The very unusual properties of the resolvent, heat kernel, and zeta function for the operator $-d^
The very unusual properties of the resolvent, heat kernel, and zeta function for the operator $-d^
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运算符 $-d^ 的解析、热核和 zeta 函数的非常不寻常的属性
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
Jinsung Park
中科院分区：
文献类型：
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作者：
K. Kirsten;Paul Loya;Jinsung Park
本文分析了有限区间上算子−d2 scindr 2 −1 scin（4 r2）的预解式、热核和谱zeta函数。这些谱函数的结构性质强烈依赖于所选择的自伴实现的运营商，一个选择是必要的，因为奇异的潜力。只有弗里德里希实现的标准性质重现，所有其他实现高度非标准性质观察。特别地，对于k <$N，我们在热核的小t渐近展开式中发现了像（logt）−k这样的项。此外，zeta函数具有作为对数分支点的s=0。
In this paper we analyze the resolvent, the heat kernel and the spectral zeta function of the operator −d2∕dr2−1∕(4r2) over the finite interval. The structural properties of these spectral functions depend strongly on the chosen self-adjoint realization of the operator, a choice being made necessary because of the singular potential present. Only for the Friedrichs realization standard properties are reproduced, for all other realizations highly nonstandard properties are observed. In particular, for k∊N we find terms like (logt)−k in the small-t asymptotic expansion of the heat kernel. Furthermore, the zeta function has s=0 as a logarithmic branch point.