Geometric and computational spectral theory : Seminaire de Mathematiques Superieures, June 15-26, 2015, Centre de Recherches Mathematiques, Université de Montréal, Montréal, Québec, Canada / Alexandre Girouard [and five others], editors.

Format:

Book

Contributor:

Girouard, Alexandre, 1976- editor.

Series:

Contemporary mathematics (American Mathematical Society). 0271-4132 700

Contemporary mathematics, 700 0271-4132

Language:

English

Subjects (All):

Metric spaces--Congresses.

Metric spaces.

Spectral geometry--Congresses.

Spectral geometry.

Geometry, Differential--Congresses.

Geometry, Differential.

Physical Description:

1 online resource (287 pages) : illustrations (some color).

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island ; Montreal, Quebec : American Mathematical Society : Centre de Recherches Mathematiques, 2017.

Summary:

The book is a collection of lecture notes and survey papers based on the mini-courses given by leading experts at the 2015 Séminaire de Mathématiques Supérieures on Geometric and Computational Spectral Theory, held from June 15-26, 2015, at the Centre de Recherches Mathématiques, Université de Montréal, Montréal, Quebec, Canada. The volume covers a broad variety of topics in spectral theory, highlighting its connections to differential geometry, mathematical physics and numerical analysis, bringing together the theoretical and computational approaches to spectral theory, and emphasizing the interplay between the two.

Contents:

Cover

Title page

Contents

Preface

The spectrum of the Laplacian: A geometric approach

1. Introduction, basic results and examples

2. Variational characterization of the spectrum and simple applications

3. Lower bounds for the first nonzero eigenvalue

4. Estimates on the conformal class

5. Another geometric method to construct upper bounds and applications.

6. The conformal spectrum

Acknowledgments

References

An elementary introduction to quantum graphs

1. Introduction

2. Schrödinger equation on a metric graph

3. Dirichlet condition

4. Interlacing inequalities

5. Secular determinant

6. Symmetry and isospectrality

7. Magnetic Schrödinger operator and nodal count

8. Concluding remarks

A free boundary approach to the Faber-Krahn inequality

2. Setting the variational framework

3. Proof of the Faber-Krahn inequality

4. Further remarks: higher order eigenvalues

Acknowledgements

Some nodal properties of the quantum harmonic oscillator and other Schrödinger operators in ℝ²

1. Introduction and main results

2. A reminder on Hermite polynomials

3. Stern-like constructions for the harmonic oscillator in the case -odd

4. Proof of Theorem 1.3

5. Proof of Theorem 1.4

6. Eigenfunctions with "many" nodal domains, proof of Theorem 1.6

7. On bounds for the length of the nodal set

Numerical solution of linear eigenvalue problems

2. Background in Numerical Linear Algebra

3. Small to moderate-sized matrices

4. Large and sparse matrices

5. Conclusions

Finite element methods for variational eigenvalue problems

2. Problem setting

3. Galerkin approximation

4. The finite element method

5. Saddle point problems.

Appendix A. Programs for experiments

Appendix B. Selected problems

Computation of eigenvalues, spectral zeta functions and zeta-determinants on hyperbolic surfaces

1. The method of particular solutions

2. The Method of Particular Solutions in a Geometric Context

3. Hyperbolic Surfaces and Teichmüller Space

4. The Method of Particular Solutions for Hyperbolic Surfaces

5. Heat Kernels, Spectral Asymptotics, and Zeta functions

6. The Selberg Trace Formula

7. Completeness of a Set of Eigenvalues

Scales, blow-up and quasimode constructions

2. A short introduction to manifolds with corners and resolutions

3. Generalities on quasimode constructions

the main steps

4. Regular perturbations

5. Adiabatic limit with constant fibre eigenvalue

6. Adiabatic limit with variable fibre eigenvalue

7. Adiabatic limit with ends

8. Summary of the quasimodes constructions

Scattering for the geodesic flow on surfaces with boundary

2. Geometric background

3. Scattering map, length function and -ray transform

4. Resolvents and boundary value problems for transport equations

5. Injectivity of X-ray transform for tensors

6. Some references

Acknowledgement

Back Cover.

Notes:

Includes bibliographical references at the end of each chapters.

Description based on print version record.

ISBN:

1-4704-4258-2