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Nonselfadjoint operators --- Congresses

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Algebras, Linear --- Nonselfadjoint operators --- Spectral theory (Mathematics) --- Algèbre linéaire --- Opérateurs non auto-adjoints --- Spectre (Mathématiques)

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This monograph provides a comprehensive treatment of expansion theorems for regular systems of first order differential equations and n-th order ordinary differential equations.In 10 chapters and one appendix, it provides a comprehensive treatment from abstract foundations to applications in physics and engineering. The focus is on non-self-adjoint problems. Bounded operators are associated to these problems, and Chapter 1 provides an in depth investigation of eigenfunctions and associated functions for bounded Fredholm valued operators in Banach spaces. Since every n-th orde

Boundary value problems. --- Nonselfadjoint operators. --- Eigenvalues. --- Differential equations. --- 517.91 Differential equations --- Differential equations --- Matrices --- Non-self-adjoint operators --- Operators, Non-self-adjoint --- Operators, Nonselfadjoint --- Linear operators --- Boundary conditions (Differential equations) --- Functions of complex variables --- Mathematical physics --- Initial value problems

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This book is devoted to the study of pseudo-di?erential operators, with special emphasis on non-selfadjoint operators, a priori estimates and localization in the phase space. We have tried here to expose the most recent developments of the theory with its applications to local solvability and semi-classical estimates for non-selfadjoint operators. The?rstchapter,Basic Notions of Phase Space Analysis,isintroductoryand gives a presentation of very classical classes of pseudo-di?erential operators, along with some basic properties. As an illustration of the power of these methods, we give a proof of propagation of singularities for real-principal type operators (using aprioriestimates,andnotFourierintegraloperators),andweintroducethereader to local solvability problems. That chapter should be useful for a reader, say at the graduate level in analysis, eager to learn some basics on pseudo-di?erential operators. The second chapter, Metrics on the Phase Space begins with a review of symplectic algebra, Wigner functions, quantization formulas, metaplectic group and is intended to set the basic study of the phase space. We move forward to the more general setting of metrics on the phase space, following essentially the basic assumptions of L. H¨ ormander (Chapter 18 in the book [73]) on this topic.

Metric spaces. --- Nonselfadjoint operators. --- Pseudodifferential operators. --- Pseudodifferential operators --- Nonselfadjoint operators --- Metric spaces --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Applied Mathematics --- Geometry --- Calculus --- Phase space (Statistical physics) --- Operators, Pseudodifferential --- Pseudo-differential operators --- Space, Phase (Statistical physics) --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Analysis. --- Operator theory --- Generalized spaces --- Global analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- 517.1 Mathematical analysis --- Mathematical analysis

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The asymptotic distribution of eigenvalues of self-adjoint differential operators in the high-energy limit, or the semi-classical limit, is a classical subject going back to H. Weyl of more than a century ago. In the last decades there has been a renewed interest in non-self-adjoint differential operators which have many subtle properties such as instability under small perturbations. Quite remarkably, when adding small random perturbations to such operators, the eigenvalues tend to distribute according to Weyl's law (quite differently from the distribution for the unperturbed operators in analytic cases). A first result in this direction was obtained by M. Hager in her thesis of 2005. Since then, further general results have been obtained, which are the main subject of the present book. Additional themes from the theory of non-self-adjoint operators are also treated. The methods are very much based on microlocal analysis and especially on pseudodifferential operators. The reader will find a broad field with plenty of open problems.

Nonselfadjoint operators. --- Spectral theory (Mathematics) --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Non-self-adjoint operators --- Operators, Non-self-adjoint --- Operators, Nonselfadjoint --- Linear operators --- Functions of complex variables. --- Differential equations, partial. --- Differential Equations. --- Operator theory. --- Functions of a Complex Variable. --- Several Complex Variables and Analytic Spaces. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Operator Theory. --- 517.91 Differential equations --- Differential equations --- Partial differential equations --- Complex variables --- Elliptic functions --- Functions of real variables --- Differential equations. --- Partial differential equations.