A Guide to Spectral Theory

1. Front Matter

   Pages i-xx

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2. A FIRST LOOK AT SPECTRAL THEORY

   • Christophe Cheverry, Nicolas Raymond

   Pages 1-15

3. UNBOUNDED OPERATORS

   • Christophe Cheverry, Nicolas Raymond

   Pages 17-64

4. SPECTRUM

   • Christophe Cheverry, Nicolas Raymond

   Pages 65-84

5. COMPACT OPERATORS

   • Christophe Cheverry, Nicolas Raymond

   Pages 85-100

6. FREDHOLM THEORY

   • Christophe Cheverry, Nicolas Raymond

   Pages 101-121

7. SPECTRUM OF SELF-ADJOINT OPERATORS

   • Christophe Cheverry, Nicolas Raymond

   Pages 123-159

8. HILLE–YOSIDA AND STONE’S THEOREMS

   • Christophe Cheverry, Nicolas Raymond

   Pages 161-171

9. ABOUT THE SPECTRAL MEASURE

   • Christophe Cheverry, Nicolas Raymond

   Pages 173-201

10. TRACE-CLASS AND HILBERT-SCHMIDT OPERATORS

    • Christophe Cheverry, Nicolas Raymond

    Pages 203-228

11. SELECTED APPLICATIONS OF THE FUNCTIONAL CALCULUS

    • Christophe Cheverry, Nicolas Raymond

    Pages 229-241

12. Back Matter

    Pages 243-258

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This textbook provides a graduate-level introduction to the spectral theory of linear operators on Banach and Hilbert spaces, guiding readers through key components of spectral theory and its applications in quantum physics.  Based on their extensive teaching experience, the authors present topics in a progressive manner so that each chapter builds on the ones preceding. Researchers and students alike will also appreciate the exploration of more advanced applications and research perspectives presented near the end of the book.
Beginning with a brief introduction to the relationship between spectral theory and quantum physics, the authors go on to explore unbounded operators, analyzing closed, adjoint, and self-adjoint operators. Next, the spectrum of a closed operator is defined and the fundamental properties of Fredholm operators are introduced. The authors then develop the Grushin method to execute the spectral analysis of compact operators. The chapters that follow are devoted to examining Hille-Yoshida and Stone theorems, the spectral analysis of self-adjoint operators, and trace-class and Hilbert-Schmidt operators. The final chapter opens the discussion to several selected applications. Throughout this textbook, detailed proofs are given, and the statements are illustrated by a number of well-chosen examples. At the end, an appendix about foundational functional analysis theorems is provided to help the uninitiated reader.


A Guide to Spectral Theory: Applications and Exercises is intended for graduate students taking an introductory course in spectral theory or operator theory. A background in linear functional analysis and partial differential equations is assumed; basic knowledge of bounded linear operators is useful but not required. PhD students and researchers will also find this volume to be of interest, particularly the research directions provided in later chapters.

• spectral theory applications in quantum physics
• spectral theory of linear operators
• spectral theory of unbounded operators
• spectral theory applications
• Fredholm operators spectral theory

“Throughout the text there are numerous exercises, often with complete solutions or helpful hints … . The brief notes at the end of each chapter provide useful additional comments and references to the literature. Overall, this is an interesting and engagingly written text on spectral theory and its applications. Readers with the requisite background knowledge will find in this book a wealth of interesting applications of classical spectral theory, often to problems of current interest in research.” (David Seifert, zbMATH 1505.47001, 2023)

• University of Rennes 1, Rennes, France

  Christophe Cheverry

• University of Angers, Angers, France

  Nicolas Raymond

Christophe Cheverry is a mathematics professor at the University of Rennes 1. He works in the area of analysis, partial differential equations, and mathematical physics. Nicolas Raymond is a mathematics professor at the University of Angers. His research is focused on semiclassical spectral theory and partial differential equations.

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