I have been thinking about the application of a time-dependent harmonic oscillator. The problem finding time-dependent harmonic oscillator itself is long-standing and fascinating one in many fields, e.g., quantum optics. In the context of that, an interesting approach done by Lewis and Riesenfeld is famous as the LR invariant method.
The point is introducing a dynamically invariant operator “I” which satisfies dI/dt = 0 for a given, time-dependent Hamiltonian. Using this method, we can obtain a lot of exact solutions for time-dependent harmonic oscillator systems.
What I am trying now is to apply these solutions to noncommutative solitons that are generated from the projection operators in the famous, ordinary (time-independent) harmonic oscillator system in quantum mechanics. It would be very easy to imagine that we can construct time-dependent noncommutative solitons from the time-dependent projection operators. It is straightforward to realize that, but the physical interpretation of that is crucial. That is to say, we need the physical interpretation for using a different Hamiltonian from the ordinary one.
In the construction of noncommutative solitons, all we use is assignment of the space coordinates (x, y) and the coordinates (x, p) in phase space in quantum mechanics. In that sense, we can use any Hamiltonian as long as the canonical commutation relation [x, y]= i. I want to understand this issue more fundamentally.