On the Possibility of the Implicit Renormalization of the Casimir Energy

We propose a procedure for renormalizing the Casimir energy that makes the steps that are used in the standard renormalization procedure, that is, regularization, subtraction, and deregularization, implicit. The proposed procedure is based on the calculation of a set of convergent sums, each of which is related to the initial divergent sum of the non-renormalized Casimir energy. Next, we construct a system of linear equations that relates this set of convergent sums to the renormalized Casimir energy. The unknown renormalized Casimir energy is obtained as a result of solving this system of equations. In this case, both the calculations of the convergent sums and the subsequent solution of the system of linear equations are performed with a certain (generally speaking, arbitrary) ordered accuracy; thus, the result is also approximate. The proposed procedure is, first, more computationally effective than the standard one, and, second, applicable not only to the problems where a transcendental equation for the spectrum can be written, but also to the problems where the spectrum is known only numerically.