Heisenberg uniqueness pairs and the Klein-Gordon equation. (English) Zbl 1227.42002

Hedenmalm, Haakan; Montes-Rodríguez, Alfonso

Ann. Math. (2) 173, No. 3, 1507-1527 (2011).

Summary: A Heisenberg uniqueness pair (HUP) is a pair \((\Gamma,\Lambda)\), where \(\Gamma\) is a curve in the plane and \(\Lambda\) is a set in the plane, with the following property: any finite Borel measure \(\mu\) in the plane supported on \(\Gamma\), which is absolutely continuous with respect to arc length, and whose Fourier transform \(\widehat\mu\) vanishes on \(\Lambda\), must automatically be the zero measure. We prove that when \(\Gamma\) is the hyperbola \(x_1x_2=1\), and \(\Lambda\) is the lattice-cross
\[ \Lambda=\big(\alpha\mathbb Z\times\{0\}\big)\cup \big(\{0\}\times\beta\mathbb Z\big), \]
where \(\alpha,\beta\) are positive reals, then \((\Gamma,\Lambda)\) is an HUP if and only if \(\alpha\beta\leq1\); in this situation, the Fourier transform \(\widehat\mu\) of the measure solves the one-dimensional Klein-Gordon equation. Phrased differently, we show that
\[ e^{\pi i\alpha nt},e^{\pi i\beta n/t}\in\mathbb Z, \]
span a weak-star dense subspace in \(L^\infty(\mathbb R)\) if and only if \(\alpha\beta\leq1\). In order to prove this theorem, some elements of linear fractional theory and ergodic theory are needed, such as the Birkhoff ergodic theorem. An idea parallel to the one exploited by Makarov and Poltoratski (in the context of model subspaces) is also needed. As a consequence, we solve a problem on the density of algebras generated by two inner functions raised by Matheson and Stessin.

MSC:

42A10	Trigonometric approximation
42B10	Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
11K50	Metric theory of continued fractions
31B35	Connections of harmonic functions with differential equations in higher dimensions
43A15	\(L^p\)-spaces and other function spaces on groups, semigroups, etc.
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

Keywords:

composition operator; ergodic theory; inversion; Klein-Gordon equation; trigonometric system

References:

[1]	J. Aaronson, An Introduction to Infinite Ergodic Theory, Providence, RI: Amer. Math. Soc., 1997, vol. 50. · Zbl 0882.28013
[2]	A. F. Beardon, The Geometry of Discrete Groups, New York: Springer-Verlag, 1983, vol. 91. · Zbl 0528.30001
[3]	F. Cellarosi, ”Renewal-type limit theorem for continued fractions with even partial quotients,” Ergodic Theory Dynam. Systems, vol. 29, iss. 5, pp. 1451-1478, 2009. · Zbl 1193.37013
[4]	J. Gilman and B. Maskit, ”An algorithm for \(2\)-generator Fuchsian groups,” Michigan Math. J., vol. 38, iss. 1, pp. 13-32, 1991. · Zbl 0724.20033
[5]	V. Havin and B. Jöricke, The Uncertainty Principle in Harmonic Analysis, New York: Springer-Verlag, 1994, vol. 28. · Zbl 0827.42001
[6]	W. Heisenberg, ”Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Z. Physik, vol. 43, pp. 172-198, 1927. · JFM 53.0853.05
[7]	C. Kraaikamp and A. Lopes, ”The theta group and the continued fraction expansion with even partial quotients,” Geom. Dedicata, vol. 59, iss. 3, pp. 293-333, 1996. · Zbl 0841.58049
[8]	N. Makarov and A. Poltoratski, ”Meromorphic inner functions, Toeplitz kernels and the uncertainty principle,” in Perspectives in Analysis, New York: Springer-Verlag, 2005, vol. 27, pp. 185-252. · Zbl 1118.47020
[9]	A. L. Matheson and M. I. Stessin, ”Cauchy transforms of characteristic functions and algebras generated by inner functions,” Proc. Amer. Math. Soc., vol. 133, iss. 11, pp. 3361-3370, 2005. · Zbl 1087.46019
[10]	F. Schweiger, ”Continued Fractions with odd and even partial quotients,” Arbeitsberichte Math. Institut Universtät Salzburg, vol. 4, pp. 59-70, 1982.
[11]	F. Schweiger, ”On the approximation by continued fractions with odd and even partial quotients,” Arbeitsberichte Math. Institut Universtät Salzburg, vol. 1-2, pp. 105-114, 1984.