We consider semiclassical self-adjoint operators whose symbol, defined on a two-dimensional symplectic manifold, reaches a non-degenerate minimum $b_0$ on a closed curve. We derive a classical and quantum normal form which allows us, in addition to the complete integrability of the system, to obtain eigenvalue asymptotics in a window $(-\infty,b_0+\epsilon]$ for $\epsilon > 0$ independent on the semiclassical parameter. These asymptotics are obtained in two complementary settings: either a symmetry of the system under translation along the curve, or a Morse hypothesis reminiscent of Helffer-Sjostrand's "miniwell" situation.

中文翻译：

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相空间环上半经典井的均匀谱渐近

我们考虑半经典自伴随算子，其符号定义在二维辛流形上，在闭合曲线上达到非退化最小值 $b_0$。我们推导出一个经典的和量子的范式，除了系统的完全可积性之外，它允许我们在窗口 $(-\infty,b_0+\epsilon]$ 中获得特征值渐近，对于 $\epsilon > 0$ 独立于半经典参数。这些渐近性是在两个互补设置中获得的：要么是系统沿曲线平移的对称性，要么是让人想起 Helffer-Sjostrand 的“微型井”情况的莫尔斯假设。