Resonances for nonanalytic potentials

Abstract

We consider semiclassical Schrödinger operators on ‫ޒ‬ n , with C ∞ 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 ‫ޒ‬ n .

Tosio Kato's Work on Non--Relativistic Quantum Mechanics

“…The basic papers include Jensen [272], Sigal [530], Cycon [97], and Nakamura [439,440]. Some approaches for non-analytic potentials include Cattaneo-Graf-Hunziker [82], Cancelier-Martinez-Ramond [77] and Martinez-Ramond-Sjöstrand [426]. There is an enormous literature on the theory of resonances from many points of view.…”

The spectrum of the scattering matrix near resonant energies in the semiclassical limit

“…Due to tunnelling, these bound states in fact become resonances with exponentially small (in the semiclassical regime) imaginary part; see e.g. [4,9,12,16,8,17,15]. Resonances produced in this way are called shape resonances.…”

“…We will call the positive eigenvalues of H int the resonant energies. As mentioned above, under additional assumptions one can prove that for each resonant energy E res there exists a resonance E res of H with |E res − Re E res | and |Im E res | exponentially small in the semiclassical regime, see [4,9,12,16,17,15]. However, it is technically convenient for us to work with real resonant energies E res rather than with complex resonances E res .…”

“…Thus, although resonances provide motivation for our work and are key to interpreting our results, we will say nothing about them and instead refer to resonant energies. In fact (although this is merely a technical point) we do not assume that the resolvent of H admits an analytic continuation and so the existence of resonances under our assumptions cannot be guaranteed; see [15] for a detailed analysis of this issue.…”