Advertisement

Israel Journal of Mathematics

, Volume 221, Issue 2, pp 779–802 | Cite as

Non-accretive Schrödinger operators and exponential decay of their eigenfunctions

  • D. Krejčiřík
  • N. Raymond
  • J. Royer
  • P. Siegl
Article

Abstract

We consider non-self-adjoint electromagnetic Schrödinger operators on arbitrary open sets with complex scalar potentials whose real part is not necessarily bounded from below. Under a suitable sufficient condition on the electromagnetic potential, we introduce a Dirichlet realisation as a closed densely defined operator with non-empty resolvent set and show that the eigenfunctions corresponding to discrete eigenvalues satisfy an Agmon-type exponential decay.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. A. Abramov, A. Aslanyan and E. B. Davies, Bounds on complex eigenvalues and resonances, J. Phys. A: Math. Gen. 34 (2001), 57–72.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    S. Agmon, Bounds on exponential decay of eigenfunctions of Schrödinger Operators, in Schrödinger operators (Como, 1984), Lecture Notes in Mathematics, Vol. 1159, Springer, Berlin, 1985, pp. 1–38.Google Scholar
  3. [3]
    Y. Almog and B. Helffer, Global stability of the normal state of superconductors in the presence of a strong electric current, Comm. Math. Phys. 330 (2014), 1021–1094.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Y. Almog and B. Helffer, On the spectrum of non-selfadjoint Schrödinger operators with compact resolvent, Communications in Partial Differential Equations 40 (2015), no. 8, 1441–1466.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    S. Bögli, P. Siegl and C. Tretter, Approximations of spectra of Schrödinger operators with complex potential on Rd, Communications in Partial Differential Equations, 2017.Google Scholar
  6. [6]
    A. S. Bonnet-Ben Dhia, P. Ciarlet, Jr. and C. M. Zwölf, Time harmonic wave diffraction problems in materials with sign-shifting coefficients, Journal of Computational and Applied Mathematics 234 (2010), no. 6, 1912–1919.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    J.-M. Bouclet and J. Royer, Local energy decay for the damped wave equation, J. Funct. Anal. 266 (2014), no. 2, 4538–4615.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    R. C. Brown and D. B. Hinton, Two separation criteria for second order ordinary or partial differential operators, Academy of Sciences of the Czech Republic. Mathematical Institute. Mathematica Bohemica 124 (1999), no. 2-3, 273–292.MathSciNetzbMATHGoogle Scholar
  9. [9]
    M. Cappiello, T. Gramchev and L. Rodino, Entire extensions and exponential decay for semilinear elliptic equations, Journal d’Analyse Mathématique 111 (2010), 339–367.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    E. B. Davies, Spectral Theory and Differential Operators, Cambridge Studies in Advanced Mathematics, Vol. 42, Cambridge University Press, Cambridge, 1995.Google Scholar
  11. [11]
    A. Dufresnoy, Un exemple de champ magnétique dans Rd, Duke Mathematical Journal 50 (1983), no. 3, 729–734.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    N. Dunford and J. T. Schwartz, Linear Operators, Spectral Theory, Self Adjoint Operators in Hilbert Space, Part 2, John Wiley & Sons, Inc., New York, 1988.zbMATHGoogle Scholar
  13. [13]
    D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford University Press, Oxford, 1987.zbMATHGoogle Scholar
  14. [14]
    W. D. Evans and A. Zettl, Dirichlet and separation results for Schrödinger-type operators, Proceedings of the Royal Society of Edinburgh. Section A. Mathematical and Physical Sciences 80 (1978), no. 1-2, 151–162.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    W. N. Everitt and M. Giertz, Inequalities and separation for Schrödinger type operators in L2(Rn), Proceedings of the Royal Society of Edinburgh. Section A. Mathematical and Physical Sciences 79 (1977/78), no. 3-4, 257–265.MathSciNetzbMATHGoogle Scholar
  16. [16]
    P. Exner, Open Quantum Systems and Feynman Integrals, D. Reidel Publishing Company, Dordrecht, 1985.CrossRefzbMATHGoogle Scholar
  17. [17]
    L. Fanelli, D. Krejčiřík and L. Vega, Spectral stability of Schrödinger operators with subordinated complex potentials, J. Spectr. Theory (2016).Google Scholar
  18. [18]
    I. Gohberg, S. Goldberg and M. A. Kaashoek, Classes of Linear Operators, Vol. 1, Birkhäuser Verlag, Basel, 1990.CrossRefzbMATHGoogle Scholar
  19. [19]
    L. Grubísíc, V. Kostrykin, K. A. Makarov and K. Veselić, Representation theorems for indefinite quadratic forms revisited, Mathematika 59 (2013), 169–189.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    B. Helffer and A. Mohamed, Caractérisation du spectre essentiel de l’opérateur de Schrödinger avec un champ magnétique, Université de Grenoble. Annales de l’Institut Fourier 38 (1988), no. 2, 95–112. MR 949012CrossRefzbMATHGoogle Scholar
  21. [21]
    T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1966.CrossRefzbMATHGoogle Scholar
  22. [22]
    D. Krejčiřík and P. Siegl, Elements of spectral theory without the spectral theorem, in Non-selfadjoint Operators in Quantum Physics: Mathematical Aspects (F. Bagarello, J.-P. Gazeau, F. H. Szafraniec, and M. Znojil, eds.), Wiley-Interscience, New York, 2015, 432 pp.Google Scholar
  23. [23]
    A. Persson, Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator, Mathematica Scandinavica 8 (1960), 143–153.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    N. Raymond, Bound States of the Magnetic Schrödinger Operator, EMS Tracts in Mathematics, European Mathematical Society (EMS), Zürich, 2017.CrossRefzbMATHGoogle Scholar
  25. [25]
    A. Regensburger, Ch. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Parity-time synthetic photonic lattices, Nature 488 (2012), 167–171.CrossRefGoogle Scholar
  26. [26]
    Y. Sibuya, Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient, North-Holland Publishing Co., Amsterdam, 1975.zbMATHGoogle Scholar
  27. [27]
    P. Siegl and D. Krejčiřík, On the metric operator for the imaginary cubic oscillator, Phys. Rev. D 86 (2012), 121702(R).CrossRefGoogle Scholar
  28. [28]
    J. Sjöstrand, Asymptotic distribution of eigenfrequencies for damped wave equations, Publ. RIMS, Kyoto Univ. 36 (2000), 573–611.CrossRefzbMATHGoogle Scholar
  29. [29]
    A. F. M. ter Elst, M. Sauter and H. Vogt, A generalisation of the form method for accretive forms and operators, Journal of Functional Analysis 269 (2015), 705–744.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2017

Authors and Affiliations

  • D. Krejčiřík
    • 1
    • 2
  • N. Raymond
    • 3
  • J. Royer
    • 4
  • P. Siegl
    • 5
    • 6
  1. 1.Department of Theoretical PhysicsNuclear Physics Institute ASCRŘežCzech Republic
  2. 2.Department of Mathematics, Faculty of Nuclear Sciences and Physical EngineeringCzech Technical University in PraguePrague 2Czech Republic
  3. 3.IRMARUniversité de Rennes 1, Campus de BeaulieuRennes cedexFrance
  4. 4.Institut de mathématiques de ToulouseUniversité Toulouse 3Toulouse cedex 9France
  5. 5.Mathematical InstituteUniversity of BernBernSwitzerland
  6. 6.ŘežCzech Republic

Personalised recommendations