Abstract
We consider theinverted harmonic potential and present infinitely degenerate non-stationary orthogonal states. We discuss that a non-interacting gas under the inverted harmonic potential has an infinite entropy at the absolute zero temperature. . We derive energy eigenvalues for a trapped particle and construct bounded eigenstates for the semi-infinite system. We obtain a new free particle wave packet using the eigenstates for the inverted harmonic potential. It has unique self-focusing feature and can be used as focusing beam without a lens in optical systems where paraxial approximation is used.
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