\etab}{{\bar\eta}}
\newcommand{\bgam}{{\bar\gamma}}
\newcommand{\balpbbet}{{(\bar\alpha\bar\beta)}}

\newcommand{\sigmax}{\sigma_{max}}

%
%  Paulimatrizen
%
\newcommand{\paue}{\sigma_1}
\newcommand{\pauz}{\sigma_2}
\newcommand{\paud}{\sigma_3}
\newcommand{\pauv}{\vecgr{\sigma}}
\newcommand{\paui}{\sigma_i}
\newcommand{\pauj}{\sigma_j}
\newcommand{\pauk}{\sigma_k}



%-------------------------------------------------------------------
%
%   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{\hpvi}{\mbox{$H_{\sssty\mathrm PV}^{\sssty (i)}$}}
\newcommand{\hpvez}{\mbox{$H_{\sssty\mathrm PV}^{\sssty (1,2)}$}}

\newcommand{\hcm}{\mbox{$H_{c.m.}$}}
\newcommand{\hext}{\mbox{$H_{ext}$}}
\newcommand{\Hngam}{\mbox{$H_0^{(\gam)}$}}
\newcommand{\Hrad}{\mbox{$H_{rad}$}}
\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{\HnINT}{H_0^{\sssty INT}}
\newcommand{\HPVINT}{H_{\sssty\mathrm PV}^{\sssty INT}}
\newcommand{\HEXT}{H^{\sssty EXT}}

\newcommand{\qbar}{\mbox{$\bar{q}$}}

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

\newcommand{\qA}{q_{\sssty \!A}}
\newcommand{\ql}{q_{\ell}}

%
%   schwache Ladungen, Weinberg-Winkel, Bohrradius
%
\newcommand{\GAq}{\mbox{$G_{\sssty A}^{\sssty (q)}$}}

\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{\cC}{{\cal C}}
\newcommand{\cD}{{\cal D}}
\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{\wdele}{\mbox{$\sqrt{\delta_1}$}}
\newcommand{\wdelz}{\mbox{$\sqrt{\delta_2}$}}
\newcommand{\wdeli}{\mbox{$\sqrt{\delta_i}$}}
\newcommand{\wdelez}{\mbox{$\sqrt{\delta_{1,2}}$}}
\newcommand{\wudele}{\sqrt{\delta_1}}
\newcommand{\wudelz}{\sqrt{\delta_2}}
\newcommand{\wudelez}{\sqrt{\delta_{1,2}}}

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

\newcommand{\Odelez}{\cO(\dele^2,\dele\delz,\delz^2)}


%
%  geometrische Phasen, Spuren der geom Phasenmatr., Eigenwerte
%
\newcommand{\ggeo}{\gamma^{geom}}
\newcommand{\ggeoa}{\gamma^{geom}_\alpha}
\newcommand{\ggeoap}{\gamma^{geom}_{\alpha,+}}
\newcommand{\ggeoapn}{\gamma^{geom}_{\alpha,+,0}}
\newcommand{\ggeoape}{\gamma^{geom}_{\alpha,+,1}}
\newcommand{\ggeoam}{\gamma^{geom}_{\alpha,-}}
\newcommand{\ggeoamn}{\gamma^{geom}_{\alpha,-,0}}
\newcommand{\ggeoame}{\gamma^{geom}_{\alpha,-,1}}
\newcommand{\ggeoapm}{\gamma^{geom}_{\alpha,\pm}}
\newcommand{\ggeoapmn}{\gamma^{geom}_{\alpha,\pm,0}}
\newcommand{\ggeoapme}{\gamma^{geom}_{\alpha,\pm,1}}
\newcommand{\ggeoapmz}{\gamma^{geom}_{\alpha,\pm,2}}
\newcommand{\ggeoapmi}{\gamma^{geom}_{\alpha,\pm,i}}
\newcommand{\unggeo}{\ungam^{geom}}

