Related papers: Topologically non-trivial quantum layers
Under several geometric conditions imposed below, the existence of the discrete spectrum below the essential spectrum is shown for the Dirichlet Laplacian on the quantum layer built over a spherically symmetric hypersurface with a pole…
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 structure of the spectrum of the three-dimensional Dirichlet Laplacian in the 3D polyhedral layer of fixed width is studied. It appears that the essential spectrum is defined by the smallest dihedral angle that forms the boundary of the…
Motivated by the theory of quantum waveguides, we investigate the spectrum of the Laplacian, subject to Dirichlet boundary conditions, in a curved strip of constant width that is defined as a tubular neighbourhood of an infinite curve in a…
We investigate the spectrum of the three-dimensional Dirichlet Laplacian in a prototypal infinite polyhedral layer, that is formed by three perpendicular quarter-plane walls of constant width joining each other. Alternatively, this domain…
We study the Laplace operator with Dirichlet or Neumann boundary condition on polygons in the Euclidean plane. We prove that almost every simply connected polygon with at least four vertices has simple spectrum. We also address the more…
We study spectral properties of Dirichlet Laplacian on the conical layer of the opening angle $\pi-2\theta$ and thickness equal to $\pi$. We demonstrate that below the continuum threshold which is equal to one there is an infinite sequence…
We consider a nonrelativistic quantum particle constrained to a curved layer of constant width built over a non-compact surface embedded in $R^3$. We suppose that the latter is endowed with the geodesic polar coordinates and that the layer…
We derive a lower bound to the spectral threshold of the Dirichlet Laplacian in tubular neighbourhoods of constant radius about complete surfaces. This lower bound is given by the lowest eigenvalue of a one-dimensional operator depending on…
We investigate spectral properties of the Laplacian in $L^2(Q)$, where $Q$ is a tubular region in $\mathbb{R}^3$ of a fixed cross section, and the boundary conditions combined a Dirichlet and a Neumann part. We analyze two complementary…
We derive a number of spectral results for Dirac operators in geometrically nontrivial regions in $\mathbb{R}^2$ and $\mathbb{R}^3$ of tube or layer shapes with a zigzag type boundary using the corresponding properties of the Dirichlet…
We consider a quantum particle constrained to a curved layer of a constant width built over an infinite smooth surface. We suppose that the latter is a locally deformed plane and that the layer has the hard-wall boundary. Under this…
Motivated by a question of Rubel, we consider the problem of characterizing which noncompact hypersurfaces in $\RR^n$ can be regular level sets of a harmonic function modulo a $C^\infty$ diffeomorphism, as well as certain generalizations to…
In this paper we develop an abstract theory for the Codazzi equation on surfaces, and use it as an analytic tool to derive new global results for surfaces in the space forms ${\bb R}^3$, ${\bb S}^3$ and ${\bb H}^3$. We give essentially…
We present a general construction of embedded minimal and constant mean curvature surfaces in $\mathbb{S}^n$ and one-phase free boundaries joined by a smooth interpolation by capillary hypersurfaces. This framework recovers all known…
We introduce the Plaque Topology on the inverse limit of a branched covering self-map of a Riemann surface of a finite degree greater than one. We present the notions of regular and irregular points in the setting of this Plaque Inverse…
The note introduces a novel concept of non-Abelian patchworking arising as real locus of non-Abelian complex-phase tropical hypersurfaces, the theory of which is now developed enough to allow the proposed spin-off. Although, non-Abelian…
Spacelike surfaces in the Lorentz-Minkowski space L^3 can be endowed with two different Riemannian metrics, the metric inherited from L^3 and the one induced by the Euclidean metric of R^3. It is well known that the only surfaces with zero…
We consider the Dirichlet Laplacian in tubular neighbourhoods of complete non-compact Riemannian manifolds immersed in the Euclidean space. We show that the essential spectrum coincides with the spectrum of a planar tube provided that the…
We study entanglement spectra of gapped states on the surfaces of symmetry-protected topological phases. These surface states carry anomalies that do not allow them to be terminated by a trivial state. Their entanglement spectra are…