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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