Quantum mechanical resonance and limiting absorption: The many body problem

Abstract

We introduce a notion of quantum mechanical resonance that does not rely on analytic continuation of resolvent or scattering matrix and relate it to slow temporal decay of certain distinguished resonant states. We proceed to prove existence of resonances for the generalized many body Schrόdinger operator for a rather large class of potentials containing Coulomb and Yukawa, but also nonsymmetric and nonanalytic potentials with Coulomblike singularities at the origin and certain differentiability and decay properties.

Nonexponential decay laws in perturbation theory of near threshold eigenvalues

“…[23], [24], [8], [25], [2], [6], [16], and reference therein) and at the level of rigorous mathematical physics (see e.g. [21], [5], [4], [1], [3], [9], [20], [27], [13], [14], [15], and references therein). The common wisdom, following the computation by Dirac of the decay rate in second order time-dependent perturbation theory (the famous Fermi golden rule), or the Wigner-Weisskopf approximation in scattering theory, is that the decay law is exponential.…”

Resonances for nonanalytic potentials

“…In [Orth 1990;Gérard and Sigal 1992;Soffer and Weinstein 1998;Jensen and Nenciu 2006], the point of view is quite different from ours, while in [Cancelier et al 2005], the definition is based on the use of an almost-analytic extension of the potential and seems to strongly depend both on the choice of this extension and on the complex distortion.…”

“…There are a small number of works about the definition of resonances for nonanalytic potentials, for example, [Orth 1990;Gérard and Sigal 1992;Soffer and Weinstein 1998;Cancelier et al 2005;Jensen and Nenciu 2006]. In [Orth 1990;Gérard and Sigal 1992;Soffer and Weinstein 1998;Jensen and Nenciu 2006], the point of view is quite different from ours, while in [Cancelier et al 2005], the definition is based on the use of an almost-analytic extension of the potential and seems to strongly depend both on the choice of this extension and on the complex distortion.…”

Закон Распада Квазистационарного Состояния Оператора Шредингера Для Кристаллической Пленки

“…В имеющейся литературе наличие экспоненциального распада квазистационарно-го (метастабильного) состояния доказывается для различных классов операторов, с помощью метода комплексного скейлинга [4], метода Шура-Лившица-Фешбаха-Грушина [5] (оператор H ε заменяется "матрицей" в прямой сумме im P 0 ⊕ im(I − P 0 ), где P 0 -ортогональный проектор на собственное подпространство ker(H 0 − λ 0 )); ис-пользуется также неравенство Мурра [6].…”