Fixed-ℏ vanishing of the quantum Lyapunov exponent

Prove or disprove that, for the one-dimensional quantum system with Hamiltonian H(x,p)=p^2+x^2+W(x), where W is smooth and compactly supported, and with the Carlen–Maas 1-Wasserstein distance d_CM,1, the fixed-ℏ quantum Lyapunov exponent defined by the long-time supremum over distinct quantum states is zero.

Background

The paper explains that, at fixed ℏ, the dynamically relevant low-energy sector of a confining quantum Hamiltonian is finite dimensional. On this sector, the trace norm and the Kantorovich–Rubinstein norm associated with d_CM,1 are equivalent, while unitary Schrödinger evolution preserves the trace norm. Consequently, the transport-distance expansion ratio is uniformly bounded in time for states confined to that sector.

The authors further argue that high-energy dynamics for a compactly supported perturbation of the harmonic oscillator are asymptotically governed by the stable harmonic oscillator and therefore should not produce exponential expansion. They propose that finite-dimensionality and compactness may generically obstruct a fixed-ℏ theory of quantum chaos, motivating the conjecture that the naive fixed-ℏ exponent vanishes.