For generic Z2-symmetric two-dimensional potentials, the first three quantum Birkhoff normal form layers determine the potential's full Taylor series up to spatial inversion.
Colin de Verdi\`ere, A semi-classical inverse problem II: reconstruction of the potential, in Geometric Aspects of Analysis and Mechanics, Progress in Mathematics, vol
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Generic Spectral Determination of Semiclassical Schr\"odinger Operators with $\mathbb Z_2$-Symmetry
For generic Z2-symmetric two-dimensional potentials, the first three quantum Birkhoff normal form layers determine the potential's full Taylor series up to spatial inversion.