Phase transition in a periodic tubular structure

arXiv2303.00872MaRDI QIDQ6509012

Alexander V. Kiselev, K. Ryadovkin

Abstract: We consider an $varepsilon$-periodic ($varepsilono0$) tubular structure, modelled as a magnetic Laplacian on a metric graph, which is periodic along a single axis. We show that the corresponding Hamiltonian admits norm-resolvent convergence to an ODE on $mathbbR$ which is fourth order at a discrete set of values of the magnetic potential (emph{critical points}) and second-order generically. In a vicinity of critical points we establish a mixed-order asymptotics. The rate of convergence is also estimated. This represents a physically viable model of a phase transition as the strength of the (constant) magnetic field increases.

Mathematics Subject Classification ID

Asymptotic distributions of eigenvalues in context of PDEs (35P20) Dilations, extensions, compressions of linear operators (47A20) Asymptotic expansions of solutions to ordinary differential equations (34E05) Multiple scale methods for ordinary differential equations (34E13) Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices (81Q35)

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