Propagator for inverted harmonic potential.

[FedEx]

Hello.

I was trying to find out the propagator for the inverted SHO (something like tachyon oscilltor) and turns out that it remains unitary only for very short times. Which didnt make much sense to me. I tried looking at the usual SHO propagator, and that too seems to be not Unitary! ( I tried checking it by doing ∫U(x,x') U*(x',x'') dx' and see if that equals the dirac delta. I found that it blows up at x=x'' as it should but for x ≠ x' it is not zero)

Ofcourse I might(should) be making a mistake somewhere. But even an initial glance at the propagator for usual SHO would hint that there is something it fishy, since it contains terms like sin(ωt).

Thanks

 

[azntoon]

in advance.The propagator for the inverted SHO is a tricky one that requires careful consideration of certain conditions and assumptions. The unitary nature of the propagator depends on the time scale that you are considering. For short times, the propagator is unitary, however, for longer times, the propagator's unitary nature may be compromised. This can be seen in the usual SHO propagator as well. As you have noted, the sin(ωt) term in the propagator suggests that the system is not always unitary. This is because the oscillatory behavior of the sin(ωt) term leads to phase shifts which can result in the system not being described by a unitary transformation. To ensure that the system remains unitary, it is important to consider time-dependent boundary conditions.