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Generic Spectral Determination of Semiclassical Schrödinger Operators with ℤ2-Symmetry

  • Qiaoling Wei
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Abstract

Consider the two-dimensional semiclassical Schrödinger operator Pℏ=−ℏ2/2Δ+V on ℝ2, where V has a nondegenerate well at the origin, its harmonic frequencies are rationally independent, and V is ℤ2-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)↦ 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.

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Cite

BibTeXRIS
BibTeX
@misc{wei2026generic,
  title = {{Generic Spectral Determination of Semiclassical Schrödinger Operators with $\mathbb Z_2$-Symmetry}},
  author = {Qiaoling Wei},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.11111},
  eprint = {2608.11111},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.11111},
}
RIS
TY  - UNPB
TI  - Generic Spectral Determination of Semiclassical Schrödinger Operators with ℤ_2-Symmetry
AU  - Qiaoling Wei
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2608.11111
UR  - https://arxiv.org/abs/2608.11111
AB  - Consider the two-dimensional semiclassical Schrödinger operator P_ℏ=−ℏ^2/2Δ+V on ℝ^2, where V has a nondegenerate well at the origin, its harmonic frequencies are rationally independent, and V is 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)↦ 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.
ER  -