In this note we prove that the maximally defined operator associated with the Dirac-type differential expression M(Q) = i ( d dx Im −Q −Q − d dx Im ) , where Q represents a symmetric m × m matrix (i.e., Q(x) = Q(x) a.e.) with entries in L loc (R), is J -self-adjoint, where J is the antilinear conjugation defined by J = σ1C, σ1 = ( 0 Im Im 0 ) and C(a1, . . . , am, b1, . . . , bm) = (a1, . . . ,...

The Glazman-Povzner-Wienholtz theorem states that the completeness of a manifold, when combined with semiboundedness Schr\"odinger operator $-\Delta + q$ and suitable local regularity assumptions on $q$, guarantees its essential self-adjointness. Our aim is to extend this result operators graphs. We first obtain corresponding for metric graphs, allowing in particular distributional potentials $...

For 20th century mathematicians, the role of Cantor's sets has been that of the ideally featureless canvases on which all needed algebraic and geometrical structures can be painted. (Certain passages in Cantor's writings refer to this role.) Clearly, the resulting contradiction, 'the points of such sets are distinct yet indistinguishable', should not lead to inconsistency. Indeed, the productiv...

We provide a shorter and more transparent proof of a result by I. Oleinik [27, 28, 29]. It gives a sufficient condition of the essential self-adjointness of a Schrödinger operator on a non-compact Riemannian manifold with a locally bounded potential in terms of the completeness of the dynamics for a related classical system. The simplification of the proof given by I. Oleinik is achieved by an ...

We present new family of exact analytic solutions for three anyons in a harmonic potential (or in free space) in terms of generalized harmonics on S 3 , which supplement the known solutions. The new solutions satisfy the hard-core condition when α = 1 3 , 1 (α being the statistical parameter) but otherwise, have finite non-vanishing two-particle colliding probability density , which is consiste...

In this paper we consider a second order, Sturm-Liouville-type boundary-value operator of the form Lu := −[pu∇]∆ + qu, on an arbitrary, bounded time-scale T, for suitable functions p, q, together with suitable boundary conditions. We show that, with a suitable choice of domain, this operator can be formulated in the Hilbert space L(Tκ), in such a way that the resulting operator is self-adjoint,...

It is widely appreciated that the iterative solution of linear systems of equations with large sparse matrices is much easier when the matrix is symmetric. It is equally advantageous to employ symmetric iterative methods when a nonsymmetric matrix is self-adjoint in a non-standard inner product. Here, general conditions for such self-adjointness are considered. In particular, a number of known ...

We give sufficient conditions for the essential self-adjointness of perturbed biharmonic operators acting on sections a Hermitian vector bundle over Riemannian manifold with additional assumptions, such as lower semi-bounded Ricci curvature or bounded sectional curvature. In case curvature, we formulate our results in terms completeness metric that is conformal to original one, via factor depen...