(a) Let $\{|n\rangle\}$ be a basis of eigenstates of a non-degenerate Hamiltonian $H$, with corresponding eigenvalues $\left\{E_{n}\right\}$. Write down an expression for the energy levels of the perturbed Hamiltonian $H+\lambda \Delta H$, correct to second order in the dimensionless constant $\lambda \ll 1$.

(b) A particle travels in one dimension under the influence of the potential

$$V(X)=\frac{1}{2} m \omega^{2} X^{2}+\lambda \hbar \omega \frac{X^{3}}{L^{3}}$$

where $m$ is the mass, $\omega$ a frequency and $L=\sqrt{\hbar / 2 m \omega}$ a length scale. Show that, to first order in $\lambda$, all energy levels coincide with those of the harmonic oscillator. Calculate the energy of the ground state to second order in $\lambda$.

Does perturbation theory in $\lambda$ converge for this potential? Briefly explain your answer.