Abstract:
The effective range approximation is generalized to the case of the Dirac equation. Formulas are obtained that express the parameters in the expansion of the $S$ matrix as $\varepsilon\to -mc^2$ in terms of the wave function of the level at the critical point. These formulas are used to investigate the entry of levels of the discrete spectrum into the lower continuum. The critical charge of the nucleus for muons is calculated (the case $R\gg\hbar/mc$).
Citation:
V. D. Mur, V. S. Popov, “Bound states near the limit of the lower continuum”, TMF, 27:2 (1976), 204–216; Theoret. and Math. Phys., 27:2 (1976), 429–438
\Bibitem{MurPop76}
\by V.~D.~Mur, V.~S.~Popov
\paper Bound states near the limit of the lower continuum
\jour TMF
\yr 1976
\vol 27
\issue 2
\pages 204--216
\mathnet{http://mi.mathnet.ru/tmf3319}
\transl
\jour Theoret. and Math. Phys.
\yr 1976
\vol 27
\issue 2
\pages 429--438
\crossref{https://doi.org/10.1007/BF01051234}
Linking options:
https://www.mathnet.ru/eng/tmf3319
https://www.mathnet.ru/eng/tmf/v27/i2/p204
This publication is cited in the following 19 articles:
N. K. Dulaev, D. A. Telnov, V. M. Shabaev, Y. S. Kozhedub, X. Ma, I. A. Maltsev, R. V. Popov, I. I. Tupitsyn, “Three-dimensional calculations of positron creation in supercritical collisions of heavy nuclei”, Phys. Rev. D, 111:1 (2025)
N. K. Dulaev, D. A. Telnov, V. M. Shabaev, Y. S. Kozhedub, I. A. Maltsev, R. V. Popov, I. I. Tupitsyn, “Angular and energy distributions of positrons created in subcritical and supercritical slow collisions of heavy nuclei”, Phys. Rev. D, 109:3 (2024)
R. V. Popov, V. M. Shabaev, I. A. Maltsev, D. A. Telnov, N. K. Dulaev, D. A. Tumakov, “Spontaneous vacuum decay in low-energy collisions of heavy nuclei beyond the monopole approximation”, Phys. Rev. D, 107:11 (2023)
Krylov K.S. Mur V.D. Fedotov A.M., “On the Resonances Near the Continua Boundaries of the Dirac Equation With a Short-Range Interaction”, Eur. Phys. J. C, 80:3 (2020), 270
Pivnyak G. Pevzner M. Medvedev A. Caseres Cabana E. Bak A. Bajerski A. Smolinski A., “A Study on the Static Field of a Point Charge in Three-Dimensional Electrodynamics”, J. Phys. Commun., 4:7 (2020), 075020
R. V. Popov, V. M. Shabaev, D. A. Telnov, I. I. Tupitsyn, I. A. Maltsev, Y. S. Kozhedub, A. I. Bondarev, N. V. Kozin, X. Ma, G. Plunien, T. Stöhlker, D. A. Tumakov, V. A. Zaytsev, “How to access QED at a supercritical Coulomb field”, Phys. Rev. D, 102:7 (2020)
K S Krylov, V D Mur, “On solutions of the Dirac equation in a strong disappearing at infinity electrostatic field”, J. Phys.: Conf. Ser., 1238:1 (2019), 012042
Sergey Godunov, Bruno Machet, Mikhail Vysotsky, V.E. Volkova, Y.V. Zhezher, D.G. Levkov, V.A. Rubakov, V.A. Matveev, “Resonances in positron scattering on a supercritical nucleus and spontaneous production of e+e- pairs”, EPJ Web Conf., 191 (2018), 02018
Kuleshov V.M. Mur V.D. Fedotov A.M. Lozovik Yu.E., “Coulomb Problem For Z > Z(Cr) in Doped Graphene”, J. Exp. Theor. Phys., 125:6 (2017), 1144–1162
Kuleshov V.M. Mur V.D. Narozhny N.B., “Coulomb Problem For Graphene With Supercritical Impurity”, V International Conference on Problems of Mathematical and Theoretical Physics and Mathematical Modelling, Journal of Physics Conference Series, 788, IOP Publishing Ltd, 2017, UNSP 012044
S. I. Godunov, B. Machet, M. I. Vysotsky, “Resonances in positron scattering on a supercritical nucleus and spontaneous production of $e^{+}e^{-}$ e + e - pairs”, Eur. Phys. J. C, 77:11 (2017)
B. M. Karnakov, V. D. Mur, S. V. Popruzhenko, V. S. Popov, “Current progress in developing the nonlinear ionization theory of atoms and ions”, Phys. Usp., 58:1 (2015), 3–32
V. M. Kuleshov, V. D. Mur, N. B. Narozhny, A. M. Fedotov, Yu. E. Lozovik, V. S. Popov, “Coulomb problem for a $Z>Z_{\rm cr}$ nucleus”, Phys. Usp., 58:8 (2015), 785–791
V. M. Kuleshov, V. D. Mur, N. B. Narozhny, A. M. Fedotov, Yu. E. Lozovik, “Coulomb problem for graphene with the gapped electron spectrum”, JETP Letters, 101:4 (2015), 264–270
Eugene B. Kolomeisky, Joseph P. Straley, Hussain Zaidi, “Fermion space charge in narrow band-gap semiconductors, Weyl semimetals, and around highly charged nuclei”, Phys. Rev. B, 88:16 (2013)
H. HOGREVE, “BOUND, VIRTUAL AND RESONANCE STATES OF THE THREE-DIMENSIONAL DIRAC AND SCHRÖDINGER SQUARE WELL”, Int. J. Mod. Phys. E, 22:11 (2013), 1350079
N Szpak, R Schützhold, “Optical lattice quantum simulator for quantum electrodynamics in strong external fields: spontaneous pair creation and the Sauter–Schwinger effect”, New J. Phys., 14:3 (2012), 035001
S. P. Andreev, B. M. Karnakov, V. D. Mur, “Energy spectrum of a particle in potentials with strongly differing ranges”, Theoret. and Math. Phys., 64:2 (1985), 838–846
V. D. Mur, V. S. Popov, “Coulomb problem with short-range interaction: Exactly solvable model”, Theoret. and Math. Phys., 65:2 (1985), 1132–1140
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