Discrete Spectrum of Quantum Tubes

A quantum tube is essentially a tubular neighborhood about an immersed complete manifold in some Euclidean space. To be more precise, let Σ →֒ R, k ≥ 1, n = dim(Σ), be an isometric immersion, where Σ is a complete, noncompact, orientable manifold. Then consider the resulting normal bundle TΣ over Σ, and the submanifold F = {(x, ξ)|x ∈ Σ, |ξ < r} ⊂ TΣ for r small enough. The quantum tube is defined as the Riemannian manifold (F, f (dsE)), where dsE is the Euclidean metric in R n+k and the map f is defined by f(x, ξ) = x + ξ. If k = 1, then the quantum tube is also called the quantum layer. The immersion of Σ means that the resulting image of F under f in R can have intersections. Moreover, since Σ can have quite complicated topology in general, f(F ) can too. However, by doing our analysis on F directly (with the pull-back metric), these complications are naturally bypassed (cf. [1, 5]).