Boundary Conditions Matter: On The Spectrum Of Infinite Quantum Graphs

Abstract

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.

Embedded trace operator for infinite metric trees

“…While quantum graphs represent by now a well-established theory [5,6,19], the most attention was concentrated on the study of regular configurations with suitable lower bounds on the edge lengths and other parameters: in that case it is known that gluing conditions at the nodes are sufficient to define a self-adjoint operator or a non-self-adjoint one with good properties [26]. More recent papers [13,14,17,20,24,36,40,55] initiated the discussion of the most general quantum graphs, which shows that in many cases additional "boundary conditions at the external boundary" must be imposed. It should be noted that the notion of boundary for general graphs is not obvious, which is a well-known issue for both metric and discrete infinite graphs [14,31,33,37,41,59]; we recall that metric and discrete graphs show a number of common features [3,4,15,37,39,48], and in case of equilateral metric graphs even a kind of unitary equivalence between respective Laplacians can be established [49,50].…”