Approach to energy eigenvalues and eigenfunctions from nonperturbative regions of eigenfunctions.

We study the approach to energy eigenvalues and eigenfunctions of Hamiltonian matrices with band structure from diagonalization of their truncated matrices. Making use of a generalization of Brillouin-Wigner perturbation theory, it is shown that in order to obtain approximate energy eigenvalues and eigenfunctions the sizes of truncated matrices should be larger than the nonperturbative regions of the eigenfunctions by several band width of the Hamiltonian matrix, with the nonperturbative regions being able to be estimated before the eigenfunctions are known. This prediction is checked numerically by the Wigner-band random-matrix model, which shows that 99% of eigenfunctions can be obtained when the sizes of truncated matrices are larger than those of the nonperturbative regions of the eigenfunctions by three band widths of the Hamiltonian matrix, on average.