Self-adjoint extensions of the Laplace-Beltrami operator and unitaries at the boundary

Abstract

We construct in this article a class of closed semi-bounded quadratic forms on the space of square integrable functions over a smooth Riemannian manifold with smooth boundary. Each of these quadratic forms specifies a semi-bounded self-adjoint extension of the Laplace-Beltrami operator. These quadratic forms are based on the Lagrange boundary form on the manifold and a family of domains parametrized by a suitable class of unitary operators on the boundary that will be called admissible. The corresponding quadratic forms are semi-bounded below and closable. Finally, the representing operators correspond to semi-bounded self-adjoint extensions of the Laplace-Beltrami operator. This family of extensions is compared with results existing in the literature and various examples and applications are discussed.

References in corpus (7)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• On Self-adjoint extensions and symmetries in Quantum Mechanics
• Hardy inequalities for Robin Laplacians
• Spectral asymptotics for Robin problems with a discontinuous coefficient
• Finite element method to solve the spectral problem for arbitrary self-adjoint extensions of the Laplace-Beltrami operator on manifolds with a boundary
• On the many Dirichlet Laplacians on a non-convex polygon and their approximations by point interactions
• Boundary pairs associated with quadratic forms

Cited by in corpus (24)

• Mode solutions for a Klein-Gordon field in anti-de Sitter spacetime with dynamical boundary conditions of Wentzell type
• On Self-adjoint extensions and symmetries in Quantum Mechanics
• The Topology and Geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators
• Fundamental solutions for the wave operator on static Lorentzian manifolds with timelike boundary
• Edge States at Phase Boundaries and Their Stability
• Spectral enclosures for non-self-adjoint extensions of symmetric operators
• Self-adjoint extensions and unitary operators on the boundary
• Boundary Dynamics Driven Entanglement
• Finite element method to solve the spectral problem for arbitrary self-adjoint extensions of the Laplace-Beltrami operator on manifolds with a boundary
• Dirac-like operators on the Hilbert space of differential forms on manifolds with boundaries
• Quantum Control at the Boundary
• On $\mathbb{Z}$-invariant self-adjoint extensions of the Laplacian on quantum circuits
• Boundaries without boundaries
• Quantum cavities with alternating boundary conditions
• On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain
• Representation of non-semibounded quadratic forms and orthogonal additivity
• Hearing the shape of a quantum boundary condition
• Quantum controllability on graph-like manifolds through magnetic potentials and boundary conditions
• On global approximate controllability of a quantum particle in a box by moving walls
• Quantum magnetic billiards: boundary conditions and gauge transformations
• On a sharper bound on the stability of non-autonomous Schrödinger equations and applications to quantum control
• Global approximate controllability of quantum systems by form perturbations and applications
• Isospectrality and non-locality of generalized Dirac combs
• Quantum Control at the Boundary: an application to quantum circuits