George A. Hagedorn, Sam L. Robinson
Bohr-Sommerfeld quantization Rules in the Semiclassical Limit
(193K, gzipped postscript)

ABSTRACT.  We study one-dimensional quantum mechanical systems in the semiclassical 
limit. We construct a lowest order quasimode $\psi(\hbar )$ for the 
Hamiltonian $H(\hbar)$ when the energy $E$ and Planck's constant $\hbar$ 
satisfy the appropriate Bohr-Sommerfeld conditions. This means that 
$\psi(\hbar)$ is an approximate solution of the Schr\"{o}dinger equation 
in the sense that 
$$ 
\left\| \left[ H(\hbar )-E\right] \psi(\hbar )\right\| 
\leq C\hbar^{3/2}\left\| \psi(\hbar ) \right\| . 
$$ 
It follows that $H(\hbar)$ has some spectrum within a distance 
$C\hbar^{3/2}$ of $E$. Although the result has a long history, 
our time-dependent construction technique is novel and elementary.