Quantum inverted harmonic potential

We consider a non-interacting gas under the inverted harmonic potential and present infinitely degenerate non-stationary orthogonal states. discuss that it has an infinite entropy at absolute zero temperature. show uncertainty in position of particle can be as there exists solution which asymptotes to Dirac delta function. obtain new free wave packet using eigenstates for potential. It unique self-focusing feature used focusing beam without lens optical systems where paraxial approximation is used.