Adiabatic Theorem without a Gap Condition

  title={Adiabatic Theorem without a Gap Condition},
  author={J. Avron and Alexander Elgart},
  journal={Communications in Mathematical Physics},
  • J. Avron, A. Elgart
  • Published 25 May 1998
  • Physics, Mathematics
  • Communications in Mathematical Physics
Abstract:We prove the adiabatic theorem for quantum evolution without the traditional gap condition. All that this adiabatic theorem needs is a (piecewise) twice differentiable finite dimensional spectral projection. The result implies that the adiabatic theorem holds for the ground state of atoms in quantized radiation field. The general result we prove gives no information on the rate at which the adiabatic limit is approached. With additional spectral information one can also estimate this… 
A Note on the Adiabatic Theorem Without Gap Condition
We simplify the proof of the adiabatic theorem of quantum mechanics without gap condition of Avron and Elgart by providing an elementary solution of the ‘commutator equation’. In addition, a minor
Adiabatic approximation in open quantum systems
We generalize the standard quantum adiabatic approximation to the case of open quantum systems. We define the adiabatic limit of an open quantum system as the regime in which its dynamical
An adiabatic theorem for resonances
We prove a robust extension of the quantum adiabatic theorem. The theorem applies to systems that have resonances instead of bound states and to systems for which just an approximation to a bound
We prove an adiabatic theorem for the evolution of spectral data under a weak additive perturbation in the context of a system without an intrinsic time scale. For continuous functions of the
Necessary and sufficient condition for quantum adiabatic evolution by unitary control fields
We decompose the quantum adiabatic evolution as the products of gauge invariant unitary operators and obtain the exact nonadiabatic correction in the adiabatic approximation. A necessary and
Adiabatic evolution for systems with infinitely many eigenvalue crossings
We formulate an adiabatic theorem adapted to models that present an instantaneous eigenvalue experiencing an infinite number of crossings with the rest of the spectrum. We give an upper bound on the
Adiabatic Theorems for Generators of Contracting Evolutions
We develop an adiabatic theory for generators of contracting evolution on Banach spaces. This provides a uniform framework for a host of adiabatic theorems ranging from unitary quantum evolutions
Adiabatic perturbation theory in quantum dynamics
Introduction.- First-order adiabatic theory.- Space-adiabatic perturbation theory.- Applications and extensions.- Quantum dynamics in periodic media.- Adiabatic decoupling without spectral gap.-
Weakly Nonlinear Time-Adiabatic Theory
We revisit the time-adiabatic theorem of quantum mechanics and show that it can be extended to weakly nonlinear situations, that is to nonlinear Schrödinger equations in which either the nonlinear


Adiabatic theorems and applications to the quantum hall effect
We study an adiabatic evolution that approximates the physical dynamics and describes a natural parallel transport in spectral subspaces. Using this we prove two folk theorems about the adiabatic
The Adiabatic Theorem of Quantum Mechanics
We prove the adiabatic theorem for quantum evolution without the traditional gap condition. We show that the theorem holds essentially in all cases where it can be formulated. In particular, our
Adiabatic theorem without a gap condition: Two-level system coupled to quantized radiation field
We prove an adiabatic theorem for the ground state of the Dicke model in a slowly rotating magnetic field and show that for weak electron-photon coupling, the adiabatic time scale is close to the
Adiabatic theorems for dense point spectra
We prove adiabatic theorems in situations where the Hamiltonian has dense point spectrum. The gap condition of the standard adiabatic theorems is replaced by an appropriate condition on the
Linear adiabatic theory. Exponential estimates
A general adiabatic expansion is written down. The basic result is that, under appropriate smoothness conditions, the adiabatic estimations can be pushed up to exponential order. The results imply
Precise exponential estimates in adiabatic theory
General adiabatic evolutions associated to Hamiltonians, which admit a holomorphic extension with respect to the time variable in a complex strip, and whose spectrum satisfies a gap condition are
Power-law corrections to the Kubo formula vanish in quantum Hall systems
In first order perturbation theory conductivity is given by the Kubo formula, which in a Quantum Hall System equals the first Chern class of a vector bundle. We apply the adiabatic theorem to show
Renormalization Group Analysis of Spectral Problems in Quantum Field Theory
Abstract In this paper we present a self-contained and detailed exposition of the new renormalization group technique proposed in [1, 2]. Its main feature is that the renormalization group
On the Existence and Uniqueness of Ground States of a Generalized Spin-Boson Model
A generalization of the standard spin-boson model is considered. The HamiltonianH(α) of the model with a coupling parameterα∈Racts in the tensor product H⊕Fbof a Hilbert space H and the boson