Spectral estimates for Dirichlet Laplacian on tubes with exploding twisting velocity

Abstract

We study the spectrum of the Dirichlet Laplacian on an unbounded twisted tube with twisting velocity exploding to infinity. If the tube cross section does not intersect the axis of rotation, then its spectrum is purely discrete under some additional conditions on the twisting velocity (D. Krejčiřík, 2015). In the current work we prove a Berezin type upper bound for the eigenvalue moments.Mathematics subject classification (2010): 35P20, 35P15, 81Q10, 81Q37.

Quantum strips in higher dimensions

“…Open Problem 5.7. Solid tubes with asymptotically diverging twisting represent a new class of models which lead to previously unobserved phenomena like the annihilation of the essential spectrum [23] and establishing a non-standard Weyl's law for the accumulation of eigenvalues at infinity remains open (a first step in this direction has been recently taken in [1] by establishing a Berezin-type upper bound for the eigenvalue moments). The case of twisted strips with |Θ ′ (s)| → ∞ as |s| → ∞ is rather different for some essential spectrum is always present, but related questions about the accumulation of discrete eigenvalues remain open, too (cf.…”

Spectral analysis of sheared nanoribbons

“…In 2008 Ekholm, Kovařík and one of the present authors [9] observed that the geometric deformation of twisting has a quite opposite effect on the energy spectrum of an electron confined to three-dimensional tubes, for it creates an effectively repulsive interaction (see [12] for an overview of the two reciprocal effects). More specifically, twisting the tube locally gives rise to Hardy-type inequalities and a stability of quantum transport, the effect becomes stronger in globally twisted tubes [4] and in extreme situations it may even annihilate the essential spectrum completely [13] (see also [2]). The repulsive effect remains effective even under modification of the boundary conditions [3].…”

“…Remark 2). This phenomenon is reminiscent of three-dimensional waveguides with asymptotically diverging twisting [13,2].…”

The asymptotic behaviour of the heat equation in a sheared unbounded strip

“…To prove the rest of the proposition we use Theorem 3 which was mentioned above. In our case V = H 1 1 and H = H 1 . We show that a s (•, •) satisfies all three assumptions of Theorem 3.…”

“…First, we show that the sesquilinear form a s is well defined with the domain D(a s ) := H 1 1 for any fixed s ∈ [0, ∞) and thus it is continuous. Using the fact that σ s is bounded for every finite s we only have to show that for every v ∈ H 1 1 we have yv ∈ H 1 . For v ∈ C ∞ 0 (Ω) we obtain…”