Space-adiabatic perturbation theory

Abstract

We study approximate solutions to the time-dependent S c hr odinger equation i"@ t t (x)=@t = H(x ;i"r x ) t (x) with the Hamiltonian given as the Weyl quantization of the symbol H(q p ) taking values in the space of bounded operators on the Hilbert space H f of fast \internal" degrees of freedom. By assumption H(q p ) has an isolated energy band. Using a method of Nenciu and Sordoni NeSo] we p r o ve that interband transitions are suppressed to any order in ". As a consequence, associated to that energy band there exists a subspace of L 2 (R d H f ) almost invariant under the unitary time evolution. We d e v elop a systematic perturbation scheme for the computation of e ective Hamiltonians which g o vern approximately the intraband time evolution. As examples for the general perturbation scheme we discuss the Dirac and Born-Oppenheimer type Hamiltonians and we reconsider also the timeadiabatic theory.

Coherent states, quantum gravity, and the Born-Oppenheimer approximation. I. General considerations

“…The adiabatic Born-Oppenheimer approximation of (2.6), improved by the Berry-Simon connection 35,36 or Mead potential 37 , consists in ignoring the inter-band terms of the operator D, which yields (in electronic units m e = 1, me mnuc = ε 2 ) 10) where D n is the diagonal part of D in n-th electronic band. Additionally, a perturbative expansion in the adiabatic parameter ε of the eigenvalues E and eigenvector coefficients Ψ E n,dn (Q) is performed 38 .…”

“…Fortunately, these techniques also lift the restriction of the above considerations to couplings of slow and fast degrees of freedom via mutually commuting operators in the slow sector. Semi-classical approximations to the dynamics of the slow variables can be obtained by ε-dependent pseudo-differential techniques (Egorov's theorem 10,11,40 ), yielding in zeroth order in ε classical dynamics governed by a Hamiltonian with potential energy given by the electronic energy associated with the band (Peierls substitution), i.e.…”

“…To this end, we continue and generalise certain ideas and proposals for the extraction of quantum field theory on curved spacetimes 3-6 from models of loop quantum gravity present in the literature [7][8][9] , and try to put them into a rigorous and (computationally) effective mathematical framework. The latter is provided by space adiabatic perturbation theory as developed by Panati, Teufel and Spohn in 10,11 which is a mathematically precise formulation, by means of pseudodifferential calculus, of the intuitive content of the Born-Oppenheimer approximation 12 in a timedependent setting. Namely, the quantum system under consideration is split into slow and fast subsystems, and (partially) dequantised in the slow sector (deformation quantisation).…”

Adiabatic approximation for a two-level atom in a light beam

“…This means that the convergence to the BornOppenheimer Hamiltonian as ε → 0 has to be uniform w.r.t τ x ∈ (0, 1] . This last point requires to reconsider carefully the work of [36] by following the uniformity w.r.t τ of the estimates given by Weyl-Hörmander calculus ( [23,10]) for τ -dependent metrics which have uniform structural constants. This is done for the low frequency part in Section 2 while the basic tools of semiclassical calculus are reviewed and adapted in the Appendix A.…”

“…In [36], the adiabatic approximation is completely justified for bounded symbols or when global elliptic properties of the complete matricial symbol allow to reduce to this case after spectral truncation. Unfortunately, it is not the case here, because the eigenvalues of the symbol of the linear part H Lin are ε 2δ τ x τ y |p| 2 + V ε,τ (x, y) ± Ω( τx τy x) .…”