An Efficient Chebyshev–Lanczos Method for Obtaining Eigensolutions of the Schrödinger Equation on a Grid

been employed in bound state calculations, where the spectrum of the Hamiltonian can be obtained from the solution A grid method for obtaining eigensolutions of bound systems is presented. In this, the block–Lanczos method is applied to a Chebyof the TDSE. The problem usually encountered is the shev approximation of exp(2H/D), where D is the range of eigenvaldiagonalization of the Hamiltonian H in (1). Since this in ues we are interested in. With this choice a preferential convergence practice cannot be done, it is impossible to compute the of the eigenvectors corresponding to low-lying eigenvalues of H is action of the propagator e2 on an arbitrary wave function achieved. The method is used to solve a variety of one-, two-, and three-dimensional problems. To apply the kinetic energy operator [2] and, thus, the time-evolution is accomplished using we use the fast sine transform instead of the fast Fourier transform, various approximations to the exponential. We mention thus fullfilling, a priori, the box boundary conditions. We further here the Crank–Nicolson scheme [3–5] and the Kosloff extend the Chebyshev approximation to treat general functions of and Tal-Ezer method [6]. For a detailed discussion we refer matrices, thus allowing its application to cases for which no analytithe reader to the review article of de Raedt [2]. The existing cal expressions of the expansion coefficients are available. Q 1996