Spectral asymptotics for two-dimensional Dirac operators in thin waveguides

2022/07/18 by Borrelli, William, Kerraoui, Nour, Ourmières-Bonafos, Thomas
#35P05 #81Q10 #81Q15 #81Q37 #82D77 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2207.08700

Abstract

We consider the two-dimensional Dirac operator with infinite mass boundary conditions posed in a tubular neighborhood of a C⁴-planar curve. Under generic assumptions on its curvature κ, we prove that in the thin-width regime the splitting of the eigenvalues is driven by the one dimensional Schrödinger operator on L²(\mathbb R) L_e := -(d²)/(ds²) - (κ²)/(π²) with a geometrically induced potential. The eigenvalues are shown to be at distance of order ε from the essential spectrum, where 2ε is the width of the waveguide. This is in contrast with the non-relativistic counterpart of this model, for which they are known to be at a finite distance.

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