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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.

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The book is a graduate text on unbounded self-adjoint operators on Hilbert space and their spectral theory with the emphasis on applications in mathematical physics (especially, Schrödinger  operators) and analysis (Dirichlet and Neumann Laplacians, Sturm-Liouville operators, Hamburger moment problem) . Among others, a number of advanced special topics   are treated on a text book level  accompanied by numerous illustrating examples and exercises. The main themes of the book are the following: - Spectral integrals and  spectral decompositions of self-adjoint and normal operators - Perturbations of self-adjointness and of spectra of self-adjoint operators - Forms and operators - Self-adjoint extension theory :boundary triplets, Krein-Birman-Vishik theory of positive self-adjoint extension.

Selfadjoint operators. --- Hilbert space. --- Banach spaces --- Hyperspace --- Inner product spaces --- Operators, Selfadjoint --- Self-adjoint operators --- Linear operators --- Selfadjoint operators --- Hilbert space --- Operator theory --- Spectral theory (Mathematics) --- Opérateurs auto-adjoints --- Hilbert, Espaces de --- Opérateurs, Théorie des --- Théorie spectrale (mathématiques) --- Functional analysis. --- Mathematical physics. --- Operator theory. --- Functional Analysis. --- Mathematical Methods in Physics. --- Operator Theory. --- Mathematical Physics. --- Theoretical, Mathematical and Computational Physics. --- Functional analysis --- Physical mathematics --- Physics --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Mathematics --- Opérateurs auto-adjoints. --- Hilbert, Espaces de. --- Opérateurs, Théorie des. --- Mathematics. --- Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Opérateurs auto-adjoints. --- Opérateurs, Théorie des. --- Théorie spectrale (mathématiques)