arXiv subjects

David Krejcirik

Publications and source records attributed to David Krejcirik.

112 records · Page 7Linked to original sources

arXiv2002

Quantum strips on surfaces

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 two-dimensional Riemannian manifold. Under the assumption that the strip is asymptotically straight in a suitable sense, we localise the essential spectrum and find sufficient conditions which guarantee the existence of geometrically induced bound states. In particular, the discrete spectrum exists for non-negatively curved strips which are studied in detail. The general results are used to recover and revisit the known facts about quantum strips in the plane. As an example of non-positively curved quantum strips, we consider strips on ruled surfaces.

math-ph↗

arXiv2001

Bound States in Mildly Curved Layers

It has been shown recently that a nonrelativistic quantum particle constrained to a hard-wall layer of constant width built over a geodesically complete simply connected noncompact curved surface can have bound states provided the surface is not a plane. In this paper we study the weak-coupling asymptotics of these bound states, i.e. the situation when the surface is a mildly curved plane. Under suitable assumptions about regularity and decay of surface curvatures we derive the leading order in the ground-state eigenvalue expansion. The argument is based on Birman-Schwinger analysis of Schroedinger operators in a planar hard-wall layer.

math-ph↗

arXiv2001

Bound 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 $R^3$. 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.

math-ph↗

arXiv2000

Waveguides coupled through a semitransparent barrier: a Birman-Schwinger analysis

The paper is devoted to a model of a mesoscopic system consisting of a pair of parallel planar waveguides separated by an infinitely thin semitransparent boundary modeled by a transverse delta interaction. We develop the Birman-Schwinger theory for the corresponding generalized Schroedinger operator. The spectral properties become nontrivial if the barrier coupling is not invariant with respect to longitudinal translations, in particular, there are bound states if the barrier is locally more transparent in the mean and the coupling parameter reaches the same asymptotic value in both directions along the guide axis. We derive the weak-coupling expansion of the ground-state eigenvalue for the cases when the perturbation is small in the supremum and the L^1-norms. The last named result applies to the situation when the support of the leaky part shrinks: the obtained asymptotics differs from that of a double guide divided by a pierced Dirichlet barrier. We also derive an upper bound on the number of bound states.

math-ph↗