Two ways of organizing scalar product in the boundary state method

Introduction. The influence of two ways of organizing scalar product on the convergence rate of the solution in the energy method of boundary states is considered. The method is based on the spaces of internal and boundary states which are conjugated through isomorphism. Both spaces are orthonormalized using one scalar product or another. The desired state is expanded in the Fourier series according to the elements of the orthonormalized basis; and the coefficients of this linear combination are determined. The two methods differ in the assignment of scalar products and the calculation of the Fourier coefficients.Materials and Methods. In relation to the method of boundary states, a new theory of organizing a scalar product in the spaces of internal and boundary states is proposed. Computational algorithms are constructed for its practical implementation. In the traditional (first) approach, the internal energy of elastic deformation is used as an orthogonalizer in the space of internal states...