Journal of Statistical Physics

, Volume 166, Issue 3–4, pp 1029–1035 | Cite as

On the Spectral Gap of a Square Distance Matrix

  • Xinyu Cheng
  • Dong LiEmail author
  • David Shirokoff
  • Brian Wetton


We consider a square distance matrix which arises from a preconditioned Jacobian matrix for the numerical computation of the Cahn–Hilliard problem. We prove strict negativity of all but one associated eigenvalues. This solves a conjecture in Christieb et al. (J Comput Phys 257:193–215, 2014).


Distance matrix Eigenvalue Solvability 



X. Cheng, D. Li and B. Wetton were supported in part by NSERC Discovery grants. This work was supported by a Grant from the Simons Foundation (\(\#359610\), David Shirokoff).


  1. 1.
    Bandegi, M., Shirokoff, D.: Approximate global minimizers to pairwise interaction problems via convex relaxation. arXiv:1511.03354
  2. 2.
    Christieb, A., Jones, J., Promislow, K., Wetton, B.: High accuracy solutions to energy gradient flows from material science models. J. Comput. Phys. 257, 193–215 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Li, D., Hickernell, F.J.: Trigonometric spectral collocation methods on lattices. Recent advances in scientific computing and partial differential equations (Hong Kong, 2002), pp. 121–132. Contemp. Math., 330, Amer. Math. Soc., Providence (2003)Google Scholar
  4. 4.
    Li, D., Qiao, Z.: On second order semi-implicit Fourier spectral methods for 2D Cahn-Hilliard equations. J. Sci. Comput. (2016). doi: 10.1007/s10915-016-0251-4
  5. 5.
    Li, D., Qiao, Z., Tang, T.: Characterizing the stabilization size for semi-implicit Fourier-spectral method to phase field equations. SIAM J. Numer. Anal. 54, 1653–1681 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Onsager, L.: The effects of shape on the interaction of colloidal particles. Ann. N. Y. Acad. Sci. 51, 627–659 (1049)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Xinyu Cheng
    • 1
  • Dong Li
    • 1
    Email author
  • David Shirokoff
    • 2
  • Brian Wetton
    • 1
  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.Department of Mathematical SciencesNew Jersey Institute of TechnologyNewarkUSA

Personalised recommendations