Time-independent Green's function of a quantum simple harmonic oscillator system and solutions with additional generic delta-function potentials

The one-dimensional time-independent Green's function G ₀ of a quantum simple harmonic oscillator (SHO) system ($V_0\left(x\right)=m{\omega }^2{x}^2/2$) 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 SHO with an generic delta-function potential at an arbitrary site, i.e. $V_1\left(x\right)\propto \delta (x-a)$. 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 SHO with two or more generic delta-function potentials. We give the solutions of the case with two additional delta-functions for illustration. (paper)