非厄米动力学的量子绝热定理
The Quantum Adiabatic Theorem for Non-Hermitian Dynamics
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中文总结 AI 辅助
研究一类有限维非厄米薛定谔动力学的量子绝热定理，通过构造含时度量算符及戴森映射，利用标准方法和围道预解式估计得出绝热估计，包含拉回的厄米绝热误差和投影比较误差，并以两能级模型说明。
中文摘要 AI 辅助
我们为一类有限维非厄米薛定谔动力学建立了定量绝热估计。假定原始非厄米哈密顿量可对角化，具有实谱和非交叉本征值。接着构造一个动态兼容的含时度量算符，其正平方根定义了一个戴森映射。相关的戴森变换哈密顿量是厄米的。当此变换哈密顿量满足厄米绝热假设时，标准预解式投影方法给出厄米表示中的显式绝热估计。通过戴森映射将此估计拉回得到原始非厄米表示中的近似。然后利用围道预解式估计将所得的戴森拉回投影与原始非厄米哈密顿量的谱投影进行比较。最终界包含两部分贡献：拉回的厄米绝热误差和投影比较误差。给出一个两能级非厄米模型来说明定理的假设以及最终估计中出现的两部分贡献。
英文摘要
We establish a quantitative adiabatic estimate for a class of finite-dimensional non-Hermitian Schrödinger dynamics. The original non-Hermitian Hamiltonian is assumed to be diagonalizable with real spectrum and non-crossing eigenvalues. We then construct a dynamically compatible time-dependent metric operator, and its positive square root defines a Dyson map. The associated Dyson-transformed Hamiltonian is Hermitian. When this Dyson-transformed Hamiltonian satisfies the Hermitian adiabatic assumption, the standard resolvent-projection method gives an explicit adiabatic estimate in the Hermitian representation. Pulling this estimate back through the Dyson map gives an approximation in the original non-Hermitian representation. The resulting Dyson-pulled-back projections are then compared with the spectral projections of the original non-Hermitian Hamiltonian by using a contour-resolvent estimate. The final bound contains two contributions: the pulled-back Hermitian adiabatic error and the projection-comparison error. A two-level non-Hermitian model is presented to illustrate the hypotheses of the theorem and the two contributions appearing in the final estimate.
