 98604 George A. Hagedorn, Sam L. Robinson
 BohrSommerfeld quantization Rules in the Semiclassical Limit
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Sep 14, 98

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Abstract. We study onedimensional 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 BohrSommerfeld 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 timedependent construction technique is novel and elementary.
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