Spectral Theory - (Graduate Texts in Mathematics) by David Borthwick (Hardcover)

Book Synopsis

This textbook offers a concise introduction to spectral theory, designed for newcomers to functional analysis. Curating the content carefully, the author builds to a proof of the spectral theorem in the early part of the book. Subsequent chapters illustrate a variety of application areas, exploring key examples in detail. Readers looking to delve further into specialized topics will find ample references to classic and recent literature.

Beginning with a brief introduction to functional analysis, the text focuses on unbounded operators and separable Hilbert spaces as the essential tools needed for the subsequent theory. A thorough discussion of the concepts of spectrum and resolvent follows, leading to a complete proof of the spectral theorem for unbounded self-adjoint operators. Applications of spectral theory to differential operators comprise the remaining four chapters. These chapters introduce the Dirichlet Laplacian operator, Schrödinger operators, operators on graphs, and the spectral theory of Riemannian manifolds.

Spectral Theory offers a uniquely accessible introduction to ideas that invite further study in any number of different directions. A background in real and complex analysis is assumed; the author presents the requisite tools from functional analysis within the text. This introductory treatment would suit a functional analysis course intended as a pathway to linear PDE theory. Independent later chapters allow for flexibility in selecting applications to suit specific interests within a one-semester course.

From the Back Cover

This textbook offers a concise introduction to spectral theory, designed for newcomers to functional analysis. Curating the content carefully, the author builds to a proof of the spectral theorem in the early part of the book. Subsequent chapters illustrate a variety of application areas, exploring key examples in detail. Readers looking to delve further into specialized topics will find ample references to classic and recent literature.

Beginning with a brief introduction to functional analysis, the text focuses on unbounded operators and separable Hilbert spaces as the essential tools needed for the subsequent theory. A thorough discussion of the concepts of spectrum and resolvent follows, leading to a complete proof of the spectral theorem for unbounded self-adjoint operators. Applications of spectral theory to differential operators comprise the remaining four chapters. These chapters introduce the Dirichlet Laplacian operator, Schrödinger operators, operators on graphs, andthe spectral theory of Riemannian manifolds.

Spectral Theory offers a uniquely accessible introduction to ideas that invite further study in any number of different directions. A background in real and complex analysis is assumed; the author presents the requisite tools from functional analysis within the text. This introductory treatment would suit a functional analysis course intended as a pathway to linear PDE theory. Independent later chapters allow for flexibility in selecting applications to suit specific interests within a one-semester course.

Review Quotes

"The student will benefit from the many illustrative examples worked out in the book. ... The author succeeds, and the hope is that after working on some of the examples presented, the student will want to explore further applications. Additionally, the instructor may also find inspiration for individual study topics that don't require extensive prerequisites." (Valentin Keyantuo, Mathematical Reviews, February, 2023)

"This is an excellent textbook, which shall be a very useful tool for anyone who is oriented to the applications of functional analysis, especially to partial differential equations." (Panagiotis Koumantos, zbMATH 1444.47001, 2020)

About the Author

David Borthwick is Professor and Director of Graduate Studies in the Department of Mathematics at Emory University, Georgia, USA. His research interests are in spectral theory, global and geometric analysis, and mathematical physics. His monograph Spectral Theory of Infinite-Area Hyperbolic Surfaces appears in Birkhäuser's Progress in Mathematics, and his Introduction to Partial Differential Equations is published in Universitext.