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.
Abstract The problem of the harmonic oscillator with a centrally located delta function potential can be exactly solved in one dimension, where the eigenfunctions are expressed as superpositions of the Hermite polynomials or as confluent hypergeometric functions in general. The eigenfunctions obtained exactly are difficult to visualise and hence, to gain more insight, one can attempt to use model wave functions which are explicitly and simply expressed. Here, we apply the variational method to verify how closely one can approach the exact ground state eigenvalues using such trial wave functions. We obtain the estimates of the ground state energies, which are closer to the exact values in comparison to earlier approximate results for both the repulsive and attractive delta potentials.