Background: So I know from my lecture that if I am given a wave-function, which describes a particle and a potential V(x) in which the particle is in, I can use V(x) in the time-independent Schrödinger-equation to obtain the systems Hamiltonian.

Now I am given in some other example a thermally isolated system with a Hamiltonian

$$H(t) = H_0 + V(t)\tag{1}$$

where $H_0$ is the systems interal Hamiltonian and V(t) is some interaction "Hamiltonian".

My question: Can I make any statements about the potential this system is in by looking at this Hamiltonian from eq. (1)? Can I say that it is a harmonic potential or anything at all?

Thanks in advance!

Bonus knowledge: The total Hamiltonian $H(t)$ is cyclic in the sense that $V(t)$ is only non-zero in a time intervall $[0+\delta t,t\delta-\tau]$ so $H(0)=H(\tau)$.


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