The spectrum problem is a fundamental problem in Riemannian geometry. For a compact Riemannian manifold, the Laplacian only has eigenvalues with finite multiplicity. The set of these eigenvalues is called pure point spectrum. For complete non-compact manifold, the spectrum is more complicated: in general, it may contain both pure point spectrum and essential spectrum. And for a large class of complete non-compact manifolds, the set of pure point spectrum is an empty set. For example, the pure point spectrum of the n dimensional Euclidean space with standard metric is an empty set.

The concept of quantum layer was introduced in the study of mesoscopic physics. A quantum layer is a kind of complete non-compact manifold with boundary. It is defined as follows:

Suppose that Σ is a complete non-compact embedded surface in R³. We assume that the second fundamental form of Σ goes to zero at infinity. Let a be a small positive number. The quantum layer Ω is defined as the set of points in R³ such that the distance to Σ is no more than a.

It is suprising to see that for a large class of surfaces Σ, the set of pure point spectrum is not empty. Furthermore, the ground state of the Laplacian exists[1]. Needless to say, this result is very useful in physics.

Let's go back to mathematics. Based on the work of Duclos, Exner, Krejcirik [1], we make the following Quantum Layer Conjecture:

Let Σ be a complete non-compact embedded surface in R³ which is asymptotically flat in the sense that the second fundamental form of it goes to zero at infinity. Let Ω be the quantum layer defined with respect to Σ. Let K be the Gauss curvature of Σ. If

   

Then the ground state of Ω exists.

The conjecture was proved under the additional assumption that the total Gauss curvarure

is non-positive. Thus in order to prove the conjecture, we may assume that the total Gauss curvature is positive. In this case, the topology of Σ is quite simple: it is differmorphic to R². Interestingly, this is the most difficult situation. We need to know the asymptotic properties of Σ at infinity. Unfortunately, other than minimal surfaces, we know quite little about that.

In [2], under the assumption that Σ is convex, the conjecture was proved. Futher results were discussed in [3], in which a lot of cases were studied and proved. Thus we are not far from proving the conjecture from [3].

References:

[1] P. Duclos and P. Exner and D. Krejcirik, Bound States in Curved Quantum Layers, Comm. Math. Phys., 223(1), 2001, 13-28.

[2] C. Lin and Z. Lu, Existence of Bound States for Layers Built over Hypersurfaces in Rⁿ⁺¹, J. Funct. Anal., 244, 2007, 1-25.

[3] Z. Lu, On the ground state of quantum layers, preprint, arXiv:0708.1563.