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## Homework Statement

Let's consider a harmonic oscillator with a harmonic perturbation:

[tex]H = \frac{p^2}{2} + \frac{x^2}{2} + a \frac{x^2}{2}.[/tex]

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 [itex]E(a)[/itex]. This series is known to be a convegent one (when [itex]|a| < 1[/itex]) with sum equal to the exact answer ([itex]1/2 \sqrt{1+a}[/itex]). We want to comfirm this fact by a direct calculation.

## Homework Equations

and## 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?

Thank you in advance.

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