AI 中文总结
针对具有ℤ₂对称性的二维半经典薛定谔算子，证明其量子伯克霍夫正则形前三层可确定势的泰勒级数，低能谱可确定势（差空间反演），一般条件依赖势的10阶jet。
AI 中文摘要
考虑二维半经典薛定谔算子$P_\hbar=-\frac{\hbar^2}{2}\Delta+V$，其定义在$\mathbb{R}^2$上，其中$V$在原点处有非退化井，其调和频率有理无关，且$V$具有$\mathbb{Z}_2$对称性。我们证明，一般情况下，量子伯克霍夫正则形（QBNF）的前三层可确定$V$在原点处的完整泰勒级数，仅差不可避免的空间反演$V(x)\mapsto V(-x)$。该一般条件仅依赖$V$的10阶jet。因此，对于在原点处具有唯一孤立全局井的实解析势，低能半经典谱一般可确定$V$，仅差空间反演。
英文摘要
Consider the two-dimensional semiclassical Schrödinger operator $P_\hbar=-\frac{\hbar^2}{2}Δ+V$ 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.
Comments40 pages
