Shell model calculation has a long history over half century. Many advanced approaches have been developed while exact diagonalization is most basic and is still of primary importance. As exact diagonalization, the Lanczos method [1] is widely used and has been refined, combining with M-scheme shell model calculations. There was, however, no great progress for exact diagonalization except the Lanczos and its similar methods.

In this contribution, we present a new diagonalization method for large-scale M-scheme shell model calculations; filter diagonalization [2]. This is based on the Sakurai and Sugiura (SS) method [3] with shift algorithm for linear equations and is alternative for the Lanczos method which has been used over half century.

In the SS method, by considering a moment defined by the energy eigenvalues, large-scale shell model problem is reduced into small-scale one. By the Cauchy’s contour integral, we can solve energy eigenvalues and wave functions in a given energy interval, which is quite unique compared to the Lanczos method. The SS method demands, however, heavy computations because it converts one diagonalization problem to solve a large number of linear equations. To overcome this difficulty, we combine the SS method with shift algorithm of linear equations, which is very recently used in all-to-all propagators in lattice quantum chromodynamics (QCD) [4].

In this contribution, we present an algorithm of filter diagonalization in the M-scheme shell model calculations and demonstrate how it works in the large-scale problem.


Reference:
(1) C. Lanczos, J. Res. Nat. Bur. Stand, 45, 255 (1950)
(2) T. Mizusaki, K. Kaneko, M.Honma and T. Sakurai, submitted to Phys. Rev. C.
(3) T. Sakurai, H. Sugiura, J Comput Appl Math 2003, 159, 119.
(4) H. Ohno, Y. Kuramashi, T. Sakurai and H. Tadano, submitted to JSIAM Letters, hep-lat/1004.0292.