Self-adjoint extension schemes and modern applications to quantum Hamiltonians

Abstract

This monograph contains revised and enlarged materials from previous lecture notes of undergraduate and graduate courses and seminars delivered by both authors over the last years on a subject that is central both in abstract operator theory and in applications to quantum mechanics: to decide whether a given densely defined and symmetric operator on Hilbert space admits a unique self-adjoint realisation, namely its operator closure, or whether instead it admits an infinite multiplicity of distinct self-adjoint extensions, and in the latter case to classify them and characterise their main features (operator and quadratic form domains, spectrum, etc.) This is at the same time a very classical, well established field, corresponding to the first part of the monograph, and a territory of novel, modern applications, a selection of which, obviously subjective to some extent, but also driven by a pedagogical criterion, is presented in depth in the second part. A number of models are discussed, which are receiving today new or renewed interest in mathematical physics, in particular from the point of view of realising certain operators of interests self-adjointly, classifying their self-adjoint extensions as actual quantum Hamiltonians, studying their spectral and scattering properties, and the like, but also from the point of view of intermediate technical questions that have theoretical interest per se, such as characterising the corresponding operator closures and adjoints.