Inequalities and Operators Involving Magnetic Fields

  • Alexander A. Balinsky
  • W. Desmond Evans
  • Roger T. Lewis
Part of the Universitext book series (UTX)


In classical mechanics the motion of charged particles depends only on electric and magnetic fields E, B which are uniquely described by Maxwell’s equations:
$$\displaystyle{\nabla \cdot \mathbf{E} = 4\pi \rho,}$$
$$\displaystyle{\nabla \cdot \mathbf{B} = 0,}$$
$$\displaystyle{\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t},}$$
$$\displaystyle{\nabla \times \mathbf{B} = 4\pi \mathbf{J} + \frac{\partial \mathbf{E}} {\partial t}.}$$


Gauge Transformation Zero Mode Connected Domain Magnetic Potential Form Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 3.
    Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in quantum theory. Phys. Rev. 115, 485–491 (1959)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 4.
    Aharonov, Y., Bohm, D.: Further considerations on electromagnetic potentials in quantum theory. Phys. Rev. 123, 1511–1524 (1961)MathSciNetCrossRefGoogle Scholar
  3. 5.
    Ahlfors, L.V.: Complex Analysis. An Introduction to the Theory of Analytic Functions of One Complex Variable. International Series in Pure and Applied Mathematics, 3rd edn. McGraw-Hill Book Co., New York (1978)Google Scholar
  4. 13.
    Avron, J., Herbst, I., Simon, B.: Schrödinger operators with magnetic fields. I. General Interactions. Duke Math. J. 45(4), 847–883 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 14.
    Balinsky, A.A.: Hardy type inequalities for Aharonov-Bohm magnetic potentials with multiple singularities, Math. Res. Lett. 10, 169–176 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 18.
    Balinsky, A., Evans, W.D., Lewis, R.T.: On the number of negative eigenvalues of Schrodinger operators with an Aharonov-Bohm magnetic field. Proc. R. Soc. Lond. 457, 2481–2489 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 19.
    Balinsky, A., Evans, W.D., Lewis, R.T.: Sobolev, Hardy and CLR inequalities associated with Pauli operators in \(\mathbb{R}^{3}\). J. Phys. A 34(5), L19–L23 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 22.
    Balinsky, A., Laptev, A., Sobolev, A.: Generalized Hardy inequality for the magnetic Dirichlet forms. J. Stat. Phys. 116(114), 507–521 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 24.
    Bargmann, V.: On the number of bound states in a central field of force. Proc. Nat. Acad. Sci. USA 38, 961–966 (1952)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 25.
    Batelaan, H., Tonomura, A.: The Aharonov–Bohm effects: variations on a subtle theme. Phys. Today 62(9), 38–43 (2009)CrossRefGoogle Scholar
  11. 27.
    Benguria, R.D., Van Den Bosch, H.: A criterion for the existence of zero modes for the Pauli operator with fastly decaying fields. J. Math. Phys. 56, 052104 (2015)MathSciNetCrossRefGoogle Scholar
  12. 36.
    Chambers, R.G.: Shift of an electron interference pattern by enclosed magnetic flux. Phys. Rev. Lett. 5(3), (1960)Google Scholar
  13. 48.
    Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Oxford University Press, Oxford (1987) [OX2 GDP]zbMATHGoogle Scholar
  14. 49.
    Edmunds, D.E., Evans, W.D.: Hardy Operators, Function Spaces, and Embeddings. Springer Monographs in Mathematics. Springer, Berlin/Heidelberg/New York (2004)zbMATHCrossRefGoogle Scholar
  15. 50.
    Ehrenberg, W., Siday, R.: The refractive index in electron optics and the principles of dynamics. Proc. Phys. Soc. B 62, 821 (1949)CrossRefGoogle Scholar
  16. 66.
    Frohlich, J., Lieb, E., Loss, M.: Stability of Coulomb systems with magnetic fields I. The one-electron atom. Commun. Math. Phys. 104(2), 251–270 (1986)MathSciNetCrossRefGoogle Scholar
  17. 76.
    Helffer, B., Mohamed, A.: Caractérisation du spectre essentiel de l’opérateur de Schrödinger avec un champ magnétique. Ann. Inst. Fourier 38(2), 95–112 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 79.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Springer, Berlin/Heidelberg (1983)zbMATHGoogle Scholar
  19. 84.
    Kato, T.: Schrödinger operators with singular potentials. Isr. J. Math. 13(1–2), 135–148 (1972)CrossRefGoogle Scholar
  20. 88.
    Krantz, S.: Complex Analysis: The Geometric Viewpoint. Carus Mathematical Monographs, vol. 23. Mathematical Association of America, Washington, DC (1990)Google Scholar
  21. 89.
    Kregar, A.: Aharonov-Bohm Effect, University of Ljubljana, Department of Physics, March 2011Google Scholar
  22. 95.
    Landau, L.D., Lifshitz, E.M.: Quantum Mechanics (Nonrelativistic Theory). Pergamon, Oxford (1977)Google Scholar
  23. 96.
    Laptev, A.: Spectral inequalities for partial differential equations and their applications. AMS/IP Stud. Adv. Math. 51, 629–643 (2012)MathSciNetGoogle Scholar
  24. 97.
    Laptev, A., Netrusov, Yu.: On the negative eigenvalues of a class of Schrödinger operators. Differential Operators and Spectral Theory. Am. Math. Soc. Transl. 2 189, 173–186 (1999)MathSciNetGoogle Scholar
  25. 99.
    Laptev, A., Weidl, T.: Hardy inequalities for magnetic Dirichlet forms. Oper. Theory: Adv. Appl. 108, 299–305 (1999)MathSciNetGoogle Scholar
  26. 111.
    Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence (2001)Google Scholar
  27. 132.
    Schmidt, K.M.: A short proof for Bargmann-type inequalities. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458(2027), 2829–2832 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 135.
    Shen, Zh.: Eigenvalue asymptotics and exponential decay of eigenfunctions of Schrödinger operators with magnetic fields, Trans. Am. Math. Soc. 348, 4465–4488 (1996)zbMATHCrossRefGoogle Scholar
  29. 141.
    Thaller, B.: The Dirac Equation. Springer, Berlin (1992)CrossRefGoogle Scholar
  30. 149.
    Wen, G.-C.: Conformal Mappings and Boundary-Value Problems. Translations of Mathematical Monographs, vol. 166. American Mathematical Society, Providence (1992)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Alexander A. Balinsky
    • 1
  • W. Desmond Evans
    • 1
  • Roger T. Lewis
    • 2
  1. 1.School of MathematicsCardiff UniversityCardiffUK
  2. 2.Department of MathematicsUniversity of Alabama at BirminghamBirminghamUSA

Personalised recommendations