Dirac operator coupled to bosons
https://doi.org/10.17586/2220-8054-2016-7-2-332-339
Abstract
We consider a model of point-like interaction between electrons and bosons in a cavity. The electrons are relativistic and are described by a Dirac operator on a bounded interval while the bosons are treated by second quantization. The model fits into the extension theory of symmetric operators. Our main technical tool to handle the model is the so-called boundary triplet approach to extensions of symmetric operators. The approach allows explicit computation of the Weyl function.
Supporting agencies: This work was partially financially supported by the Government of the Russian Federation (grant 074-U01), by Ministry of Science and Education of the Russian Federation (GOSZADANIE 2014/190, Projects No 14.Z50.31.0031 and No. 1.754.2014/K), by grants MK-5001.2015.1 and MK-5161.2016.1 of the President of the Russian Federation, by grant 16-11-10330 of Russian Science Foundation. The first author acknowledge hospitality and financial support of WIAS Berlin. The second author thanks the European Research Council and the DFG supporting the preparation of the paper via ERC-2010-AdG no. 267802 (Analysis of Multiscale Systems Driven by Functionals) and Grant NE 1439/3-1 (Development of Methods in the Theory of Self-adjoint Extensions), respectively.
About the Authors
A. A. Boitsev
ITMO University
Russian Federation
Russian Federation
Kronverkskiy, 49, St. Petersburg, 197101
H. Neidhardt
Weierstrass Institute for Applied Analysis and Stochastics
Germany
Germany
Mohrenststr. 39, D-10117, Berlin
I. Y. Popov
ITMO University
Russian Federation
Russian Federation
Kronverkskiy, 49, St. Petersburg, 197101
References
1. Pan L., Fu X., Zhou G. Electron dwell time through a quantum wire under a electromagnetic field irradiation. Phys. Lett. A, 2007, 368(1-2), P. 97–100.
2. Popov I.Y. Operator extensions theory model for electromagnetic field-electron interaction. J. Math. Phys., 2012, 53, P. 063505.
3. Neidhardt H., Wilhelm L., Zagrebnov V.A. A new model for quantum dot light emitting-absorbing devices. J. Math. Phys., Anal., Geom., 2014, 10(3), P. 350–385.
4. Neidhardt H., Wilhelm L., Zagrebnov V.A. A new model for quantum dot light emitting-absorbing devices: proofs and supplements. Nanosystems: Phys. Chem. Math., 2015, 6(1), P. 6–45.
5. Albeverio S., Brasche J.F., Malamud M.M., Neidhardt H. Inverse spectral theory for symmetric operators with several gaps: scalar-type Weyl functions. J. Funct. Anal., 2005, 228(1), P. 144–188,
6. Boitsev A.A. ,Neidhardt H., Popov I.Y. Weyl function for sum of operators tensor products. Nanosystems: Phys. Chem. Math., 2013, 4(6), P. 747–757.
7. Carlone R., Malamud M., Posilicano A. On the spectral theory of Gesztesy-ˇSeba realizations of 1-D Dirac operators with point interactions on a discrete set. J. Differential Equations, 2013, 254(9), P. 3835–3902.
8. Derkach V.A., Malamud M.M. On the Weyl function and Hermite operators with lacunae. Dokl. Akad. Nauk SSSR, 1987, 293(5), P. 1041–1046.
9. Derkach V.A., Malamud M.M. Generalized resolvents and the boundary value problems for Hermitian operators with gaps. J. Funct. Anal., 1991, 95(1), P. 1–95.
10. Derkach V.A., Malamud M.M. The extension theory of Hermitian operators and the moment problem. J. Math. Sci., 1995, 73(2), P. 141–242.
11. Gorbachuk V.I., Gorbachuk M.L. Boundary value problems for operator differential equations, volume 48 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991.
12. Koˇcube˘ıA. N. Extensions of symmetric operators and of symmetric binary relations. Mat. Zametki, 1975, 17, P. 41–48.
13. Malamud M.M. Some classes of extensions of a Hermitian operator with lacunae. Ukra¨ın. Mat. Zh., 1992, 44(2), P. 215–233.
14. Malamud M.M., Neidhardt H. Sturm-Liouville boundary value problems with operator potentials and unitary equivalence. J. Differential Equations, 2012, 252(11), P. 5875–5922.
15. Schm¨udgen K. Unbounded self-adjoint operators on Hilbert space, 2012, volume 265 of Graduate Texts in Mathematics. Springer, Dordrecht.
Review
For citations:
Boitsev A.A., Neidhardt H., Popov I.Y. Dirac operator coupled to bosons. Nanosystems: Physics, Chemistry, Mathematics. 2016;7(2):332-339. https://doi.org/10.17586/2220-8054-2016-7-2-332-339
Views:
5
Email this article 