## An Introduction to Spectral TheoryA brief and accessible introduction to the spectral theory of linear second order elliptic differential operators. By introducing vital topics of abstract functional analysis where necessary, and using clear and simple proofs, the book develops an elegant presentation of the theory while integrating applications of basic real world problems involving the Laplacian. Suitable for use as a self-contained introduction for beginners or as a one-semester student text; contains some 25 examples and 60 exercises, most with detailed hints. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

Compact selfadjoint operators | 25 |

Some examples and applications | 53 |

Integral representation | 75 |

Spectral decomposition of the Laplaciaii | 101 |

Further spectral properties | 125 |

Eigenvalues of the Laplacian | 149 |

Some applications to hydrodynamics | 163 |

Bibliography | 191 |

### Common terms and phrases

arbitrary Banach space belongs boundary conditions boundary value problem bounded domain chapter characteristic subspace compact operator compact self-adjoint operator compact set concludes the proof Consequently continuous function convergent sequence Corollary corresponding Definition denote differential equations differential operator eigenfunctions eigenvalue eigenvector elements equicontinuous equivalent essential spectrum Example Exercise exists a sequence follows Fourier transform Fredholm function f Hilbert space Hilbert-Schmidt theorem Holder inequality implies inner product integral equation interval kernel Laplace operator Laplacian last relation Lemma Let us consider let us define linear functional linear operator non-trivial normed space obtain Obviously orthogonal orthonormal permutable polynomial positive operator projection operator properties Remark self-adjoint operator sequence xn singular solution space H Sturm-Liouville problem Suppose symmetric theorem 16 uniformly unique vector verify weak derivatives weak solution weakly convergent Weyl sequence zero