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The misterious 'inverted' harmonic oscillator

  1. May 20, 2010 #1
    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 ??
  2. jcsd
  3. May 20, 2010 #2
    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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