Mats-Erik Pistol | Isospectral quantum graphs

Days on Diffraction 2024. Mini-symposium “Inverse Problems”. Wednesday, 12 June, 2024
Mats-Erik Pistol (Lund University)
Isospectral quantum graphs
Quantum graphs are defined by having a Laplacian defined on the edges of a metric graph with boundary conditions (typically Neumann-Kirchhoff) on each vertex such that the resulting operator is self-adjoint. There are few known examples of pairs of non-isomorphic but isospectral quantum graphs. We have found all sets of isospectral but non-isomorphic equilateral connected quantum graphs with at most nine vertices. This includes thirteen isospectral triplets and one isospectral set of four. One set has the loop as a member. We also present several different combinatorial methods to generate arbitrarily large sets of isospectral graphs, including infinite graphs in different dimensions. As part of this we have found a method to determine if two vertices have the same Titchmarsh-Weyl M-function.
All of this has been done using computer algebra which is done symbolically, and thus exactly. Our program also allows visualisation of eigenfunctions and can handle a large set of self-adjoint boundary conditions. We find that several sets of graphs that are isospectral under both Neumann and Dirichlet
boundary conditions as well as under more general, $\delta$-type and $\delta_{s\prime}$-type, boundary conditions. Our software is open-source and can be inspected by the community. Most of our results can be verified by hand.