On Self-Adjoint Extensions and Symmetries in Quantum Mechanics

Abstract

Given a unitary representation of a Lie group G on a Hilbert space H, we develop the theory of G-invariant self-adjoint extensions of symmetric operators both using von Neumann's theorem and the theory of quadratic forms. We also analyze the relation between the reduction theory of the unitary representation and the reduction of the G-invariant unbounded operator. We also prove a G-invariant version of the representation theorem for quadratic forms.The previous results are applied to the study of G-invariant self-adjoint extensions of the Laplace-Beltrami operator on a smooth Riemannian manifold with boundary on which the group G acts. These extensions are labeled by admissible unitaries U acting on the L 2 -space at the boundary and having spectral gap at −1. It is shown that if the unitary representation V of the symmetry group G is traceable, then the self-adjoint extension of the Laplace-Beltrami operator determined by U is G-invariant if U and V commute at the boundary. Various significant examples are discussed at the end.

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

“…It is not difficult to prove the following theorem: (see the proof in the case of Laplace operators in [Ib14c]). …”

“…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.…”

“…then V defines also a continuous unitary representation on H 1 (Ω) with its corresponding Sobolev scalar product (see for instance [Ib14c]). Now we can extend the property of equivariant Dirac operators expressed by Eq.…”

“…This problem will be stated and partially solved (see also [Ib14c]) and some examples will be exhibited. Particular attention will be paid to the space of self-adjoint extensions of the quotient Dirac operator of a Dirac operator with symmetry and its characterization as the symplectic manifold of fixed points of the self-adjoint Grassmannian.…”

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

“…In what follows we will motivate the definition of G-invariant symmetric operators and show what role plays the G-invariance in the characterization of self-adjoint extensions. For further details we refer to [ILPP14].…”