Generic Spectral Determination of Semiclassical Schrödinger Operators with ${\mathbb{Z}}_2$-Symmetry

Consider the two-dimensional semiclassical Schrödinger operator $P_{\mathrm{\hslash }}=-\frac{{\mathrm{\hslash }}^2}{2}\mathrm{\Delta }+V$ on ${\mathbb{R}}^2$, where $V$ has a nondegenerate well at the origin, its harmonic frequencies are rationally independent, and $V$ is ${\mathbb{Z}}_2$-symmetric. We prove that, generically, the first three layers of the quantum Birkhoff normal form (QBNF) determine the full Taylor series of $V$ at the origin, up to the unavoidable spatial inversion $V(x)\mapsto V(-x)$. The generic condition depends only on the jet of $V$ through order ten. Consequently, for real analytic potentials with a unique isolated global well at the origin, the low-lying semiclassical spectrum generically determines $V$ up to spatial inversion.