An Algorithmic Approach for the
Symmetric Perturbed Eigenvalue Problem

Application to the solution of a Schrödinger equation
by a kp-Perturbation Method

Claude-Pierre Jeannerod (1), Nicolas Maillard (2), and Eckhard Pflügel (3)

(1) Claude-Pierre Jeannerod LMC-IMAG - (2) Nicolas Maillard LMC-IMAG, Observatoire de Grenoble - (3) Eckhard Pflügel LMC-IMAG

Full article postscript version

Abstract: In semi-conductor theory, applying the kp-method to the monodimensional Schrödinger equation leads to a symmetric perturbed eigenvalue problem [8], i.e. to the diagonalization of a matrix tex2html_wrap_inline1252 depending on a small parameter tex2html_wrap_inline1254 , symmetric tex2html_wrap_inline1256 . The eigenelements of tex2html_wrap_inline1252 are expansions in fractional powers of tex2html_wrap_inline1254 (Puiseux series).

Usually, physicists solve this problem by using Schrödinger perturbation formulas under some restrictive conditions, which make perturbed eigenvector symbolic approximation impossible. This is illustrated by the modified Kane matrix [5].

To solve this problem completely from a symbolic computing point of view, we consider the symmetric perturbed eigenvalue problem in the case of analytic perturbations [2, 9]. We first review the classical characteristic polynomial approach, showing why it may be not optimal. We also present a direct matricial algorithm [7]: transforming the analytic matrix tex2html_wrap_inline1252 into its so-called q-reduced form allows to recover the information we need for the eigenvalues.

This alternative method, as well as the classical one, can be described in terms of the Newton polygon. However, our approach uses only a finite number of terms of tex2html_wrap_inline1252 and is more suitable for large matrices and a low approximation order.

Besides, we show that the q-reduction process can simultaneously provide symbolic approximations of both the perturbed eigenvalues and eigenvectors.

The implementation of this algorithm in MAPLE is used to diagonalize the modified Kane matrix up to a given order.

Keywords: perturbation of eigenvalues, symmetric analytic matrices, Newton-Puiseux polygon, q-reduced form of a matrix depending on a parameter, MAPLE package, Schrödinger equation, semi-conductor, kp-method, quantum well.


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