通过能量算符δ函数作用于完全随机格点波函数,构造了可用于直接计算给定范围[Emin,Emax]内能量本征值和本征函数的局域子空间.在非正交局域基下详细推导了交迭积分和哈密顿算符在分立位置表象中的表示,讨论了广义本征值问题的解法.以Morse势和Henon-Heiles势的多个能量范围为例检验了算法.

An direct theoretical calculation of eigenvalues and eigenfunctions in any arbitrary range of energy is proposed. The calculation produce a small local basis set (nonorthogonal) by applying the Chebyshev polynomials expansion of the Greens operator on a completely random grid wavefunction. The analytic expressions of matrix element of Hamilton H, overlap S are then derived directly in this nonorthoganal basis set, and the corresponding generalized eigenvalue problem is solved to give the eigenvalues and eigenfunctions within range of [ E min,E max ]. The calculation requires a minimal storage because a discrete position grid representation of the H,S is used. Two instructive examples, the one dimension Morses potential and two dimension Henon Heiles potential are considered for four energy ranges in the present study. The method proves to be efficient, straightforward and simple.