SEMICLASSICAL THEORY OF SPECTRAL RIGIDITY
SEMICLASSICAL THEORY OF SPECTRAL RIGIDITY
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DOI：
10.1098/rspa.1985.0078
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发表时间：
1985-01-01
影响因子：
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通讯作者：
BERRY, MV
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文献类型：
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作者：
BERRY, MV
一组量子能级的光谱刚性ε（L）是在L平均能级间距范围内光谱阶梯与最佳拟合直线的均方差。在半经典极限（λ →0）下，得到了λ（L）作为经典周期轨道上的和的公式.当L <$Lmax，其中Lmax ~<$-（N-1）为N个自由度的系统时，由于很长经典轨道的性质，<$（L）表现出下列普适性态：如果系统是经典可积的（所有周期轨道填充环面），<$（L）<$1/5L（如在不相关的（泊松）特征值序列中）;如果系统是典型的混沌（所有周期轨道孤立且不稳定），并且没有对称性，<$（L）<$InL/2π2+Dif 1 <$L <$Lmax（如随机矩阵理论的高斯幺正系综）;如果系统是混沌的，并且具有时间反演对称性，则<$L = InL/π2+Eif 1 <$L <$Lmax（如高斯正交系综）。当L <$Lmax时，<$L（L）在一个由短经典轨道确定的值处非普适饱和，对于可积系统，该值为<$N-1阶，对于混沌系统，该值为In（<$N-1）阶.这些结果是通过使用谱密度的π-轨道展开，以及长轨道强度的经典求和规则和限制其贡献干扰方式的半经典求和规则得到的。对于两个例子，我们详细研究了π（L）：矩形台球（可积）和黎曼zeta函数（假设它的零点是一个未知量子系统的特征值，该系统的未知经典极限是混沌的）。
The spectral rigidity ⊿(L) of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range ofLmean level spacings. In the semiclassical limit (ℏ→0), formulae are obtained giving ⊿(L) as a sum over classical periodic orbits. WhenL≪Lmax, whereLmax~ℏ-(N-1) for a system ofNfreedoms, ⊿(L) is shown to display the following universal behaviour as a result of properties of very long classical orbits: if the system is classically integrable (all periodic orbits filling tori), ⊿(L)═1/5L(as in an uncorrelated (Poisson) eigenvalue sequence); if the system is classically chaotic (all periodic orbits isolated and unstable) and has no symmetry, ⊿(L) ═ InL/2π2+Dif 1≪L≪Lmax(as in the gaussian unitary ensemble of random-matrix theory); if the system is chaotic and has time-reversal symmetry, ⊿(L) = InL/π2+Eif 1 ≪L≪Lmax(as in the gaussian orthogonal ensemble). WhenL≫Lmax, ⊿(L) saturates non-universally at a value, determined by short classical orbits, of order ℏ–(N–1)for integrable systems and In (ℏ-1) for chaotic systems. These results are obtained by using the periodic-orbit expansion for the spectral density, together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere. For two examples ⊿(L) is studied in detail: the rectangular billiard (integrable), and the Riemann zeta function (assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic).