On the maximal scarring for quantum cat map eigenstates

Abstract

We consider the quantized hyperbolic automorphisms on the 2-dimensional torus (or generalized quantum cat maps), and study the localization properties of their eigenstates in phase space, in the semiclassical limit. We prove that if the semiclassical measure corresponding to a sequence of normalized eigenstates has a pure point component (phenomenon of ``strong scarring''), then the weight of this component cannot be larger than the weight of the Lebesgue component, and therefore admits the sharp upper bound 1/2.

14 pages, uses the AMS article style

Cited by in corpus (14)

• From dual-unitary to quantum Bernoulli circuits: Role of the entangling power in constructing a quantum ergodic hierarchy
• Entropy of semiclassical measures in dimension 2
• Weyl's law and quantum ergodicity for maps with divided phase space
• Selective amplification of scars in a chaotic optical fiber
• Entropic bounds on semiclassical measures for quantized one-dimensional maps
• Semiclassical measures for higher dimensional quantum cat maps
• Using the Hadamard and related transforms for simplifying the spectrum of the quantum baker's map
• Optical Scar in a chaotic fibre
• Scarred eigenstates for arithmetic toral point scatterers
• The full delocalization of eigenstates for the quantized cat map
• Entropy of quantum limits for symplectic linear maps of the multidimensional torus
• Quantum Leaks
• Quantisations of piecewise affine maps on the torus and their quantum limits
• Maximal scarring for eigenfunctions of quantum graphs