On the adiabatic theorem when eigenvalues dive into the continuum

Abstract

We consider a reduced two-channel model of an atom consisting of a quantum dot coupled to an open scattering channel described by a three-dimensional Laplacian. We are interested in the survival probability of a bound state when the dot energy varies smoothly and adiabatically in time. The initial state corresponds to a discrete eigenvalue which dives into the continuous spectrum and re-emerges from it as the dot energy is varied in time and finally returns to its initial value. Our main result is that for a large class of couplings, the survival probability of this bound state vanishes in the adiabatic limit. At the end of the paper we present a short outlook on how our method may be extended to cover other classes of Hamiltonians; details will be given elsewhere.the existence of the so-called spontaneous (adiabatic) pair creation in linear quantum electrodynamics [10,11,15], as well as for memory effects in quantum mesoscopic transport [2,3,4,9].Turning this heuristics into a mathematical statement proved to be a hard problem and boiled down to a proof of various aspects of the adiabatic theorem in the case where the eigenvalue hits the threshold of the continuous spectrum. Accordingly, the existence results are very limited and the proofs are rather technical and often need further assumptions [3,10,11].Our main result states that the survival probability vanishes in the adiabatic limit, i.e. the adiabatic theorem breaks down, for a large class of couplings between the quantum dot and the open channel, when the bound state dives into the continuous spectrum during the adiabatic tuning of E. In addition, a detailed spectral analysis of the model is given and a 'threshold adiabatic theorem' is proved.Our model is considerably simpler than the one in [11] and allows for rather straightforward dispersive estimates, while the threshold analysis is self-contained. In spite of the relative simplicity of the model, our method is quite robust and may be generalized to cover a larger class of operators. In Section 7 we present a short outlook on the Dirac and N-body Schrödinger case with N ≥ 2. Details will be given elsewhere.Below we present the setting, formulate the results, and comment on them. Sections 2-6 contain the proofs of the statements in the main theorem.

A solvable model of the breakdown of the adiabatic approximation

“…Under the assumption that λ(t) remains as an (embedded) eigenvalue of H(t) for −L ≤ t ≤ L, then a general argument has been established and a result similar to (1.3) is obtained ( [10]). Moreover, the result has been applied by Dürr-Pickl [5] to the Dirac equation to explain the adiabatic pair creation and by Cornean-Jensen-Knörr-Nenciu [3] to specific finite rank perturbations of Schrödinger equations. However, if H(t) has no embedded eigenvalues for −L ≤ t ≤ L and the eigenvalue λ(t) "melts away into the continuum", then there is no general theory to deal with the problem; it is even not clear what is meant by the adiabatic approximation.…”