An Inverse Theorem for Partially Symmetric Two-Dimensional Semiclassical Schr\"odinger Operators

We prove a formal local inverse spectral result for a two-dimensional semiclassical Schr\"odinger operator whose potential well possesses a single reflection symmetry. After a harmonic linear normalization, the potential can be written as $V(x_1,x_2)=\frac{1}{2}(v_1x_1^2+v_2x_2^2)+\sum_{j+2k\ge 3}a_{j,2k}\,x_1^j x_2^{2k}$, with $v_1/v_2\notin\mathbb{Q}$. The operator can be brought into a quantum Birkhoff normal form whose Weyl symbol is a formal series $B \equiv H_2 + \sum_{2r+k+\ell \ge 2} b_{r,k,\ell}\, \hbar^{2r} \Omega_1^k \Omega_2^\ell$. If the coefficient $a_{30}$ of the cubic term $x_1^3$ is non-zero, then the first two layers of the quantum Birkhoff normal form (i.e., the coefficients $b_{0,k,\ell}$ and $b_{1,k,\ell}$) uniquely determine the full Taylor series of $V$, once the sign of $a_{30}$ and the transverse-line data $\{a_{1,2k}\}_{k\ge 1}$ are prescribed.