论文标题
准集成量子系统中Sobolev规范的增长
Growth of Sobolev norms in quasi integrable quantum systems
论文作者
论文摘要
我们证明了一个抽象的结果，给出了$ \ langle t \ rangle^\ varepsilon $ umper bount to $ {i} \ dotψ= h_0ψ+ v（t）ψ$的时间依赖的sobolev norms的增长。在这里，$ H_0 $被认为是陡峭的量子集成系统的哈密顿量，并且是订单$ {\ tt d}> 1 $的伪差操作员； $ v（t）$是一个时间依赖的pseudoDifferential运算符的家族，无界，但顺序$ {\ tt b} <{\ tt d} $。然后，将抽象定理应用于维度2中量子Anharmonic振荡器的扰动，以及在具有可集成的大地测量流的歧管上的Laplacian的扰动，尤其是Zoll歧管，旋转不变的表面和躺椅。证明基于经典nekhoroshev定理的证明的量子版本。
We prove an abstract result giving a $\langle t \rangle^\varepsilon$ upper bound on the growth of the Sobolev norms of a time-dependent Schrödinger equation of the form ${i} \dot ψ= H_0 ψ+ V (t)ψ$. Here $H_0$ is assumed to be the Hamiltonian of a steep quantum integrable system and to be a pseudodifferential operator of order ${\tt d} > 1$; $V (t)$ is a time-dependent family of pseudodifferential operators, unbounded, but of order ${\tt b} < {\tt d}$. The abstract theorem is then applied to perturbations of the quantum anharmonic oscillators in dimension 2 and to perturbations of the Laplacian on a manifold with integrable geodesic flow, and in particular Zoll manifolds, rotation invariant surfaces and Lie groups. The proof is based on a quantum version of the proof of the classical Nekhoroshev theorem.
