Representation of non-semibounded quadratic forms and orthogonal additivity

Abstract

A representation theorem for non-semibounded Hermitian quadratic forms in terms of a (non-semibounded) self-adjoint operator is proven. The main assumptions are closability of the Hermitian quadratic form, the direct integral structure of the underlying Hilbert space and orthogonal additivity. We apply this result to several examples, including the position operator in quantum mechanics and quadratic forms invariant under a unitary representation of a separable locally compact group. The case of invariance under a compact group is also discussed in detail.

Substantial changes with respect to v1. Introduction of the notion of strong representability and substantial simplification of the proofs in Section 4. To appear in J. Math. Anal. Appl

References in corpus (6)

• Spectral gaps and discrete magnetic Laplacians
• 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
• On the theory of self-adjoint extensions of symmetric operators and its applications to Quantum Physics

Cited by in corpus (2)

• Quantum controllability on graph-like manifolds through magnetic potentials and boundary conditions
• Global approximate controllability of quantum systems by form perturbations and applications