Continued fraction calculation of the eigenvalues of tridiagonal matrices arising from the Schroedinger equation (original) (raw)

Finite-dimensional approximations of the resolvent of an infinite band matrix and continued fractions

Sbornik: …, 2007

In this paper we study the approximability of the resolvent of an operator generated by a band matrix by means of the resolvents of its finite-dimentional sections. For bounded perturbations of selfadjoint matrices a positive result in a large domain is obtained. We apply it to tridiagonal complex matrices in order to establish convergence conditions for Chebyshev continued fraction on sets of the complex domain. In the particular case of compact perturbation, this result is sharpened and the connection between the poles of the limit function and the eigenvalues of the tridiagonal matrix is shown. * Visiting Proffesor at Univ. Carlos

Matrix continued fractions and Expansions of the Error Function

Zenodo (CERN European Organization for Nuclear Research), 2022

In this paper we recall some results and some criteria on the convergence of matrix continued fractions. The aim of this paper is to give some properties and results of continued fractions with matrix arguments. Then we give continued fraction expansions of the error function erf(A) where A is a matrix. At the end, some numerical examples illustrating the theoretical results are discussed.

Computation of multiple eigenvalues of infinite tridiagonal matrices

Mathematics of Computation, 2003

In this paper, it is first given as a necessary and sufficient condition that infinite matrices of a certain type have double eigenvalues. The computation of such double eigenvalues is enabled by the Newton method of two variables. The three-term recurrence relations obtained from its eigenvalue problem (EVP) subsume the well-known relations of (A) the zeros of Jν (z); (B) the zeros of zJ ν (z) + HJν (z); (C) the EVP of the Mathieu differential equation; and (D) the EVP of the spheroidal wave equation. The results of experiments are shown for the three cases (A)-(C) for the computation of their "double pairs".

Matrix continued fraction and expansions of the Gauss hypergeometric function

New Trends in Mathematical Science, 2021

The aim of this paper is to give some properties and results of continued fractions with matrix arguments. Then we give continued fraction expansion of the Gauss hypergeometric function. At the end, some numerical examples illustrating the theoretical results are discussed.

Relaxation dynamics of a particle in the presence of an external potential: exact solution in terms of matrix continued fractions

Physica A: Statistical Mechanics and its Applications, 1994

Exact expressions for the Laplace transform of the after effect function arising from the solution of the Langevin or underlying Fokker-Planck equation for Brownian motion in an external potential are obtained as a sum of products of infinite matrix continued fractions. This is accomplished by reducing the scalar multiterm recurrence relations associated with the Fokker-Planck equation to matrix three term recurrence relations. The solution is illustrated by considering the problem of dielectric relaxation of a single axis rotator subjected to both constant and crystalline anisotropy fields.

Matrix continued fractions related to first-order linear recurrence systems

We introduce a matrix continued fraction associated with the first-order linear recurrence system Y k = θ k Y k−1 . A Pincherle type convergence theorem is proved. We show that the n-th order linear recurrence relation and previous generalizations of ordinary continued fractions form a special case. We give an application for the numerical computation of a non-dominant solution and discuss special cases where θ k is constant for all k and the limiting case where lim k→+∞ θ k is constant. Finally the notion of adjoint fraction is introduced which generalizes the notion of the adjoint of a recurrence relation of order n.

The continued fraction method in the theory of slow electron scattering by molecules and molecular ions

Chemical Physics Letters, 1986

Analytical possibilities of the continued fraction method are demonstrated by solving two problems connected with a description of slow electron resonance scattering by positive molecular ions. Using the rotation adiabatic approximation, exact solutions of the multiquantum vibrational transition problem have been found for the fractional-linear approximation of the vibrational adiabatic K matrix. The equivalence of different formulations of the multichannel quantum defect method has been shown. The existence of autoionization states with anomalously small decay widths has been established. The role of simultaneous rotational-vibrational transitions in molecular Rydberg spectra and autoionization is analyzed.

Matrix continued fraction expansion of Bessel function

New Trends in Mathematical Science, 2016

The aim of this paper is to provide some results and applications of continued fractions with matrix arguments. First, we recall some properties of matrix functions with real coefficients. Afterwards, we give a matrix continued fraction expansion of the Bessel function.

Matrix methods for radial Schrödinger eigenproblems defined on a semi-infinite domain

Applied Mathematics and Computation, 2014

In this paper, we discuss numerical approximation of the eigenvalues of the one-dimensional radial Schrödinger equation posed on a semi-infinite interval. The original problem is first transformed to one defined on a finite domain by applying suitable change of the independent variable. The eigenvalue problem for the resulting differential operator is then approximated by a generalized algebraic eigenvalue problem arising after discretization of the analytical problem by the matrix method based on high order finite difference schemes. Numerical experiments illustrate the performance of the approach.

Anharmonic oscillator and the analytic theory of continued fractions

Physical Review D, 1978

We study anharmonic oscillators of the type ax'+ bx'+ cx using the theory of continued fractions. Introducing a new set of coupling constants (depending on a, b, and c) in terms of which the associated difference equation simplifies, we write the Green's function of the theory in terms of an infinite continued fraction of the Stieltjes type, whose poles give the energy eigenvalues. We prove that this continued fraction converges where the corresponding perturbation series in the dominant coupling diverges. We obtain the analytic structure of the Green's function in the complex plane of this coupling constant. A scale transformation allows us to study the analyticity of the Green's function for ax '+ cx6 oscillators in the energy plane.