arXiv subjects
Publications and source records attributed to Nabila Torki-Hamza.
In this paper we consider the discrete Laplacian acting on 1-forms and we study its spectrum relative to the spectrum of the 0-form Laplacian. We show that the non zero spectrum can coincide for these Laplacians with the same nature. We examine the characteristics of 0-spectrum of the 1-form Laplacian compared to the cycles of graphs.
The notions of magnetic difference operator defined on weighted graphs or magnetic exterior derivative are discrete analogues of the notionof covariant derivative on sections of a fibre bundle and its extension on differential forms. In this paper, we extend this notion to certain 2-simplicial complexes called triangulations, in a manner compatible with changes of gauge. Then we study the magnetic Gauss-Bonnet operator naturally defined in this context and introduce the geometric hypothesis of $χ-$completeness which ensures the essential self-adjointness of this operator. This gives also the essential self-adjointness of the magnetic Laplacian on triangulations. Finally we introduce an hypothesis of bounded curvature for the magnetic potential which permits to characterize the domain of the self-adjoint extension.
We consider a non self-adjoint Laplacian on a directed graph with non symmetric weights on edges. We give a criterion for the m-accretiveness and the m-sectoriality of this Laplacian. Our results are based on a comparison of this operator with its symmetric part for which we can apply dierent results concerning essential self-adjointness of a symmetric Laplace operator on an innite graph. This gives results on the heat operator related to our non-symmetric Laplacian.
We consider a non self-adjoint Laplacian on a directed graph with non symmetric edge weights. We give necessary conditions for this Laplacian to be sectorial. We introduce a special self-adjoint operator and compare its essential spectrum with that of the non self-adjoint Laplacian considered.
We propose a general condition, to ensure essential self-adjointness for the Gauß-Bonnet operator, based on a notion of completeness as Chernoff. This gives essential self-adjointness of the Laplace operator both for functions or 1-forms on infinite graphs. This is used to extend Flanders result concerning solutions of Kirchhoff's laws.
We consider weighted graphs, we equip them with a metric structure given by a weighted distance, and we discuss essential self-adjointness for weighted graph Laplacians and Schrödinger operators in the metrically non complete case.
We introduce the weighted graph Laplacian and the notion of Schrödinger operator on a locally finite weighted graph. Concerning essential self-adjointness, we extend Wojciechowski's and Dodziuk's results for graphs with vertex constant weight. The main result in this work states that on any metrically complete weighted graph with bounded degree, the weighted graph Laplacian is essentially self-adjoint and the same holds for the Schrödinger operator provided the associated quadratic form is bounded from below. We construct for the proof a strictly positive and harmonic function which allows us to write any Schrödinger operator as a weighted graph Laplacian modulo a unitary transform.
We define the magnetic Schrödinger on an infinite graph by the data of a magnetic field, some weights on vertices and some weights on edges . We discuss essential self-adjointness of this operator for graphs of bounded degree. The main result is a discrete version of a result of two authors of the present paper.
We introduce the weighted graph Laplacian and the notion of Schrödinger operator on a locally finite weighted graph . Concerning essential self-adjointness, we extend Wojciechowski's and Dodziuk's results for graphs with vertex constant weight. The main result in this work states that on any metrically complete weighted graph with bounded degree, the Laplacian is essentially self-adjoint and the same holds for the Schrödinger operator provided the associated quadratic form is bounded from below. We construct for the proof a strictly positive and harmonic function which allows us to write any Schrödinger operator as a weighted graph Laplacian modulo a unitary transform.