Hardy Inequalities in Globally Twisted Waveguides

Scattering in Twisted Waveguides

We consider a twisted quantum waveguide, i.e., a domain of the form Ωθ := rθω×R where ω ⊂ R 2 is a bounded domain, and rθ = rθ(x3) is a rotation by the angle θ(x3) depending on the longitudinal variable x3. We investigate the nature of the essential spectrum of the Dirichlet Laplacian Hθ, self-adjoint in L (Ωθ), and consider related scattering problems. First, we show that if the derivative of ...

Resonances in Twisted Quantum Waveguides

In this paper we consider embedded eigenvalues of a Schrödinger Hamiltonian in a waveguide induced by a symmetric perturbation. It is shown that these eigenvalues become unstable and turn into resonances after twisting of the waveguide. The perturbative expansion of the resonance width is calculated for weakly twisted waveguides and the influence of the twist on resonances in a concrete model i...

Hardy Inequalities in Function Spaces

Critical Hardy–Sobolev Inequalities

Abstract We consider Hardy inequalities in IRn, n ≥ 3, with best constant that involve either distance to the boundary or distance to a surface of co-dimension k < n, and we show that they can still be improved by adding a multiple of a whole range of critical norms that at the extreme case become precisely the critical Sobolev norm. Résumé Nous étudions des inegalités de Hardy dans IRn, n ≥ 3,...

Orlicz-hardy Inequalities

We relate Orlicz-Hardy inequalities on a bounded Euclidean domain to certain fatness conditions on the complement. In the case of certain log-scale distortions of Ln, this relationship is necessary and sufficient, thus extending results of Ancona, Lewis, and Wannebo. 0. Introduction Suppose Ω ⊂ R is a bounded domain and let d(x) = dist(x, ∂Ω). We consider integral Hardy inequalities (0.1) ∀u ∈ ...