Good books and lecture notes to learn pseudo-differential operators and spectral theory

Question

I am looking for a list of good books and lecture notes to learn pseudo-differential operators and spectral theory (for infinite dimensions.)

I am familiar with introductory functional analysis, Sobolev spaces and Fourier Analysis. Books inclined towards partial differential equations is a plus. In particular, I intend to study fractional Laplacian operators.

Edit: Please note that it need not be a single book covering both the topics.

Answer 1

Take a look at this graduate text text:

Spectral theory of linear differential operators and comparison algebras By Heinz Otto Cordes, Cordes Heinz Otto · 1987

I think this book covers every topic you mentioned.

https://books.google.com/books?id=6WlLjznD2zwC&printsec=frontcover&dq=heinz+otto+cordes&hl=en&newbks=1&newbks_redir=1&sa=X&ved=2ahUKEwi8zJeg6cz2AhVbm2oFHaMcDRkQ6AF6BAgLEAI

Answer 2

For some references with specific material on both spectral theory and $\Psi$DO's, see

1. Shubin's book "Pseudodifferential Operators and Spectral Theory" sounds perfect for you.
2. Michael Taylor's 2nd PDE book contains two large chapters on both topics. He also has a more extensive textbook on pseudodifferential operators and nonlinear PDE.
3. Dimassi and Sjostrand's "Spectral Asymptotics in the Semi-Classical Limit" covers both.
4. Grigis and Sjostrand's "Microlocal Analysis for Differential Operators: An Introduction" covers both.
5. Zworski's "Semiclassical Analysis" covers both and has many applications to PDE theory.

Also, see Textbook/monograph for microlocal analysis.