We give a condition of essential self-adjointness for magnetic Schrödinger operators on non-compact Riemannian manifolds with a given positive smooth measure which is fixed independently of the metric. This condition is related to the classical completeness of a related classical hamiltonian without magnetic field. The main result generalizes the result by I. Oleinik [46, 47, 48], a shorter and...

Let M be a complete Riemannian manifold and D : C∞ 0 (E) → C∞ 0 (F ) a first order differential operator acting between sections of the hermitian vector bundles E, F . Moreover, let V : C∞(E) → L∞ loc (E) be a self–adjoint zero order differential operator. We give a sufficient condition for the Schrödinger operator H = DD + V to be essentially self–adjoint. This generalizes recent work of I. Ol...

We specify the structure of completely positive operators and quantum Markov semigroup generators that are symmetric with respect to a family inner products, also providing new information on order extreme points in some previously studied cases.

We study representations of the enveloping algebra of a Lie group G which are induced by representations of a Lie subgroup H, assuming that G=H is reductive. Such representations describe the superselection sectors of a quantum particle moving on G=H. It is found that the representatives of both the generators and the quadratic Casimir operators of G have a natural geometric realization in term...

The time reversal of a completely-positive, nonequilibrium discrete-time quantum Markov evolution is derived via a suitable adjointness relation. Space-time harmonic processes are introduced for the forward and reverse-time transition mechanisms, and their role for relative entropy dynamics is discussed.

We give some characterizations of self-adjointness and symmetricity of operator monotone functions by using the Barbour transform f → t+f 1+f and show that there are many non-symmetric operator means between the harmonic mean ! and the arithmetic mean ∇. Indeed, we show that there exists a non-symmetric operator mean between any two symmetric operator means.

We analyze the discretization of nonlinear parabolic equations in Hilbert spaces by both implicit and implicit–explicit multistep methods and establish local stability under best possible and best possible linear stability conditions, respectively. Our approach is based on suitable quantifications of the non-self-adjointness of linear elliptic operators and a discrete perturbation argument.

Recent work in the literature has studied fourth-order elliptic operators on manifolds with boundary. This paper proves that, in the case of the squared Laplace operator, the boundary conditions which require that the eigenfunctions and their normal derivative should vanish at the boundary lead to self-adjointness of the boundary-value problem. On studying, for simplicity, the squared Laplace o...