A note on the critical points of the localization landscape

Abstract

Let \(\Omega \subset {{\mathbb{C}}}\) be a bounded domain. In this note, we use complex variable methods to study the number of critical points of the function \(v=v_\Omega\) that solves the elliptic problem \(\Delta v = -2\) in \(\Omega ,\) with boundary values \(v=0\) on \(\partial \Omega .\) This problem has a classical flavor but is especially motivated by recent studies on localization of eigenfunctions. We provide an upper bound on the number of critical points of v when \(\Omega\) belongs to a special class of domains in the plane, namely, domains for which the boundary \(\partial \Omega\) is contained in \(\{z:|z|^2 = f(z) + \overline{f(z)}\},\) where \(f^{\prime}(z)\) is a rational function. We furnish examples of domains where this bound is attained. We also prove a bound on the number of critical points in the case when \(\Omega\) is a quadrature domain. The note concludes with the statement of some open problems and conjectures.

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Notes

  1. 1.

    Among complex analysts, the index theorem is often referred to as the “generalized argument principle” [34, Sect. 2.5]. Alternatively, one can use Morse theory instead of index theory.

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Acknowledgements

We thank Alexandre Eremenko, Dmitry Khavinson, and Svitlana Mayboroda for helpful comments.

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Correspondence to Erik Lundberg.

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Lundberg, E., Ramachandran, K. A note on the critical points of the localization landscape. Complex Anal Synerg 7, 12 (2021). https://doi.org/10.1007/s40627-021-00075-y

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Keywords

  • Critical points
  • Elliptic equations
  • Torsion function
  • Localization of eigenfunctions
  • Anti-holomorphic dynamics
  • Morse theory