We study, in general, infinite rank singular perturbations of self-adjoint operators. For a given unbounded self-adjoint operator A acting on a separable Hilbert space B and, generally speaking, an unbounded self-adjoint operator G from <$E B sub +2 (A)> into <$E B sub -2 (A)> with the property <$E R(G) Bar~inter~B~=~{0}> we construct a family of regularizations of the formal expression <$E A~+~G> in terms of skew unbounded projections onto <$E B sub +2>, define corresponding self-adjoint in B realizations of this expression and describe their domains and resolvents. This new approach, based on extension theory of symmetric operators with the exit into rigged Hilbert spaces, allows to develop infinite-dimensional version of singular perturbations of self-adjoint operators and in the case of finite rank perturbations to get results close to the corresponding ones obtained by S. Albeverio and P. Kurasov. For a nonnegative self-adjoint operator A we establish for the first time necessary and sufficient conditions on G that guarantee the existence of nonnegative self-adjoint operators among the realizations of <$E A~+~G>, as well as when those nonnegative realizations contain the Krein-von Neumann nonnegative extremal self-adjoint extension. Singular perturbations of the Laplace operator in <$E bold roman R sup 3> by delta potentials are considered.

Індекс рубрикатора НБУВ: В162.4

Рубрики:

Оператори у гільбертовому просторі

Шифр НБУВ: Ж41243 Пошук видання у каталогах НБУВ 

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