Magnetic cochains Laplacians and their essential self-adjointness

In this paper, we introduce the notion of oriented triangular faces F, the notion of edges potential P-E and the notion of triangular faces potential P-F in a connected oriented locally finite graph (V, E) in order to construct a new framework that's we call the magnetic weighted 2-simplicial complex M = (V, E, F, P-E, P-F). On this new magnetic weighted framework, we introduce the magnetic 0-cochains set, the magnetic 1-cochains set, the magnetic 2-cochains set and the magnetic cochains set. After that, we construct the magnetic 0-cochains Laplacian, the magnetic 1-cochains Laplacian, the magnetic 2-cochains Laplacian and the magnetic cochains Laplacian. Finally, we ensure essential self-adjointness for our new magnetic cochains Laplacians using the Stieltjes vectors.