The Dirac equation for a massive spin-12 field in a central potential V in three dimensions is studied without fixing a priori the functional form of V . The second-order equations for the radial parts of the spinor wave function are shown to involve a squared Dirac operator for the free case, whose essential self-adjointness is proved by using the Weyl limit point-limit circle criterion, and a...

In this paper we consider linear ordinary elliptic differential operators with smooth coefficients on finite quantum graphs. We discuss criteria for the operator to be self-adjoint. This involves conditions on matrices representative of the boundary conditions at each vertex. The main point is the development of a Maple application to test these conditions.

In this article, linear operators satisfying anti-commutation relations are considered. It is proven that an anti-commutative type of the Glimm-Jaffe-Nelson commutator theorem follows.

Let M be a complete Riemannian manifold and let Ω•(M) denote the space of differential forms on M . Let d : Ω(M) → Ω(M) be the exterior differential operator and let ∆ = dd + dd be the Laplacian. We establish a sufficient condition for the Schrödinger operator H = ∆ + V (x) (where the potential V (x) : Ω(M) → Ω(M) is a zero order differential operator) to be self-adjoint. Our result generalizes...

Yang–Mills theory in four dimensions is studied by using the Coulomb gauge. The Coulomb gauge Hamiltonian involves integration of matrix elements of an operator P built from the Laplacian and from a first-order differential operator. The operator P is studied from the point of view of spectral theory of pseudo-differential operators on compact Riemannian manifolds, both when self-adjointness ho...

Our main result is a local limit law for the empirical spectral distribution of the anticommutator of independent Wigner matrices, modeled on the local semicircle law. Our approach is to adapt some techniques from one of the recent papers of Erdös-Yau-Yin. We also use an algebraic description of the law of the anticommutator of free semicircular variables due to Nica-Speicher, a self-adjointnes...

(1.1) i∂tψ(t) = Hψ(t) . For all initial conditions ψ(0) ∈ H, the unique solution is given by ψ(t) = e−itHψ(0), for all t ∈ R. Due to Stone’s Theorem e−itH is a strongly continuous one-parameter group of unitary operators on H, and the self-adjointness of H is the necessary and sufficient condition for the existence of a unique continuous solution for all times. An alternative description of thi...