Can we trust the relationship between resonance poles and lifetimes?

Abstract

We show that the shape resonances induced by a one dimensional well of delta functions disappear as soon as a small constant electric field is applied. In particular, in any compact subset of {z : Rez > 0, Im z < 0} there are no resonances if the non-zero field is small enough. In contrast to the lack of convergence of the lifetimes computed from the widths of the resonances we show that the "experimental lifetimes" are continuous at zero field. The shape resonances are replaced by an infinite set of other resonances whose location and number we analyze.

Corrigendum: Quantum resonances and time decay for a double-barrier model (2016J. Phys. A: Math. Theor.49175301)

“…It agrees with the expansion given in equation (1.4), lemma B.2, by [2]. Consequently, also the statement of theorem 3 by [3] must be amended (the proof of the theorem follows the same line as in [3]).…”

Asymptotics of resonances for 1D Stark operators

“…Finally we note the that Herbst and Mavi [8] considered resonances of the Stark operator perturbed by delta-potentials. 1.5.…”

Resonances in the one dimensional Stark effect in the limit of small field

“…It is easy to see that K 0,f (z) := V 1 (H f,0 − z) −1 V 2 has an entire analytic continuation from the upper half plane to C (see (3) and the discussion below) and thus since the analytic continuation of K 0,f (z) is compact I −V 1 (H f −z) −1 V 2 has a meromorphic continuation to C. We define a resonance as a pole of this meromorphic continuation. Note the resolvent equation…”

“…The potential f x represents an electric field of strength f in the negative x direction acting on a particle of charge 1 and mass 1/2. There have been several simple models considered which contribute to the understanding of the existence and non-existence of resonances with an emphasis on the fate of pre-existing resonances of H = −d 2 /dx 2 + V (x) (see [2,3,4]) as f → 0. None of these references treat the model given by H f above although in [3] a specific model is treated where V is replaced by the sum of two delta functions.…”

“…There have been several simple models considered which contribute to the understanding of the existence and non-existence of resonances with an emphasis on the fate of pre-existing resonances of H = −d 2 /dx 2 + V (x) (see [2,3,4]) as f → 0. None of these references treat the model given by H f above although in [3] a specific model is treated where V is replaced by the sum of two delta functions. In this paper we are not only interested in pre-existing resonances which mostly disappear in the limit f → 0, rather we attempt to calculate the limiting behavior of all the resonances of H f for small f > 0 (at least those in a compact set of the plane).…”