  For a quadratic Markov branching process (QMBP), we show that the decay
parameter is equal to the first eigenvalue of a Sturm-Liouville operator
associated with the PDE that the generating function of the transition
probability satisfies. The proof is based on the spectral properties of the
Sturm-Liouville operator. Both the upper and lower bounds of the decay
parameter are given explicitly by means of a version of Hardy inequality. Two
examples are provided to illustrate our results. The important quantity, the
Hardy index, which is closely linked with the decay parameter of QMBP, is
deeply investigated and estimated.
