[Paper Review] Quantum unique ergodicity

Steve Zelditch|arXiv (Cornell University)|Jan 27, 2003

Quantum chaos and dynamical systems5 references3 citations

TL;DR

This paper establishes that in quantum uniquely ergodic (QUE) systems, off-diagonal matrix elements of pseudodifferential operators between eigenfunctions tend to zero when eigenvalue gaps vanish. The key result shows that quasi-modes with a bounded number of essential frequencies and singular classical limits cannot exist in QUE systems, implying such systems—like the stadium and Donnelly’s surfaces—cannot be QUE if they support these localized quasi-modes.

ABSTRACT

This short note proves that a Laplacian cannot be quantum uniquely ergodic if it possesses a quasimode of order zero which (i) has a singular limit, and (ii) is a linear combination of a uniformly bounded number of eigenfunctions (modulo an o(1) error). Bouncing ball quasimodes of the stadium are believed to have this property (E.J. Heller et al) and so are analogous quasimodes recently constructed by H. Donnelly on certain non-positively curved surfaces. The main ingredient is the proof that all sequences of off-diagonal matrix elements of QUE systems with vanishing spectral gaps tend to zero.

Motivation & Objective

• To analyze the asymptotic behavior of off-diagonal matrix elements in quantum unique ergodic (QUE) systems.
• To investigate the compatibility of quasi-modes with singular classical limits (e.g., scarring on periodic orbits) and the QUE property.
• To establish a criterion for ruling out QUE in systems that support quasi-modes with few essential frequencies and non-Liouville limits.
• To provide a theoretical basis for understanding the non-QUE nature of systems like the stadium and Donnelly’s surfaces based on quasi-mode structure.

Proposed method

• Uses weak* limits of measures $ d ilde{\Phi}_{i,j} $ associated with matrix elements $ \langle A\varphi_i, \varphi_j \rangle $ to analyze asymptotic behavior.
• Applies the invariance of weak* limits under the geodesic flow, known from prior work on quantum ergodicity.
• Shows that any weak* limit of $ d\tilde{\Phi}_{i,j} $ must be a constant multiple of Liouville measure $ dL $ due to ergodicity and absolute continuity.
• Proves the constant must be zero when $ i \neq j $, using orthogonality and the identity operator in the limit.
• Applies this to quasi-modes by decomposing them into eigenfunctions with bounded frequency sets $ \Lambda_k $, assuming $ n(k) \leq C $.
• Derives a contradiction if such a quasi-mode has a singular limit measure $ d\mu \neq dL $, under the QUE assumption.

Experimental results

Research questions

• RQ1 Can quasi-modes with a bounded number of essential frequencies and a singular classical limit exist in a quantum uniquely ergodic system?
• RQ2 What is the relationship between eigenvalue clustering and the failure of quantum unique ergodicity in systems with localized quasi-modes?
• RQ3 How does the vanishing of off-diagonal matrix elements under small eigenvalue gaps constrain the structure of quasi-modes in QUE systems?
• RQ4 To what extent can the soft scarring criterion—based on singular limits of quasi-mode matrix elements—be used to rule out QUE?
• RQ5 Are systems like the Bunimovich stadium and Donnelly’s non-positively curved surfaces necessarily non-QUE if their quasi-modes have few essential frequencies?

Key findings

• For any QUE system, off-diagonal matrix elements $ \langle A\varphi_i, \varphi_j \rangle $ tend to zero as $ |\lambda_i - \lambda_j| \to 0 $, even when $ i \neq j $.
• Any weak* limit of the measures $ d\tilde{\Phi}_{i,j} $ associated with such matrix elements must be a constant multiple of Liouville measure $ dL $.
• The constant multiple is zero for $ i \neq j $, due to orthogonality of eigenfunctions and the trace condition with $ A = I $.
• A quasi-mode $ \{\psi_k\} $ of order 0 with uniformly bounded $ n(k) \leq C $ and a singular classical limit $ d\mu \neq dL $ cannot exist in a QUE system.
• This implies that if the stadium or Donnelly’s surfaces support quasi-modes with few essential frequencies and scarring, they are not QUE.
• The result provides a theoretical basis for the widely held belief that such systems are not QUE, based on the structure of their quasi-modes.