Existence of bound states for layers built over hypersurfaces in Rn+1
نویسندگان
چکیده
The existence of discrete spectrum below the essential spectrum is deduced for the Dirichlet Laplacian on tubular neighborhoods (or layers) about hypersurfaces in Rn+1, with various geometric conditions imposed. This is a generalization of the results of Duclos, Exner, and Krejčiřík (2001) in the case of a surface in R3. The key to the generalization is the notion of parabolic manifolds. An interesting case in R3—that of the layer over a convex surface—is also investigated. © 2006 Elsevier Inc. All rights reserved.
منابع مشابه
EXISTENCE OF BOUND STATES FOR LAYERS BUILT OVER HYPERSURFACES IN Rn+1 CHRISTOPHER LIN AND ZHIQIN LU
In their study of the spectrum of quantum layers [6], Duclos, Exner, and Krejčǐŕık proved the existence of bound states for certain quantum layers . Part of their motivations to study the quantum layers is from mesoscopic physics. From the mathematical point of view, a quantum layer is a noncompact noncomplete manifold. For such a manifold, the spectrum of the Laplacian (with Dirichlet or Neuma...
متن کاملBound States in Curved Quantum Layers
We consider a nonrelativistic quantum particle constrained to a curved layer of constant width built over a non-compact surface embedded in R. We suppose that the latter is endowed with the geodesic polar coordinates and that the layer has the hard-wall boundary. Under the assumption that the surface curvatures vanish at infinity we find sufficient conditions which guarantee the existence of ge...
متن کاملExistence of Convex Hypersurfaces with Prescribed Gauss-kronecker Curvature
Let f(x) be a given positive function in Rn+1. In this paper we consider the existence of convex, closed hypersurfaces X so that its GaussKronecker curvature at x ∈ X is equal to f(x). This problem has variational structure and the existence of stable solutions has been discussed by Tso (J. Diff. Geom. 34 (1991), 389–410). Using the Mountain Pass Lemma and the Gauss curvature flow we prove the ...
متن کاملA sharp upper bound for the first eigenvalue of the Laplacian of compact hypersurfaces in rank-1 symmetric spaces
M |H| 2, where H is the mean curvature of the hypersurface M. These inequalities of Bleecker–Weiner and Reilly are also sharp for geodesic spheres in Rn. Since then, Reilly’s inequality has been extended to hypersurfaces in other simply connected space forms (see [7] and [8] for details and related results). While trying to understand these results, we noticed that one can obtain a similar shar...
متن کاملLaguerre Geometry of Hypersurfaces in Rn
Laguerre geometry of surfaces in R is given in the book of Blaschke [1], and have been studied by E.Musso and L.Nicolodi [5], [6], [7], B. Palmer [8] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in Rn. For any umbilical free hypersurface x : M → Rn with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invarian...
متن کامل