  The one-dimensional time-independent Green's function $G_0$ of a quantum
simple harmonic oscillator system ($V_0(x)=m \omega^2 x^2/2$) can be obtained
by solving the equation directly. It has a compact expression, which gives
correct eigenvalues and eigenfunctions easily. The Green's function $G$ with an
additional delta-function potential can be obtained readily. The same technics
of solving the Green's function $G_0$ can be used to solve the eigenvalue
problem of the simple harmonic oscillator with an generic delta-function
potential at an arbitrary site, i.e. $V_1(x)\propto \delta(x-a)$. The
Wronskians play an important and interesting role in the above studies.
Furthermore, the approach can be easily generalized to solve the quantum system
of a simple harmonic oscillator with two or more generic delta-function
potentials. We give the solutions of the case with two additional
delta-functions for illustration.
