We study the dynamic of scalar bosons in the presence of Aharonov-Bohm magnetic field. First, we give the differential equation that governs this dynamic. Secondly, we use variational techniques to show that the following Schrödinger-Newton equation: , where A is an Aharonov-Bohm magnetic potential, has a unique ground-state solution.
References
[1]
Lieb, E.H. and Seiringer, R. (2010) Stability of Matter in Quantum Mechanics. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511819681
[2]
Lieb, E.H. and Yau, H.T. (1987) The Chandrasekhar Theory of Stellar Collapse as the Limit of Quantum Mechanics. Communications in Mathematical Physics, 112, 147-174. https://doi.org/10.1007/BF01217684
[3]
Lions, P.-L. (1987) Solutions of Hartree-Fock Equations for Coulomb Systems. Communications in Mathematical Physics, 109, 33-97. https://doi.org/10.1007/BF01205672
[4]
Madsen, M.S. and Liddle, A.R. (1990) The Cosmological Formation of Boson Stars. Physics Letters B, 251, 507-510. https://doi.org/10.1016/0370-2693(90)90788-8
[5]
Melgaard, M. and Zongo, F. (2022) Solitary Waves and Excited States for Boson Stars. Analysis and Applications, 20, 285-302. https://doi.org/10.1142/S0219530521500147
[6]
Sharma, R., Karmakar, S. and Mukherjee, S. (2008) Boson Star and Dark Matter. ArXiv: 0812.3470.
[7]
Aharonov, Y. and Bohm, D. (1959) Significance of Electromagnetic Potentials in the Quantum Theory. Physical Review, 115, 485-491. https://doi.org/10.1103/PhysRev.115.485
[8]
Ehrenberg, W. and Siday, R.E. (1949) The Refractive Index in Electron Optics and the Principles of Dynamics. Proceedings of the Physical Society, London, Section B, 62, 8-21. https://doi.org/10.1088/0370-1301/62/1/303
[9]
Tonomura, A., et al. (1982) Observation of Aharonov-Bohm Effect by Electron Holography. Physical Review Letters, 48, 1443-1446. https://doi.org/10.1103/PhysRevLett.48.1443
[10]
Tonomura, A., et al. (1986) Evidence for Aharonov-Bohm Effect with Magnetic Field Completely Shielded from Electron Wave. Physical Review Letters, 56, 792-795. https://doi.org/10.1103/PhysRevLett.56.792
[11]
Bandyopadhyay, A. and Mallik, S. (2017) Rho Meson Decay in the Presence of a Magnetic Field. The European Physical Journal C, 77, Article No. 771. https://doi.org/10.1140/epjc/s10052-017-5357-9
[12]
Piccinelli, G., Jaber-Urquiza, J. and Sànchez, A. (2019) Magnetic Effect on the Decay Process of Neutral Scalar Boson to Charged Fermions. Astronomical Notes, 340, 230-233. https://doi.org/10.1002/asna.201913595
[13]
Piccinelli, G. and Sànchez, A. (2017) Scalar Boson Decay in Presence of Magnetic Field. Astronomical Notes, 338, 1136-1141. https://doi.org/10.1002/asna.201713450
[14]
Tsai, W. and Erber, T. (1974) Photon Pair Creation in Intense Magnetic Fields. Physical Review D, 10, 492-499. https://doi.org/10.1103/PhysRevD.10.492
[15]
Sogut, K., Yanar, H. and Havare, A. (2017) Production of Dirac Particles in External Electromagnetic Fields. Acta Physica Polonica B, 48, 1493-1505. https://doi.org/10.5506/APhysPolB.48.1493
[16]
Kawaguchi, M. and Matsuzaki, S. (2017) Lifetime of Rho Meson in Correlation with Magnetic-Dimensional Reduction. The European Physical Journal A, 53, Article No. 68. https://doi.org/10.1140/epja/i2017-12254-1
[17]
Boumali, A. and Aounallah, H. (2018) Exact Solutions of Scalar Bosons in the Presence of the Aharonov-Bohm and Coulomb Potentials in the Gravitational Field of Topological Defects. Advances in High Energy Physics, 2018, Article ID: 1031763. https://doi.org/10.1155/2018/1031763
[18]
Bagrov, V.G., Gitman, D.M. and Tlyachev, V.B. (2001) Solutions of Relativistic Wave Equations in Superpositions of Aharonov, Bohm Magnetic and Electric Fields. Journal of Mathematical Physics, 42, 1933-1959. https://doi.org/10.1063/1.1353182
[19]
Castro, L.B. and Silva, E.O. (2015) Relativistic Quantum Dynamics of Vector Bosons in Aharonov-Bohm Potential. ArXiv: 1507.07790.
