### Preise

Der AHP-Preis 2015 wurde an Ira Herbst und Juliane Rama für ihre Arbeit

"Instability for Pre-existing Resonances under a small constant Electric Field"

verliehen. (Dieser Preis wird jedes Jahr für den bemerkenswertesten Artikel in der Zeitschrift Annales Henri Poincaré verliehen.)

# Tunneling for a class of Difference Operators: Complete Asymptotics

#### Autoren: Markus Klein, Elke Rosenberger (2018)

We analyze a general class of diﬀerence operators <tex>H_\varepsilon = T_\varepsilon +V_\varepsilo</tex> on <tex>\ell^2((\varepsilon\mathbb{Z})^d)</tex>, where
<tex>V_\varepsilon</tex> is a multi-well potential and <tex>\varepsilon</tex> is a small parameter. We derive full asymptotic expansions
of the prefactor of the exponentially small eigenvalue splitting due to interactions between two
“wells” (minima) of the potential energy, i.e., for the discrete tunneling eﬀect. We treat both
the case where there is a single minimal geodesic (with respect to the natural Finsler metric
induced by the leading symbol<tex> h_0(x, ξ)</tex> of <tex>H_\varepsilon</tex>) connecting the two minima and the case where
the minimal geodesics form an k+1 dimensional manifold, k ≥ 1. These results on the tunneling
problem are as sharp as the classical results for the Schrödinger operator in [Helﬀer, Sjöstrand,
1984]. Technically, our approach is pseudodiﬀerential and we adapt techniques from [Helﬀer,
Sjöstrand, 1988] and [Helﬀer, Parisse, 1994] to our discrete setting.

Zeitschrift:
Annales Henri Poincare
Verlag:
Springer Verlag
Seiten:
3511-3559
Band:
19(11)

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