This book provides a systematic summary and condensation of research on infinite-dimensional Hamiltonian operator spectrum theory over the past thirty years, and offers simple and concise proofs for some new achievements. The book first introduces Hamiltonian systems, both finite-dimensional and infinite-dimensional, laying the foundation for the subsequent introduction of infinite-dimensional Hamiltonian operator spectrum theory. Chapter 2 presents the infinite-dimensional Hamiltonian operator and systematically elaborates on its spectral properties. Chapters 3 and 4 focus on the completeness of the characteristic function system and the symplectic self-adjointness of infinite-dimensional Hamiltonian operators, respectively, achieving improvements and deepening of the relevant content. Chapters 5 and 6 introduce the numerical range and the theory of indefinite metric spaces related to infinite-dimensional Hamiltonian operators, reflecting the broader application prospects of such operators and the novelty of the book's scope.This book will be useful for senior undergraduate students, graduate students, and teachers specializing in mathematics. To read this book, readers are expected to have knowledge of mathematical analysis and advanced algebra, including matrix theory and some basic knowledge of operator theory.
