A local quantum version of the Kolmogorov theorem

TL;DR

This paper extends the classical Kolmogorov theorem into a quantum setting, demonstrating the persistence of eigenvalue structures for a class of perturbed quantum harmonic oscillators with diophantine frequencies.

Contribution

It introduces a quantum analogue of the Kolmogorov theorem, showing eigenvalue stability under small perturbations for certain quantum operators with diophantine frequencies.

Findings

Eigenvalues near zero follow a quantization formula involving perturbed frequencies.

Existence of a small perturbation threshold ensuring eigenvalue structure persistence.

Quantum eigenvalues can be explicitly approximated in the perturbed regime.

Abstract

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.