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...

Strong and Markov uniqueness problems in L 2 for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on a{priori estimates is used. The extension of the problem to the L p-setting is discussed. As a direct application essential self{ adjointness and strong uniqueness in L p is proved for the generator (with initial domain the bounded smooth cylinder functions) o...

We introduce here a generalization of the modified Bernstein polynomials for Jacobi weights using the q-Bernstein basis proposed by G.M. Phillips to generalize classical Bernstein Polynomials. The function is evaluated at points which are in geometric progression in ]0, 1[. Numerous properties of the modified Bernstein Polynomials are extended to their q-analogues: simultaneous approximation, p...

The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition of P T symmetry, one obtains new infinite classes of complex Hamiltonians whose spectra are also real and positive. These P T symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase spa...

A linear operator on a Hilbert space , in the classical approach of von Neumann, must be symmetric to guarantee self-adjointness. However, it can shown that symmetry could omitted by using criterion for graph and adjoint graph. Namely, S is densely defined closed if only . In more general setup, we consider relations instead operators prove this situation similar result holds. We give necessary...

We show that the particle motion in Bohmian mechanics, given by the solution of an ordinary differential equation, exists globally: For a large class of potentials the singularities of the velocity field and infinity will not be reached in finite time for typical initial values. A substantial part of the analysis is based on the probabilistic significance of the quantum flux. We elucidate the c...

In manifolds with spatial boundary, BRST formalism can be used to quantize gauge theories. We show that, in a U(1) gauge theory, only a subset of all the boundary conditions allowed by the self-adjointness of the Hamiltonian preserves BRST symmetry. Hence, the theory can be quantized using BRST formalism only when that subset of boundary conditions is considered. We also show that for such boun...