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:

  • spectral geometry
    • spectra of differential operators on manifolds
    • isoperimetric inequalities
    • spectral optimisation
    • nodal sets and hot spots of eigenfunctions

  • non-self-adjoint operators
    • spectral and pseudospectral properties
    • Schrodinger and Dirac operators
    • complex electromagnetic fields
    • damped wave equation

  • mesoscopic physics, nanostructures
    • spectral and scattering properties
    • quantum waveguides, layers, tubes
    • geometrically induced bound states
    • Hardy-type inequalities

In detail:

I am mainly (but not necessarily) interested in a mathematical study of problems, coming both from modern as well as classical physics, where significant features of geometry as regards physical properties play a crucial role.

PANORAMA

  1. Optimal shapes of vibrating membranes

    Miriam Bareket's conjecture from 1977 states that the disk maximises the principal frequency of any membrane of fixed area with elastically supported edges (modelled by the Laplacian with attractive Robin boundary conditions). In my paper with Pedro Freitas [Advances in Mathematics (2015)], we disprove the conjecture by showing that the annulus is more optimal. This provides the first known example where the extremal domain for the lowest eigenvalue of the Laplace operator is not a ball.
    membrane
  2. Twisting versus bending in quantum waveguides

    In 1989 Pavel Exner and Petr Seba discovered a surprising fact that bending acts as an attractive interaction: it gives rise to bound states (discrete eigenvalues) of the quantum Hamiltonian (Dirichlet Laplacian) in an unbounded tube. In my paper with Thomas Ekholm and Hynek Kovarik [Archive for Rational Mechanics and Analysis (2008)], we show that twisting acts as a repulsive interaction: it produces geometrically induced Hardy-type inequalities. The result has important consequences for the stability of quantum transport and a faster decay of the heat flow or the Brownian motion in twisted tubes [Journal de Mathematiques Pures et Appliquees (2010), with Enrique Zuazua].
    twisted and bent waveguide

  3. The Brownian traveller on manifolds

    Which geometry is better to travel in? The simplest non-trivial setting to analyse this question in a stochastic setting is that of the Laplace-Beltrami operator in a tubular neighbourhood of an infinite curve in a two-dimensional surface, subject to Dirichlet boundary conditions. In my paper with Martin Kolb [Journal of Spectral Theory (2014)], we argue (based on the large-time behaviour of the heat equation) that it is the negative/positive ambient curvature which is good/bad for the Brownian motion (mathematically, due to the existence of Hardy-type inequalities or discrete spectra, respectively).

    ants


  4. Non-self-adjoint spectral theory

    Studying non-self-adjoint operators is like being a vet rather than a doctor: one has to acquire a much wider range of knowledge, and to accept that one cannot expect to have as high a rate of success when confronted with particular cases.
    [Brian Davies (2007)]

    This very pertinent judgment has been a leitmotif for my research on spectral and pseudospectral properties of non-self-adjoint operators in quantum mechanics and elsewhere. It has also been a motto for the conference series, which I have been organising with Lyonell Boulton and Petr Siegl in various places in Europe and America since 2010.

    To me, the transfer from the self-adjoint paradise to non-self-adjoint torments was like opening Pandora's box...


    Pandora
    Arthur Rackham: Pandora's box

    An important success with particular cases is given by my paper with Luca Fanelli and Lucrezia Cossetti [Communications in Mathematical Physics (2020)], where we develop the method of multipliers to obtain absence of eigenvalues of relativistic operators of Pauli and Dirac types. The case of Schrodinger operators with complex potentials is covered by my papers with Luca Fanelli and Luis Vega [Journal of Spectral Theory & Journal of Functional Analysis (2018)].

    In my papers with Petr Siegl and Tho Nguyen Duc [Journal of Functional Analysis (2019 & 2022)], we develop a first systematic construction of non-semi-classical pseudomodes for Schrodinger and Dirac operators. We cover discontinuous potentials, solving thus an open problem from a 2015 AIM workshop.

  5. Quasi-self-adjoint quantum mechanics

    In my (most cited) paper with Petr Siegl [Physical Review D (2012)], we disprove quasi-self-adjointness of the imaginary cubic (or Bender's) oscillator. More specifically, we establish the existence of a generalised metric operator (based on the completeness of eigenfunctions) which is however necessarily singular (due to the absence of Riesz-basis property).

    In another highly cited paper [Journal of Physics A (2006), with Hynek Bila and Miloslav Znojil], I introduce a new class of quasi-self-adjoint toy models: The operator acts as the one-dimensional Laplacian and the non-self-adjointness is introduced via complex Robin boundary conditions only. The simplicity allows for closed formulae of the metric operator as well as of the self-adjoint Hamiltonian. At the same time, the multidimensional generalisations lead to peculiar spectral properties.

    PT-symmetric waveguide
    Eigenvalues in a waveguide with complex Robin 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 Milos Tater.)


Last modified: 9 November 2023