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- Canto-Martín, Francisco, et al.
(author)
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Perron-Frobenius operators and the Klein-Gordon equation
- 2014
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In: Journal of the European Mathematical Society (Print). - 1435-9855 .- 1435-9863. ; 16:1, s. 31-66
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Journal article (peer-reviewed)abstract
- For a smooth curve Gamma and a set Lambda in the plane R-2, let AC(Gamma; Lambda) be the space of finite Borel measures in the plane supported on Gamma, absolutely continuous with respect to arc length and whose Fourier transform vanishes on Lambda. Following [12], we say that (Gamma, Lambda) is a Heisenberg uniqueness pair if AC(Gamma; Lambda) = {0}. In the context of a hyperbola Gamma, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets Gamma of a collection of solutions to the Klein-Gordon equation. In this work, we mainly address the issue of finding the dimension of AC(Gamma; Lambda) when it is nonzero. We will fix the curve Gamma to be the hyperbola x(1)x(2) = 1, and the set Lambda = Lambda(alpha,beta) to be the lattice-cross Lambda(alpha,beta) = (alpha Zeta x {0}) boolean OR ({0} x beta Z), where alpha, beta are positive reals. We will also consider Gamma(+), the branch of x(1)x(2) = 1 where x(1) > 0. In [12], it is shown that AC(Gamma; Lambda(alpha,beta)) = {0} if and only if alpha beta <= 1. Here, we show that for alpha beta > 1, we get a rather drastic "phase transition": AC(Gamma; Lambda(alpha,beta)) is infinite-dimensional whenever alpha beta > 1. It is shown in [13] that AC(Gamma(+); Lambda(alpha,beta)) = {0} if and only if alpha beta < 4. Moreover, at the edge alpha beta = 4, the behavior is more exotic: the space AC(Gamma(+); Lambda(alpha,beta)) is one-dimensional. Here, we show that the dimension of AC(Gamma(+); Lambda(alpha,beta)) is infinite whenever alpha beta > 4. Dynamical systems, and more specifically Perron-Frobenius operators, play a prominent role in the presentation.
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| 2. |
- Hedenmalm, Håkan, 1961-, et al.
(author)
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The Klein-Gordon equation, the Hilbert transform, and dynamics of Gauss-type maps
- 2020
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In: Journal of the European Mathematical Society (Print). - : European Mathematical Society - EMS - Publishing House GmbH. - 1435-9855 .- 1435-9863. ; 22:6, s. 1703-1757
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Journal article (peer-reviewed)abstract
- We study the uncertainty principle associated with the Klein-Gordon equation. As in the previous work [Ann. of Math. 173 (2011)], we consider vanishing along a lattice-cross. The following variants appear naturally: (1) vanishing only along "half" of the lattice-cross, where the "half" is defined as being on the boundary of a quarter-plane, and (2) that the function vanishes on the whole lattice-cross, but we require the function to have Fourier transform supported by one of the two branches of the hyperbola. In case (1) the critical phenomenon is whether the given condition forces the function to vanish on the quarter-plane in question. Here it turns out to be crucial whether the quarter-plane is space-like or time-like, and in short the answer is yes for space-like and no for time-like. The analysis brings us quite far, involving the orbit of the Hilbert kernel under the iterates of the transfer operator, and uses methods from the theory of totally positive matrices as well as Hurwitz zeta functions, and is partially postponed to a separate publication. In case (2), the critical phenomenon occurs at another density, and the dynamics then comes from the standard Gauss transformation t bar right arrow 1/t mod Z on the interval [0, 1]. In the intermediate range of the density of the lattice-cross, we obtain unique extendability of the Fourier transform from one branch of the hyperbola to the other.
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