Matrix algorithms for solving (in)homogeneous bound state equations.
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In particular, one has to deal with linear, homogeneous integral equations which, in sophisticated model setups, use numerical representations of the solutions of other integral equations as part of their input.These can be solved very efficiently using well-known matrix algorithms for eigenvalues (in the homogeneous case) and the solution of linear systems (in the inhomogeneous case).This is valuable insight, in particular for the study of baryons in a three-quark setup and more involved systems.
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Affiliation: Institut für Physik, Universität Graz, Universitätsplatz 5, 8010 Graz, Austria.
ABSTRACT
In the functional approach to quantum chromodynamics, the properties of hadronic bound states are accessible via covariant integral equations, e.g. the Bethe-Salpeter equation for mesons. In particular, one has to deal with linear, homogeneous integral equations which, in sophisticated model setups, use numerical representations of the solutions of other integral equations as part of their input. Analogously, inhomogeneous equations can be constructed to obtain off-shell information in addition to bound-state masses and other properties obtained from the covariant analogue to a wave function of the bound state. These can be solved very efficiently using well-known matrix algorithms for eigenvalues (in the homogeneous case) and the solution of linear systems (in the inhomogeneous case). We demonstrate this by solving the homogeneous and inhomogeneous Bethe-Salpeter equations and find, e.g. that for the calculation of the mass spectrum it is as efficient or even advantageous to use the inhomogeneous equation as compared to the homogeneous. This is valuable insight, in particular for the study of baryons in a three-quark setup and more involved systems. No MeSH data available. Related in: MedlinePlus |
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Mentions: The results are given in Figs. 6 and 7, where the sensitivity of both algorithms to the initial conditions and the comparison of the efficiency in the ideal case are shown. From Fig. 6 it is clear that the Arnoldi factorization is less sensitive to a change in initial conditions than the simple iteration, and that it is in general more efficient. Even in the ideal case (cf. Fig. 7) for the first eigenvalue the advanced algorithm is 36% more efficient (7 iterations compared to 11), and becomes even more advantageous for an increasing number of eigenvalues. |
View Article: PubMed Central - PubMed
Affiliation: Institut für Physik, Universität Graz, Universitätsplatz 5, 8010 Graz, Austria.
No MeSH data available.