Title: The vacuum energy with non-ideal boundary conditions via an approximate functional equation Show Chinese title

Title: 通过近似函数方程的非理想边界条件下的真空能

Abstract: We discuss the vacuum energy of a quantized scalar field in the presence of classical surfaces, defining bounded domains $\Omega \subset {\mathbb{R}}^{d}$, where the field satisfies ideal or non-ideal boundary conditions. For the electromagnetic case, this situation describes the conductivity correction to the zero-point energy. Using an analytic regularization procedure, we obtain the vacuum energy for a massless scalar field at zero temperature in the presence of a slab geometry $\Omega=\mathbb R^{d-1}\times[0, L]$ with Dirichlet boundary conditions. To discuss the case of non-ideal boundary conditions, we employ an asymptotic expansion, based on an approximate functional equation for the Riemann zeta-function, where finite sums outside their original domain of convergence are defined. Finally, to obtain the Casimir energy for a massless scalar field in the presence of a rectangular box, with lengths $L_{1}$ and $L_{2}$, i.e., $\Omega=[0,L_{1}]\times[0,L_{2}]$ with non-ideal boundary conditions, we employ an approximate functional equation of the Epstein zeta-function. Show Chinese abstract

Abstract: 我们讨论在存在经典表面的情况下，量子标量场的真空能量，定义有界区域$\Omega \subset {\mathbb{R}}^{d}$，其中场满足理想或非理想边界条件。 对于电磁情况，这种情况描述了零点能的电导率修正。 使用一种解析正则化过程，我们得到了在具有狄利克雷边界条件的板状结构$\Omega=\mathbb R^{d-1}\times[0, L]$存在下，零温度质量为零的标量场的真空能量。 为了讨论非理想边界条件的情况，我们采用了一种渐近展开，该展开基于黎曼zeta函数的近似函数方程，在其原始收敛域之外的有限和被定义。 最后，为了得到在具有非理想边界条件的长方体（长度为$L_{1}$和$L_{2}$，即$\Omega=[0,L_{1}]\times[0,L_{2}]$）存在下质量为零的标量场的卡西米尔能量，我们采用了埃普斯坦zeta函数的近似函数方程。