1994/11/02 by Alain Joye, Joye, Alain
Mathematics · Physics and Astronomy · #FOS: Mathematics #Functional Analysis (math.FA) #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #funct-an #math.FA
paper · pdf · doi:10.48550/arxiv.funct-an/9411001
17 pages, LaTex
arxiv created 1994/11/02 · openalex publication_date 1994/11/02 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The adiabatic approximation in quantum mechanics is considered in the case where the self-adjoint hamiltonian H0(t), satisfying the usual spectral gap assumption in this context, is perturbed by a term of the form εH1(t). Here ε→ 0 is the adiabaticity parameter and H1(t) is a self-adjoint operator defined on a smaller domain than the domain of H0(t). Thus the total hamiltonian H0(t)+εH1(t) does not necessarily satisfy the gap assumption, ∀ ε>0. It is shown that an adiabatic theorem can be proven in this situation under reasonnable hypotheses. The problem considered can also be viewed as the study of a time-dependent system coupled to a time-dependent perturbation, in the limit of large coupling constant.