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Zbl 0834.35107
Joye, Alain
An adiabatic theorem for singularly perturbed Hamiltonians. (English)
[J] Ann. Inst. Henri Poincaré, Phys. Théor. 63, No.2, 231-250 (1995). ISSN 0246-0211

Summary: The adiabatic approximation in quantum mechanics is considered in the case where the selfadjoint Hamiltonian $H_0 (t)$, satisfying the usual spectral gap assumption in this context, is perturbed by a term of the form $\varepsilon H_1 (t)$. Here $\varepsilon\to 0$ is the adiabaticity parameter and $H_1 (t)$ is a selfadjoint operator defined on a smaller domain than the domain of $H_0 (t)$. Thus the total Hamiltonian $H_0 (t)+ \varepsilon H_1 (t)$ does not necessarily satisfy the gap assumption, $\forall \varepsilon >0$. \par It is shown that an adiabatic theorem can be proved in this situation under reasonable 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.

MSC 2000:: *35Q40 PDE from quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory
81Q15 Perturbation theories for operators and differential equations
35B25 Singular perturbations (PDE)

Keywords: adiabatic approximation; Hamiltonian; gap assumption; time-dependent perturbation

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