KAM theory at the Quantum resonance

The paper announces a Gevrey quantum KAM/normal-form result at partial resonances, giving a spectral lattice formula and scarring, but key parts are either misstated or under-specified. Most notably, the “exponentially small in h” remainder is written as O(ε exp(−c h^{1/(α−1)})), which does not decay as h→0 and contradicts the claimed exponential smallness; the natural Gevrey rate is exp(−c h^{−1/(α−1)}). This sign/power issue is repeated in the quantum normal-form section and the main spectral formula (see the abstract and main claims with the O(ε exp(−c h^{1/(α−1)})) term, and the quantum normal-form Proposition with the same rate ). Moreover, the operator is set on L^2(R^l) with translation-invariant principal part H0(hD_x); discrete spectrum and Weyl counting invoked later (on a d-dimensional torus) are not justified without additional compactness or confining assumptions, creating a mismatch of frameworks (R^l vs. T^d) in the scarring section (e.g., the Weyl-law counting and torus microlocalization arguments in Section 6 ). The model solution corrects the exponential rate and outlines a standard Gevrey QBNF/KAM approach with appropriate references, but it also implicitly assumes spectral discreteness/ellipticity or periodic boundary conditions without stating them, and uses projection from quasimodes to true eigenfunctions in a way that fails if the spectrum is continuous. Hence both are incomplete: the paper by incorrect exponent and missing spectral hypotheses; the model by missing global spectral assumptions.