On the Convergence of the Perturbation Method. I

The perturbation method is one of the most important methods of approximation in quantum mechanics as well as in some fields of classical mechanics. But the question of its convergence has not yet been fully discussed from the systematic point of view. Many of the discussions hitherto given on subject are based on plausibility considerations and draw no decisive conclusion. Among a few works of mathematical character, we should mention those of Wilson and Rellich. The papers of Wilson are rather unsystematic and mainly concerned with bounded operators, which restricts the fields of its application. Rellich's study is the most complete one in the mathematical sense, and treats the case of the so-called regular operators in which the formal series of perturbation are proved to be convergent for sufficiently small value of the parameter. His results are applicable to many problems, especially in classical mechanics, but are still restricted considerably in application, for it is rather usual that the perturbation method is valid only in the sense of asymptotic expansion but not in the sense of power expansion, and in such cases the perturbation cannot be regular in Rellich's sense. On the basis of the variational principle in a generalized sense, the writer developed a theory of the perturbation method regarded as an asymptotic expansion, which is much wider in scope of application than Rellich's regular perturbation. Before entering into this subject, however, it is worth while first to treat the regular perturbation, for the formal part of the perturbation method is completely determined in this case and, moreover, it has itself an important application in quantum mechanics of atoms as we shall show below. But since the original method of Rellich is somewhat complicated and abstruse, we will give in this paper an improved and much simplified treatment of the regular perturbation based on the use of resolvents and contour integrals. Moreover, our method allows us to give explicit formulas representing eigen-values and eigen-vectors as far as any order of the perturbation. Also the estimation of the convergence radii is much improved. We restrict ourselves, however, to a brief outline of the theory together with some illustrative examples and refer the readers to another paper of the writer regarding more detailed and rigorous treatment of the problem. More general case (asymptotic expansion) and the perturbation containing the time will be discussed in subsequent papers.