Higher Analysis - ISBN:3335003217

Higher Analysis

Date: 23 May 2011, 17:49

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This book arose from lectures repeatedly given by the author at the Friedrich Schiller University Jena. It is intended for students who have a working knowledge of the calculus as well as the theory of ordinary differential equations. We give a brief description of the characteristic features of this book. It is divided into two parts:
The first part consists of Chapters 1—4. The first two chapters contain a systematic introduction to the theory of Banach and Hilbert spaces to the usual extent. In parallel with this we develop the theory of distributions, including the Sobolev spaces. Chapter 3 starts with extended considerations on distributions (tensor products and convolutions). Taking this as a basis, partial differential equations are studied, including fundamental solutions, boundary and initial value problems. In Chapter 4 the main emphasis is placed on the spectral theory of self-adjoint operators in the Hilbert space.
The second part consists of Chapters 5 — 7. Based on the theory of distributions, Sobolev's embedding theorems, and the spectral theory, in Chapter 5 we develop a theory of special functions and the corresponding ordinary differential operators. Chapter 6 contains an introduction to the theory of elliptic differential operators as well as relatively extensive investigations of surface harmonics and separation setups. The concluding Chapter 7 combines the various techniques developed so far in this book. The spectral theory of quantum mechanical operators is in the centre of discussion. In particular we study the various aspects of the hydrogen atom, including the explicit determination of eigenvalues and eigenfunctions.

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