We consider, for h , E > 0, resolvent estimates for the semiclassical Schrödinger operator − h 2 Δ + V − E. Near infinity, the potential takes the form V = V L + V S , where V L is a long range potential which is Lipschitz with respect to the radial variable, while V S = O (  x  − 1 ( log  x  ) − ρ ) for some ρ > 1. Near the origin,  V  may behave like  x  − β , provided 0 ⩽ β < 2 ( 3 − 1 ). We find that, for any ρ ˜ > 1, there are C , h 0 > 0 such that we have a resolvent bound of the form exp ( C h − 2 ( log ( h − 1 ) ) 1 + ρ ˜ ) for all h ∈ ( 0 , h 0 ]. The hdependence of the bound improves if V S decays at a faster rate toward infinity.
 NSFPAR ID:
 10338847
 Date Published:
 Journal Name:
 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
 Volume:
 380
 Issue:
 2225
 ISSN:
 1364503X
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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