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.
1. Hamiltonian Systems.- 1.1 Finite-dimensional Hamiltonian Systems and
Hamiltonian Matrices.- 1.2 Linear Infinite-dimensional Hamiltonian Canonical
Systems and Infinite-dimensional Hamiltonian Operators.- 1.3 Hamiltonian
Canonical Systems and the Pseudo-division Algorithm of Multivariate
Polynomial Matrices.- 1.4 Monic Factorization of Bivariate Polynomial
Matrices.- 1.5 Matrix Method of Solving Characteristic Sequences of the
System of Polynomials.- 2 Spectra of Infinite-dimensional Hamiltonian
Operators.- 2.1 Spectra of Linear Operators.- 2.2 Spectra of Diagonal
Infinite-dimensional Hamiltonian Operators.- 2.3 Spectra of Skew-diagonal
Infinite-dimensional Hamiltonian Operators.- 2.4 Spectra of Upper Triangular
Infinite-dimensional Hamiltonian Operators.- 2.5 Spectra of Non-negative
Hamiltonian Operators.- 2.6 Spectra of the General Infinite-dimensional
Hamiltonian Operators.- 3 Completeness of Eigenvector System of
Infinite-dimensional Hamiltonian Operators.- 3.1 Infinite-dimensional
Symplectic Spaces.- 3.2 Symplectic Orthogonality of Eigenvector System of
Infinite-dimensional Hamiltonian Operators.- 3.3 Completeness of
Eigenfunction System of 2×2 Infinite-dimensional Hamiltonian Operators.- 3.4
Completeness of the Eigenfunction System of 4×4 Infinite-dimensional
Hamiltonian Operators.- 4 Symplectic Self-adjointness of Infinite-dimensional
Hamiltonian Operators.- 4.1 Definition of Symplectic Self-adjoint Operators.-
4.2 The Adjoint of the Sum of Two Operators.- 4.3 The Adjoint of Product of
Operators.- 4.4 Characterization of Symplectic Self-adjointness by Using the
Spectral Set of Infinite-dimensional Hamiltonian Operators.- 5 Numerical
Range Theory of Infinite-dimensional Hamiltonian Operators.- 5.1 Numerical
Range and Its Definition.- 5.2 Numerical Range of Infinite-dimensional
Hamiltonian Operators.- 5.3 Numerical Radius of Infinite-dimensional
Hamiltonian Operators.- 5.4 Quadratic Numerical Range of Infinite-dimensional
Hamiltonian Operators.- 5.5 Quadratic Numerical Radius of
Infinite-dimensional Hamiltonian Operator.- 5.6 Essential Numerical Range of
Infinite-dimensional Hamiltonian Operators.- 6 Spectral Theory of
Infinite-dimensional Hamiltonian Operators in Complete Indefinite Inner
Product Spaces.- 6.1 Krein Spaces.- 6.2 Spectra of Infinite-dimensional
Hamiltonian Operators in Krein Spaces.- 6.3 numerical Range in Krein Spaces.
Professor Alatancang Chen is an expert on operator theory, and he has been working on this area for more than 30 years. He has achieved remarkable research results and rich writing experience, has published more than 200 articles and 6 monographs, solving mathematical problems such as spectral symmetry of infinite dimensional Hamiltonian operators and completeness of root vector system. He has extended the traditional method of separating variables to non-self-adjoint operator cases, providing a theoretical basis for constructing the method of separating variables in Hamiltonian systems. The Chinese version of this book was published by Higher Education Press in 2023 and has been applied to graduate education.
