Scars in the semiclassical limit
For a specified Anosov quantum system, bound the mass that a semiclassical measure of exact eigenstates can assign to an unstable periodic orbit, and determine whether the bounds are attainable by exceptional subsequences.
Problem statement
For a specified Anosov quantum system, bound the mass that a semiclassical measure of exact eigenstates can assign to an unstable periodic orbit, and determine whether the bounds are attainable by exceptional subsequences.
Lean 4
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Background
A quantum scar is an enhancement of eigenfunction density along an unstable classical periodic orbit, first observed by Heller. In the semiclassical limit, the question is which invariant measures can arise as weak limits ('semiclassical measures') of eigenfunction mass, and how much of that mass can sit on periodic orbits despite classical ergodicity.
What is known: Entropy bounds constrain semiclassical concentration for Anosov flows, with constants and hypotheses depending on the geometry; a universal unqualified one-half rule would overstate them. Explicit partially scarred eigenstates exist for certain quantum cat maps. A quantum map is not a Laplace operator on a negatively curved surface, so those constructions do not settle M8.
Quantum ergodicity already excludes a positive-density family converging to a fixed non-Liouville measure under its usual hypotheses. Meaningful residuals are exceptional zero-density sequences, sharp quantitative concentration bounds, or quasimodes with an explicitly specified spectral width. Finite-energy visual enhancement need not persist as a scarred limiting measure. This is distinct from many-body scars (B7).
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
- [1]Heller, 'Bound-state eigenfunctions of classically chaotic Hamiltonian systems: Scars of periodic orbits', PRL 53, 1515 (1984) link
- [2]Anantharaman & Nonnenmacher, 'Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold', Ann. Inst. Fourier 57 (2007) link
- [3]Faure, Nonnenmacher & De Bièvre, 'Scarred eigenstates for quantum cat maps of minimal periods', Comm. Math. Phys. 239 (2003) link
- [4]Dyatlov, quantum chaos and semiclassical measures (2023) link
Recorded progress
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