Heriot-Watt Mathematics Report Series
HWM07-33, 29 Aug 2007

Trends and Tricks in Spectral Theory

L Boulton and M Levitin


Abstract

The spectra of operators, under different names, manifest themselves almost everywhere in our everyday life. The colour of light we see is related to the spectra of atoms and molecules. The tones and overtones of musical instruments we hear are determined by the spectra of strings and drums. The resonances produced by cars running over a bridge are predicted by the spectral analysis of beams and suspension cables.

The spectral analysis of differential operators is an area of research that has been active for over one hundred years. In these lecture notes we have deliberately picked only some particular “trends” in this theory. Many good surveys are available, and it makes no sense even to attempt to compete against them. In this book however we provide a rough overview of the theoretical aspects, and focus mostly on describing techniques and “tricks” that have been successfully used to solve some long standing problems, several of which are among the most important ones in mathematical analysis in the last century.

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