Bound states ofn-dimensional harmonic oscillator decorated with Dirac delta functions

Abstract

Bound state solutions of the Schrodinger equation have been investigated for n-dimensional (n ≥ 2) harmonic oscillator potential decorated with any finite number (P) of Dirac delta functions. The potential is radially symmetric and given as , where σis are arbitrary real numbers, r1 < r2 < ⋅⋅⋅ < rP and ri (0, + ∞). We have demonstrated that addition of Dirac delta functions lifts the accidental degeneracies of n-dimensional harmonic oscillator energy levels and leaves only the degeneracy due to the radial symmetry. Explicit forms of bound state eigenfunctions and the eigenvalue equation are given for n, l values, where n is the space dimension and l is the degree of n-dimensional spherical harmonics. We have shown that, for given n and l, there are a countably infinite number of bound state energy levels which are continuous functions of ω, σis and at most P of them can be negative.