A review on quantum graphs: From mathematical foundations to applications

Quantum graphs provide a rigorous framework for modelling quantum dynamics on network-like structures, where edges represent one-dimensional wires and vertices encode interaction conditions. Since their introduction as models for wave propagation and molecular systems, quantum graphs have grown to become an essential tool in mathematical physics, providing information on spectral, transport, and scattering phenomena. The mathematical foundations of quantum graphs are reviewed in this review, covering self-adjoint operators, metric graph formalisms, and vertex conditions that control the behaviour of wave functions. Next, we discuss important developments in spectrum theory that make quantum graphs strong models for quantum chaos and universal spectral statistics, including resonance features, eigenfunction statistics, and spectral gap optimisation. Stability analyses and inverse spectrum problems broaden the theoretical scope of the framework, while extensions to leaky, transparent, and random quantum graphs show how versatile it is. The survey emphasises the function of quantum graphs as a link between discrete and continuous models by combining these viewpoints. We conclude by describing open difficulties in the areas of spectral invariants and random graph models.

کلیدواژه ها:

Quantum graphs ، Spectral theory ، Quantum chaos ، Scattering ، Resonance