Inverted Harmonic Oscillator Research Articles

Entropy and uncertainty of squeezed quantum open systems

We dene the entropy S and uncertainty function of a squeezed system interacting with a thermal bath, and study how they change in time by following the evolution of the reduced density matrix in the influence functional formalism. As examples, we calculate the entropy of two exactly solvable squeezed systems: an inverted harmonic oscillator and a scalar eld mode evolving in an inflationary universe. For the inverted oscillator with weak coupling to the bath, at both high and low temperatures, S ! r, where r is the squeeze parameter. In the de Sitter case, at high temperatures, S ! (1 c)r where c = 0=H, 0 being the coupling to the bath and H the Hubble constant. These three cases conrm previous results based on more ad hoc prescriptions for calculating entropy. But at low temperatures, the de Sitter entropy S! (1=2 c)r is noticeably dierent. This result, obtained from a more rigorous approach, shows that factors usually ignored by the conventional approaches, i.e., the nature of the environment and the coupling strength betwen the system and the environment, are important.

The Riemann Zeta Function and the Inverted Harmonic Oscillator

The Riemann zeta function has phase jumps ofπevery time it changes sign as the parameter t in the complex arguments=1/2+itis varied. We show analytically that as the real part of the argument is increased toσ>1/2, the memory of the zeros fades only gradually through a Lorentzian smoothing of the density of the zeros. The corresponding trace formula, forσ⪢1, is of the same form as that generated by a one-dimensional harmonic oscillator in one direction, along with an inverted oscillator in the transverse direction. It is pointed out that Lorentzian smoothing of the level density for more general dynamical systems may be done similarly. The Gutzwiller trace formula for the simple saddle plus oscillator model is obtained analytically, and is found to agree with the quantum result.