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Spectral Asymptotics for Fractional Laplacians

  • Victor IvriiEmail author
Chapter

Abstract

In this article we consider fractional Laplacians which seem to be of interest to probability theory. This is a rather new class of operators for us but our methods works (with a twist, as usual). Our main goal is to derive a two-term asymptotics since one-term asymptotics is easily obtained by R. Seeley’s method.

Key words and phrases

Fractional Laplacians spectral asymptotics 

2010 Mathematics Subject Classification:

35P20 58J50 

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Bibliography

  1. [1]
    R. Bañuelos and T. Kulczycki, Trace estimates for stable processes, Probab. Theory Related Fields 142:313–338 (2008).MathSciNetCrossRefGoogle Scholar
  2. [2]
    R. Bañuelos., T. Kulczycki T. and B. Siudeja., On the trace of symmetric stable processes on Lipschitz domains. J. Funct. Anal. 257(10):3329–3352 (2009).MathSciNetCrossRefGoogle Scholar
  3. [3]
    R. M. Blumenthal and R. K. Getoor, Sample functions of stochastic processes with stationary independent increments. J. Math. Mech., 10:493–516 (1961).Google Scholar
  4. [4]
    R. Frank, and L. Geisinger, Refined Semiclassical asymptotics for fractional powers of the Laplace operator. arXiv:1105.5181, 1–35, (2013).MathSciNetCrossRefGoogle Scholar
  5. [5]
    G. Grubb, Local and nonlocal boundary conditions for \(\mu \)-transmission and fractional elliptic pseudodifferential operators, Analysis and Part. Diff. Equats., 7(71):649–1682 (2014).Google Scholar
  6. [6]
    G. Grubb, Fractional Laplacians on domains, a development of Hörmander’s theory of \(\mu \)-transmission pseudodifferential operators, Adv. Math. 268:478–528 (2015).Google Scholar
  7. [7]
    G. Grubb, Spectral results for mixed problems and fractional elliptic operators, J. Math. Anal. Appl. 421(2):1616–1634 (2015).MathSciNetCrossRefGoogle Scholar
  8. [8]
    G. Grubb and L. Hörmander, The transmission property. Math. Scand., 67:273–289 (1990).MathSciNetCrossRefGoogle Scholar
  9. [9]
    V. Ivrii, Microlocal Analysis and Sharp Spectral Asymptotics, available online at http://www.math.toronto.edu/ivrii/monsterbook.pdf
  10. [10]
    V. Ivrii, Spectral asymptotics for Dirichlet to Neumann operator, arXiv:1802.07524, 1–14, (2018).
  11. [11]
    M. Kwaśnicki, Eigenvalues of the fractional laplace operator in the interval. J. Funct. Anal., 262(5):2379–2402 (2012).MathSciNetCrossRefGoogle Scholar
  12. [12]
    R. Musina, A. I. Nazarov, On fractional Laplacians. Comm. Part. Diff. Eqs. 39(9):1780–1790 (2014).MathSciNetCrossRefGoogle Scholar
  13. [13]
    R. Musina, L. Nazarov. On fractional Laplacians–2. Annales de l’Institut Henri Poincare. Non Linear Analysis, 33(6):1667–1673 (2016).Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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