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.
Similar content being viewed by others
Notes
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.
References
Aharonov, D., Shapiro, H.S.: Domains on which analytic functions satisfy quadrature identities. J. Anal. Math. 30, 39–73 (1976)
Alessandrini, G., Magnanini, R.: The index of isolated critical points and solutions of elliptic equations in the plane. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19(4), 567–589 (1992)
Alessandrini, G., Magnanini, R.: Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions. SIAM J. Math. Anal. 25(5), 1259–1268 (1994)
Arnold, D.N., David, G., Jerison, D., Mayboroda, S., Filoche, M.: Effective confining potential of quantum states in disordered media. Phys. Rev. Lett. 116, 056602 (2016)
Arnold, D., David, G., Filoche, M., Jerison, D., Mayboroda, S.: Computing spectra without solving eigenvalue problems. SIAM J. Sci. Comput. 41(1), B69–B92 (2019)
Bergweiler, W., Eremenko, A.: On the number of solutions of a transcendental equation arising in the theory of gravitational lensing. Comput. Methods Funct. Theory 10(1), 303–324 (2010)
Bergweiler, W., Eremenko, A.: Green’s function and anti-holomorphic dynamics on a torus. Proc. Am. Math. Soc. 144(7), 2911–2922 (2016)
Bergweiler, W., Eremenko, A.: On the number of solutions of some transcendental equations. Anal. Math. Phys. 8(2), 185–196 (2018)
Bleher, P.M., Homma, Y., Ji, L.L., Roeder, R.K.W.: Counting zeros of harmonic rational functions and its application to gravitational lensing. Int. Math. Res. Not. 2014(8), 2245–2264 (2013)
Branner, B., Fagella, N.: Quasiconformal surgery in holomorphic dynamics, volume 141 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2014). With contributions by X. Buff, S. Bullett, A.L. Epstein, P. Haïssinsky, C. Henriksen, C.L. Petersen, K.M. Pilgrim, T. Lei and M. Yampolsky
Chai, C.-L., Lin, C.-S., Wang, C.-L.: Mean field equations, hyperelliptic curves and modular forms: I. Camb. J. Math. 3(1–2), 127–274 (2015)
Crowe, W.D., Hasson, R., Rippon, P.J., Strain-Clark, P.E.D.: On the structure of the Mandelbar set. Nonlinearity 2(4), 541–553 (1989)
Davis, P.J.: The Schwarz Function and Its Applications. The Carus Mathematical Monographs, No. 17. The Mathematical Association of America, Buffalo (1974)
Draisma, J., HorobeŢ, E., Ottaviani, G., Sturmfels, B., Thomas, R.R.: The Euclidean distance degree of an algebraic variety. Found. Comput. Math. 16(1), 99–149 (2016)
Filoche, M., Mayboroda, S.: Strong localization induced by one clamped point in thin plate vibrations. Phys. Rev. Lett. 103, 254301 (2009)
Filoche, M., Mayboroda, S.: Universal mechanism for Anderson and weak localization. Proc. Natl Acad. Sci. USA 109(37), 14761–14766 (2012)
Fleeman, M., Khavinson, D.: Approximating ${{\overline{z}}}$ in the Bergman space. In: Recent Progress on Operator Theory and Approximation in Spaces of Analytic Functions. Contemporary Mathematics, vol. 679, pp. 79–90. American Mathematical Society, Providence (2016)
Fleeman, M., Lundberg, E.: The Bergman analytic content of planar domains. Comput. Methods Funct. Theory 17(3), 369–379 (2017)
Fleeman, M., Simanek, B.: Torsional rigidity and Bergman analytic content of simply connected regions. Comput. Methods Funct. Theory 19(1), 37–63 (2019)
Gustafsson, B., Shapiro, H.S.: What is a quadrature domain? In: Quadrature Domains and Their Applications. Operator Theory: Advances and Applications, vol. 156, pp. 1–25. Birkhäuser, Basel (2005)
Khavinson, D., Lundberg, E.: Transcendental harmonic mappings and gravitational lensing by isothermal galaxies. Complex Anal. Oper. Theory 4(3), 515–524 (2010)
Khavinson, D., Neumann, G.: On the number of zeros of certain rational harmonic functions. Proc. Am. Math. Soc. 134(4), 1077–1085 (2006)
Khavinson, D., Świa̧tek, G.: On the number of zeros of certain harmonic polynomials. Proc. Am. Math. Soc. 131(2):409–414 (2003)
Lawrence, D.J.: A Catalog of Special Plane Curves. Dover, Garden City (1972)
Lee, S.-Y., Makarov, N.G.: Topology of quadrature domains. J. Am. Math. Soc. 29(2), 333–369 (2016)
Lin, C.-S., Wang, C.-L.: Elliptic functions, Green functions and the mean field equations on tori. Ann. Math. (2) 172(2), 911–954 (2010)
Lin, C.-S., Wang, C.-L.: Mean field equations, hyperelliptic curves and modular forms: II. J. Éc. Polytech. Math. 4, 557–593 (2017)
Magnanini, R.: An introduction to the study of critical points of solutions of elliptic and parabolic equations. Rend. Ist. Mat. Univ. Trieste 48, 121–166 (2016)
Matsumoto, Y.: An Introduction to Morse Theory. Translations of Mathematical Monographs, vol. 208. American Mathematical Society, Providence (2002). Translated from the 1997 Japanese original by K. Hudson and M. Saito, Iwanami Series in Modern Mathematics
Mukherjee, S., Nakane, S., Schleicher, D.: On multicorns and unicorns II: bifurcations in spaces of antiholomorphic polynomials. Ergod. Theory Dyn. Syst. 37(3), 859–899 (2017)
Muskhelishveli, N.: Some Basic Problems of the Mathematical Theory of Elasticity. Springer, Cham (1977)
Nakane, S., Schleicher, D.: On multicorns and unicorns. I. Antiholomorphic dynamics, hyperbolic components and real cubic polynomials. Int. J. Bifurc. Chaos Appl. Sci. Eng. 13(10), 2825–2844 (2003)
Rhie, S.H.: n-point gravitational lenses with 5(n-1) images (2003). arxiv.org/abs/astro-ph/0305166
Sheil-Small, T.: Complex Polynomials. Cambridge Studies in Advanced Mathematics, vol. 75. Cambridge University Press, Cambridge (2002)
Steinerberger, S.: Localization of quantum states and landscape functions. Proc. Am. Math. Soc. 145(7), 2895–2907 (2017)
Wilmshurst, A.S.: The valence of harmonic polynomials. Proc. Am. Math. Soc. 126(7), 2077–2081 (1998)
Acknowledgements
We thank Alexandre Eremenko, Dmitry Khavinson, and Svitlana Mayboroda for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Accepted:
Published:
DOI: https://doi.org/10.1007/s40627-021-00075-y