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Unbounded harmonic functions on homogeneous manifolds of negative curvature

Richard Penney, Roman Urban (2002)

Colloquium Mathematicae

We study unbounded harmonic functions for a second order differential operator on a homogeneous manifold of negative curvature which is a semidirect product of a nilpotent Lie group N and A = ℝ⁺. We prove that if F is harmonic and satisfies some growth condition then F has an asymptotic expansion as a → 0 with coefficients from 𝓓'(N). Then we single out a set of at most two of these coefficients which determine F. Then using asymptotic expansions we are able to prove some theorems...

Uncertainty principles for the Schrödinger equation on Riemannian symmetric spaces of the noncompact type

Angela Pasquale, Maddala Sundari (2012)

Annales de l’institut Fourier

Let X be a Riemannian symmetric space of the noncompact type. We prove that the solution of the time-dependent Schrödinger equation on X with square integrable initial condition f is identically zero at all times t whenever f and the solution at a time t 0 > 0 are simultaneously very rapidly decreasing. The stated condition of rapid decrease is of Beurling type. Conditions respectively of Gelfand-Shilov, Cowling-Price and Hardy type are deduced.

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