Author	Hausner, Melvin. author
Title	Spectral Theory of Differential Operators [electronic resource] : Self-Adjoint Differential Operators / by V. A. Il'in
Imprint	Boston, MA : Springer US, 1995
Connect to	http://dx.doi.org/10.1007/978-1-4615-1755-9
Descript	XII, 390 p. online resource

In this fully-illustrated textbook, the author examines the spectral theory of self-adjoint elliptic operators. Chapters focus on the problems of convergence and summability of spectral decompositions about the fundamental functions of elliptic operators of the second order. The author's work offers a novel method for estimation of the remainder term of a spectral function and its Riesz means without recourse to the traditional Carleman technique and Tauberian theorem apparatus

1. Expansion in the Fundamental System of Functions of the Laplace Operator -- 1.1 Fundamental Systems of Functions and Their Properties -- 1.2 Fractional Kernels -- 1.3 Estimate for the Remainder Term of a Spectral Function in the Metric L2 and the Resulting Corollaries -- 1.4 Exact Conditions for the Localization and Uniform Convergence of Expansions with Respect to an Arbitrary FSF in the Sobolev-Liouville Classes -- 1.5 On the Potential Generalization of the Theory -- Comments on Chapter 1 -- 2. Spectral Decompositions Corresponding to an Arbitrary Self-Adjoint Nonnegative Extension of the Laplace Operator -- 2.1 Self-Adjoint Nonnegative Extensions of Elliptic Operators. Ordered Spectral Representations of the Space L2. Classes of Differentiate Functions of N Variables -- 2.2 Formulation and Analysis of Main Results -- 2.3 Certain Properties of the Fundamental Functions of an Arbitrary Ordered Spectral Representation in the Space L2 -- 2.4 Proof of Negative Theorem 2.1 -- 2.5 Proof of Positive Theorem 2.3 -- 2.6 Estimate for the Remainder Term of the Riesz Means of a Spectral Function in the Metric L2 -- 2.7 Estimate for the Remainder Term of the Riesz Means of a Spectral Function in the Metric L2 -- Comments on Chapter 2 -- 3. On the Riesz Equisummability of Spectral Decompositions in the Classical and the Generalized Sense -- 3.1 On the Riesz Equisummability of Spectral Decompositions in the Classical Sense -- 3.2 On the Riesz Equisummability of Spectral Decompositions in the Generalized Sense -- Comments on Chapter 3 -- 4. Self-Adjoint Nonnegative Extensions of an Elliptic Operator of Second Order -- 4.1 Ancillary Propositions about Fundamental Functions -- 4.2 Theorems of Negative Type -- 4.3 Theorems of Positive Type -- Comments on Chapter 4 -- Appendix 1. Conditions for the Uniform Convergence of Multiple Trigonometric Fourier Series with Spherical Partial Sums -- Appendix 2. Conditions for the Uniform Convergence of Decompositions in Eigenfunctions of the First, Second, and Third Boundary-Value Problems for an Elliptic Operator of Second Order -- Epilogue -- References