Harmonic Oscillator with a Step and its Isospectral Properties

We investigate the one-dimensional Schr\"{o}dinger equation for a harmonic oscillator with a finite jump $a$ at the origin. The solution is constructed by employing the ordinary matching-of-wavefunctions technique. For the special choices of $a$, $a=4\ell$ ($\ell=1,2,\ldots$), the wavefunctions can be expressed by the Hermite polynomials. Moreover, we explore isospectral deformations of the potential via the Darboux transformation. In this context, infinitely many isospectral Hamiltonians to the ordinary harmonic oscillator are obtained.