Abstract
We consider two families of realizations of the 2p×2p–Dirac differential expression with point interactions on a discrete set X = {x n }n=1∞ ⊂ ℝ on a half–line (line) and generalize certain results from [10] to the matrix case. We show that these realizations are always self-adjoint. We investigate the nonrelativistic limit as the velocity of light tends to infinity.
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References
N. Arrizabalaga, A. Mas, and L. Vega, “Shell Interactions for Dirac Operators,” J. Math. Pures Appl. (9), 102 (4), 617–639 (2014).
N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis (Oliver and Boyd Ltd, Edinburgh, London, 1965).
S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics (Sec. Edition, AMS Chelsea Publ., 2005).
S. Albeverio, A. Kostenko, and M. M. Malamud, “Spectral Theory of Semi-Bounded Sturm–Liouville Operators with Local Interactions on a Discrete Set,” J. Math. Phys. 51, p. 24 (2010).
H. Arnbak, P. L. Christiansen, and Yu. B. Gaididei, “Non-Relativistic and Relativistic Scattering by Short–Range Potentials,” Philos. Trans. R. Soc. Lond. Ser. A 369, 1228–1244 (2011).
J. Behrndt, P. Exner, M. Holzmann, and V. Lotoreichik, “On the Spectral Properties of Dirac Operators with Electrostatic δ-Shell Interactions,” J. Math. Pures Appl. [in press], arXiv:1609.00608v1, p. 32 (2016).
S. Benvegn`u and L. D¸abrowski, “Relativistic Point Interaction,” Lett. Math. Phys. 30, 159–167 (1994).
I. N. Braeutigam, “Limit–Point Criteria for the Matrix Sturm–Liouville Operators and Its Powers,” Opuscula Math. 37 (1), 5–19 (2017).
I. N. Braeutigam, K. A. Mirzoev, and T. A. Safonova, “On Deficiency Index for Some Second Order Vector Differential Operators,” Ufa Math. J. 9 (1), 18–28 (2016).
R. Carlone, M. Malamud, and A. Posilicano, “On the Spectral Theory of Gesztesy–Šeba Realizations of 1D Dirac Operators with Point Interactions on a Discrete Set,” J. Differential Equations 254 (9), 3835–3902 (2013).
V. A. Derkach, S. Hassi, M. M. Malamud, and H. S. V. de Snoo, “Boundary Triplets and Weyl Functions. Recent Developments,” LMS Lecture Notes 404, 161–220 (2012).
V. A. Derkach, and M. M. Malamud, “Generalised Resolvents and the Boundary Value Problems for Hermitian Operators with Gaps,” J. Funct. Anal. 95, 1–95 (1991).
P. Exner, Seize ans après, Appendix to “Solvable Models in Quantum Mechanics,” by S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden (Sec. Edition, AMS Chelsea Publ., 2004).
F. Gesztesy and P. Šeba, “New Analytically Solvable Models of Relativistic Point Interactions,” Lett. Math. Phys. 13, 345–358 (1987).
V. I. Gorbachuk and M. L. Gorbachuk, Boundary Value Problems for Operator Differential Equations, Mathematics and Its Applications (Soviet Series, 48, Kluwer Academic Publishers Group, Dordrecht, 1991).
T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin-Heidelberg, New York, 1966).
A. S. Kostenko and M. M. Malamud, “1D Schrödinger Operators with Local Point Interactions on a Discrete Set,” J. Differential Equations 249, 253–304 (2010).
A. S. Kostenko, M. M. Malamud, and D. D. Natyagailo, “Matrix Schrödinger Operator with δ-Interactions,” Math. Notes 100 (1), 48–64 (2016).
A. G. Kostyuchenko and K. A. Mirzoev, “Generalized Jacobi Matrices and Deficiency Numbers of Ordinary Differential Operators with Polynomial Coefficients,” Funct. Anal. Appl. 33, 30–45 (1999).
M. G. Krein, “Infinite J–Matrices and the Matrix Moment Problem,” RAS USSR 69 (3), 125–128 (1949).
M. Lesch and M. M. Malamud, “On the Deficiency Indices and Self-Adjointness of Symmetric Hamiltonian Systems,” JDE 189 (2), 556–615 (2003).
M. M. Malamud and H. Neidhardt, “Sturm–Liouville Boundary Value Problems with Operator Potentials and Unitary Equivalence,” J. Differential Equations 252, 5875–5922 (2012).
C. R. de Oliveira, Intermediate Spectral Theory and Quantum Dynamics (Birkhäuser, Basel, 2009).
K. Schmüdgen, Unbounded Self-Adjoint Operators on Hilbert Space (Springer-Verlag, New York, 2012).
B. Thaller, The Dirac Equation, Texts and Monographs in Physics (Springer, 1992).
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Dedicated with great pleasure to Edvard Tsekanovskii on the occasion of his 80-th birthday
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Budyika, V., Malamud, M. & Posilicano, A. Nonrelativistic limit for 2p × 2p–Dirac operators with point interactions on a discrete set. Russ. J. Math. Phys. 24, 426–435 (2017). https://doi.org/10.1134/S1061920817040021
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DOI: https://doi.org/10.1134/S1061920817040021