Wave functions for pentadiagonal matrices in the weak coupling limit

Abstract

We consider a pentadiagonal matrix which will be described in this paper. We demonstrate practical methods for obtaining weak coupling expressions for the lowest eigenvector in terms of the parameters in the matrix, v and w. It is found that the expressions simplify if the wave function coefficients are put in the denominator.

Matrix Model: Emergence of a Quantum Number in the Strong Coupling Regime

“…We have shown in this and previous works [1][2][3][4] [5] we can get very interesting outcomes from rather simple matrices-both in the weak coupling and the strong coupling limits. In this paper, we demonstrated the symmetric patterns among the values and introduced a quantum number to classify the odd-even results under strong coupling.…”

“…In previous works [3][4] [5] we considered perturbation theory in the weak coupling limit. That meant that the v's and w's were small compared to nE.…”

“…where v = v/E. Selected results for the pentadiagonal case [5] are: a(0, 0) = 1, a(0, 1) = −v , a(0, 2) = v 2 /2 − w /2, a(0, 3) = −v 2 /6 + v w /2. We were able to get analytic expressions for the a(m, n)'s in terms of the v's and w's.…”

“…In previous works we studied the properties of simple tridiagonal and pentadiagonal matrices [1][2][3][4] [5]. We here show the complete pentadiagonal matrix with 2 parameters, v and w. We can reduce it to a tridiagonal case by simply setting w to zero.…”

Comparisons of matrices with different elements but identical eigenvalues

“…With different eigenfunctions we would have different observable properties such as static electromagnetic moments, transition rates, half lives, etc. To answer this question we turn to simple matrices that were previously studied [3][4][5][6][7][8]. Mainly we consider tridiagonal matrices with one parameter [v], but also briefly consider pentadiagonal matrices with 2 parameters [v, w].…”