Abstract
We deal with Dirac operators with external homogeneous magnetic fields. Hardy-type inequalities related to these operators are investigated: for a suitable class of transversal magnetic fields, we prove a Hardy inequality with the same best constant as in the free case. This leaves naturally open an interesting question whether there exist magnetic fields for which a Hardy inequality with a better constant than the usual one, in connection with the well known diamagnetic phenomenon arising in non-relativistic models.
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References
Arrizabalaga, N.: Distinguished self-adjoint extensions of Dirac operators via Hardy–Dirac inequalities. J. Math. Phys. 52(9), 092301 (2011)
Avron, J., Simon, B.: A counterexample to the paramagnetic conjecture. Phys. Lett. A 79(12), 41–42 (1979/1980)
Balinsky, A., Laptev, A., Sobolev, A.: Generalized Hardy inequality for the magnetic Dirichlet forms. J. Stat. Phys. 116(1–4), 507–521 (2004)
Dolbeault, J., Duoandikoetxea, J., Esteban, M.J., Vega, L.: Hardy-type estimates for Dirac operators. Ann. Sci. École Norm. Sup. (4) 40(6), 885–900 (2007)
Dolbeault, J., Esteban, M.J., Loss, M.: Relativistic hydrogenic atoms in strong magnetic fields. Ann. Henri Poincaré 8(4), 749–779 (2007)
Dolbeault, J., Esteban, M.J., Loss, M.: Characterization of the critical magnetic field in the Dirac-Coulomb equation. J. Phys. A 41(18), 185303 (2008)
Dolbeault, J., Esteban, M.J., Loss, M., Vega, L.: An analytical proof of Hardy-like inequalities related to the Dirac operator. J. Funct. Anal. 216(1), 1–21 (2004)
Dolbeault, J., Esteban, M.J., Séré, E.: On the eigenvalues of operators with gaps. Application to Dirac operators. J. Funct. Anal. 174, 208–226 (2000)
Erdos, L.: Recent developments in quantum mechanics with magnetic fields. Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday. In: Proceedings of Symposia in Pure Mathematics. 76, Part 1. American Mathematical Society, Providence, RI, pp. 401–428, 2007.
Herbst, I.: Barry Simon’s work on electric and magnetic fields and the semi-classical limit. Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, 443–461, Proceedings of Symposia Pure Mathematics. 76, Part 1. American Mathematical Society, Providence, RI, 2007.
Hogreve, H., Schrader, R., Seiler, R.: A conjecture on the spinor functional determinant. Nuclear Phys. B 142(4), 525–534 (1978)
Laptev, A., Weidl, T.: Hardy inequalities for magnetic Dirichlet forms. Mathematical results in quantum mechanics (Prague, 1998). Oper. Theory Adv. Appl. Birkhäuser 108, 299–305 (1999)
Lieb, E.H., Loss, M.: Analysis, Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence (2001)
Simon, B.: An abstract Katos inequality for generators of positivity improving semigroups. Indiana Univ. Math. J. 26(6), 1067–1073 (1977)
Thaller, B.: The Dirac Equation, Texts and Monographs in Physics. Springer, Berlin (1992)
Acknowledgments
The first and third authors were supported by the Italian Project FIRB 2012: “Dispersive dynamics: Fourier Analysis and Variational Methods”
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Fanelli, L., Vega, L. & Visciglia, N. Relativistic Hardy Inequalities in Magnetic Fields. J Stat Phys 154, 866–876 (2014). https://doi.org/10.1007/s10955-014-0915-0
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DOI: https://doi.org/10.1007/s10955-014-0915-0