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

(3) |

(4) |

(5) |

The first five rows and columns of the matrix
are

Obviously we do not wish to solve this problem for an infinite matrix. Fortunately, for small enough , 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.