EXTENSION OF TENSOR PRODUCT FOR OPERATORS ON THE DIRAC OPERATOR EXAMPLE

A. A. Boitsev, H. Neidhardt, I. Y. Popov


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Abstract

The paper deals with extension method for the operator which is a sum of tensor products. Boundary triplets approach is used. One of the operators is considered to be densely defined and symmetric with equal deficiency indices, the other one is considered to be bounded and self- adjoint. For self-adjoint extensions construction of the mentioned operator, its boundary triplet is constructed in terms of boundary triplet of symmetric operator. Gamma-field and the Weyl function are obtained using the boundary triplet of symmetric operator. Formulas, connecting gamma-field and the Weyl function of symmetric operator with gamma-field and the Weyl function of the studied operator make it possible to use generic resolvent Krein-type formula for all self-adjoint extensions in this case as well. Theoretical results are applied to the Dirac operator, interesting from the physical point of view. Boundary triplet, gamma-field and the Weyl function are constructed for the Dirac operator. The self-adjoint extensions are obtained by Krein formula. Received results can be useful for correct description of quantum systems interaction.


Keywords: boundary triplets approach, Dirac operator, self-adjoint extensions
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