Page 150 - Compact Numerical Methods For Computers

The generalised symmetric matrix eigenvalue problem 139

are not orthonormal, so that a generalised eigenproblem arises. Because of the
nature of the operations these eigenproblems are symmetric.
In order to find suitable functions f (x) to expand f as
j
(11.24)

we note that the oscillator which has coefficients k = 1, k = 0 in its potential has
2
4
exact solutions which are polynomials multiplied by
2
exp(-0·5 x ).
Therefore, the basis functions which will be used here are
j-1 2
f (x) = Nx exp(-ax ) (11.25)
i
where N is a normalising constant.
The approximation sought will be limited to n terms. Note now that
2
Hf (x) = N exp(–ax )[–(j – 1)(j – 2)x j-3 + 2a(2j – 1)x j- 1
j
2 j +1 j+3
+( k -4a )x + k x ]. (11.26)
2
4
The minimisation of the Rayleigh quotient with respect to the coefficients c gives
j
the eigenproblem
Ac = eBc (11.27)
where
(11.28)

and

(11.29)

These integrals can be decomposed to give expressions which involve only the
integrals

for m odd

for m even (11.30)

for m = 0.
2
The normalising constant N has been chosen to cancel some awkard constants in
the integrals (see, for instance, Pierce and Foster 1956, p 68).
Because of the properties of the integrals (11.30) the eigenvalue problem
(11.27) reduces to two smaller ones for the even and the odd functions. If we set a
parity indicator w equal to zero for the even case and one for the odd case,