Numerical Analysis Problem

The numerical analysis problem we must solve is to find the eigenvalues of an infinite banded matrix given by

$$h^{\rm anh}_{ij} = (i+1/2)\delta_{ij} + \epsilon g_{ij}$$ (3)

for $i,j = 0,1,\ldots{}$, where

$$\delta_{ij} = \left\{\begin{array}{ll} 1 & {\rm for }\ \ \ i=j \\ 0 & {\rm otherwise} \end{array}\right.$$ (4)

and

$$g_{ij} = \left\{\begin{array}{ll} \frac{3}{2}(m^2 + m + \f... ...or } i=j\pm 4\\ 0 & {\rm otherwise} \end{array}\right.$$ (5)

where $m = \min(i,j)$.

The first five rows and columns of the matrix $h^{\rm anh}_{ij}$ are

$$\left( \begin{array}{ccccc} \frac{1}{2}+ \frac{3\epsilon}{4}... ...& 0 & \frac{9}{2} + \frac{123 \epsilon}{4} \end{array}\right)$$

Obviously we do not wish to solve this problem for an infinite matrix. Fortunately, for small enough $\epsilon$, the expansion in the harmonic oscillator basis is convergent, so we may consider a truncated problem, stopping at a finite number of rows and columns. We may get a numeric measure of the effect of truncation by increasing the size of the truncated matrix and observing the change in the eigenvalues.

For a discussion of this problem and a solution using the Lanczos method of tridiagonalization, see S.S.M. Wong, Computational Methods in Physics and Engineering (Prentice Hall, Englewood Cliffs, New Jersey, 1992), pp 293ff.