Time-Dependent Frequency Harmonic Oscillator

[canbula]

Homework Statement

Consider an harmonic oscillator with time-dependent frequency as:
\omega (t)=\omega_0 * \exp^{- \lambda t}
Find the time dependence of the ground state energy of this oscillator for \lambda << 1 situation.

Homework Equations

H=H_{0} + V(t)
H_{0} = \frac{p^2}{2m} + \frac{1}{2} m \omega_{0}^{2} x^{2}
and if we use the power series expansion for \lambda << 1 we get
V(t) = - \frac{1}{2} m \omega_{0}^2 \lambda t x^{2}

The Attempt at a Solution

I know that I should use the time-dependent perturbation theory, but I am not good at it. So I need some help to solve this problem.

 

[Thaakisfox]

What are the relevant equations for time dependent perturbation theory?

 

[canbula]

sorry I added them to my original post

 

[Thaakisfox]

next, how do we express the energy eigenvalues in time-dependent perturbation theory? ;)