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The study of arithmetic differential operators is a novel and promising area of mathematics. This complete introduction to the subject starts with the basics: a discussion of p-adic numbers and some of the classical differential analysis on the field of p-adic numbers leading to the definition of arithmetic differential operators on this field. Buium's theory of arithmetic jet spaces is then developed succinctly in order to define arithmetic operators in general. Features of the book include a comparison of the behaviour of these operators over the p-adic integers and their behaviour over the unramified completion, and a discussion of the relationship between characteristic functions of p-adic discs and arithmetic differential operators that disappears as soon as a single root of unity is adjoined to the p-adic integers. This book is essential reading for researchers and graduate students who want a first introduction to arithmetic differential operators over the p-adic integers.

Differential operators. --- Arithmetic functions. --- p-adic numbers. --- Numbers, p-adic --- Number theory --- p-adic analysis --- Functions, Arithmetic --- Functions of complex variables --- Operators, Differential --- Differential equations --- Operator theory

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Periodic differential operators have a rich mathematical theory as well as important physical applications. They have been the subject of intensive development for over a century and remain a fertile research area. This book lays out the theoretical foundations and then moves on to give a coherent account of more recent results, relating in particular to the eigenvalue and spectral theory of the Hill and Dirac equations. The book will be valuable to advanced students and academics both for general reference and as an introduction to active research topics.

Mathematics. --- Operator algebras. --- Singularities (Mathematics). --- Differential operators --- Differential equations --- Operator theory --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Differential operators. --- Operators, Differential --- Operator theory. --- Differential equations. --- Ordinary Differential Equations. --- Operator Theory. --- 517.91 Differential equations --- Functional analysis --- Math --- Science --- Differential Equations.

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This is the first monograph devoted to the Sturm oscillatory theory for infinite systems of differential equations and its relations with the spectral theory. It aims to study a theory of self-adjoint problems for such systems, based on an elegant method of binary relations. Another topic investigated in the book is the behavior of discrete eigenvalues which appear in spectral gaps of the Hill operator and almost periodic Schrödinger operators due to local perturbations of the potential (e.g., modeling impurities in crystals). The book is based on results that have not been presented in other

Spectral theory (Mathematics) --- Differential operators. --- Selfadjoint operators. --- Hilbert space. --- Operator theory. --- Functional analysis --- Banach spaces --- Hyperspace --- Inner product spaces --- Operators, Selfadjoint --- Self-adjoint operators --- Linear operators --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Operators, Differential --- Differential equations --- Operator theory

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This book provides a comprehensive introduction to modern global variational theory on fibred spaces. It is based on differentiation and integration theory of differential forms on smooth manifolds, and on the concepts of global analysis and geometry such as jet prolongations of manifolds, mappings, and Lie groups. The book will be invaluable for researchers and PhD students in differential geometry, global analysis, differential equations on manifolds, and mathematical physics, and for the readers who wish to undertake further rigorous study in this broad interdisciplinary field.

Operator theory --- Banach spaces. --- Interpolation. --- Function spaces. --- Differential operators. --- Banach spaces --- Interpolation --- Function spaces --- Differential operators --- 517.982 --- 517.982 Linear spaces with topology and order or other structures --- Linear spaces with topology and order or other structures --- Approximation theory --- Numerical analysis --- Spaces, Function --- Functional analysis --- Operators, Differential --- Differential equations --- Functions of complex variables --- Generalized spaces --- Topology

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Operator theory --- Linear operators --- 517.98 --- Hilbert space --- #TCPW W8.0 --- Linear maps --- Maps, Linear --- Operators, Linear --- Banach spaces --- Hyperspace --- Inner product spaces --- Functional analysis and operator theory --- Linear operators. --- Hilbert space. --- 517.98 Functional analysis and operator theory --- Operators(Self Adjoint-) --- Scattering theory --- Operators(Differential-) --- Operators in hilbert space(Linear-)