Scattering matrices for perturbations of Laplace operator by infinite sums of zero-range potentials
DOI:
https://doi.org/10.3842/umzh.v78i7-8.9170Keywords:
unbounded operator, B-Weyl, Left and right B-Weyl operators, Left and right Drazin invertible operators, Left and right B-Weyl spectra, Left and right Drazin spectra.Abstract
UDC 517.983, 517.984
We study the resolvents and scattering matrices for the following pairs of unbounded self-adjoint operators in $\mathbb{L}_{2}(\mathbf{R}_{3})$: the standard Laplace operator $A$ and its self-adjoint perturbation, which rigorously realizes the formal sum of $A$ with linear combinations of infinitely many zero-range potentials. By using Krein’s resolvent formula for the perturbed operator, we establish sufficient conditions for the resolvent difference of the analyzed pair of operators to be a nuclear operator. Under these conditions, we deduce and analyze an explicit formula for the corresponding scattering matrices.
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