Bound states of n-dimensional harmonic oscillator decorated with Dirac delta functions

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 V(r) = 1/2m omega(2)r(2) - h(2)/2m Sigma(i=1)(P) sigma(i)delta(r - r(i)), where sigma(i)s are arbitrary real numbers, r(1) < r(2) < (...) < r(p) and r(i) epsilon (0, + infinity). 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 omega, sigma(i)s and at most P of them can be negative.