Eigenvalue Asymptotics in a Twisted Waveguide

We consider a twisted quantum wave guide i.e. a domain of the form Ωθ := rθω × R where ω is a connected open and bounded subset of R2 and rθ = rθ(x3) is a rotation by the angle θ(x3) depending on the longitudinal variable x3. We are interested in the spectral analysis of the Dirichlet Laplacian H acting in Ωθ. We suppose that the derivative θ̇ of the rotation angle can be written as θ̇(x3) = β − ε(x3) with a positive constant β and ε(x3) ∼ L|x3|, |x3| → ∞. We show that if L > 0 and α ∈ (0, 2), or if L > L0 > 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. AMS 2000 Mathematics Subject Classification: 35J10, 81Q10, 35P20