In view of recent development in perturbation theory, supplementary notes and a supplementary bibliography are added at the end of the new edition. Little change has been made in the text except that the para- graphs V- 4.5, VI- 4.3, and VIII- 1.4 have been completely rewritten, and a number of minor errors, mostly typographical, have been corrected. The author would like to thank many readers who brought the errors to his attention. Due to these changes, some theorems, lemmas, and formulas of the first edition are missing from the new edition while new ones are added. The new ones have numbers different from those attached to the old ones which they may have replaced. Despite considerable expansion, the bibliography i" not intended to be complete. Berkeley, April 1976 TosIO RATO Preface to the First Edition This book is intended to give a systematic presentation of perturba- tion theory for linear operators. It is hoped that the book will be useful to students as well as to mature scientists, both in mathematics and in the physical sciences.