Jacob Muller: An introduction to quantum graphs via Euler–Bernoulli beam theory

Abstract:
A quantum graph is a metric graph endowed with a certain differential operator acting on functions on the graph which satisfy some prescribed conditions at the vertices. In keeping with the name ‘quantum’, the majority of efforts in the field have gone into the study of magnetic Schrödinger operators, particularly Laplacian operators. Fundamental results are therefore lacking for many other operators, despite having obvious real-world applications. For example, the simple construction of a planar network of beams which oscillate orthogonally to the plane can be modelled by a quantum graph with a bi-Laplacian operator (the fourth derivative). Here, the edges of the graph represent the beams and the functions measure their transverse deflection. With this example in mind, we’ll begin with a brief introduction to the subject of quantum graphs. We shall then examine how some of the well-known results from the established theory of Laplacian quantum graphs (regarding classification, spectral properties and scattering, for instance) compare with the corresponding results for bi-Laplacian graphs.