In this paper, we further Meirong Zhang, et al.’s work by computing the
number of weighted eigenvalues for Sturm-Liouville equations, equipped
with general integrable potentials and Dirac weights, under Dirichlet
boundary condition. We show that, for a Sturm-Liouville equation with a
general integrable potential, if its weight is a positive linear
combination of $n$ Dirac Delta functions, then it has at most $n$
(may be less than $n$, or even be $0$) distinct real Dirichlet
eigenvalues, or every complex number is a Dirichlet eigenvalue; in
particular, under some sharp condition, the number of Dirichlet
eigenvalues is exactly $n$. Our main method is to introduce the
concepts of characteristics matrix and characteristics polynomial for
Sturm-Liouville problem with Dirac weights, and put forward a general
and direct algorithm used for computing eigenvalues. As an application,
a class of inverse Dirichelt problems for Sturm-Liouville equations
involving single Dirac distribution weights is studied.