Helmholtz operators on infinite graphs

Varadha Raj Manivannan; Madhu Venkataraman

doi:10.1016/j.heliyon.2024.e24761

Helmholtz operators on infinite graphs

Heliyon. 2024 Jan 18;10(2):e24761. doi: 10.1016/j.heliyon.2024.e24761. eCollection 2024 Jan 30.

Authors

Varadha Raj Manivannan¹, Madhu Venkataraman¹

Affiliation

¹ Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore - 632 014, Tamil Nadu, India.

Abstract

The Helmholtz equation in its simplest form is $Δ u (a) = - k^{2} u (a)$ . In this note, we study a generalized discrete version of this equation on an infinite graph, by using potential-theoretic methods.

Keywords: 31A30; 31C05; 31C20; Cartesian product; Choquet theorem; HH-hyperbolic; HH-parabolic; HH-superharmonic; Helmholtz infinite network.