Spectral Theory: Almost Mathieu Operator, Bauer-Fike Theorem, Decomposition of Spectrum (Functional Analysis), Dirac Spectrum, Dirichle

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 43. Chapters: Almost Mathieu operator, Bauer-Fike theorem, Decomposition of spectrum (functional analysis), Dirac spectrum, Dirichlet eigenvalue, Discrete spectrum (physics), Essential spectrum, Hearing the shape of a drum, Heat kernel, Isospectral, Krein-Rutman theorem, Lax pair, Numerical range, Polyakov formula, Projection-valued measure, Proto-value functions, Ramanujan graph, Rayleigh-Faber-Krahn inequality, Rigged Hilbert space, Schur-Horn theorem, Selberg zeta function, Spectral abscissa, Spectral asymmetry, Spectral geometry, Spectral graph theory, Spectral radius, Spectral theorem, Spectral theory of compact operators, Spectral theory of ordinary differential equations, Spectrum of a C*-algebra, Sturm-Liouville theory, Transfer operator, Transform theory, Weyl law. Excerpt: In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl generalized the classical Sturm-Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh-Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups. Spectral theory for second order ordinary differential equations on a compact interval was developed by Jacques Charles Francois Sturm and Joseph Liouville in the nineteenth century and is now known as Sturm-Liouville theory. In modern language it is an application of the spectral theorem for compact operators due to David Hilbert. In his dissertation, published in 1910, Hermann Weyl extended this theory to second order ordinary differential equations with singularities at the endpoints of the interval, now allowed to be infinite or semi-infinite. He simultaneously developed a spectral theory adapted to these special operators and introduced boundary conditions in terms of his celebrated dichotomy between limit points and limit circles. In the 1920s John von Neumann established a general spectral theorem for unbounded self-adjoint operators, which Kunihiko Kodaira used to streamline Weyl's method. Kodaira also generalised Weyl