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...

Let S be the orthogonal sum of infinitely many pairwise unitarily equivalent symmetric operators with non-zero deficiency indices. Let J be an open subset of R. If there exists a self-adjoint extension S0 of S such that J is contained in the resolvent set of S0 and the associated Weyl function of the pair {S, S0} is monotone with respect to J , then for any self-adjoint operator R there exists ...

In this paper, we examine the semiclassical behavior of scattering data a non-self-adjoint Dirac operator with fairly smooth—but not necessarily analytic—potential decaying at infinity. particular, using ideas and methods from work Langer Olver [Philos. Trans. R. Soc. London, Ser. A 278(1279), 137–174 (1975)], provide rigorous analysis coefficients, Bohr–Sommerfeld condition for location eigenv...

We study weighted composition operators on Hilbert spaces of analytic functions on the unit ball with kernels of the form (1− 〈z, w〉)−γ for γ > 0. We find necessary and sufficient conditions for the adjoint of a weighted composition operator to be a weighted composition operator or the inverse of a weighted composition operator. We then obtain characterizations of self-adjoint and unitary weigh...

We develop the general a priori error analysis of residual-free bubble nite element approximations to non-self-adjoint elliptic problems of the form ("A+C)u = f subject to homogeneous Dirichlet boundary condition, where A is a symmetric second-order elliptic operator, C is a skew-symmetric rst-order diierential operator, and " is a positive parameter. Optimal-order error bounds are derived in v...

We characterize possible spectra of rank-one perturbations B a self-adjoint operator A with discrete spectrum and, in particular, prove that the may include any number real or non-real eigenvalues arbitrary algebraic multiplicity

1. Cautionary example 2. Criterion for essential self-adjointness 3. Examples of essentially self-adjoint operators 4. Appendix: Friedrichs' canonical self-adjoint extensions 5. The following has been well understood for 70-120 years, or longer, naturally not in contemporary terminology. The differential operator T = d 2 dx 2 on L 2 [a, b] or L 2 (R) is a prototypical natural unbounded operator...

A self-adjoint operator A in a Krĕın space (K, [ · , · ]) is called partially fundamentally reducible if there exist a fundamental decomposition K = K+[+̇]K− (which does not reduce A) and densely defined symmetric operators S+ and S− in the Hilbert spaces (K+, [ · , · ]) and (K−,−[ · , · ]), respectively, such that each S+ and S− has defect numbers (1, 1) and the operator A is a self-adjoint ext...

Abstract In this paper we study the almost everywhere convergence of the expansions related to the self-adjoint extension of the Laplace operator. The sufficient conditions for summability is obtained. For the orders of Riesz means, which greater than critical index N−1 2 we established the estimation for maximal operator of the Riesz means. Note that when order α of Riesz means is less than cr...