Abstract
We study the energy levels of a single particle in a homogeneous magnetic field and in an axially symmetric external potential. For potentials that are superharmonic off the central axis, we find a general ‘pseudoconcave’ ordering of the ground state energies of the Hamiltonian restricted to the sectors with fixed angular momentum. The physical applications include atoms and ions in strong magnetic fields. There the energies are monotone increasing and concave in angular momentum. In the case of a periodic chain of atoms, the pseudoconcavity extends to the entire lowest band of Bloch functions.
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References
Adams, R. A.: Sobolev Spaces, Academic Press, New York, 1975.
Avron, J., Herbst, I. and Simon, B.: Schrödinger operators with magnetic fields. I. General interaction, Duke Math. J. 45 (1978), 847–883.
Avron, J. E., Herbst, I. W. and Simon, B.: Schrödinger operators with magnetic fields III. Atoms in homogeneous magnetic field, Comm. Math. Phys. 79 (1981), 529–572.
Baumgartner, B.: Level comparison theorems, Ann. Phys. (NY) 168 (1985), 484–526.
Baumgartner, B.: Perturbation of supersymmetric systems in quantum mechanics, In: A. Boutet de Monvel et al. (eds), Recent Developments in QuantumMechanics, Kluwer Acad. Publ., Dordrecht, 1991, pp. 195–208.
Baumgartner, B., Grosse, H. and Martin, A.: Order of levels in potential models, Nuclear Phys. B 254 (1985), 528–42.
Baumgartner, B. and Seiringer, R.: Atoms with bosonic ‘electrons’ in strong magnetic fields, arXiv:math-ph/0007007, to be published in Ann. Henri Poincaré.
Baumgartner, B., Solovej, J. P. and Yngvason, J.: Atoms in strong magnetic fields: The high field limit at fixed nuclear charge, Comm. Math. Phys. 212 (2000), 703–724.
Grosse, H. and Stubbe, J.: Splitting of Landau levels in the presence of external potentials, Lett. Math. Phys. 34 (1995), 59–68.
Hainzl, C. and Seiringer, R.: A discrete density matrix theory for atoms in strong magnetic fields, arXiv:math-ph/0010005, to appear in Comm. Math. Phys.
Johnsen, K., Rögnvaldsson, Ö. and Yngvason, J.: in preparation.
Lavine, R. and O'Carroll, M.: Ground state properties and lower bounds for energy levels of a particle in a uniform magnetic field and external potential, J. Math. Phys. 18 (1977), 1908–12.
Lieb, E. H. and Loss, M.: Analysis, Amer. Math. Soc., Providence, 1997.
Lieb, E. H., Solovej, J. P. and Yngvason, J.: Asymptotics of heavy atoms in high magnetic fields: I. Lowest Landau band regions, Comm. Pure Appl. Math. 47, (1994), 513–591.
Reed, M. and Simon, B.: Methods ofModernMathematical Physics IV, Academic Press, New York, 1978.
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Baumgartner, B., Seiringer, R. On the Ordering of Energy Levels in Homogeneous Magnetic Fields. Letters in Mathematical Physics 54, 213–226 (2000). https://doi.org/10.1023/A:1010978807635
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DOI: https://doi.org/10.1023/A:1010978807635