Localization of Discrete Spectrum of Multiparticle Schrödinger Operators
نویسنده
چکیده
In the case of oneand two-particle Schrödinger opera tors / / , uppe r bounds on the multiplicity of eigenvalues have been given in [1 4 ] , They generalize results of Birman [5], Schwinger [6], and others (for a review see Simon [7]), who gave global bounds on the n u m b e r of eigenvalues, i.e. bounds on the total n u m b e r of eigenvalues of H below some — x, x positive. Mult ipart icle systems admi t a wealth of processes, e.g. capture processes, b r eakup processes, rearrangements and excitations, which cannot occur or occur only rudimentar i ly in oneand two-particle systems. This dynamical complicat ion is also reflected in the resolvent equat ion: E.g. the kernel of the homogeneous Lippman-Schwinger equat ion and the kernel of the Rollnik equat ion is not compact for particle number N bigger than 2. Therefore a direct transcription to the N particle case is feasible nei ther for the global bounds [ 5 7 ] nor for the local bounds [1 -4 ] , Thus, the results on global bounds on the number of eigenvalues of mult ipar t ic le systems appeared relatively late: Yafaev [8] gave bounds on the total n u m b e r of bound states of some threepart icle systems, whose interaction potentials are short range and negative, using the Faddeev equations. Simon [9] and Klaus and Simon [10] obtained bounds on the total number of eigenvalues for such potentials for the general N-body case, if the essential spect rum of the Schrödinger opera tor is given by two
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