Closed form expressions for the Green’s function of a quantum graph—a scattering approach

Lawrie, Tristan; Gnutzmann, Sven; Tanner, Gregor K.

Abstract

In this work we present a three step procedure for generating a closed form expression of the Green’s function on both closed and open finite quantum graphs with general self-adjoint matching conditions. We first generalize and simplify the approach by Barra and Gaspard (2001 Phys. Rev. E 65 016205) and then discuss the validity of the explicit expressions. For compact graphs, we show that the explicit expression is equivalent to the spectral decomposition as a sum over poles at the discrete energy eigenvalues with residues that contain projector kernel onto the corresponding eigenstate. The derivation of the Green’s function is based on the scattering approach, in which stationary solutions are constructed by treating each vertex or subgraph as a scattering site described by a scattering matrix. The latter can then be given in a simple closed form from which the Green’s function is derived. The relevant scattering matrices contain inverse operators which are not well defined for wave numbers at which bound states in the continuum exists. It is shown that the singularities in the scattering matrix related to these bound states or perfect scars can be regularised. Green’s functions or scattering matrices can then be expressed as a sum of a regular and a singular part where the singular part contains the projection kernel onto the perfect scar.

Citation

Lawrie, T., Gnutzmann, S., & Tanner, G. K. (2023). Closed form expressions for the Green’s function of a quantum graph—a scattering approach. Journal of Physics A: Mathematical and Theoretical, 56(47), Article 475202. https://doi.org/10.1088/1751-8121/ad03a5

• Journal Article Type: Article
• Acceptance Date: Oct 16, 2023
• Online Publication Date: Nov 30, 2023
• Publication Date: Nov 24, 2023
• Deposit Date: Oct 23, 2023
• Publicly Available Date: Oct 23, 2023
• Journal: Journal of Physics A: Mathematical and Theoretical
• Print ISSN: 1751-8113
• Electronic ISSN: 1751-8121
• Publisher: IOP Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 56
• Issue: 47
• Article Number: 475202
• DOI: https://doi.org/10.1088/1751-8121/ad03a5
• Keywords: General Physics and Astronomy, Mathematical Physics, Modeling and Simulation, Statistics and Probability, Statistical and Nonlinear Physics
• Public URL: https://nottingham-repository.worktribe.com/output/26262381
• Publisher URL: https://iopscience.iop.org/article/10.1088/1751-8121/ad03a5