  Using the Green's function associated with the one-dimensional Schroedinger
equation it is possible to establish a hierarchy of sum rules involving the
eigenvalues of confining potentials which have only a boundstate spectrum. For
some potentials the sum rules could lead to divergences. It is shown that when
this happens it is possible to examine the separate sum rules satisfied by the
even and odd eigenstates of a symmetric confining potential and by subtraction
cancel the divergences exactly and produce a new sum rule which is free of
divergences. The procedure is illustrated by considering symmetric power law
potentials and the use of several examples. One of the examples considered
shows that the zeros of the Airy function and its derivative obey a sum rule
and this sum rule is verified. It is also shown how the procedure may be
generalised to establish sum rules for arbitrary symmetric confining
potentials.
