论文标题
量子星图的多重特征函数
Multifractal eigenfunctions for quantum star graphs
论文作者
论文摘要
我们证明，在某些指定的情况下，量子星图的本征函数表现出多重分子的自相似结构。在半经典状态下，当光谱参数和顶点倾向于无穷大时，我们得出了由一组键长的键函数的Mellin变换的渐近条件，该键是由一组键长，该键长导致与特征功能相关的Renyi Entropy渐近。我们应用此结果表明，可以构建满足多重尺度定律定律的简单量子星图。在低频制度中，我们通过根据与键长集相关的ZETA函数来计算Renyi熵来证明多纹状体。在某些算术情况下，分形指数D_Q满足围绕Q = 1/4的对称关系，这是由Zeta函数的功能方程式产生的。从某种意义上说，我们的结果类似于作者最近证明算术SEBA台球的多重尺度缩放定律。但是，与在这种情况下不同，我们不需要满足算术条件，也不需要依赖精致的算术估计。
We prove that the eigenfunctions of quantum star graphs exhibit multifractal self-similar structure in certain specified circumstances. In the semiclassical regime, when the spectral parameter and the number of vertices tend to infinity, we derive an asymptotic condition for the Mellin transform of a specified function arising from the set of bond lengths which yields an asymptotic for the Renyi entropy associated with an eigenfunction. We apply this result to show that one may construct simple quantum star graphs which satisfy a multifractal scaling law. In the low frequency regime we prove multifractality by computing the Renyi entropy in terms of a zeta function associated with the set of bond lengths. In certain arithmetic cases the fractal exponent D_q satisfies a symmetry relation around q=1/4 which arises from the functional equation of the zeta function. Our results are, in some sense, analogous to the multifractal scaling law that the authors recently proved for arithmetic Seba billiards. However, unlike in that case, we do not require arithmetic conditions to be satisfied, and nor do we rely on delicate arithmetic estimates.
