## 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. |

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### Contents

Compact selfadjoint operators | 25 |

Some examples and applications | 53 |

Integral representation | 75 |

Spectral decomposition of the Laplacian | 101 |

Further spectral properties | 125 |

Eigenvalues of the Laplacian | 149 |

Some applications to hydrodynamics | 163 |

Bibliography | 191 |

### Common terms and phrases

a e H arbitrary Banach space boundary conditions boundary value problem bounded domain bounded set 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 essential spectrum Example Exercise follows Fourier transform function u(x Hilbert space Hilbert-Schmidt theorem Hölder Hölder inequality implies inequality inner product integral equation interval kernel L2 Q Laplace operator Laplacian last relation Lemma Let us consider let us define linear continuous linear functional linear operator non-trivial normed space obtain orthogonal orthonormal permutable polynomial positive operator projection operator properties self-adjoint operator singular solution space H Sturm-Liouville problem Suppose theorem 16 uniformly unique vector verify weak derivatives weak solution weakly convergent Weyl sequence zero