00-520 Vladimir GEORGESCU, Andrei IFTIMOVICI

Riesz-Kolmogorov Compacity Criterion, Ruelle Theorem and Lorentz Convergence on Locally Compact Abelian Groups (296K, Postscript) Dec 31, 00

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Abstract. We prove a version of Ruelle's theorem (concerning the description of the pure point and continuous spectral subspaces of a hamiltonian in terms of bound and scattering states) valid for an arbitrary self-adjoint operator $H$ in $L^2(X)$, the configuration space $X$ being an arbitrary abelian locally compact group. This version follows from the Riesz-Kolmogorov theorem which, in our presentation, gives a description of relatively compact sets of states solely in terms of phase space properties of the system. We also replace in Ruelle's theorem the convergence in the Ces\`aro mean by convergence in Lorentz sense, which is sharper than any convergence in the mean.

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