1110.6807 (Zhiqin Lu et al.)

On the discrete spectrum of quantum layers    [PDF]

Zhiqin Lu, Julie Rowlett

Consider a quantum particle trapped between a curved layer of constant width built over a complete, non-compact, $\mathcal C^2$ smooth surface embedded in $\mathbb{R}^3$. We assume that the surface is asymptotically flat in the sense that the second fundamental form vanishes at infinity, and that the surface is not totally geodesic. This geometric setting is known as {\it a quantum layer.} We consider the quantum particle to be governed by the Dirichlet Laplacian as Hamiltonian. Our work concerns the existence of bound states with energy beneath the essential spectrum, which is mathematically equivalent to the existence of {\it discrete spectrum.} We first prove that if the Gauss curvature is integrable, and the surface is {\it weakly $\kappa$-parabolic,} then the discrete spectrum is non-empty. This result implies that if the total Gauss curvature is {\it non-positive,} then the discrete spectrum is non-empty. We next prove that if the Gauss curvature is {\it non-negative,} then the discrete spectrum is non-empty. Finally, we prove that if the surface is {\it parabolic,} then the discrete spectrum is non-empty if the layer is sufficiently thin.

View original: http://arxiv.org/abs/1110.6807