Weyl Laws for Partially Open Quantum Maps

Probabilistic Weyl Laws for Quantized Tori

“…S ρ classes for the torus. The S ρ classes for the quantization of the torus have already been considered in [16] and we refer to that paper for more detailed results such as the sharp Gårding inequality. Here we continue with a self-contained presentation.…”

Resonances in open quantum maps

“…Since the resonances counted in [98] were in a range that contained Γ , the fractional exponents observed are probably due to insufficient statistics or the semiclassical regime not been reached yet. Another result presented in [100,101] is that, for strongly chaotic systems, the width of the distribution of decay rates around Γ decays like (ln N ) −1 . This was proven for a toy model and numerically observed (albeit not quite clearly) for the baker map and a perturbed cat map.…”

“…However, it was later proved [100,101] that in the semiclassical limit the resonances tend to cluster at a particular value of the decay rate. More specifically, let R(q, p) denote the reflection probability at point (q, p).…”

“…It was shown in [100,101] that the number of resonances with decay rate inside a fixed interval containing Γ scales with N 1 in the semiclassical limit. In other words, most resonances have this decay rate.…”

Wave-packet evolution in non-Hermitian quantum systems

“…The semiclassical properties of non-Hermitian dynamics have recently been approached from various directions. Examples include the study of ray dynamics of absorbing wave equations for weak non-Hermiticities [9]; the mean-field approximation for a non-Hermitian manyparticle system [6]; the quantum classical correspondence for open quantum maps in the chaotic regime [10]; and complex extensions of the quantum probability distribution [11]. In [12] a coherent state approximation has been applied to non-Hermitian systems, and a generalized canonical structure involving a metric gradient flow has been identified.…”