Magnetic interpretation of the nodal defect on graphs

Yves Colin de Verdière (1)

1 IF - Institut Fourier

In this note, we present a natural proof of a recent and surprising result of Gregory Berkolaiko (arXiv 1110.5373) interpreting the "Courant nodal defect" of a Schrödinger operator on a finite graph as a Morse index associated to the deformations of the operator by switching on a magnetic field. This proof is inspired by a nice paper of Miroslav Fiedler published in 1975.

Mots clés: Operator on graphs; Schrödinger operators with magnetic fields; Morse index; Courant nodal Theorem; Hill's operator

Domaines: Physique mathématique [math-ph]; Combinatoire [math.CO]; Théorie spectrale [math.SP]

Yves Colin de Verdière. Magnetic interpretation of the nodal defect on graphs. Analysis & Partial Differential Equations, 2013, 6 (5), pp.1235--1242. ⟨10.2140/apde.2013.6.1235⟩. ⟨hal-00656768v2⟩