98-604 George A. Hagedorn, Sam L. Robinson
Bohr-Sommerfeld quantization Rules in the Semiclassical Limit (193K, gzipped postscript) Sep 14, 98
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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.

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