David Krejcirik's page:

Professional Interests

Looking-Glass House
Then she began looking about, and noticed that what could be seen from the old room was quite common and uninteresting, but that all the rest as different as possible. For instance, the pictures on the wall next the fire seemed to be all alive, and the very clock on the chimney-piece (you know you can only see the back of it in the Looking-glass) had got the face of a little old man, and grinned at her.

In general:

  • mathematical physics
  • quantum mechanics
  • spectral theory

In particular:

  • mesoscopic physics, nanostructures
    • spectral and scattering properties of quantum waveguides, layers
    • geometrically induced bound states and Hardy-type inequalities
    • Laplacian in tubular neighbourhoods of submanifolds
    • differential geometry of tubes
    • Dirichlet, Neumann, Robin or combinations of these boundary conditions
    • singular (delta) interactions
    • Mourre's theory as a tool for analysing the essential spectrum
  • spectral geometry
    • spectra of differential operators on manifolds
    • isoperimetric inequalities, estimates of eigenvalues
    • singular asymptotic expansions for eigenvalues
    • nodal sets of eigenfunctions
  • wave equation with indefinite damping
    • spectral analysis of the non-self-adjoint generator
    • stability/instability of solutions
  • quantum mechanics with non-Hermitian operators
    • spectral theory for non-self-adjoint operators
    • PT-symmetry
    • non-Hermitian Robin-type boundary conditions

In detail:

I am mainly interested in the mathematical study of problems, coming both from modern as well as classical physics, where the significant features of geometry as regards physical properties play a crucial role.
  1. Quantum waveguides

    Modern experimental techniques of mesoscopic physics make it possible to fabricate tiny semiconductor structures (often called nanostructures) of various shapes devised and reproducible in the laboratory and yet small enough to exhibit quantum effects. The physical nature of these systems enables one to describe the quantum dynamics by a simple model in which a free particle (with an effective mass) is confined to a spatial region. Borrowing the terminology from classical systems, one then speaks of quantum waveguides or layers in the case of regions being infinitely stretched tubes.

    From the mathematical point of view, and more generally, one therefore deals with

    the Laplace operator in a tubular neighbourhood
    of a (non-compact) submanifold of a Riemannian manifold,

    subject to suitable boundary conditions. I am mainly interested in the spectrum of the Laplacian, which has the physical meaning of the outcomes of measuring of the particle energy in the waveguide. However, this is no easy task whenever the geometry is non-trivial, i.e. curved. Furthermore, very little is known about the structure of the spectrum of the Laplacian in non-compact manifolds with boundary.

    Probably the most interesting results of my research for tubes in the Euclidean space, subject to Dirichlet boundary conditions, can be summarized in the following statements:

    • bending acts as an attractive interaction (i.e. the Laplacian possesses discrete eigenvalues),
    • twisting acts as a repulsive interaction (i.e. there exists a Hardy-type inequality for the Laplacian).

    twisted and bent waveguide

    I have also been interested in quantitative properties of the eigenvalues, consequences of the Hardy-type inequality on the transport in the waveguide, different geometries, other types of boundary conditions, etc.

  2. Quantum traveller on manifolds

    The ambient manifold of a quantum waveguide is usually identified with the flat Euclidean space. This restriction is obviously due to physical reasons, however, at least from the mathematical point of view, one may be interested equally in the situations when it is a general Riemannian manifold.

    I have been interested in the simplest non-trivial case, i.e. if the configuration space of the waveguide is

    a tubular neighbourhood of an infinite curve
    in a two-dimensional surface

    (not necessarily embedded in the three-dimensional Euclidean space), subject to Dirichlet boundary conditions. This also provides an elementary setting to study the conceptual question of the effect of ambient curvature on quantum transport, containing already the main features of the problem. My principal results can be summarized as follows:

    • positive curvature hurts the transport (i.e. the Laplacian possesses discrete eigenvalues),
    • negative curvature improves the transport (i.e. there exists a Hardy-type inequality for the Laplacian).

    twisted and bent waveguide

  3. Spectral geometry

    The trend of spectral geometry is to study the interplay between the shape of a region, boundary conditions and the spectrum of an associated differential operator (Laplace, Schrodinger, etc.). Such problems have been considered for more than a century, with classical motivations coming from theories of elasticity, acoustics, electromagnetism, etc, and with the modern one coming from mesoscopic physics, inter alia.

