The Complex WKB Method

  • Johannes Sjöstrand
Part of the Pseudo-Differential Operators book series (PDO, volume 14)


In this chapter we shall study the exponential growth and asymptotic expansions of exact solutions of second-order differential equations in the semi-classical limit. As an application, we establish a Bohr-Sommerfeld quantization condition for Schrödinger operators with real-analytic complex-valued potentials.


  1. 23.
    N. Boussekkine, N. Mecherout, \(\mathcal {P}\mathcal {T}\) -symmetry and Schrödinger operators – the simple well case. Math. Nachr. 289(1), 13–27 (2016), French version at
  2. 44.
    M.V. Fedoryuk, Asymptotic Analysis. Linear Ordinary Differential Equations (Springer, Berlin, 1993)CrossRefGoogle Scholar
  3. 101.
    N. Mecherout, N. Boussekkine, T. Ramond, J. Sjöstrand, \(\mathcal {P}\mathcal {T}\) -symmetry and Schrödinger operators. The double well case. Math. Nachr. 289(7), 854–887 (2016).
  4. 107.
    D.V. Nekhaev, A.I. Shafarevich, Semiclassical limit of the spectrum of the Schrödinger operator with complex periodic potential. Mat. Sb. 208(10), 126–148 (2017)MathSciNetCrossRefGoogle Scholar
  5. 115.
    P. Redparth, Spectral properties of non-self-adjoint operators in the semiclassical regime. J. Differ. Equ. 177(2), 307–330 (2001)MathSciNetCrossRefGoogle Scholar
  6. 127.
    A.A. Shkalikov, Spectral portraits of the Orr-Sommerfeld operator at large Reynolds numbers (Russian). Sovrem. Mat. Fundam. Napravl. 3, 89–112 (2003), Translation in J. Math. Sci. (N. Y.) 124(6), 5417–5441 (2004)Google Scholar
  7. 128.
    A.A. Shkalikov, Eigenvalue asymptotics of perturbed self-adjoint operators. Methods Funct. Anal. Topol. 18(1), 79–89 (2012)MathSciNetzbMATHGoogle Scholar
  8. 157.
    A. Voros, Spectre de l’Équation de Schrödinger et Méthode BKW. Publications Mathématiques d’Orsay 81, vol. 9, 75 pp. (Université de Paris-Sud, Département de Mathématique, Orsay, 1982)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Johannes Sjöstrand
    • 1
  1. 1.Université de Bourgogne Franche-ComtéDijonFrance

Personalised recommendations