A Special Theory of Quantum Nonlinear Stability

This paper develops the quantum version of a classical theory of nonlinear stability for systems in the Trojan universality class. These systems show unusually long-lived coherence near certain elliptic equilibria in two-degree-of-freedom Hamiltonian dynamics, protected by an arithmetic gap in the frequencies, a nonzero twist in the normal form, and the structure of the Birkhoff normal form itself. The work restricts attention to the local neighborhood of one equilibrium and to the Weyl quantization of a polynomial or analytic germ, remaining strictly two-dimensional and, except for a brief treatment of dissipation, unitary. Because the Moyal bracket of a quadratic symbol with any other symbol coincides exactly with the classical Poisson bracket, the homological equations that generate the normal form are identical in the classical and quantum settings. The quantum normal form is therefore simply the classical normal form plus corrections that appear only in even powers of the Planck constant. For states carrying only a modest number of quanta, this normal form supplies explicit quasimodes whose residuals shrink rapidly enough to guarantee spectral localization and survival times that grow faster than any power of the Planck constant whenever the frequency ratio is irrational. The same normal form identifies the quartic Birkhoff coefficients with the familiar Dunham anharmonicity constants of molecular spectroscopy, shows that transition energies are universal through a definite order, and demonstrates that the twist determinant itself can be read from a handful of measured energy levels. For equilibria of mixed Krein signature the quantized quadratic Hamiltonian has pure-point dense spectrum below the classical Routh threshold and purely absolutely continuous spectrum at and above that threshold; energy therefore ceases to confine states exactly where linear stability is lost. Resonant sectors remain finite-dimensional only for difference resonances. A third-order sum resonance without twist yields a Jacobi operator that is not essentially self-adjoint, while a nonzero twist restores a compact resolvent and discrete spectrum. Once a resonance island shrinks below one Planck cell it can support at most one quantum state, cutting the classical resonance web off at a finite order fixed by the value of the Planck constant. Finally, under a standard Lindblad model a mode of negative signature grows from the vacuum at the classical rate, so dissipation limits its lifetime even at zero temperature. Numerical checks on model systems confirm each quantitative claim. The exponential (Nekhoroshev-type) regime is left conditional on a Gevrey remainder bound, and the possible quantum role of thin chaos and dynamical tunnelling is stated only as a conjecture. Within its deliberately narrow scope the theory shows that the structural conditions defining the Trojan class survive quantization and become, in several cases, directly measurable spectral properties.