A local quantum version of the Kolmogorov theorem

Details A local quantum version of the Kolmogorov theorem A local quantum version of the Kolmogorov theorem

2004-06-14

arxiv.org/abs/math-ph/0406026v1

Physics

Mathematical Physics

18 pages

Scientific paper

10.1007/s00220-005-1299-4

Consider in $L^2 (\R^l)$ the operator family $H(\epsilon):=P_0(\hbar,\omega)+\epsilon Q_0$. $P_0$ is the quantum harmonic oscillator with diophantine frequency vector $\om$, $Q_0$ a bounded pseudodifferential operator with symbol holomorphic and decreasing to zero at infinity, and $\ep\in\R$. Then there exists $\ep^\ast >0$ with the property that if $|\ep|<\ep^\ast$ there is a diophantine frequency $\om(\ep)$ such that all eigenvalues $E_n(\hbar,\ep)$ of $H(\ep)$ near 0 are given by the quantization formula $E_\alpha(\hbar,\ep)= {\cal E}(\hbar,\ep)+\la\om(\ep),\alpha\ra\hbar +|\om(\ep)|\hbar/2 + \ep O(\alpha\hbar)^2$, where $\alpha$ is an $l$-multi-index.

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