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.
No comments:
Post a Comment