    A special emphasis is put on the so-called isoperimetric inequalities, which are aimed to estimate a not easily accessible physical quantity (e.g., principal frequency of a drum, ground-state energy of a nanostructure device) on the basis of easily accessible geometrical data (e.g., area, perimeter, curvature). I am mainly interested in such upper and lower bounds for the low-lying eigenvalues of the Laplace operator, notably on non-compact or non-complete manifolds, subject to various boundary conditions.

    2nd circular eigenfunction 6th circular eigenfunction
    Vibrational modes of a circular membrane
    (corresponding to the 2nd and 6th eigenfunction of the Dirichlet Laplacian)
    the nodal set is composed by the stationary points in the interior of the membrane

    Probably the most beautiful (i.e. easily stated but apparently difficult to prove/disprove) conjecture I have encountered in my research is the nodal-line conjecture of Payne's from 1967: "the nodal set of a second eigenfunction of the Dirichlet Laplacian for any planar region cannot consist of a closed curve". In other words, the second fundamental frequency of any drum corresponds to vibrations of a membrane with stationary points hitting the boundary. It is known that the conjecture holds for convex regions and that there exist counterexamples for multiply-connected regions. The conjecture is still open for simply-connected regions. Moreover, very little is known in higher dimensions. In collaboration with P.Freitas, we have proved that

    • the nodal-line conjecture does not hold for unbounded regions,
    • the nodal-line conjecture holds for thin curved tubes (of arbitrary cross-section and in any dimension).

    It should be stressed that it is for the first time when the conjecture has been proved for non-convex regions without any symmetry conditions.

  4. Damped wave equation

    The wave equation with a damping term is a mathematical model for a variety of evolution processes in many areas of physics: electromagnetism (telegraph equation), relativistic quantum mechanics, cosmology (Klein-Gordon equation in a curved space-time), viscoelasticity, etc. Positive damping corresponds to a dissipation, while negative "damping" models a supply of energy into the system.

    essential spectrum
    The essential spectrum of the damped wave operator in an unbounded tube as a function of increasing uniform negative damping

    I am interested in the issue of time-stability/instability of solutions in the difficult situation of indefinite damping, notably in the context of spectral analysis of the (non-self-adjoint) generator associated with the damped wave equation in unbounded regions.

  5. Quantum mechanics with non-Hermitian operators

    Although quantum mechanic is conceptually a self-adjoint theory, there are numbers of physical problems that require the analysis of non-self-adjoint operators (e.g., the study of resonances via the method of complex scaling, derivation of the famous Landau-Zener formula for the adiabatic transition probability between eigenstates of a time-dependent two-level system, etc.). Another strong impetus to go beyond the self-adjointness in quantum mechanics comes from the recent observation that there exist a huge class of operators whose spectrum is real (therefore in principle measurable) as a consequence of certain (physical) symmetries rather than the (mathematical) self-adjointness. It is often (but not always!) the case if the system is PT-symmetric, i.e. the Schrodinger equation is invariant under a simultaneous change of spatial reflection and time reversal.

    However, in contrast to the well understood theory of self-adjoint operators, the non-self-adjoint theory can be quite different and is certainly less developed. The former is much easier to analyse because of the existence of the spectral theorem.

    Motivated by the lack of powerful techniques based on the spectral theorem (and also by the absence of many rigorous results in the literature on PT-symmetric quantum mechanics), I have recently introduced and studied a new class of non-Hermitian models, whose novelty consists in that the Hamiltonian of the system acts simply as the free Hamiltonian and a non-trivial PT-symmetric interaction is introduced via Robin-type boundary conditions with complex coupling.

    PT-symmetric waveguide
    Broken PT-symmetry in a waveguide with non-Hermitian boundary conditions.
    The emergence of real eigenvalues (blue and green balls) and complex conjugated pairs of eigenvalues (red and cyan balls) from the continuous spectrum (thick white line) and their trajectories (with apparent collisions) in the complex plane as a boundary coupling parameter increases.
    (Collaboration with M.Tater.)

Last modified: 11 January 2008