\newcommand{\etaa}{\eta_{\alp}}
\newcommand{\etaan}{\eta_{\alp,0}}
\newcommand{\etaae}{\eta_{\alp,1}}
\newcommand{\etaaz}{\eta_{\alp,2}}
\newcommand{\etaai}{\eta_{\alp,i}}
\newcommand{\etaaj}{\eta_{\alp,j}}
\newcommand{\etaapm}{\eta_{\alp,\pm}}
\newcommand{\etaanpm}{\eta_{\alp,0,\pm}}
\newcommand{\etaaepm}{\eta_{\alp,1,\pm}}

\newcommand{\kapa}{\kappa_{\alp}}
\newcommand{\kapan}{\kappa_{\alp,0}}
\newcommand{\kapae}{\kappa_{\alp,1}}
\newcommand{\kapaz}{\kappa_{\alp,2}}
\newcommand{\kapai}{\kappa_{\alp,i}}
\newcommand{\kapaj}{\kappa_{\alp,j}}


%
%  allg. Quantenzahlen
%
\newcommand{\FdB}{|F_3|}



%
%   n=2 Unterraum Matrizen
%
\newcommand{\unA}{\mbox{$\underline{A\!}\,$}}

\newcommand{\unC}{\mbox{$\underline{C\!}\,$}}
\newcommand{\unCt}{\mbox{$\underline{C\!\!\:t\!}\,$}}
\newcommand{\unCtil}{\mbox{$\widetilde{{\underline{C\!}\,}}$}}

\newcommand{\unBtil}{\mbox{$\widetilde{{\underline{B\!}\,}}$}}

\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{\unDi}{\mbox{$\underline{D\!}\,$}_i}
\newcommand{\unDv}{\mbox{$\underline{\vec{D}}$}}
%\newcommand{\unDv}{\underline{\mbox{$\vec{D}$}}}
\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{\unFi}{{\underline{F\!}\,}_i}
\newcommand{\unFj}{{\underline{F\!}\,}_j}
\newcommand{\unFk}{{\underline{F\!}\,}_k}
\newcommand{\unFv}{\mbox{$\underline{\vec{F}}$}}

\newcommand{\unGa}{\mbox{$\underline{\Gamma}$}}
\newcommand{\unGam}{\mbox{$\underline{\Gamma}$}}
\newcommand{\ungam}{\mbox{$\underline{\gamma}$}}

\newcommand{\unH}{\mbox{$\underline{\cal H}$}}
\newcommand{\unhext}{\mbox{${\underline{H\!}\,}_{ext}$}}
\newcommand{\unHh}{\mbox{${\underline{\hat H\!}\,}$}}
\newcommand{\unHhh}{\mbox{$\doublehat{{\underline{H\!}\,}}$}}
\newcommand{\unHtil}{\mbox{$\widetilde{{\underline{H\!}\,}}$}}
\newcommand{\unKh}{\mbox{${\underline{\hat K\!}\,}$}}
\newcommand{\unKhh}{\mbox{$\doublehat{{\underline{K\!}\,}}$}}
%
\newcommand{\unmuv}{\mbox{$\underline{\muv}$}}

\newcommand{\unMr}{\mbox{$\underline{M\!}\,$}}
\newcommand{\unM}{\mbox{$\underline{\cal M\!}\,$}}
\newcommand{\uncM}{\mbox{$\underline{\cal M\!}\,$}}
\newcommand{\unMn}{\mbox{$\underline{\cal M\!}\,_{\sssty 0}$}}
\newcommand{\unMniKn}{\mbox{$\underline{\cal M\!}\,_{\sssty 0}^\iKn$}}
\newcommand{\unMniKe}{\mbox{$\underline{\cal M\!}\,_{\sssty 0}^\iKe$}}
\newcommand{\unMniKi}{\mbox{$\underline{\cal M\!}\,_{\sssty 0}^\iKi$}}
\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{\unrN}{\mbox{$\underline{N}$}}

