量子图顶点附近的局域谱密度和真空能
Local Spectral Density and Vacuum Energy near a Quantum Graph Vertex
摘要：
在一个顶点处的delta相互作用在区间上推广了Robin（广义Neumann）边界条件。研究一个单一的顶点与N个无限的线索足以确定这样一个顶点上的状态密度等的局部化效果的所有标准的初值问题，如波动方程，有关的积分核（格林函数）的图形可以很容易地构造从相应的基本格林函数的实直线。从结果中可以得到谱投影核、局部谱密度和局部能量密度。能量密度，这是指一个解释的图形作为一个域的量化标量场，是一个系数的渐近展开的格林函数的椭圆问题，涉及图形哈密顿量;该扩展包含频谱/几何信息超出了研究热内核扩展。
英文摘要：
The delta interaction at a vertex generalizes the Robin (generalized Neumann) boundary condition on an interval. Study of a single vertex with N infinite leads suffices to determine the localized effects of such a vertex on densities of states, etc. For all the standard initial-value problems, such as that for the wave equation, the pertinent integral kernel (Green function) on the graph can be easily constructed from the corresponding elementary Green function on the real line. From the results one obtains the spectral-projection kernel, local spectral density, and local energy density. The energy density, which refers to an interpretation of the graph as the domain of a quantized scalar field, is a coefficient in the asymptotic expansion of the Green function for an elliptic problem involving the graph Hamiltonian; that expansion contains spectral/geometrical information beyond that in the much-studied heat-kernel expansion.
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