Customer Reviews

Applied Analysis by the Hilbert Space Method: An Introduction with Applications to the Wave, Heat, and Schrodinger Equations (Dover Books on Mathematics)

5 star:		(3)
4 star:		(0)
3 star:		(0)
2 star:		(0)
1 star:		(0)

 

 

This is an interesting book which combines Hilbert spaces, differential equations, and Fourier analysis at very elementary undergraduate level. This textbook may have the best elementary introduction to Schrodinger equation without being a functional analysis or quantum mechanic course. Its applications to Laplace operators and Sturm Liouville problems are very well motivated and developed. This is not a standard textbook, its content and examples are not easily obtained from other sources in the same context. Their early editions were very expensive and hard to find, so this new paperback reedition is welcome.

This book is for a one year course on functional analysis. It would replace a traditional course course on mathematical physics with advantage.The first two chapters are on ordinary differential equations. They are very good.The third chapter is on Hilbert spaces beggining with the vibrating string , fourier series , etc, thus motivating the last section on Hilbert Spaces.The fourth chapter is on eigenvalues and eigenfunctions of second order differential operators The remaining of the book covers the Schrodinger equation in one dimension , Bessel's Fuctions , Eigenfunctions of the Laplacian and the Fourier transform .The book explain difficult concepts in simple terms. Things like Lebesque integral,Absolutely continuos functions, etc Everything is carefully explained.Let me quote this passage on page 419: "The elements of this Hilbert Space are functions of two variables.The Hilbert spaces we considered in earlier chapters consisted in functions of one variable. Does this confuses us? Not us!"In fact the book is almost perfect

This is an outstanding book. It skims the theory -- he is quite explicit about this -- but shows well how the methods work. You could call this "why separation of variable works". If you read more in functional analysis this book will serve as great mathematical "motivation". All his explanations are very clear, and there are a lot of examples.