Boundary Conditions Matter: On The Spectrum Of Infinite Quantum Graphs

We develop a comprehensive spectral geometric theory for two distinguished self-adjoint realisations of the Laplacian, the so-called Friedrichs and Neumann extensions, on infinite metric graphs. We present a new criterion to determine whether these extensions have compact resolvent or not, leading to concrete examples where this depends on the chosen extension. In the case of discrete spectrum, under additional metric assumptions, we also extend known upper and lower bounds on Laplacian eigenvalues to metric graphs that are merely locally finite. Some of these bounds are new even on compact graphs.