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

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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

1.     Krein M.G., Langer G.K. Defective subspaces and generalized resolvents of an Hermitian operator in the space . Functional Analysis and Its Applications, 1971, vol. 5, no. 3, pp. 217–228. doi: 10.1007/BF01078128
2.     Naimark M.A. Lineinye Differentsial'nye Operatory [Linear Differential Operators]. Moscow, Nauka Publ., 1969, 528 p.
3.     Baumgartel H., Wollenberg M. Mathematical Scattering Theory. Berlin, Academie-Verlag, 1983.
4.     Derkach V.A., Malamud M.M. On the Weyl function and Hermite operators with lacunae. Dokl. Ak. Nauk SSSR, 1987, vol. 293, no. 5, pp. 1041–1046.
5.     Derkach V.A., Malamud M.M. Generalized resolvents and the boundary value problems for Hermitian operators with gaps. Journal of Functional Analysis, 1991, vol. 95, no. 1, pp. 1–95.
6.     Derkach V.A., Malamud M.M. The extension theory of Hermitian operators and the moment problem. Journal of Mathematical Sciences, 1995, vol. 73, no. 2, pp. 141–242. doi: 10.1007/BF02367240
7.     Gorbachuk V.I., Gorbachuk M.L. Boundary Value Problems for Operator Differential Equations. Kluwer, Dordrecht, 1990, 364 p.
8.     Malamud M.M. Some classes of extensions of a Hermitian operator with lacunae. Ukraine Mat. Zh., 1992, vol. 44, no. 2, pp. 215–233.
9.     Malamud M.M., Neidhardt H. Sturm-liouville boundary value problems with operator potentials and unitary equivalence. Journal of Differential Equations, 2012, vol. 252, no. 11, pp. 5875–5922. doi: 10.1016/j.jde.2012.02.018
10.  Smudgen K. Unbounded Self-Adjoint Operators on Hilbert Space. Springer, 2012, 432 p.
11.  Malamud M.M., Malamud S.M. Spectral theory of operator measures in a Hilbert space. Algebra i Analiz, 2003, vol. 15, no. 3, pp. 1–77.
12.  Gorbachuk M.L. Self-adjoint boundary problems for a second-order differential equation with unbounded operator coefficient. Functional Analysis and Its Applications, 1971, vol. 5, no. 1, pp. 9–18. doi: 10.1007/BF01075842
13.  Birman M.S. Existence conditions for wave operators. Izv. Akad. Nauk SSSR, 1963, no. 27, pp. 883–906.
14.  Malamud M.M., Neidhardt H. On the Kato-Rosenblum and the Weyl-Neuman theorems. Doklady Mathematics, 2010, vol. 81, no. 3, pp. 368–372.
15.  Boitsev A.A., Neidhardt H., Popov I.Yu. Weyl function for sum of operators tensor product. Nanosystems: Physics, Chemistry, Mathematics, 2013, vol. 4, no. 6, pp. 747–759.

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