AI 中文总结
该研究针对具有单一反射对称性的二维半经典薛定谔算子，证明了其形式化局部逆谱结果，在给定相关条件下可通过量子Birkhoff标准形前两层唯一确定势的完整泰勒级数。
AI 中文摘要
我们针对具有单一反射对称性的二维半经典薛定谔算子，证明了一个形式化的局部逆谱结果。经调和线性归一化后，势可写为$V(x_1,x_2)=\frac{1}{2}(v_1x_1^2+v_2x_2^2)+\sum_{j+2k\ge 3}a_{j,2k}\\,x_1^j x_2^{2k}$，其中$v_1/v_2\notin\mathbb{Q}$。该算子可化为量子Birkhoff标准形，其Weyl符号为形式级数$B \equiv H_2 + \sum_{2r+k+\ell \ge 2} b_{r,k,\ell}\\, \hbar^{2r} \Omega_1^k \Omega_2^\ell$。若三次项$x_1^3$的系数$a_{30}$非零，则在给定$a_{30}$的符号及横向线数据$\{a_{1,2k}\}_{k\ge 1}$后，量子Birkhoff标准形的前两层（即系数$b_{0,k,\ell}$和$b_{1,k,\ell}$）可唯一确定$V$的完整泰勒级数。
英文摘要
We study a two-dimensional semiclassical Schrödinger operator whose potential admits one reflection symmetry. Under a rational independence assumption on the harmonic frequencies and a nonvanishing condition \(a_{30}\ne0\), we show that the first two layers of the quantum Birkhoff normal form, together with the sign of \(a_{30}\) and the transverse data \(\{a_{1,2k}\}_{k\ge1}\), determine the Taylor series of the potential at the bottom of the well. The proof is constructive: the \(\hbar^2\)-layer gives a triangular recursion for odd-degree terms, while the classical layer recovers even-degree terms from their resonant projections.
Comments14 pages; Revised version. Reorganized the exposition of the triangular recursion argument to improve clarity. Refinements in wording and general English polishing
