  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.
