Quantum graphs. I: Some basic structures. (English) Zbl 1063.81058
Summary: A quantum graph is a graph equipped with a self-adjoint differential or pseudo-differential Hamiltonian. Such graphs have been studied recently in relation to some problems of mathematics, physics and chemistry. The paper has a survey nature and is devoted to the description of some basic notions concerning quantum graphs, including the boundary conditions, self-adjointness, quadratic forms, and relations between quantum and combinatorial graph models.
MSC:
81Q99 | General mathematical topics and methods in quantum theory |
34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
47E05 | General theory of ordinary differential operators |
47N50 | Applications of operator theory in the physical sciences |
82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |