一维安德森局域化中的反射共振：有限长度统计、维格纳时间延迟与边界本征函数
Reflection Resonances in the One-Dimensional Anderson Localization: Finite-Length Statistics, Wigner Time Delay, and Boundary Eigenfunctions
AI总结：
该研究分析一维安德森局域化有限长度无序样品的反射共振，关联共振密度与反射系数，推导维格纳时间延迟统计等结果，通过计算验证相关规律。
AI中文摘要：
我们研究长度为L的有限一维无序样品的反射共振极点Z_j=E_j-iΓ_j（Γ_j>0），该样品一端耦合到半无限引线，处于L≫ℓ_L≫k⁻¹的区域，其中ℓ_L为局域化长度，k=√E。关键步骤是将共振密度与复能量E+iΓ（对应均匀吸收）下的反射系数关联起来。精确的有限链Kac-Rice和Poincaré-Lelong恒等式将极点计数简化为反射强度的有限长度扩散。对于完美接触透明度，密度在Γ_L=k/ℓ_L处从局域化控制的Γ⁻¹定律交叉到宽共振的Γ⁻²定律，与Fyodorov和Meibohm的L→∞结果一致。有限长度在Γ_ultra=(e^(1-γ_E)/2)Γ_L e^(-L/ℓ_L)处截断Γ⁻¹区域。我们导出了超窄共振交叉形状作为显式移动前沿；对于接触透明度𝒯<1，该尺度移至𝒯Γ_ultra。相同的反射过程产生有限长度维格纳时间延迟统计，并在弱吸收极限下产生Comtet-Texier分布。我们表明，在相应闭合狄利克雷样品的本征值处，维格纳延迟与归一化本征函数的平方边界导数成反比。该量也给出本征值对引线接触端狄利克雷边界位移的响应，因此是本征模式对该边界施加的力；我们获得其有限长度分布。最后，有限透明度的共振密度满足单通道Moldauer-Simonius求和规则，将其完美耦合发散与Γ⁻²尾部关联起来。直接晶格和光谱计算验证了交叉和前沿常数。
英文摘要：
We study reflection-resonance poles $Z_j=E_j-iΓ_j$, $Γ_j>0$, of a finite one-dimensional disordered sample of length $L$, coupled at one end to a semi-infinite lead, in the regime $L\gg\ell_L\gg k^{-1}$, where $\ell_L$ is the localization length and $k=\sqrt{E}$. The key step is to relate the resonance density to the reflection coefficient at the complex energy $E+iΓ$, corresponding to uniform absorption. Exact finite-chain Kac--Rice and Poincaré--Lelong identities reduce pole counting to a finite-length diffusion of the reflected intensity. For the perfect contact transparency the density crosses over from the localization-controlled $Γ^{-1}$ law to the broad-resonance $Γ^{-2}$ law at $Γ_L=k/\ell_L$, in agreement with $L\to\infty$ result of Fyodorov and Meibohm. Finite length cuts off the $Γ^{-1}$ regime at $Γ_{\rm ultra}=\frac{e^{1-γ_{\rm E}}}{2}Γ_L e^{-L/\ell_L}$. We derive the ultranarrow resonances crossover shape as an explicit moving front; for contact transparency $\mathcal T<1$ this scale shifts to $\mathcal TΓ_{\rm ultra}$. The same reflection process yields the finite-length Wigner time-delay statistics and, in the weak-absorption limit, the Comtet--Texier distribution. We show that at eigenvalues of the corresponding closed Dirichlet sample the Wigner delay is inversely proportional to the squared boundary derivative of the normalized eigenfunction. This quantity also gives the eigenvalue response to displacement of the Dirichlet boundary at the lead-contact end, and hence the force exerted by the eigenmode on that boundary; we obtain its finite-length distribution. Finally, the finite-transparency resonance density obeys the single-channel Moldauer--Simonius sum rule, linking its perfect-coupling divergence to the $Γ^{-2}$ tail. Direct lattice and spectral computations test the crossover and front constant.
