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? ;)