Eigenvalue Asymptotics in a Twisted Waveguide

Abstract

Abstract We consider a twisted quantum wave guide i.e., a domain of the form Ωθ: = r θ ω × ℝ where ω ⊂ ℝ2 is a bounded domain, and r θ = r θ(x 3) is a rotation by the angle θ(x 3) depending on the longitudinal variable x 3. We are interested in the spectral analysis of the Dirichlet Laplacian H acting in L 2(Ωθ). We suppose that the derivative of the rotation angle can be written as (x 3) = β − ϵ(x 3) with a positive constant β and ϵ(x 3) ∼ L|x 3|−α, |x 3| → ∞. We show that if L > 0 and α ∈ (0,2), or if L > L 0 > 0 and α = 2, then there is an infinite sequence of discrete eigenvalues lying below the infimum of the essential spectrum of H, and obtain the main asymptotic term of this sequence. Keywords: Eigenvalue asymptoticsSchrödinger operatorsWaveguidesMathematics Subject Classification: 35J1081Q1035P20 Acknowledgments P. Briet and G. Raikov were partially supported by the Chilean Scientific Foundation Fondecyt under Grant 1090467. H. Kovařík was partially supported by the German Research Foundation (DFG) under Grant KO 3636/1-1. G. Raikov was partially supported by Núcleo Científico ICM P07-027-F "Mathematical Theory of Quantum and Classical Magnetic Systems".

Locations

• Communications in Partial Differential Equations - View
• arXiv (Cornell University) - View - PDF