Publications (14) View all
-
Article: Uniqueness of solutions to Schr\"odinger equations on 2-step nilpotent Lie groups
[show abstract] [hide abstract]
ABSTRACT: Let g=g_1+g_2, [g,g] =g_2, be a nilpotent Lie algebra of step 2, V_1,..., V_m a basis of g_1 and L=\sum_{j,k} a_{jk} V_j V_k be a left-invariant differential operator on G=exp (g), where the coefficients a_{jk} form a real, symmetric mxm-matrix. It is shown that if a solution w(t,x) to the Schr\"odinger equation \partial_t w(t,g)=i Lw(t,g), w(0,g)=f(g), satisfies a suitable Gaussian type estimate at time t= 0 and at some time t=T\ne 0, then w=0 . The proof is based on Hardy's uncertainty principle and explicit computations within Howe's oscillator semigroup. Our results extend work by Ben Said and Thangavelu in which the authors study the Schr\"odinger equation associated to the sub-Laplacian on the Heisenberg group.07/2012; -
Article: Flat orbits, minimal ideals and spectral synthesis
[show abstract] [hide abstract]
ABSTRACT: Let G = exp \mathfrakg{\mathfrak{g}} be a connected, simply connected, nilpotent Lie group and let ω be a continuous symmetric weight on G with polynomial growth. In the weighted group algebra L1w(G){L^{1}_{\omega}(G)} we determine the minimal ideal of given hull {pl¢ Î [^(G)] | l¢ Î l + \mathfrakn^}{\{\pi_{l'} \in \hat{G} | l' \in l + \mathfrak{n}^{\perp}\}}, where \mathfrakn{\mathfrak{n}} is an ideal contained in \mathfrakg(l){\mathfrak{g}(l)}, and we characterize all the L ∞(G/N)-invariant ideals (where N = exp\mathfrakn{N = {\rm exp}\, \mathfrak{n}}) of the same hull. They are parameterized by a set of G-invariant, translation invariant spaces of complex polynomials on N dominated by ω and are realized as kernels of specially built induced representations. The result is particularly simple if the co-adjoint orbit of l is flat. KeywordsNilpotent Lie group-Irreducible representation-Co-adjoint orbit-Flat orbit-Minimal ideal-Spectral synthesis-Weighted group algebra Mathematics Subject Classification (2000)22E30-22E27-43A20Monatshefte für Mathematik 04/2012; 160(3):271-312. · 0.62 Impact Factor -
Article: Beurling-Fourier algebras on compact groups: spectral theory
[show abstract] [hide abstract]
ABSTRACT: For a compact group $G$ we define the Beurling-Fourier algebra $A_\omega(G)$ on $G$ for weights $\omega$ defined on the dual $\what G$ and taking positive values. The classical Fourier algebra corresponds to the case $\omega$ is the constant weight 1. We study the Gelfand spectrum of the algebra realizing it as a subset of the complexification $G_{\mathbb C}$ defined by McKennon and Cartwright and McMullen. In many cases, such as for polynomial weights, the spectrum is simply $G$. We discuss the questions when the algebra $A_\omega(G)$ is symmetric and regular. We also obtain various results concerning spectral synthesis for $A_\omega(G)$.03/2011; -
Article: The Paley–Wiener theorem for certain nilpotent Lie groupsJean
[show abstract] [hide abstract]
ABSTRACT: We generalize the classical Paley–Wiener theorem to special types of connected, simply connected, nilpotent Lie groups: First we consider nilpotent Lie groups whose Lie algebra admits an ideal which is a polarization for a dense subset of generic linear forms on the Lie algebra. Then we consider nilpotent Lie groups such that the co-adjoint orbits of all the elements of a dense subset of the dual of the Lie algebra * are flat (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)Mathematische Nachrichten 09/2009; 282(10):1423 - 1442. · 0.68 Impact Factor -
Article: The C*-alegbras of the Heisenberg Group and of thread-like Lie groups
[show abstract] [hide abstract]
ABSTRACT: We describe the C*-algebras of the Heisenberg group H_n, n\geq 1, and the thread-like Lie group G_N, N\geq 3, in terms of C*-algebras of operator fields.05/2009;
