Spectral Theory and Differential OperatorsCambridge University Press, 1995 - 182 oldal This book is an introduction to the theory of partial differential operators. It assumes that the reader has a knowledge of introductory functional analysis, up to the spectral theorem for bounded linear operators on Banach spaces. However it describes the theory of Fourier transforms and distributions as far as is needed to analyse the spectrum of any constant coefficient partial differential operator. A completely new proof of the spectral theorem for unbounded self-adjoint operators is followed by its application to a variety of second order elliptic differential operators, from those with discrete spectrum to Schrödinger operators acting on L2(RN), The book contains a detailed account of the application of variational methods to estimate the eigenvalues of operators with measurable coefficients defined by the use of quadratic form techniques.This book could be used either for self-study or as a course text, and aims to lead the reader to the more advanced literature on the subject. |
Tartalomjegyzék
II | 1 |
III | 7 |
IV | 15 |
V | 17 |
VI | 19 |
VII | 22 |
VIII | 24 |
IX | 32 |
XXVIII | 99 |
XXIX | 101 |
XXX | 104 |
XXXI | 109 |
XXXII | 113 |
XXXIII | 116 |
XXXIV | 118 |
XXXV | 127 |
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analysis application associated assume Banach space bounded linear operator bounded operator bounded region closed closure Co(R coefficients compact resolvent complete orthonormal set continuous function Corollary cyclic subspaces deduce defined denote dense differential operators Dirichlet boundary conditions Dom(A Dom(H domain dominated convergence theorem eigenfunctions eigenvalue of H eigenvalues elliptic operators equal equation essential spectrum essentially self-adjoint Example form Q formula Fourier transform Friedrichs extension functions f given Hilbert space implies integral invariant L²(RN Laplacian Lemma Let f Let H linear subspace lower bounds matrix multiplication operator Neumann boundary conditions non-negative self-adjoint operator norm resolvent one-one operator H Proof Let properties prove quadratic form region in RN satisfies Schrödinger operator Section self-adjoint extensions smooth functions Spec(H spectral theorem spectrum of H subject to Dirichlet subset symmetric operator theory unitary operator zero