## Quantum harmonic oscillator with harmonic perturbation

1. The problem statement, all variables and given/known data

Let's consider a harmonic oscillator with a harmonic perturbation:
$$H = \frac{p^2}{2} + \frac{x^2}{2} + a \frac{x^2}{2}.$$
Exact solution is known, but we want to derive it using perturbation theory. More specifically, suppose we want to obtain a series for the ground state energy $E(a)$. This series is known to be a convegent one (when $|a| < 1$) with sum equal to the exact answer ($1/2 \sqrt{1+a}$). We want to comfirm this fact by a direct calculation.

2. Relevant equations and 3. The attempt at a solution

Well-knows recurrence relations of Rayleigh-Schrodinger theory are applicable, and I've manually checked first five orders. The calculations quickly become very cumbersome, and there seems to be no easy way to obtain a general expression for n-th order (known in advance from the exact answer).

What is the most straightforward way to obtain this series? Will the Kato's theory be of any help in this case?

 Sure, I use them to calculate matrix elements of perturbation. This part of calculation is indeed simple. What seems to be not simple is the calculation of the n-th order of perturbation series and not just 1st and 2nd like most textbooks do. 1st order correction is $A |2\rangle$, 2nd order - $B |2\rangle + C |4\rangle$, 3rd order - $D |2\rangle + F |4\rangle + G |6\rangle$ and so on. The number of terms grows with n. There should be an elegant way to do the calculation even in the framework of an "old" perturbation theory. I deliberately avoid using diagramms here.