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**Adiabatic perturbation theory in quantum dynamics.**
*(English)*
Zbl 1053.81003

Lecture Notes in Mathematics 1821. Berlin: Springer (ISBN 3-540-40723-5/pbk). vi, 236 p. (2003).

This book addresses the important issue of scale separation in complex systems and other situations in natural sciences. Sometimes, scales are well separated and it is not difficult to derive simple laws for some or all of the slow variables in terms of the fast ones. But even before that, the precise meaning of slow and fast variables must be clarified, and indeed may differ widely in different systems. As explained by the author, a spinning top constitutes a simple example for the kind of situations considered in the monograph while it spins at high frequency, its rotation axis precesses much slower. The orientation of the rotation axis is the slower degree of freedom in this example, while the angle of rotation with respect to the axis is the fast one. In the case of the Earth, it turns once a day while it precesses only once in 25700 years.

In the monograph, quantum mechanical systems which display such a scale separation are considered, the prototype example being molecules, i.e., systems consisting of two types of particles with very different masses (generically, electrons are lighter than nuclei by a factor of over one thousand, so that they move at least 50 times faster). Fast degrees of freedom become slaved by slow ones, which evolve autonomously, a phenomenon called adiabatic decoupling. As explained in the book, this is common to a large variety of physical systems, for which similar mathematical methods can be applied. The fast scale is the quantum mechanical time scale, involving Planck-s constant \(\hbar\). The abstract mathematical question involved is the \(\hbar \to 0\) limit of Schrödinger’s equation with a special type of Hamiltonian. In many interesting situations the complexity of the full system makes a numerical treatment of this equation impossible. Even a qualitative understanding of the dynamics cannot be based on the equation of motion alone, and one is led to find more simple effective equations of motion that yield reliable approximate solutions when \(\hbar \to 0\).

The monograph reviews an approach to adiabatic perturbation theory in quantum dynamics which goal isto find asymptotic solutions to the mentioned initial value problem, as solutions of an effective Schrödinger equation for the slow degrees of freedom alone. An insight of the approach consists in a clear separation of the adiabatic limit from the semiclassical one: it turns out that adiabatic decoupling is a necessary condition for semiclassical behavior of the slow degrees of freedom. Related with this is the emphasis on dealing with effective equations of motion, as opposed, e.g., to the direct construction of approximate solutions to the Schrödinger equation based on the WKB Ansatz or semiclassical wave packets.

After an introductory chapter (Chapter 1), the monograph contains three main parts. In Chapter 2 first order adiabatic theory for perturbations of fibered Hamiltonians is discussed. This is done with the mathematical tools of a standard course on unbounded self-adjoint operators on a Hilbert space. In Chapter 3 the general problem is considered and a theory is developed which allows for approximations to an arbitrary order in powers of \(\hbar\). Chapters 4 and 5 are devoted to applications and extensions of the general scheme, termed by the author adiabatic perturbation theory. The main mathematical tool in Chapters 3-5 are pseudodifferential operators with operator valued symbols. While all these chapters deal with adiabatic decoupling in the presence of a spectral gap for the Hamiltonian, Chapter 6 is concerned with adiabatic theory without a spectral gap. The author leaves outside the scope of the monograph a comparison of the quantum mechanical results with those of classical adiabatic theory, which is a well studied subject on its own. This would be highly interesting, and could serve as a check or a guide, in view of the novelty of the results discussed here. In an Appendix some necessary definitions and results are collected, together with references to the literature.

In the monograph, quantum mechanical systems which display such a scale separation are considered, the prototype example being molecules, i.e., systems consisting of two types of particles with very different masses (generically, electrons are lighter than nuclei by a factor of over one thousand, so that they move at least 50 times faster). Fast degrees of freedom become slaved by slow ones, which evolve autonomously, a phenomenon called adiabatic decoupling. As explained in the book, this is common to a large variety of physical systems, for which similar mathematical methods can be applied. The fast scale is the quantum mechanical time scale, involving Planck-s constant \(\hbar\). The abstract mathematical question involved is the \(\hbar \to 0\) limit of Schrödinger’s equation with a special type of Hamiltonian. In many interesting situations the complexity of the full system makes a numerical treatment of this equation impossible. Even a qualitative understanding of the dynamics cannot be based on the equation of motion alone, and one is led to find more simple effective equations of motion that yield reliable approximate solutions when \(\hbar \to 0\).

The monograph reviews an approach to adiabatic perturbation theory in quantum dynamics which goal isto find asymptotic solutions to the mentioned initial value problem, as solutions of an effective Schrödinger equation for the slow degrees of freedom alone. An insight of the approach consists in a clear separation of the adiabatic limit from the semiclassical one: it turns out that adiabatic decoupling is a necessary condition for semiclassical behavior of the slow degrees of freedom. Related with this is the emphasis on dealing with effective equations of motion, as opposed, e.g., to the direct construction of approximate solutions to the Schrödinger equation based on the WKB Ansatz or semiclassical wave packets.

After an introductory chapter (Chapter 1), the monograph contains three main parts. In Chapter 2 first order adiabatic theory for perturbations of fibered Hamiltonians is discussed. This is done with the mathematical tools of a standard course on unbounded self-adjoint operators on a Hilbert space. In Chapter 3 the general problem is considered and a theory is developed which allows for approximations to an arbitrary order in powers of \(\hbar\). Chapters 4 and 5 are devoted to applications and extensions of the general scheme, termed by the author adiabatic perturbation theory. The main mathematical tool in Chapters 3-5 are pseudodifferential operators with operator valued symbols. While all these chapters deal with adiabatic decoupling in the presence of a spectral gap for the Hamiltonian, Chapter 6 is concerned with adiabatic theory without a spectral gap. The author leaves outside the scope of the monograph a comparison of the quantum mechanical results with those of classical adiabatic theory, which is a well studied subject on its own. This would be highly interesting, and could serve as a check or a guide, in view of the novelty of the results discussed here. In an Appendix some necessary definitions and results are collected, together with references to the literature.

Reviewer: Emilio Elizalde (Barcelona)

### MSC:

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

81Q15 | Perturbation theories for operators and differential equations in quantum theory |

47G30 | Pseudodifferential operators |

35Q40 | PDEs in connection with quantum mechanics |

37N20 | Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) |