© 2020

Spectral Theory

Basic Concepts and Applications


  • Offers a concise introduction to spectral theory designed for newcomers to functional analysis

  • Illustrates a variety of applications of spectral theory to differential operators, including the Dirichlet Laplacian and Schrödinger operators

  • Incorporates a brief introduction to functional analysis, with a focus on unbounded operators and separable Hilbert spaces


Part of the Graduate Texts in Mathematics book series (GTM, volume 284)

Table of contents

  1. Front Matter
    Pages i-x
  2. David Borthwick
    Pages 1-3
  3. David Borthwick
    Pages 5-33
  4. David Borthwick
    Pages 35-65
  5. David Borthwick
    Pages 67-99
  6. David Borthwick
    Pages 101-123
  7. David Borthwick
    Pages 125-182
  8. David Borthwick
    Pages 183-223
  9. David Borthwick
    Pages 225-243
  10. David Borthwick
    Pages 245-301
  11. Back Matter
    Pages 303-338

About this book


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.


spectral theory textbook unbounded operators on separable Hilbert spaces linear maps on Hilbert spaces application-oriented approach to spectral theory spectral theory on manifolds spectral theory on non-compact manifolds operators on graphs Schrödinger operators Dirichlet Laplacian operator operator theory on Hilbert spaces functional analysis for linear PDE theory spectral theorem for unbounded self-adjoint operators spectral theory of Riemannian manifolds

Authors and affiliations

  1. 1.Department of MathematicsEmory UniversityAtlantaUSA

About the authors

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.

Bibliographic information

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“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)