We study the properties of time evolution of the 00 K K − system in spectral formulation. Within the one-pole model we find the exact form of the diagonal matrix elements of the effective Hamiltonian for this system. It appears that, contrary to the Lee-Oehme-Yang (LOY) result, these exact diagonal matrix elements are different if the total system is CPT-invariant but CP-noninvariant. The realistic model of the pair of unstable neutral particles (neutral K mesons) is discussed. The model is based on the assumed properties of the spectral function defining the evolution operator. It is assumed that this spectral function describing the mass distribution has the Breit-Wigner form. This allows one to express the ma- trix elements of the evolution operator in terms of the com- binations of exponential integral functions. Then, these matrix elements of the evolution operator are used to calculate the matrix elements of the effective Hamiltonian governing the evolution of a two-state quantum system corresponding to the K K − mesons and similar systems. The aim of the paper was to find the analytic expression for the difference of the diagonal elements of the effective Hamiltonian with the use of the exponential integral functions. As a result, the relatively complex physical model has been reduced to a form which can be immedi- ately used for computer simulations exploring the behav- iour of this very important difference with the change in time (t), given the numerical values of the parameters characterising the system (mass, lifetime etc.). Furthermore, a method for extracting asymptotic values of observable parameters (like the difference mentioned above) from general, analytic expressions, in the case of two quantum objects with very different lifetimes, has been developed. The computational methods used are of universal char- acter and might be used outside the elementary particles. As our main interest is the research of the behaviour of the exact expressions describing the system, symbolic methods were used. Additionally, some of the parts of the system are of highly oscillatory nature, so purely numerical methods might be inappropriate here.