###### Abstract  
A method for the accurate calculation of the cut-off wavenumbers of a waveguide with an arbitrary cross section and a number of inner conductors is demonstrated. Concepts of integral and infinite-matrix (summation) operator-valued functions depending nonlinearly on the frequency spectral parameter provide a secure basis for formulating the spectral problem, and the Method of Analytical Regularization is employed to implement an effective algorithm. The algorithm is based on a mathematically rigorous solution of the homogeneous Dirichlet problem for the Helmholtz equation in the interior of the waveguide, excluding the regions occupied by the inner conductor boundaries. A highly efficient method of calculating the cut-off wavenumbers and the corresponding non-trivial solutions representing the modal distribution is developed. The mathematical correctness of the problem statement, the method, and the ability to calculate the cut-off wavenumbers with any prescribed and proven accuracy provide a secure basis for treating these as \"benchmark solutions\". In this paper, we use this new approach to validate previously obtained results against our benchmark solutions. Furthermore, we demonstrate its universality in solving some new problems, which are barely accessible by existing methods. Method of Analytical Regularization; cut-off wavenumbers; waveguide with inner conductors 20242020  
justify restricted accuracy by arguing \"they are good enough for practice\"; in other words, it is highly desirable for the method to have the means of providing an internal estimate of its own accuracy. The second requirement concerns the \"resolving power\" of the method: it should not miss any modes of the spectrum, even when their cut-off frequencies lie extremely close to each other on the frequency axis. The third requirement is the ability to generate an algorithm which unerringly recognizes and correctly labels the modes corresponding to their cut-off wave numbers. This last requirement requires some explanation. In general, it is impossible to label the eigenmodes in a waveguide with inner conductors in the manner that is characteristic for empty waveguides of canonical shape; the presence of the inner conductors may produce a severe distortion of the regular electromagnetic field (EMF) oscillations of the empty waveguide eigenmodes observed in the mutually perpendicular directions of the transverse plane. But it is desirable that the perturbed oscillations and their link with the non-perturbed oscillations in the empty waveguide should be traceable. All these requirements can be fully realized if a correct mathematical treatment is performed of the boundary IE as a problem on CNs for an integral OVF of the (complex) spectral parameter and for equivalent infinite-matrix (summation) OVF constructed using the Method of Analytical Regularization (MAR). Although the MAR has been extensively used to investigate diffraction problems for variously shaped metallic screens, its implementation for the analysis of associated spectral problems has been minimal to date. Of course, this problem has been investigated by various methods already existing in computational electromagnetics (CEM). The application of the MAR approach in this paper is justified by its new capability to study the spectrum of cut-off wave numbers \(k_{c}\) in waveguides with inner conductors. A complete list of CEM methods and a comparative analysis of their effectiveness is a matter for special reviews. Nevertheless, we briefly mention some which were applied for studies of the spectrum of the cut-off wave numbers in waveguides containing inner conductors. Extensive examination has been carried out for the class of canonically shaped waveguides and inner conductors with circular, elliptical and rectangular cross sections.