Quantum resonances without analyticity

Article type: Research Article

Authors: Cancelier, Claudy | Martinez, André | Ramond, Thierry

Affiliations: Département de Mathématiques et Informatique, Université de Reims, France E-mail: claudy.cancelier@univ-reims.fr | Dipartimento di Matematica, Universita di Bologna, Italy E-mail: martinez@dm.unibo.it | Mathématiques, Université Paris Sud (UMR CNRS 8628), France E-mail: thierry.ramond@math.u-psud.fr

Abstract: We propose a definition for the resonances of Schrödinger operators with slowly decaying 𝒞∞ potentials without any analyticity assumption. Our definition is based on almost analytic extensions for these potentials, and we describe a systematic way to build such an extension that coincide with the function itself whenever it is analytic. That way, if the potential is dilation analytic, our resonances are the usual ones. We show that our resonances with negative real part are exactly the eigenvalues of the operator. We also prove that our definition coincides with the usual ones in the case of smooth exponentially decaying potentials. Then we consider semiclassical results. We show that, if the trapped set for some energy E is empty, there is no resonance in any complex vicinity of E of size O(hlog (1/h)). Finally, we investigate the semiclassical shape resonances and generalize some results of Helffer and Sjöstrand.

Keywords: resonances, almost analytic extensions, resonance-free domains, shape resonances, Schrödinger operators

Journal: Asymptotic Analysis, vol. 44, no. 1-2, pp. 47-74, 2005