quantum mechanics - Inverted Harmonic oscillator

what are the energies of the inverted Harmonic oscillator?

$$H=p^{2}-\omega^{2}x^{2}$$

since the eigenfunctions of this operator do not belong to any $L^{2}(R)$ space I believe that the spectrum will be continuous, anyway in spite of the inverted oscillator having a continuum spectrum are there discrete 'gaps' inside it?

Also if I use the notation of 'complex frequency' the energies of this operator should be

$$E_{n}= \hbar (n+1/2)i\omega$$ by analytic continuation of the frequency to imaginary values.