SPINLESS SALPETER EQUATION\\Some (Semi-) Analytical Approaches
Wolfgang Lucha
hep-ph0411069.tex
\documentclass[final]{aipproc}\layoutstyle{6x9}\begin{document}
\title{SPINLESS SALPETER EQUATION\\Some (Semi-) Analytical Approaches}
\author{Wolfgang Lucha}{address={Institute for High Energy Physics, Austrian
Academy of Sciences,\\Nikolsdorfergasse 18, A-1050 Vienna, Austria\\E-mail:
wolfgang.lucha@oeaw.ac.at}}\author{Franz F.~Sch\"oberl}{address={Institute
for Theoretical Physics, University of Vienna,\\Boltzmanngasse 5, A-1090
Vienna, Austria\\E-mail: franz.schoeberl@univie.ac.at}}\maketitle

The eigenvalue equation of a semirelativistic Hamiltonian composed of the
relativistic kinetic term of spin-0 particles and static interactions is
called spinless Salpeter equation. It is regarded as relativistic
generalization of the nonrelativistic Schr\"odinger approach, or as
approximation to the homogeneous Bethe--Salpeter equation in its
instantaneous~limit. The nonlocality inherent to this kind of operators makes
hard to find analytical solutions. Nevertheless, rigorous analytical
statements can be proved by sophisticated methods \cite{TWR}:\begin{itemize}
\item Combining minimum--maximum principle and suitable operator inequalities
allows to derive ``semianalytical'' (or even analytical) upper bounds on all
energy levels~\cite{AUB}.\item Geometrical considerations summarized under
the term ``envelope technique'' yield ``semianalytical'' expressions for both
upper and lower bounds on eigenenergies \cite{ET}. For some particular
interactions these bounds can be represented in analytical form.\end{itemize}
Resulting eigenstates must be constructed numerically anyway, for instance,
by scanning the Hilbert space variationally using standard Rayleigh--Ritz
techniques or by integrating this equation of motion by conversion to some
equivalent matrix eigenvalue problem \cite{Num}. The achieved accuracy of
these approximate solutions can then be estimated by powerful criteria
derived from generalized --- in the present case relativistic --- virial
theorems \cite{RVT}.

\bibliographystyle{aipprocl}%\bibliographystyle{aipproc}
\begin{thebibliography}{9}\bibitem{TWR}For some comprehensive review see, for
instance, W.~Lucha and F.~F.~Sch\"oberl, arXiv:hep-ph/0408184.
\bibitem{AUB}W.~Lucha and F.~F.~Sch\"oberl, Phys.~Rev.~A {\bf 54} (1996)
3790; Int.~J.~Mod.~Phys.~A {\bf 14} (1999) 2309;~Fizika B {\bf 8} (1999) 193;
J.~Math.~Phys.~{\bf 41} (2000) 1778.\bibitem{ET}R.~L.~Hall, W.~Lucha, and
F.~F.~Sch\"oberl, J.~Phys.~A {\bf 34} (2001) 5059; J.~Math.~Phys.~{\bf 42}
(2001) 5228;~Int.\ J.~Mod.~Phys.~A {\bf 17} (2002) 1931; {\it ibid}.~{\bf 18}
(2003) 2657; J.~Math.~Phys.~{\bf 43} (2002) 5913; {\it ibid}.~{\bf 43} (2002)
1237; {\it ibid}.~{\bf 44} (2003) 2724 (E); Phys.~Lett.~A {\bf 320} (2003)
127; J.~Math.~Phys.~{\bf 45} (2004) 3086.\bibitem{Num}For all the details,
consult, for example, W.~Lucha, H.~Rupprecht, and F.~F.~Sch\"oberl,
Phys.~Rev.~D {\bf 45} (1992) 1233; W.~Lucha and F.~F.~Sch\"oberl,
Phys.~Rev.~D {\bf 50} (1994) 5443; Phys.~Lett.~B {\bf 387} (1996)~573;
Phys.~Rev.~A {\bf 56} (1997) 139; Int.~J.~Mod.~Phys.~C {\bf 11} (2000) 485;
as well as the references listed therein.\bibitem{RVT}W.~Lucha and
F.~F.~Sch\"oberl, Phys.~Rev.~Lett.~{\bf 64} (1990) 2733; Mod.~Phys.~Lett.~A
{\bf 5} (1990) 2473;~Phys.\ Rev.~A {\bf 60} (1999) 5091; Int.~J.~Mod.~Phys.~A
{\bf 15} (2000) 3221.\end{thebibliography}\end{document}\endinput

hep-ph0411069.extracted_bib
\begin{thebibliography}{9}\bibitem{TWR}For some comprehensive review see, for
instance, W.~Lucha and F.~F.~Sch\"oberl, arXiv:hep-ph/0408184.
\bibitem{AUB}W.~Lucha and F.~F.~Sch\"oberl, Phys.~Rev.~A {\bf 54} (1996)
3790; Int.~J.~Mod.~Phys.~A {\bf 14} (1999) 2309;~Fizika B {\bf 8} (1999) 193;
J.~Math.~Phys.~{\bf 41} (2000) 1778.\bibitem{ET}R.~L.~Hall, W.~Lucha, and
F.~F.~Sch\"oberl, J.~Phys.~A {\bf 34} (2001) 5059; J.~Math.~Phys.~{\bf 42}
(2001) 5228;~Int.\ J.~Mod.~Phys.~A {\bf 17} (2002) 1931; {\it ibid}.~{\bf 18}
(2003) 2657; J.~Math.~Phys.~{\bf 43} (2002) 5913; {\it ibid}.~{\bf 43} (2002)
1237; {\it ibid}.~{\bf 44} (2003) 2724 (E); Phys.~Lett.~A {\bf 320} (2003)
127; J.~Math.~Phys.~{\bf 45} (2004) 3086.\bibitem{Num}For all the details,
consult, for example, W.~Lucha, H.~Rupprecht, and F.~F.~Sch\"oberl,
Phys.~Rev.~D {\bf 45} (1992) 1233; W.~Lucha and F.~F.~Sch\"oberl,
Phys.~Rev.~D {\bf 50} (1994) 5443; Phys.~Lett.~B {\bf 387} (1996)~573;
Phys.~Rev.~A {\bf 56} (1997) 139; Int.~J.~Mod.~Phys.~C {\bf 11} (2000) 485;
as well as the references listed therein.\bibitem{RVT}W.~Lucha and
F.~F.~Sch\"oberl, Phys.~Rev.~Lett.~{\bf 64} (1990) 2733; Mod.~Phys.~Lett.~A
{\bf 5} (1990) 2473;~Phys.\ Rev.~A {\bf 60} (1999) 5091; Int.~J.~Mod.~Phys.~A
{\bf 15} (2000) 3221.\end{thebibliography}