Preface

There is nothing better than the exact solution of mathematical problems, for example, equa‐ tions or integrals. There are a huge number of such relatively simple cases but the number of complex problems, which have only approximate solutions, is bigger, even tends to infinity. The approximate solutions are looked for either by analytical methods or by numerical anal‐ ysis. A powerful and widely applicable analytical approach to the obtaining of reliable ap‐ proximate solutions is the perturbation method, namely, the development of a perturbation theory of the selected problem. This general approach is well combined with numerical cal‐ culations and computer simulations.

The perturbation theories look alike and can easily be recognized through their main fea‐ tures. Their results are represented by infinite series expansions around an exact solution of a simpler problem. The latter is usually a result of a suitable reduction of the initial problem when the mathematical part, which is responsible for the lack of exact solution, is ignored. The ignored part is called "perturbation part." Under the supposition that the perturbation term has a small effect on the final result, the solution of the entire problem is represented as an infinite series in powers of some expansion parameter, which is a small factor in the ig‐ nored term. The first term in this expansion is usually labeled by the subscript "0" (zero-or‐ der term) and represents the solution of the exactly solved reduced task, whereas the other terms are in powers of the expansion parameter.

The perturbation series are infinite, but in the self-consistent theories, the magnitude of the terms in nonzero power decreases with the increase of the expansion parameter powers, and the final sum of the perturbation terms is smaller than the zero-order term. In this much desired case, the perturbation leads to a relatively small correction to the result for the exact‐ ly solvable part of the problem. This usually happens under some conditions that depend on the features of the specific task and are to be deduced within the development of the theory. In case of a number of relevant problems, both in mathematics and natural sciences, the per‐ turbation contributions are larger than the zero-order solution. This circumstance requires a more specific interpretation of the final results.

In other important cases, the perturbation series are divergent for some parameters of the theory. This situation is frequent in research problems in natural sciences, in particular, in physics. Significant efforts to extract useful information from asymptotic perturbation series are the daily concern of many theoretical physicists working on the most important physics problems, particularly in the field of quantum field theory and in the theory of phase transi‐ tions, where the interparticle interactions are relatively strong. Namely, the interaction terms in the Hamiltonian of a physical system are usually chosen as the perturbation part of perturbation expansions in quantum field theory and statistical physics, where the perturba‐ tion methods are widely used on the basis of the so-called Green's function approach. In modern theory of strongly interacting systems, the perturbation expansions are combined with ideas of scaling and renormalization, and thus these expansions are in the basis of the so-called renormalization group. The latter is a powerful tool of investigation of the effect of strong interactions in field theories.

Once introduced and highly developed in physics, perturbation methods of study are also spread in chemistry—mainly in quantum chemistry, in physical chemistry, in chemical physics, and in biophysics. In the last three–four decades, new interdisciplinary research fields appeared, for example, sociophysics and econophysics, where perturbation theories together with numerical analysis and computer simulations will undoubtedly be very im‐ portant.

The book contains seven chapters, written by noted experts and young researchers who present their recent studies of both pure mathematical problems of perturbation theories and application of perturbation methods to the study of important topics in physics, for ex‐ ample, renormalization group theory and applications to basic models in theoretical physics (Y. Takashi), the quantum gravity and its detection and measurement (F. Bulnes), atom-pho‐ ton interactions (E. G. Thrapsaniotis), treatment of spectra and radiation characteristics by relativistic perturbation theory (A. V. Glushkov et al.), and Green's function approach and some applications (Jing Huang). The pure mathematical issues are related to the problem of generalization of the boundary layer function method for bisingularly perturbed differential equations (K. Alymkulov and D. A. Torsunov) and to the development of new homotopy asymptotic methods and their applications (Baojian Hong).

> **Dimo I. Uzunov** Professor of Physics Bulgarian Academy of Sciences Sofia, Bulgaria
