Geodesic
Vacuum effects for a one-dimensional “hydrogen atom" with $Z>Z_{\mathrm{cr}}$
Teoretičeskaâ i matematičeskaâ fizika, Tome 193 (2017) no. 2, pp. 276-308 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For a supercritical Coulomb source with a charge $Z>Z_{\mathrm{cr}}$ in $1{+}1$ dimensions, we study the nonperturbative properties of the vacuum density $\rho_{\mathrm{VP}}(x)$ and the energy $\mathcal E_{\mathrm{VP}}$. We show that for corresponding problem parameters, nonlinear effects in the supercritical region can lead to behavior of the vacuum energy differing significantly from the perturbative quadratic growth, to the extent of an (almost) quadratic decrease of the form $-|\eta|Z^2$ into the negative region. We also show that although approaches for calculating vacuum expectations values and the behavior of $\rho_{\mathrm{VP}}(x)$ in the supercritical region for various numbers of spatial dimensions indeed have many common features, $\mathcal E_{\mathrm{VP}}$ for $1{+}1$ dimensions in the supercritical region nevertheless has several specific features determined by the one-dimensionality of the problem.
Mots-clés : quasi-one-dimensional Dirac–Coulomb system
Keywords: one-dimensional hydrogen atom, vacuum polarization, nonperturbative effects for $Z>Z_{\mathrm{cr}}$. 
@article{TMF_2017_193_2_a5,
     author = {Yu. S. Voronina and A. S. Davydov and K. A. Sveshnikov},
     title = {Vacuum effects for a~one-dimensional {\textquotedblleft}hydrogen atom" with $Z>Z_{\mathrm{cr}}$},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {276--308},
     year = {2017},
     volume = {193},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2017_193_2_a5/}
}
TY  - JOUR
AU  - Yu. S. Voronina
AU  - A. S. Davydov
AU  - K. A. Sveshnikov
TI  - Vacuum effects for a one-dimensional “hydrogen atom" with $Z>Z_{\mathrm{cr}}$
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2017
SP  - 276
EP  - 308
VL  - 193
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2017_193_2_a5/
LA  - ru
ID  - TMF_2017_193_2_a5
ER  - 
%0 Journal Article
%A Yu. S. Voronina
%A A. S. Davydov
%A K. A. Sveshnikov
%T Vacuum effects for a one-dimensional “hydrogen atom" with $Z>Z_{\mathrm{cr}}$
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2017
%P 276-308
%V 193
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2017_193_2_a5/
%G ru
%F TMF_2017_193_2_a5
Yu. S. Voronina; A. S. Davydov; K. A. Sveshnikov. Vacuum effects for a one-dimensional “hydrogen atom" with $Z>Z_{\mathrm{cr}}$. Teoretičeskaâ i matematičeskaâ fizika, Tome 193 (2017) no. 2, pp. 276-308. http://geodesic.mathdoc.fr/item/TMF_2017_193_2_a5/

[1] R. Loudon, “One-dimensional hydrogen atom”, Am. J. Phys., 27:9 (1959), 649–655 | DOI

[2] R. J. Elliott, R. Loudon, “Theory of the absorption edge in semiconductors in a high magnetic field”, J. Phys. Chem. Solids, 15:3–4 (1960), 196–207 | DOI | MR | Zbl

[3] C. M. Care, “One dimensional hydrogen atom with a repulsive core”, J. Phys. C: Solid State Phys., 5:14 (1972), 1799–1805 | DOI

[4] M. W. Cole, “Properties of image-potential-induced surface states of insulators”, Phys. Rev. B, 2:10 (1970), 4239–4252 | DOI

[5] M. M. Nieto, “Electrons above a helium surface and the one-dimensional Rydberg atom”, Phys. Rev. A, 61:3 (2000), 034901, 4 pp. | DOI

[6] N. B. Delone, V. P. Krainov, D. L. Shepelyanskii, “Vysokovozbuzhdennyi atom v elektromagnitnom pole”, UFN, 140:3 (1983), 355–392 | DOI | DOI

[7] R. V. Jensen, S. M. Susskind, M. M. Sanders, “Chaotic ionization of highly excited hydrogen atoms: Comparison of classical and quantum theory with experiment”, Phys. Rep., 201:1 (1991), 1–56 | DOI

[8] U.-L. Pen, T. F. Jiang, “Strong-field effects of the one-dimensional hydrogen atom in momentum space”, Phys. Rev. A, 46:7 (1992), 4297–4305 | DOI

[9] A. López-Castillo, C. R. de Oliveira, “Classical ionization for the aperiodic driven hydrogen atom”, Chaos, Solitons and Fractals, 15:5 (2003), 859–869 | DOI | Zbl

[10] F. H. L. Essler, F. Gebhard, E. Jeckelmann, “Excitons in one-dimensional Mott insulators”, Phys. Rev. B, 64:12 (2001), 125119, 15 pp. | DOI

[11] F. Wang, Y. Wu, M. S. Hybertsen, T. F. Heinz, “Auger recombination of excitons in one-dimensional systems”, Phys. Rev. B, 73:24 (2006), 245424, 5 pp. | DOI

[12] V. Canuto, D. C. Kelly, “Hydrogen atom in intense magnetic field”, Astrophys. Space Sci., 17:2 (1972), 277–291 | DOI

[13] H. Friedrich, H. Wintgen, “The hydrogen atom in a uniform magnetic field – an example of chaos”, Phys. Rep., 183:2 (1989), 37–79 | DOI | MR

[14] X. Guan, B. Li, K. T. Taylor, “Strong parallel magnetic field effects on the hydrogen molecular ion”, J. Phys. B, 36:17 (2003), 3569–3590 | DOI

