Ruled Strips with Asymptotically Diverging Twisting

Abstract

We consider the Dirichlet Laplacian in a two-dimensional strip composed of segments translated along a straight line with respect to a rotation angle with velocity diverging at infinity. We show that this model exhibits a "raise of dimension" at infinity leading to an essential spectrum determined by an asymptotic three-dimensional tube of annular cross-section. If the cross-section of the asymptotic tube is a disk, we also prove the existence of discrete eigenvalues below the essential spectrum.MSC (2010): 35P15, 58C40, 58J50, 81Q10, 81Q35.

Spectral analysis of sheared nanoribbons

“…In order to obtain a non-trivial non-negative lower bound including the case β = 0, we have to exploit the positive terms that we have neglected when coming from (14) to (15). To do so, let I ⊂ R be any bounded open interval and set Ω I 0 := I × (0, d).…”

“…Recall that E 1 (β) corresponds to the threshold of the essential spectrum if (2) holds (cf. Theorem 1), so (15) in particular ensures that there is no (discrete) spectrum below E 1 (β) (even if β = 0). Note that the Hardy weight on the right-hand side of ( 15) diverges on ∂Ω 0 .…”

“…Among the most influential theoretical results, let us quote the existence of quantum bound states due to bending in curved strips, firstly observed by Exner and Šeba [11] in 1989. The pioneering paper has been followed by a huge number of works demonstrating the robustness of the effect in various geometric settings including higher dimensions, and the research field is still active these days (see [15] for a recent paper with a brief overview in the introduction).…”

Spectrum of the Dirichlet Laplacian in waveguides with parallel cross-sections

“…Let Ω be an unbounded quantum waveguide in R n , n = 2, 3, and denote by −∆ D Ω the Dirichlet Laplacian operator in Ω. The spectrum of this operator has been extensively studied in the last years [1,2,3,4,5,6,7,8,9,11,12,13,15,17,19,20,21,23,24,25,26,27]. In fact, the subject is non-trivial since the results depend on the geometry of Ω [4,8,12,17,23,24].…”

“…The spectrum of this operator has been extensively studied in the last years [1,2,3,4,5,6,7,8,9,11,12,13,15,17,19,20,21,23,24,25,26,27]. In fact, the subject is non-trivial since the results depend on the geometry of Ω [4,8,12,17,23,24]. In the particular case where Ω is a straight waveguide, it is known that its spectrum is purely absolutely continuous and there are no discrete eigenvalues.…”