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...

The non self-adjointness of the radial momentum operator has been noted before by several authors, but the various proofs are incorrect. We give a rigorous proof that the n-dimensional radial momentum operator is not self-adjoint and has no selfadjoint extensions. The main idea of the proof is to show that this operator is unitarily equivalent to the momentum operator on L[(0,∞), dr] which is n...

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...

We study positivity, localization of binding and essential self-adjointness properties of a class of Schrödinger operators with many anisotropic inverse square singularities, including the case of multiple dipole potentials.

We extend Krupnik’s criterion of self-adjointness of the Cauchy singular integral operator to the case of finitely connected domains. The main aim of the paper is to present a new approach for proof of the criterion. Let G+ be a finitely connected domain bounded by the rectifiable curve C = ∂G+, G− = C \ clos G+ and ∞ ∈ G−. Suppose also that w(z), z ∈ C is a nonnegative weight such that w(z) 6≡...

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...

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.