Standing waves on quantum graphs
Adilbek Kairzhan, Diego Noja, and Dmitry E. Pelinovsky
TL;DR
This paper reviews the mathematical analysis of standing waves on quantum graphs, focusing on their existence and stability using variational, dynamical systems, and boundary mapping methods.
Contribution
It introduces new analytical techniques for studying the existence and stability of standing waves on quantum graphs.
Findings
Existence of standing waves established
Stability conditions derived for certain quantum graph configurations
Application of variational and dynamical systems methods to quantum graphs
Abstract
We review evolutionary models on quantum graphs expressed by linear and nonlinear partial differential equations. Existence and stability of the standing waves trapped on quantum graphs are studied by using methods of the variational theory, dynamical systems on a phase plane, and the Dirichlet-to-Neumann mappings.
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Taxonomy
TopicsSpectral Theory in Mathematical Physics · Quantum chaos and dynamical systems · advanced mathematical theories