标题： 奇异摄动与边界条件及半空间中的卡西米尔效应 显示英文标题

标题： Singular perturbations with boundary conditions and the Casimir effect in the half space

摘要： 我们研究一类非极大乘法算子在边界条件下的自伴扩张。 我们证明这些扩张对应于拉普拉斯算子的奇异一阶扰动（在\cite{AK}的意义上），即半空间上的形式拉普拉斯算子带有奇异的δ势。 这种构造是描述平坦时空中的无质量标量场与无限导电板以及点状“杂质”同时存在时相关的卡西米尔效应的适当框架。 我们使用相对的ζ行列式（如\cite{Mul}和\cite{SZ}中定义的）来正则化该模型的划分函数。 我们研究相关相对ζ函数的解析延拓，并给出了划分函数和卡西米尔力的显式结果。 显示英文摘要

摘要： We study the self adjoint extensions of a class of non maximal multiplication operators with boundary conditions. We show that these extensions correspond to singular rank one perturbations (in the sense of \cite{AK}) of the Laplace operator, namely the formal Laplacian with a singular delta potential, on the half space. This construction is the appropriate setting to describe the Casimir effect related to a massless scalar field in the flat space time with an infinite conducting plate and in the presence of a point like "impurity". We use the relative zeta determinant (as defined in \cite{Mul} and \cite{SZ}) in order to regularize the partition function of this model. We study the analytic extension of the associated relative zeta function, and we present explicit results for the partition function, and for the Casimir force.