Singular Perturbations of Differential Operators: Solvable Schrödinger-type Operators (London Mathematical Society Lecture Note Series, Series Number 271)

Differential and more general self-adjoint operators involving singular interactions arise naturally in a range of subjects such as classical and quantum physics, chemistry, and electronics. This book is a systematic mathematical study of these operators, with particular emphasis on spectral and scattering problems. The methods discussed are based on a new concept of symplectic structure of the boundary form. Suitable for researchers in analysis or mathematical physics, this volume could also be used as a text for an advanced course on the applications of analysis. Chapter 2 Generalized rank one perturbations2.1 Krein's formula for the generalized resolvents; 2.1.1 Generalized Self-adjoint extensions and generalized resolvents; 2.1.2 Generalized rank one perturbations; 2.2 Model generalized perturbations; 2.2.1 Introduction; 2.2.2 Model perturbations I: densely defined restricted operators; 2.2.3 Model perturbations II: non-densely defined construction; 2.2.4 Generalized resolvents and model generalized perturbations; 2.3 Generalized point interactions for differential operators; 2.3.1 Generalized point interaction in dimension three 1.3.1 Form bounded rank one singular perturbations1.3.2 Family of rank one form unbounded perturbations; 1.3.3 Singular rank one form unbounded perturbations of homogeneous operators; 1.3.4 Resolvent formulas; 1.4 Approximations of singular rank one perturbations; 1.4.1 Norm convergence of the approximations; 1.4.2 Strong resolvent convergence of the approximations; 1.5 Differential operators with rank one singular perturbations; 1.5.1 Point interactions in dimension three; 1.5.2 Perturbations of the first derivative operator; 1.5.3 Dirac operator with a pseudopotential 2.3.2 Generalized delta interaction in dimension oneChapter 3 Finite rank perturbations and distribution theory; 3.1 Finite rank perturbations; 3.1.1 Preliminaries; 3.1.2 Form bounded finite rank perturbations; 3.1.3 Form unbounded finite rank perturbations; 3.1.4 Generalized finite rank perturbations; 3.2 Point interactions for differential operators and distribution theory; 3.2.1 Point interactions for differential operators as finite rank perturbations; 3.2.2 Distribution theory for discontinuous test functions; 3.2.3 Differential operator of order n in one dimension 3.2.4 Second order differential operator in one dimensionChapter 4 Scattering theory for finite rank perturbations; 4.1 Scattering theory for rank one perturbations; 4.1.1 Rank one perturbations and operators with a simple spectrum; 4.1.2 Invariance of the absolutely continuous spectrum; 4.1.3 Wave operators and scattering operator for rank one perturbations; 4.2 Scattering theory for Self-adjoint extensions; 4.2.1 Wave operators for self-adjoint extensions; 4.2.2 Scattering operator for self-adjoint extensions; 4.2.3 Scattering matrix for Self-adjoint extensions Cover; Series Page; Title; Copyright; Preface; Acknowledgments; Contents; Introduction; Chapter 1 Rank one perturbations; 1.1 Bounded perturbations; 1.1.1 Resolvent analysis; 1.1.2 Infinite coupling; 1.2 Krein's formula; 1.2.1 Bounded and singular perturbations; 1.2.2 Scale of Hilbert spaces; 1.2.3 Form bounded and form unbounded perturbations; 1.2.4 Rank one perturbations and the extension theory for symmetric operators; 1.2.5 The extension theory and Krein's formula; 1.2.6 Q-function for rank one perturbations; 1.3 Singular rank one perturbations Differential (and more general self-adjoint) operators involving singular interactions arise naturally in a range of topics such as, classical and quantum physics, chemistry, and electronics. This book presents a systematic mathematical study of these operators, with particular emphasis on spectral and scattering problems. Suitable for researchers in analysis or mathematical physics, this book could also be used as a text for an advanced course on the applications of analysis. This is a systematic mathematical study of differential (and more general self-adjoint) operators 4.3 Scattering theory for finite rank perturbations