[15] H. Ruder, G. Wunner, H. Herold, F. Geyer, Atoms in Strong Magnetic Fields: Quantum Mechanical Treatment and Applications in Astrophysics and Quantum Chaos, Springer, Berlin, 2012

[16] R. Barbieri, “Hydrogen atom in superstrong magnetic fields: Relativistic treatment”, Nucl. Phys. A, 161:1 (1971), 1–11 | DOI

[17] V. P. Krainov, “Vodorodopodobnyi atom v sverkhsilnom magnitnom pole”, ZhETF, 64:3 (1973), 800–803

[18] A. E. Shabad, V. V. Usov, “Positronium collapse and the maximum magnetic field in pure QED”, Phys. Rev. Lett., 96:18 (2006), 180401, 4 pp. | DOI

[19] A. E. Shabad, V. V. Usov, “Bethe–Salpeter approach for relativistic positronium in a strong magnetic field”, Phys. Rev. D, 73:12 (2006), 125021, 16 pp. | DOI

[20] A. E. Shabad, V. V. Usov, “Electric field of a pointlike charge in a strong magnetic field and ground state of a hydrogenlike atom”, Phys. Rev. D, 77:2 (2008), 025001, 20 pp. | DOI

[21] V. N. Oraevskii, A. I. Rez, V. B. Semikoz, “Spontannoe rozhdenie pozitronov kulonovskim tsentrom v odnorodnom magnitnom pole”, ZhETF, 72:3 (1977), 820–833

[22] B. M. Karnakov, V. S. Popov, “Atom vodoroda v sverkhsilnom magnitnom pole i effekt Zeldovicha”, ZhETF, 124:5 (2003), 996–1022

[23] M. I. Vysotskii, S. I. Godunov, “Kriticheskii zaryad v sverkhsilnom magnitnom pole”, UFN, 184:2 (2014), 206–210 | DOI | DOI

[24] J. Reinhardt, W. Greiner, “Quantum electrodynamics of strong fields”, Rep. Progr. Phys., 40:3 (1977), 219–295 | DOI

[25] W. Greiner, B. Müller, J. Rafelski, Quantum electrodynamics of strong fields, Springer, Berlin, Heidelberg, 1985

[26] G. Plunien, B. Müller, W. Greiner, “The Casimir effect”, Phys. Rep., 134:2–3 (1986), 87–193 | DOI | MR

[27] V. M. Kuleshov, V. D. Mur, N. B. Narozhnyi, A. M. Fedotov, Yu. E. Lozovik, V. S. Popov, “Kulonovskaya zadacha s zaryadom yadra $Z>Z_{\mathrm{cr}}$”, UFN, 185:8 (2015), 845 | DOI | DOI

[28] J. Rafelski, J. Kirsch, B. Müller, J. Reinhardt, W. Greiner, “Probing QED vacuum with heavy ions”, arXiv: 1604.08690

[29] E. H. Wichmann, N. M. Kroll, “Vacuum polarization in a strong Coulomb field”, Phys. Rev., 101:2 (1956), 843–859 | DOI | MR

[30] M. Gyulassy, “Higher order vacuum polarization for finite radius nuclei”, Nucl. Phys. A, 244:3 (1975), 497–525 | DOI

[31] L. S. Brown, R. N. Cahn, L. D. McLerran, “Vacuum polarization in a strong Coulomb field. Induced point charge”, Phys. Rev. D, 12:2 (1975), 581–595 | DOI

[32] A. R. Neghabian, “Vacuum polarization for an electron in a strong Coulomb field”, Phys. Rev. A, 27:5 (1983), 2311–2320 | DOI

[33] W. Greiner, J. Reinhardt, Quantum Electrodynamics, Springer, Berlin, 2012 | DOI | MR

[34] P. J. Mohr, G. Plunien, G. Soff, “QED corrections in heavy atoms”, Phys. Rep. C, 293:5–6 (1998), 227–369 | DOI

[35] T. C. Adorno, D. M. Gitman, A. E. Shabad, “Coulomb field in a constant electromagnetic background”, Phys. Rev. D, 93:12 (2016), 125031, 19 pp. | DOI | MR

[36] K. A. Sveshnikov, D. I Khomovskii, “Chastitsy Shredingera i Diraka v kvaziodnomernykh sistemakh s “kulonovskim” vzaimodeistviem”, TMF, 173:2 (2012), 293–313 | DOI | DOI

[37] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, v. 1, Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Nauka, M., 1973 | MR

[38] U. Fano, “Effects of configuration interaction on intensities and phase shifts”, Phys. Rev., 124:6 (1961), 1866–1878 | DOI

[39] R. Rajaraman, Solitons and Instantons, North-Holland, Amsterdam, 1982 | MR

[40] K. Sveshnikov, “Dirac sea correction to the topological soliton mass”, Phys. Lett. B, 255:2 (1991), 255–260 | DOI

[41] A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, V. F. Weisskopf, “New extended model of hadrons”, Phys. Rev. D, 9:12 (1974), 3471–3495 | DOI | MR

[42] F. Olver (ed.), NIST Handbook of Mathematical Functions, Cambridge Univ. Press, Cambridge, 2010

[43] V. A. Yerokhin, P. Indelicato, V. M. Shabaev, “Two-loop QED corrections with closed fermion loops”, Phys. Rev. A, 77:6 (2008), 062510, 12 pp. | DOI

[44] K. Itsikson, Zh.-B. Zyuber, Kvantovaya teoriya polya, v. 2, Mir, M., 1984 | MR | MR