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.

arXiv commentTypos and minor errors corrected. Version to be published in the Springer Monographs in Mathematics series, 2022

References in corpus (19)

• Universality in Few-body Systems with Large Scattering Length
• The unitary gas in an isotropic harmonic trap: symmetry properties and applications
• Dirac Hamiltonian with superstrong Coulomb field
• Spectral Theory for Perturbed Krein Laplacians in Nonsmooth Domains
• Stability for a System of N Fermions Plus a Different Particle with Zero-Range Interactions
• Domains for Dirac-Coulomb min-max levels
• Self-adjoint extensions and stochastic completeness of the Laplace-Beltrami operator on conic and anticonic surfaces
• The Unitary Gas and its Symmetry Properties
• Self-adjoint extensions of Dirac operators with Coulomb type singularity
• Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei
• Spectral analysis and the Aharonov-Bohm~effect on certain almost-Riemannian manifolds
• Distinguished self-adjoint extensions of Dirac operators via Hardy-Dirac inequalities
• Spectral analysis of the 2+1 fermionic trimer with contact interactions
• Discrete spectra for critical Dirac-Coulomb Hamiltonians
• Energy lower bound for the unitary N+1 fermionic model
• Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling
• On the optimization of the principal eigenvalue for single-centre point-interaction operators in a bounded region
• On the Hamiltonian for three bosons with point interactions
• Closure of the Laplace-Beltrami operator on 2D almost-Riemannian manifolds and semi-Fredholm properties of differential operators on Lie manifolds

Cited by in corpus (6)

• Regularized Zero-Range Hamiltonian for a Bose Gas with an Impurity
• Minimizers of mass-constrained functionals involving a nonattractive point interaction
• Two dimensional NLS ground states with attractive Coulomb potential and point interaction
• On Schrödinger Operators Modified by $\delta$ Interactions
• Completeness Relation in Renormalized Quantum Systems
• Hamiltonians for Quantum Systems with Contact Interactions