The modified iterative method for calculating completely continuous operator eigenvalues and eigenfunctions

The modified iterative method (MIM) is discussed. The MIM is a method for obtaining the completely continuous operator eigenvalues and eigenfunctions. The authors correct the formulation and demonstration of the MIM substantive theorem, demonstrate the MIM correlation with a moment method and give some numerical results. The ordinary successive approximation method, which is used to calculate the first eigenvalue and its eigenfunction, consists of multiple iterations by the operator of any initial function for all vanishing next eigenfunctions in its decomposition. When the iterative process becomes stable the final result is obtained from the last two iterations. The MIM allow the use of all iterations with the initial function simultaneously to get the information about all eigenfunctions, not only about the first one. In this case there is a need for far fewer iterations to achieve a prescribed precision. It is also possible to separate each eigenfunction, even those which correspond to eigenvalues with closely placed magnitudes.