Ruled strips with asymptotically diverging twisting

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.