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 compact boundary. Each of these quadratic forms specifies a semibounded 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 ✩ 635 Boundary conditions the literature and various examples and applications are discussed.

The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators

“…Before making explicit the previous structures in concrete examples we notice that the previous discussion works in a similar way with the covariant Laplacian ∆ A discussed in Section 3. Thus if we are given a group acting by unitary bundle isomorphisms on an Hermitean bundle E → Ω (and by isometric diffeomorphisms on the Riemannian manifold Ω), then any unitary operator U at the boundary, (that in addition satisfies the conditions of possessing gap and being admissible, [Ib14c], that guarantee that the quadratic form constructed from the operator ∇ with boundary conditions dictated by U , read more about self-adjoint extensions determined by quadratic forms in [Ib14] and [Ib15] this volume), and that verifies the commutation relations of Theorem 19 describes a G-invariant quadratic form. The closure of this quadratic form characterizes uniquely a G-invariant self-adjoint extension of the Laplace-Beltrami operator.…”

“…We will just quote the analysis of non-local extensions of elliptic operators by Grubb [Gr68] and the theory of singular perturbations of differential operators by Albeverio and Kurasov [Al99] because of their influence on this work (see also [Ib14] for a quadratic forms based analysis of the extensions of the Laplace-Beltrami operator and [Ib12] where the reader will find a reasonable list of references on the subject).…”

On the theory of self-adjoint extensions of symmetric operators and its applications to quantum physics

“…Unfortunately, von Neumann's theorem is not suitable to perform explicit calculations. We are going to use the approach introduced in [AIM05] and further developed in [ILPP13].…”

“…[Koc75, AIM05,ILPP13], that maximally isotropic subspaces of a bilinear form like the one at the right hand side are determined uniquely by unitaries U :…”

“…An application of such ideas to control entangled states will be discussed. Lecture II will cover the recent approach developed in [ILPP13] to the theory of self-adjoint extensions of the Laplace-Beltrami operator from the point of view of their corresponding quadratic forms. New results on the theory as well as the introduction of the notion of admissible unitary operators at the boundary will be discussed.…”