Abstract
Let H 0 be a self-adjoint operator in some Hilbert space \({\fancyscript{H}}\) , and let λ0 be a (possibly degenerate) eigenvalue of H 0 embedded in its essential spectrum σ ess(H 0) with corresponding eigenprojection Π0. For small |κ|, let H(κ) be a family of perturbed Hamiltonians, which is analytic in a generalized Balslev–Combes sense. Following Hunziker’s approach in (Commun Math Phys 132:177–188, 1990), we discuss the corrections to exponential decay in
where D(κ) = Π0 + O(κ 2) (κ → 0) and h(κ) is some family of in general non self-adjoint bounded operators with Ranh(κ) = RanΠ0, leaving RanΠ0 invariant, and 0 ≤ g ≤ 1 is a cut-off function with g(λ0) = 1 and sufficiently small support. Our main result is a sharp estimate of the remainder \({\mathcal{R}(\kappa,t)}\) in terms of the Gevrey index a > 1, b > 0 of \({g\in\Gamma^{a,b}(\mathbb{R})}\) :
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Communicated by Christian Gerard.
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Klein, M., Rama, J. Almost Exponential Decay of Quantum Resonance States and Paley–Wiener Type Estimates in Gevrey Spaces. Ann. Henri Poincaré 11, 499–537 (2010). https://doi.org/10.1007/s00023-010-0036-5
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DOI: https://doi.org/10.1007/s00023-010-0036-5