Pavel Exner

CURVATURE-INDUCED BOUND STATES IN QUANTUM WAVEGUIDES IN TWO AND THREE DIMENSIONS

Dirichlet Laplacian on curved tubes of a constant cross section in two and three dimensions is investigated. It is shown that if the tube is non-straight and its curvature vanishes asymptotically, there is always a bound state below the bottom of the essential spectrum. An upper bound to the number of these bound states in thin tubes is derived. Furthermore, if the tube is only slightly bent, there is just one bound state; we derive its behaviour with respect to the bending angle. Finally, perturbation theory of these eigenvalues in any thin tube with respect to the tube radius is constructed and some open questions are formulated.

Bound states in curved quantum waveguides

A free quantum particle living on a curved planar strip Ω of a fixed width d with Dirichlet boundary conditions is studied. It can serve as a model for electrons in thin films on a cylinder‐type substrate, or in a curved quantum wire. Assuming that the boundary of Ω is infinitely smooth and its curvature decays fast enough at infinity, it is proved that a bound state with energy below the first transversal mode exists for all sufficiently small d. A lower bound on the critical width is obtained using the Birman–Schwinger technique.

Free quantum motion on a branching graph

Abstract We consider the free motion of a quantum particle on the graph consisting of three half-lines whose ends are connected. It is shown that the time evolution can be described by a Hamiltonian and the class of all admissible Hamiltonians is constructed using the theory of self-adjoint extensions. Three subclasses are discussed in detail: (a) the one-parameter family of Hamiltonians whose domains contain functions continuous at the junction, (b) the wider four-parameter family with the wavefunctions continuous between two branches of the graph only, (c) the Hamiltonian invariant under permutations of the branches. For the class (c), generalization to the graphs consisting of n half-lines is given. The scattering problem of such a branching graph is also discussed.

Convergence of spectra of graph-like thin manifolds

Abstract We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace–Beltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand, if the shrinking at the vertex parts of the manifold is sufficiently slower comparing to that of the edge parts, the limiting spectrum corresponds to decoupled edges with Dirichlet boundary conditions at the endpoints. At the borderline between the two regimes we have a third possibility when the limiting spectrum can be described by a nontrivial coupling at the vertices.

Bound states and scattering in quantum waveguides coupled laterally through a boundary window

We consider a pair of parallel straight quantum waveguides coupled laterally through a window of a width l in the common boundary. We show that such a system has at least one bound state for any l≳0. We find the corresponding eigenvalues and eigenfunctions numerically using the mode‐matching method, and discuss their behavior in several situations. We also discuss the scattering problem in this setup, in particular, the turbulent behavior of the probability flow associated with resonances. The level and phase‐shift spacing statistics shows that in distinction to closed pseudo‐integrable billiards, the present system is essentially nonchaotic. Finally, we illustrate time evolution of wave packets in the present model.

Bound States in Curved Quantum Layers

Abstract: We consider a nonrelativistic quantum particle constrained to a curved layer of constant width built over a non-compact surface embedded in ℝ3. 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 geometrically induced bound states.

Bound States in Weakly Deformed Strips and Layers

Abstract. We consider Dirichlet Laplacians on straight strips in

Topologically nontrivial quantum layers

{\Bbb R}^2

Asymptotics of eigenvalues of the Schrödinger operator with a strong δ-interaction on a loop

or layers in

Weakly coupled states on branching graphs

{\Bbb R}^3