  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.
