Calculation of eigenvalues of discrete semibounded differential operators

We consider a problem of eigenvalues of an abstract discrete semibounded operator acting in a separable Hilbert space. The existence and uniqueness of the solution, as well as a convergence of the Galerkin method with reference to the problem, are proved. Simple formulas to calculate the eigenvalues are obtained. The formulas based on Galerkin method allow to calculate eigenvalues of discrete semibounded operators with high computational efficiency. In contrast to the classical methods, the formulas sharply reduce the number of calculations. Also, the formulas allow to find eigenvalues of the operator, regardless of whether the eigenvalues with smaller numbers are known or not. The formulas solve the problem on a calculation of all necessary spectrum points of the abstract discrete semibounded operators.