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Manuscript Title: Analytical Lanczos method: quantum eigenstates of anharmonic oscillators in one or more dimensions.
Authors: M. Kaluza
Program title: LANCZOS-A
Catalogue identifier: ACTR_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 79(1994)425
Programming language: Mathematica.
Computer: IBM RISC System 6000.
Operating system: IBM AIX version 3.2.
Word size: 32
Keywords: Computer algebra, General purpose, Differential equations, Polynomial hamiltonian, Analytical lanczos Method, Quantum mechanical Eigenspectrum, Eigenfunctions, Coupled anharmonic Oscillators.
Classification: 4.3, 5.

Nature of problem:
A number of problems in chemistry, atomic, nuclear, particle and statistical physics can be reduced to accurate evaluation of eigenspectra and eigenfunctions of an anharmonic oscillator or of a system of coupled anharmonic oscillators. In this paper we present a number of accurate eigensolutions for an example of a sextic anharmonic oscillator in one dimension where up to 88 digits accuracy is achieved, and for examples of two, three and four coupled one-dimensional sextic anharmonic oscillators, where 18, 12 and 8 digit accuracy, respectively, is achieved.

Solution method:
The Lanczos method is a method of finding the approximate solutions (E,psi) of the Hamiltonian eigenproblem Hpsi = Epsi. One chooses a trial function psi1 and applies the Hamiltonian N - 1 times to obtain the Krylov space {psi1,psi2 = Hpsi1,...,psiN = HpsiN-1}. Successive Gramm-Schmidt orthonormalization of the Krylov space leads to a set of N orthonormal Lanczos vectors and a corresponding tridiagonal Lanczos N x N Hamiltonian matrix. Diagonalization of the tridiagonal Lanczos matrix gives the approximations to the N eigenvalues and eigenfunctions of H. The analytical Lanczos method is an application of the general Lanczos method to a quantum-mechanical problem with d degrees of freedom in a continuous - in this case in a coordinate [x = (x1,...,xd)] - representation with the interaction potential given in an analytical form V(x) and with kinetic energy operators represented as derivatives -nabla**2/2. All the manipulations except the diagonalization of the tridiagonal Lanczos matrix are done analytically. The general procedure is independent of the dimensionality (number of degrees of freedom d) of the system. The boundary conditions and symmetry properties of the problem are incorporated naturally in the method. An important virtue of the analytical Lanczos method is the lack of the discretization parameters. The only discrete parameter is N, the number of Lanczos vectors.

The complexity of a problem is restricted by the available memory and cpu time. In this paper we have investigated the cases of polynomial potentials for one, two, three and four degrees of freedom. Both the time requirement and the memory requirement depend on the complexity of the potential function V(x), on the number of degrees of freedom, on the choice of the working precision and on the efficiency of the polynomial multiplication. Time needed to diagonalize the tridiagonal Lanczos matrix is usually negligible as compared to the set-up time for the Lanczos matrix, and the dimension of the Lanczos matrix is never large. Tridiagonal matrices as small as 10 x 10 are diagonalized to obtain eigenvalues to precision of 10 digits in both one- and higher- dimensional problems.

Running time:
The one-dimensional test calculation with N = 12 took about 30 s. The two-dimensional test-case calculation with N = 12 took about 4.5 min. The memory requirement is proportional to the size of the last calculated Lanczos vector, which is represented as a product of a polynomial and an exponential function.