#### Vol. 2, No. 1, 2009

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Resonances for nonanalytic potentials

### André Martinez, Thierry Ramond and Johannes Sjöstrand

Vol. 2 (2009), No. 1, 29–60
##### Abstract

We consider semiclassical Schrödinger operators on ${ℝ}^{n}$, with ${C}^{\infty }$ potentials decaying polynomially at infinity. The usual theories of resonances do not apply in such a nonanalytic framework. Here, under some additional conditions, we show that resonances are invariantly defined up to any power of their imaginary part. The theory is based on resolvent estimates for families of approximating distorted operators with potentials that are holomorphic in narrow complex sectors around ${ℝ}^{n}$.

##### Keywords
resonances, Schroedinger operators, Breit–Wigner peaks
##### Mathematical Subject Classification 2000
Primary: 35B34, 35P99, 47A10, 81Q20
##### Milestones
Received: 11 May 2008
Revised: 18 December 2008
Accepted: 11 January 2009
Published: 1 February 2009
##### Authors
 André Martinez Università di Bologna Dipartimento di Matematica Piazza di Porta San Donato 5 40127 Bologna Italy http://www.dm.unibo.it/~martinez/ Thierry Ramond Département de Mathématiques Université Paris-Sud 11 UMR CNRS 8628 91405 Orsay France http://www.math.u-psud.fr/~ramond Johannes Sjöstrand IMB (UMR CNRS 5584) Université de Bourgogne 9 av. A. Savary BP 47870 21078 Dijon Cedex France