量子动力学的上界
Upper Bounds in Quantum Dynamics
摘要：
我们发展了一个通用的方法来约束薛定谔动力学下的整个波包的扩展从上面。该方法从复能量下的传递矩阵范数的下界推导出位置算子的时均矩的上界。这个一般结果适用于Fibonacci算子。我们发现，在足够大的耦合，所有的传输指数严格取值于0和1之间。这是关于反常输运的第一个严格结果。对于与三角多项式相关联的准周期势，我们证明了，所有的低输运指数，并在一个较弱的假设下的频率，所有的上输运指数消失的所有阶段，如果李雅普诺夫指数是一致有界远离零。根据赫尔曼的一个著名结果，这个假设在耦合足够大时总是成立的。对于几乎Mathieu算子的特殊情况，我们的结果适用于耦合大于2。
英文摘要：
We develop a general method to bound the spreading of an entire wavepacket under Schrödinger dynamics from above. This method derives upper bounds on time-averaged moments of the position operator from lower bounds on norms of transfer matrices at complex energies. This general result is applied to the Fibonacci operator. We find that at sufficiently large coupling, all transport exponents take values strictly between zero and one. This is the first rigorous result on anomalous transport. For quasi-periodic potentials associated with trigonometric polynomials, we prove that all lower transport exponents and, under a weak assumption on the frequency, all upper transport exponents vanish for all phases if the Lyapunov exponent is uniformly bounded away from zero. By a well-known result of Herman, this assumption always holds at sufficiently large coupling. For the particular case of the almost Mathieu operator, our result applies for coupling greater than two.
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