Solution
= Solution
The action <integral> can be evaluated exactly:
$$
I(\lambda)=\lambda\log\left(\lambda+\sqrt{\lambda^2-1}\right)
-\sqrt{\lambda^2-1}.
$$
As $\lambda\to\infty$,
$$
I(\lambda)=\lambda\log(2\lambda)-\lambda+O(\lambda^{-1}).
$$
Its leading logarithmic term is $\lambda\log\lambda$. Hence the large <eigenvalues> satisfy, to leading order,
$$
\boxed{\lambda\log\lambda
=\epsilon\left(n+\frac34\right)\pi.}
$$