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

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

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

\newcommand{\unV}{\mbox{$\underline{V\!}\,$}}

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

%
%   behutete Buchstaben
%
\newcommand{\lhat}{\hat l}

\newcommand{\Hh}{\hat H}
\newcommand{\Kh}{\hat K}
\newcommand{\Oh}{\hat O}

%\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$}}



%
%   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{\EHFS}{E_{\sssty\mathrm HFS}}
\newcommand{\EFS}{E_{\sssty\mathrm FS}}

\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{\Atil}{\widetilde A}
\newcommand{\btil}{\tilde b}
\newcommand{\Btil}{\widetilde B}
\newcommand{\ctil}{\tilde c}
\newcommand{\Ctil}{\widetilde C}
\newcommand{\Etil}{\widetilde E}
\newcommand{\etatil}{\tilde\eta}
\newcommand{\phitil}{\tilde\phi}
\newcommand{\Htil}{\widetilde H}
\newcommand{\cHtil}{\widetilde \cH}
\newcommand{\ntil}{\tilde n}
\newcommand{\cTtil}{\widetilde \cT}
\newcommand{\cRtil}{\widetilde \cR}

\newcommand{\Util}{{\widetilde U}}
\newcommand{\unUtil}{{\widetilde {\unU}}}
\newcommand{\Utilgeo}{{\widetilde U}^{geom}}
\newcommand{\Utilgeoa}{{\widetilde U}^{geom}_{\alpha}}
\newcommand{\Utilsiga}{{\widetilde U}^{\iKsig}_{\alpha}}
\newcommand{\unUtilgeo}{{\widetilde {\unU}}^{geom}}
\newcommand{\unUtildyn}{{\widetilde {\unU}}^{dyn}}
\newcommand{\unUtilsig}{{\widetilde {\unU}}^{\iKsig}}
\newcommand{\Utilpm}{{\widetilde {U}}_{\pm}}
\newcommand{\unUtilpm}{{\widetilde {\unU}}_{\pm}}
\newcommand{\unUtildynpm}{{\widetilde {\unU}}^{dyn}_{\pm}}

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

\newcommand{\EkR}{E_{\mbox{\tiny R}}}



%
% Orts-, Impuls-, etc. -vektoren
%
\newcommand{\av}{\vec{a}}
\newcommand{\Av}{\vec{A}}

\newcommand{\betv}{\vecgr{\beta}}
\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{\evemz}{(\eve\!\!-\!i\evz)}
\newcommand{\evepz}{(\eve\!\!+\!i\evz)}

\newcommand{\cEv}{\vecgr{\cal E}}
\newcommand{\cEvP}{\cEv_{\mbox{\tiny P}}}
\newcommand{\cEvR}{\cEv_{\mbox{\tiny R}}}
\newcommand{\cEvs}{\cEv^{\sssty (\sigma)}}
\newcommand{\cEsvs}{\cEv'^{\sssty (\sigma)}}
\newcommand{\cEsvz}{\cEv'^{\sssty (2)}}
\newcommand{\cEsvd}{\cEv'^{\sssty (2)}}
\newcommand{\cEvr}{\cEv^{\sssty (\rho)}}
\newcommand{\cEvt}{\cEv^{\sssty (\tau)}}
\newcommand{\cEvse}{\cEv^{\sssty (\sigma_1)}}
\newcommand{\cEvsz}{\cEv^{\sssty (\sigma_2)}}
\newcommand{\cEvsd}{\cEv^{\sssty (\sigma_3)}}
\newcommand{\cEvsv}{\cEv^{\sssty (\sigma_4)}}
\newcommand{\cEve}{\cEv^{\sssty (1)}}
\newcommand{\cEvz}{\cEv^{\sssty (2)}}
\newcommand{\cEvd}{\cEv^{\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}{\vecgr{\eta}}
\newcommand{\etavb}{{\bar{\etav}}}
\newcommand{\etavR}{\etav_{\mbox{\tiny R}}}

