The misterious 'inverted' harmonic oscillator

[zetafunction]

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

we can see inmediatly that this Hamiltonian will NOT have a BOUND state due to a 'saddle point' on (0,0) , here 'omega' is the frecuency of the Harmonic oscillations

the classical solutions are not PERIODIC $$Asinh( \omega t) +Bcosh( \omega t)$$

The Quantum counterpart is even worse since the functions will NOT be on $$L^{2} (-\infty , \infty )$$ then how is the spectrum ?

is there any webpage where this oscillator is considered ? What applications can be found ??

 

[LostConjugate]

There is no spectrum, the eigenvalues are continuous. A particle in this potential would gain momentum continuously as x goes to plus or minus infinity.

Like a ball on a hill.

It is not an oscillator.