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, and conclude the note by stating some open problems and conjectures.

13 pages, 3 figures

References in corpus (7)

• Effective confining potential of quantum states in disordered media
• n-point Gravitational Lenses with 5(n-1) Images
• Green's function and anti-holomorphic dynamics on a torus
• Torsional Rigidity and Bergman Analytic Content of Simply Connected Regions
• Mean field equations, hyperelliptic curves and modular forms: I
• On the number of solutions of some transcendental equations
• An introduction to the study of critical points of solutions of elliptic and parabolic equations

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