Pseudodifferential Operators and Spectral Theory

1. Front Matter

   Pages I-XII

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2. Foundations of ΨDO Theory

   • Mikhail A. Shubin

   Pages 1-75

3. Complex Powers of Elliptic Operators

   • Mikhail A. Shubin

   Pages 77-131

4. Asymptotic Behaviour of the Spectral Function

   • Mikhail A. Shubin

   Pages 133-173

5. Pseudodifferential Operators in ℝⁿ

   • Mikhail A. Shubin

   Pages 175-228

6. Back Matter

   Pages 229-288

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I had mixed feelings when I thought how I should prepare the book for the second edition. It was clear to me that I had to correct all mistakes and misprints that were found in the book during the life of the first edition. This was easy to do because the mistakes were mostly minor and easy to correct, and the misprints were not many. It was more difficult to decide whether I should update the book (or at least its bibliography) somehow. I decided that it did not need much of an updating. The main value of any good mathematical book is that it teaches its reader some language and some skills. It can not exhaust any substantial topic no matter how hard the author tried. Pseudodifferential operators became a language and a tool of analysis of partial differential equations long ago. Therefore it is meaningless to try to exhaust this topic. Here is an easy proof. As of July 3, 2000, MathSciNet (the database of the American Mathematical Society) in a few seconds found 3695 sources, among them 363 books, during its search for "pseudodifferential operator". (The search also led to finding 963 sources for "pseudo-differential operator" but I was unable to check how much the results ofthese two searches intersected). This means that the corresponding words appear either in the title or in the review published in Mathematical Reviews.

• Eigenvalue
• Fourier integral operator
• Pseudodifferential operator
• compactness
• manifold
• micolocal analysis spectral asymptotic

From the reviews of the second edition:

"This is the second edition of Shubin’s already classical book. It provides a fairly short, highly readable nice introduction to microlocal analysis, with emphasis on its application to spectral theory … . For anybody who holds a first course in PDO and FIO we highly recommend … . The book is very well written, in simple and direct language. From the very basics at the beginning, the reader reaches a fairly advanced graduate level at the end." (Tibor Ódor, Acta Scientiarum Mathematicarum, Vol. 73, 2007)

• Department of Mathematics, Northeastern University, Boston, USA

  Mikhail A. Shubin

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