We study bounded perturbations of an unbounded positivedefinite self-adjoint operator with discrete spectrum. The spectrum hassemi-simple eigenvalues with finite geometric multiplicity and theperturbation belongs to operator space defined by rate of the off-diagonaldecay of the operator matrix. We show that the spectral projections andthe resolvent of the perturbed operator belong to the same space asthe perturbation. These results are applied to the Hill operator and theoperator with matrix potential.We also consider the inverse problem andthe modified Galerkin method.
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