D. Krejcirik

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Publications (7)0 Total impact

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    Article: Unbounded planar domains whose second nodal line does not touch the boundary
    P. Freitas, D. Krejcirik
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    ABSTRACT: We show the existence of simply-connected unbounded planar domains for which the second nodal line of the Dirichlet Laplacian does not touch the boundary.
    12/2005;
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    Article: A lower bound to the spectral threshold in curved tubes
    P. Exner, P. Freitas, D. Krejcirik
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    ABSTRACT: We consider the Laplacian in curved tubes of arbitrary cross-section rotating together with the Frenet frame along curves in Euclidean spaces of arbitrary dimension, subject to Dirichlet boundary conditions on the cylindrical surface and Neumann conditions at the ends of the tube. We prove that the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Dirichlet Laplacian in a torus determined by the geometry of the tube. Comment: LaTeX, 13 pages; to appear in R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci
    05/2004;
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    Article: Topologically non-trivial quantum layers
    G. Carron, P. Exner, D. Krejcirik
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    ABSTRACT: Given a complete non-compact surface embedded in R^3, we consider the Dirichlet Laplacian in a layer of constant width about the surface. Using an intrinsic approach to the layer geometry, we generalise the spectral results of an original paper by Duclos et al. to the situation when the surface does not possess poles. This enables us to consider topologically more complicated layers and state new spectral results. In particular, we are interested in layers built over surfaces with handles or several cylindrically symmetric ends. We also discuss more general regions obtained by compact deformations of certain layers.
    03/2003;
  • Article: Waveguides Coupled Through a Semitransparent Barrier
    Pavel Exner, D. Krejcirík
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    ABSTRACT: 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 Schrödinger 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 L1-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.
    Reviews in Mathematical Physics - RMP. 01/2001; 13:307-334.
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    Article: Bound states in weakly deformed strips and layers
    D. Borisov, P. Exner, R. Gadylshin, D. Krejcirik
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    ABSTRACT: We consider Dirichlet Laplacians on straight strips in R^2 or layers in R^3 with a weak local deformation. First we generalize a result of Bulla et al. to the three-dimensional situation showing that weakly coupled bound states exist if the volume change induced by the deformation is positive; we also derive the leading order of the weak-coupling asymptotics. With the knowledge of the eigenvalue analytic properties, we demonstrate then an alternative method which makes it possible to evaluate the next term in the asymptotic expansion for both the strips and layers. It gives, in particular, a criterion for the bound-state existence in the critical case when the added volume is zero. Comment: LaTeX2e, 25 pages; to appear in Annales H. Poincare
    11/2000;
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    Article: Locally curved quantum layers
    P. Duclos, P. Exner, D. Krejcirik
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    ABSTRACT: 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 assumptions we prove that the particle Hamiltonian possesses geometrically induced bound states.
    11/1999;
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    Article: Quantum waveguides with a lateral semitransparent barrier: spectral and scattering properties
    P. Exner, D. Krejcirik
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    ABSTRACT: We consider a quantum particle in a waveguide which consists of an infinite straight Dirichlet strip divided by a thin semitransparent barrier on a line parallel to the walls which is modeled by a $\delta$ potential. We show that if the coupling strength of the latter is modified locally, i.e. it reaches the same asymptotic value in both directions along the line, there is always a bound state below the bottom of the essential spectrum provided the effective coupling function is attractive in the mean. The eigenvalues and eigenfunctions, as well as the scattering matrix for energies above the threshold, are found numerically by the mode-matching technique. In particular, we discuss the rate at which the ground-state energy emerges from the continuum and properties of the nodal lines. Finally, we investigate a system with a modified geometry: an infinite cylindrical surface threaded by a homogeneous magnetic field parallel to the cylinder axis. The motion on the cylinder is again constrained by a semitransparent barrier imposed on a ``seam'' parallel to the axis.
    05/1999;
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