Adiabatic evolution of 1D shape resonances: an artificial interface conditions approach

An explicit model for the adiabatic evolution of quantum observables driven by 1D shape resonances

“…Our approach consists in introducing, in addition to the complex deformation, a further modification formed by artificial interface conditions. According to the results obtained in [11], an adiabatic theorem holds for this modified system (see Theorem 7.1 in [11]), while small perturbations are produced on the relevant spectral quantities (actually the same remains true under more general assumptions). In this simplified framework, we give a reduced equation for the adiabatic evolution of the sheet density of charges accumulating around the interaction point.…”

“…The explicit character of our model and the adiabatic theorem, obtained in [11] for this class of non-selfadjoint Hamiltonians, allow to obtain a complete description of the asymptotic behaviour of A θ0 (t) as h → 0, in particular concerned with the position of the delta shaped potential well. To formulate our results, we adopt the following notation.…”

“…for all u ∈ D(H h θ0,α(t) ) (we refer to [1] for the definition of delta interaction Hamiltonians). An accurate analysis of this class of operators has been given, in [11]. It is shown that the interface conditions introduce small errors, controlled by θ 0 , with respect to the original selfadjoint model (defined by θ 0 = 0).…”

“…The main interest in introducing the artificial perturbation ∆ θ0 rests upon the fact that the corresponding Hamiltonian defines, under complex deformation, a dynamical systems of contractions. This provides us with an alternative approach to the adiabatic evolution of the shape resonances possibly associated with our model, which can be treated in terms of (adiabatic evolution of) spectral projectors for the non-selfadjoint deformed operator (for this point, we refer to Theorem 7.1 in [11]; see also the work of A. Joye [15] for the adiabatic evolution of dynamical systems without uniform time estimates on the semigroup).…”

“…Then, a different approach consists in using complex deformations, originally introduced in [2], [5]. In [11], we define a family of exterior complex deformations U θ for Hamiltonians with compactly supported potentials in (a, b)…”