First and second-order relativistic corrections to the two and higher-dimensional isotropic harmonic oscillator obeying the spinless Salpeter equation

We study the relativistic version of the $d$-dimensional isotropic quantum harmonic oscillator based on the spinless Salpeter equation. This has no exact analytical solutions. We use perturbation theory to obtain compact formulas for the first and second-order relativistic corrections; they are expressed in terms of two quantum numbers and the spatial dimension $d$. The formula for the first-order correction is obtained using two different methods and we illustrate how this correction splits the original energy into a number of distinct levels each with their own degeneracy. Previous authors obtained results in one and three dimensions and our general formulas reduce to them when $d=1$ and $d=3$ respectively. Our two-dimensional results are novel and we provide an example that illustrates why two dimensions is of physical interest. We also obtain results for the two-dimensional case using a completely independent method that employs ladder operators in polar coordinates. In total, three methods are used in this work and the results all agree.