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Semi-classical bounds for resolvents of Schrödinger operators and asymptotics for scattering phases. (English) Zbl 0561.35021

Semiclassical bounds (h\(\to 0)\) for the norm of a projected resolvent of a Schrödinger operator \(-h^ 2\Delta +V\) for positive energies and semiclassical bounds for the scattering phase are derived. These bounds are obtained provided the energy fulfills a certain ”non-trapping condition”, which means that there is an interval around E such that if the initial conditions are such that the energy lies in that interval, a classical particle (Hamilton’s equations corresponding to the above Schrödinger operator) leaves any bounded region eventually. These bounds are derived by a microlocal approximation to the resolvent. The bound on the scattering phase is an application of the bound on the resolvent. Starting point of its derivation is a consequence of the Birman-Krein trace formula to which a Tauberian argument is applied. Furthermore, it is made use of an extended functional calculus for pseudo-differential operators [B. Helffer and D. Robert, J. Funct. Anal. 53, 246-268 (1983; Zbl 0524.35103)].
Reviewer: H.Siedentop

MSC:

35J10 Schrödinger operator, Schrödinger equation
81U05 \(2\)-body potential quantum scattering theory
35P25 Scattering theory for PDEs

Citations:

Zbl 0524.35103
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References:

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