Counting Zeros of Holomorphic Functions

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


In this chapter we will generalize Proposition  3.4.6 of Hager about counting the zeros of holomorphic functions of exponential growth. In Hager and Sjöstrand (Math Ann 342(1):177–243, 2008. we obtained such a generalization, by weakening the regularity assumptions on the functions ϕ. However, due to some logarithmic losses, we were not quite able to recover Hager’s original result, and we still had a fixed domain Γ with smooth boundary.


  1. 4.
    L. Ahlfors, Complex analysis. An Introduction to the Theory of Analytic Functions of One Complex Variable, 3rd edn. International Series in Pure and Applied Mathematics (McGraw-Hill Book Co., New York, 1978)Google Scholar
  2. 15.
    P. Bleher, R. Mallison Jr., Zeros of sections of exponential sums. Int. Math. Res. Not. 2006, 1–49 (2006), Art. ID 38937Google Scholar
  3. 21.
    D. Borisov, P. Exner, Exponential splitting of bound states in a waveguide with a pair of distant windows. J. Phys. A 37(10), 3411–3428 (2004)MathSciNetCrossRefGoogle Scholar
  4. 36.
    E.B. Davies, Eigenvalues of an elliptic system. Math. Z. 243(4), 719–743 (2003)MathSciNetCrossRefGoogle Scholar
  5. 45.
    T. Fischer, Existence, Uniqueness, and Minimality of the Jordan Measure Decomposition.
  6. 49.
    D. Grieser, D. Jerison, Asymptotics of the first nodal line of a convex domain. Invent. Math. 125(2), 197–219 (1996)MathSciNetCrossRefGoogle Scholar
  7. 56.
    M. Hager, J. Sjöstrand, Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators. Math. Ann. 342(1), 177–243 (2008). MathSciNetCrossRefGoogle Scholar
  8. 59.
    B. Helffer, J. Sjöstrand, Multiple wells in the semiclassical limit. I. Commun. Partial Differ. Equ. 9(4), 337–408 (1984)MathSciNetCrossRefGoogle Scholar
  9. 72.
    M. Hitrik, J. Sjöstrand, Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions IIIa. One branching point. Canad. J. Math. 60(3), 572–657 (2008)CrossRefGoogle Scholar
  10. 93.
    B.Ya. Levin, Distribution of Zeros of Entire Functions, English translation (American Mathematical Society, Providence, 1980)Google Scholar
  11. 110.
    A. Pfluger, Die Wertverteilung und das Verhalten von Betrag und Argument einer speciellen Klasse analytischen Funktionen I, II. Comment. Math. Helv. 11, 180–214 (1938), 12, 25–65 (1939)Google Scholar
  12. 125.
    B. Shiffman, S. Zelditch, S. Zrebiec, Overcrowding and hole probabilities for random zeros on complex manifolds. Ind. Univ. Math. J. 57(5), 1977–1997 (2008)MathSciNetCrossRefGoogle Scholar
  13. 141.
    J. Sjöstrand, Counting zeros of holomorphic functions of exponential growth. J. Pseudodiffer. Operators Appl. 1(1), 75–100 (2010). MathSciNetCrossRefGoogle Scholar
  14. 146.
    J. Sjöstrand, M. Vogel, Large bi-diagonal matrices and random perturbations. J. Spectr. Theory 6(4), 977–1020 (2016). MathSciNetCrossRefGoogle Scholar
  15. 149.
    M. Sodin, Zeros of Gaussian analytic functions. Math. Res. Lett. 7, 371–381 (2000)MathSciNetCrossRefGoogle Scholar
  16. 150.
    M. Sodin, B. Tsirelson, Random complex zeroes. III. Decay of the hole probability. Isr. J. Math. 147, 371–379 (2005)zbMATHGoogle Scholar
  17. 160.
    S. Zrebiec, The zeros of flat Gaussian random holomorphic functions on ℂn, and hole probability. Mich. Math. J. 55(2), 269–284 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

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

Personalised recommendations