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Semi-classical behaviour of the scattering phase for trapping perturbations of the laplacian
Titel:
Semi-classical behaviour of the scattering phase for trapping perturbations of the laplacian
Auteur:
Bruneau, Vincent
Verschenen in:
Communications in partial differential equations
Paginering:
Jaargang 24 (1999) nr. 5-6 pagina's 1230-1251
Jaar:
1999
Inhoud:
We study the influence of trapping trajectories on the semi-classical asymptotic of the scattering phase (spectral shift function), sn(λ) associated to Schrodinger operators, if λ is a non critical energy level. Some results are known when the set of closed trajectories has Liouville measure 0 and in the case of a potential well. In this paper, for smooth and sufficiently decreasing potentials, we describe the behaviour of sn(λ+rh) in terms of the continuity properties of a certain oscillating function Q(h,r). This function. intl.oduced by V. Petkov and G. Popov to study clustering of eigenvalues, is related to the periodic trajectories. If Q(h,r) is uniformly contiuuous in r for any h ε[0,h0], we obtain a TVeyl type asynlptotic of sn(λ+rh), with a second term containing Q(h,r). On the other hand, the point of discontinuity of Q(h, r) in r may give rise to a "clustering" phenomena for the scattering phase. Hence, in contrast to non-trapping case, the derivative d/dλ (sh) can have no complete asymptotic expansion.
Uitgever:
Taylor & Francis
Bronbestand:
Elektronische Wetenschappelijke Tijdschriften
Details van artikel 113 van 135 gevonden artikelen
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