JournalsjemsVol. 16, No. 1pp. 31–66

Perron-Frobenius operators and the Klein-Gordon equation

  • Francisco Canto-Martín

    Universidad de Sevilla, Spain
  • Haakan Hedenmalm

    Royal Institute of Technology, Stockholm, Sweden
  • Alfonso Montes-Rodríguez

    Universidad de Sevilla, Spain
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Abstract

For a smooth curve Γ\Gamma and a set Λ\Lambda in the plane R2\mathbb R^2, let AC(Γ;Λ)\mathrm{AC}(\Gamma;\Lambda) be the space of finite Borel measures in the plane supported on Γ\Gamma, absolutely continuous with respect to the arc length and whose Fourier transform vanishes on Λ\Lambda. Following \cite{hh}, we say that (Γ,Λ)(\Gamma,\Lambda) is a Heisenberg uniqueness pair if AC(Γ;Λ)={0}\mathrm{AC}(\Gamma;\Lambda)=\{0\}. In the context of a hyperbola Γ\Gamma, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets Λ\Lambda of a collection of solutions to the Klein-Gordon equation. In this work, we mainly address the issue of finding the dimension of AC(Γ;Λ)\mathrm{AC}(\Gamma;\Lambda) when it is non\-zero. We will fix the curve Γ\Gamma to be the hyperbola x1x2=1x_1x_2=1, and the set Λ=Λα,β\Lambda= \Lambda_{\alpha,\beta} to be the lattice-cross

Λα,β=(αZ×{0})({0}×βZ),\Lambda_{\alpha,\beta}=\left(\alpha \mathbb Z\times\{0\}\right)\cup \left(\{0\}\times\beta \mathbb Z\right),

where α,β\alpha,\beta are positive reals. We will also consider Γ+\Gamma_+, the branch of x1x2=1x_1x_2=1 where x1>0x_1>0. In \cite{hh}, it is shown that AC(Γ;Λα,β)={0}\mathrm{AC}(\Gamma;\Lambda_{\alpha,\beta})=\{0\} if and only if αβ1\alpha\beta\le1. Here, we show that for αβ>1\alpha\beta>1, we get a rather drastic ``phase transition'': AC(Γ;Λα,β)\mathrm{AC}(\Gamma;\Lambda_{\alpha,\beta}) is infinite-dimensional whenever αβ>1\alpha\beta>1. It is shown in \cite{HM2} that AC(Γ+;Λα,β)={0}\mathrm{AC}(\Gamma_+;\Lambda_{\alpha,\beta})=\{0\} if and only if αβ<4\alpha\beta<4. Moreover, at the edge αβ=4\alpha\beta=4, the behavior is more exotic: the space AC(Γ+;Λα,β)\mathrm{AC}(\Gamma_+;\Lambda_{\alpha,\beta}) is one-dimensional. Here, we show that the dimension of AC(Γ+;Λα,β)\mathrm{AC}(\Gamma_+;\Lambda_{\alpha,\beta}) is infinite whenever αβ>4\alpha\beta>4. Dynamical systems, and more specifically Perron-Frobenius operators, will play a prominent role in the presentation.

Cite this article

Francisco Canto-Martín, Haakan Hedenmalm, Alfonso Montes-Rodríguez, Perron-Frobenius operators and the Klein-Gordon equation. J. Eur. Math. Soc. 16 (2014), no. 1, pp. 31–66

DOI 10.4171/JEMS/427