Spectral theory has for its origin the classical theory of finite matrices and, in a sense, includes all theories which may be specialized to some particular phase of that classical theory. Such a meaning for the term spectral theory is perhaps too broad, for its acceptance would force one, as is becoming increasingly evident, to admit that spectral theory embraces not a small part of such fiel...

We study spectral properties of a wide class differential operators with frozen arguments by putting them into general framework rank-one perturbation theory. In particular, we give complete characterization possible eigenvalues for these and solve the inverse problem reconstructing from resulting spectrum. This approach provides unified treatment several recent studies gives clear explanation ...

In many applications it is important to understand the spectral properties of a linear operator T : X → X, where X is some vector space over IR or I C. In the finite dimensional (complex) case linear operators may be characterised as matrices and the Jordan normal form theorem applies, providing a basis of generalised eigenvectors. If, in addition, T is normal (i.e. T and T ∗ commute) with resp...

These are notes from two lectures given in MATH 27200, Basic Functional Analysis, at the University of Chicago in March 2010. The proof of the spectral theorem for compact operators comes from [Zim90, Chapter 3]. 1. The Spectral Theorem for Compact Operators The idea of the proof of the spectral theorem for compact self-adjoint operators on a Hilbert space is very similar to the finite-dimensio...

Counting lattice points inside a ball of large radius in Euclidean space is classical problem analytic number theory, dating back to Gauss. We propose variation on this problem: studying the asymptotics measure an integer affine planes ball. The first term volume ball; we study size remainder term. While equivalent counting eigenvalues Laplace operator torus, our corresponds integrated density ...

One of the central questions in game theory deals with predicting the behavior of an agent. Here, we study the inverse of this problem: given the agents’ equilibrium behavior, what are possible utilities that motivate this behavior? We consider this problem in arbitrary normal-form games in which the utilities can be represented by a small number of parameters, such as in graphical, congestion,...

The equations that define the Lax pairs for generalized principal chiral models can be solved for any nondegenerate bilinear form on su(2). The solution is dependent on one free variable that can serve as the spectral parameter. 1 Formulation of the generalized chiral models Principal chiral models are important example of relativistically invariant field theory integrable by the inverse scatte...

Spectral Graph Theory is the study of the spectra of certain matrices defined from a given graph, including the adjacency matrix, the Laplacian matrix and other related matrices. Graph spectra have been studied extensively for more than fifty years. In the last fifteen years, interest has developed in the study of generalized Laplacian matrices of a graph, that is, real symmetric matrices with ...