  It is shown that it is possible to establish sum rules that must be satisfied
at the nodes and extrema of the eigenstates of confining potentials which are
functions of a single variable. At any boundstate energy the Schroedinger
equation has two linearly independent solutions one of which is normalisable
while the other is not. In the domain after the last node of a boundstate
eigenfunction the unnormalisable linearly independent solution has a simple
form which enables the construction of functions analogous to Green's functions
that lead to certain sum rules. One set of sum rules give conditions that must
be satisfied at the nodes and extrema of the boundstate eigenfunctions of
confining potentials. Another sum rule establishes a relation between an
integral involving an eigenfunction in the domain after the last node and a sum
involving all the eigenvalues and eigenstates. Such sum rules may be useful in
the study of properties of confining potentials. The exactly solvable cases of
the particle in a box and the simple harmonic oscillator are used to illustrate
the procedure. The relations between one of the sum rules and two-particle
densities and a construction based on Supersymmetric Quantum Mechanics are
discussed.