[20]
Lieb, E.H. (1977) Existence and Uniqueness of the Minimizing Solution of Choquard’s Nonlinear Equation. Studies in Applied Mathematics, 57, 93-105. https://doi.org/10.1002/sapm197757293
[21]
Lions, P.-L. (1980) The Choquard Equation and Related Equations. Nonlinear Analysis, 4, 1063-1073. https://doi.org/10.1016/0362-546X(80)90016-4
[22]
Kretzschmar, M. (1965) Aharonov-Bohm Scattering of a Wave Packet of Finite Extension. Zeitschrift für Physik, 185, 84-96. https://doi.org/10.1007/BF01381305
[23]
Magni, C. and Valz-Gris, F. (1995) Can Elementary Quantum Mechanics Explain the Aharonov-Bohm Effect? Journal of Mathematical Physics, 36, 177-186. https://doi.org/10.1063/1.531298
[24]
Oliveira, C.R. and Pereira, M. (2008) Mathematical Justification of the Aharonov-Bohm Hamiltonian. Journal of Statistical Physics, 133, 1175-1184. https://doi.org/10.1007/s10955-008-9631-y
[25]
De Oliveira, C.R. and Pereira, M. (2011) Impenetrability of Aharonov-Bohm Solenoids. Proof of Norm Resolvent Convergence. Letters in Mathematical Physics, 95, 41-51. https://doi.org/10.1007/s11005-010-0444-y
[26]
De Oliveira, C.R. and Romano, R.G. (2017) Aharonov-Bohm Effect without Contact with the Solenoid. Journal of Mathematical Physics, 58, Article ID: 102102. https://doi.org/10.1063/1.4992123
[27]
Hansson, A.M. (2005) On the Spectrum and Eigenfunctions of the Schrödinger Operator with Aharonov-Bohm Magnetic Field. International Journal of Mathematics and Mathematical Sciences, 23, 3751-3766. https://doi.org/10.1155/IJMMS.2005.3751
[28]
Adami, R. and Teta, A. (1998) On the Aharonov-Bohm Effect. Letters in Mathematical Physics, 43, 43-54. https://doi.org/10.1023/A:1007330512611
[29]
Melgaard, M., Ouhabaz, E.-M. and Rozenblum, G. (2004) Negative Discrete Spectrum of Perturbed Multivortex Aharonov-Bohm Hamiltonians. Annales Henri Poincaré, 5, 979-1012. https://doi.org/10.1007/s00023-004-0187-3
[30]
Pankrashkin, K. and Richard, S. (2011) Spectral and Scattering Theory for the Aharonov-Bohm Operators. Reviews in Mathematical Physics, 23, 53-81. https://doi.org/10.1142/S0129055X11004205
[31]
Lieb, E.H. and Loss, M. (2001) Analysis. Graduate Studies in Mathematics, Vol. 14. 2nd Edition, American Mathematical Society, Providence. https://doi.org/10.1090/gsm/014
[32]
Zongo, D.Y.F. (2023) Existence of Ground-State Solution for Schrödinger-Newton Equation with Aharonov-Bohm Magnetic Field. AIP Conference Proceedings, 2872, Article ID: 060029. https://doi.org/10.1063/5.0163541
[33]
Melgaard, M. and Zongo, F. (2012) Multiple Solutions of the Quasirelativistic Choquard Equation. Journal of Mathematical Physics, 53, Article ID: 033709. https://doi.org/10.1063/1.3695991