Systems with a delta-function potential

Brian P. Anderson

Minoru Takahashi

It is shown by choosing an appropriate variational wave function that the ground state energy is not affected by the repulsive delta-function potential in two and three dimensional systems. Scattering cross section by the repulsive delta-function potential is shown to be zero.

Per Chr. Hemmer, Joel L. Lebowitz

Chun-Khiang Chua, Yu-Tsai Liu, Gwo-Guang Wong

The one-dimensional time-independent Green's function G 0 of a quantum simple harmonic oscillator (SHO) system ( ) 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. . 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.

Nabakumar Bera, Kamal Bhattacharyya, Jayanta K. Bhattacharjee

We show that the δ-function potential can be exploited along with perturbation theory to yield the result of certain infinite series. The idea is that any exactly soluble potential, if coupled with a δ function potential, remains exactly soluble. We use the strength of the δ function as an expansion parameter and express the second-order energy shift as an infinite sum in perturbation theory. The analytical solution is used to determine the second-order energy shift and hence the sum of an infinite series. By an appropriate choice of the unperturbed system, we can show the importance of the continuum in the energy shift of bound states.