On the regularity of spectral tasks with two characteristic roots arbitrary multiplicity

Background. The task belonging to the class of regular spectral tasks in significantly their expanded understanding, than in sense, classical on Birkgofu-Tamarkin, is considered. Expansion concerns the main differential bunch and also regional conditions. First, - presence of two various roots of various frequency rates at the main characteristic equation. On the other hand, regional conditions belong in essence to type of any breaking-up conditions with respect for their regularity. The irregularity of such conditions in classical regional tasks is well-known. Range of a task are the numbers in the right part of the complex half-plane leaving on infinity in the direction of an imaginary axis on logarithmic removal from it. Materials and methods. The paper uses the methods of functional analysis, differential equations and algebra. Results. The construction of the resolvent of the problem is given in the form of a meromorphic function with respect to the parameter $\lambda$, - of the Green function. In the main theorem it is established that the full deduction in parameter from the resolvent attached to $(n+1)$- multiply the differentiable function (addressing in zero on the ends $0,1$ together with derivatives) is equal to this function. The specified deduction represents Fourier's number on root functions of an initial task. Conclusions. The foundations of the theory of regular spectral problems with characteristic root of arbitrary multiplicities are laid.