For a 2D potential with one reflection symmetry, a nonzero cubic term, and known transverse derivative data, the first two spectral Birkhoff layers determine the full Taylor series of the potential.
Colin de Verdi` ere and V
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An Inverse Theorem for Partially Symmetric Two-Dimensional Semiclassical Schr\"odinger Operators
For a 2D potential with one reflection symmetry, a nonzero cubic term, and known transverse derivative data, the first two spectral Birkhoff layers determine the full Taylor series of the potential.