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

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

\newcommand{\Jv}{\vec{J}}

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

\newcommand{\Lv}{\vec{L}}

\newcommand{\mv}{\vec{m}}
\newcommand{\muv}{\vecgr{\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{\Rvd}{\dot{\vec{R}}}
\newcommand{\Rvdot}{\dot{\vec{R}}}
\newcommand{\Rdot}{\dot{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{\rv}{\vec{r}}
\newcommand{\rvA}{\rv_{\sssty \!A}}
\newcommand{\rvd}{\dot{\vec{r}}}
\newcommand{\rvdot}{\dot{\vec{r}}}
\newcommand{\rvdotA}{{\dot{\vec{r}}}_{\sssty \!A}}
\newcommand{\rdot}{\dot{r}}

\newcommand{\sv}{\vec{s}}
\newcommand{\Sv}{\vec{S}}
\newcommand{\sigv}{\vecgr{\sigma}}

\newcommand{\xv}{\vec{x}}
\newcommand{\xvd}{\dot{\xv}}
\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{\UtRRn}{\Util(R,R_0)}
\newcommand{\UtgRRn}{\Util_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{\iKj}{{\sssty (j)}}
\newcommand{\iKen}{{\sssty (n)}}
\newcommand{\iKn}{{\sssty (0)}}
\newcommand{\iKe}{{\sssty (1)}}
\newcommand{\iKz}{{\sssty (2)}}
\newcommand{\iKd}{{\sssty (3)}}
\newcommand{\iKv}{{\sssty (4)}}
\newcommand{\iKvar}[1]{{\sssty (#1)}}
\newcommand{\isKvar}[1]{{'\sssty (#1)}}
\newcommand{\ikvar}[1]{\mbox{\raisebox{0.2ex}{$\ssty (#1)$}}}

\newcommand{\ieKen}{{\sssty [n]}}
\newcommand{\ieKn}{{\sssty [0]}}
\newcommand{\ieKe}{{\sssty [1]}}
\newcommand{\ieKz}{{\sssty [2]}}
\newcommand{\ieKd}{{\sssty [3]}}
\newcommand{\ieKv}{{\sssty [4]}}
\newcommand{\ieKvar}[1]{{\sssty [#1]}}

\newcommand{\igKvar}[1]{{\sssty \{#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}{\mbox{$\frac{a_1(\alp,\cE)}{a_0(\alp,\cE)}$}}
\newcommand{\aeansig}{\mbox{$\frac{a_1(\alp,\cE^\iKsig)}{a_0(\alp,\cE^\iKsig)}$}}
\newcommand{\aeansme}{\mbox{$\frac{a_1(\alp,\cE^\iKvar{\sig-1})}{a_0(\alp,\cE^\iKvar{\sig-1})}$}}

\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.}


%
%   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 ,\cEv}|$}}
\newcommand{\revrEv}{\mbox{$ | \alpha,F_3,\cEv )$}}
\newcommand{\revrsEv}{\mbox{$ | \alpha,F'_3,\cEv )$}}

\newcommand{\levlER}{\mbox{$ ( \widetilde{\alpha,F_3 ,\cEv(R)}|$}}
\newcommand{\revrER}{\mbox{$ | \alpha,F_3,\cEv(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,\cEv(R)}|$}}
\newcommand{\revrzaER}{\mbox{$ | a,\cEv(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)}\,)}


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

\newcommand{\PV}{P-Verletzung}
\newcommand{\PVD}{P-verletzend}
\newcommand{\ENV}{Energieverschiebung}
\newcommand{\HOP}{Hamiltonoperator}
\newcommand{\WKB}{W.K.B.}

\newcommand{\LEP}  {\mbox{{\sc Lep\ }}}
\newcommand{\OPAL}  {\mbox{{\sc Opal\ }}}
\newcommand{\ALEPH}  {\mbox{{\sc Aleph\ }}}
\newcommand{\CERN}   {\mbox{{\sc Cern\ }}}
\newcommand{\FERMILAB}   {\mbox{{\sc Fermilab\ }}}
\newcommand{\SLAC}   {\mbox{{\sc Slac\ }}}


