The misterious 'inverted' harmonic oscillator

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given the Hamiltonian [tex] H=p^{2}- \omega x^{2} [/tex]

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 [tex] Asinh( \omega t) +Bcosh( \omega t) [/tex]

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

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

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