Professional Interests

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 chimneypiece
(you know you can only see the back of it in the Lookingglass) 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
 nonselfadjoint 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
 Hardytype 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

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.
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 Hardytype 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].
The Brownian traveller on manifolds
Which geometry is better to travel in?
The simplest nontrivial setting to analyse this question
in a stochastic setting is that of the LaplaceBeltrami operator
in a tubular neighbourhood of an infinite curve
in a twodimensional surface,
subject to Dirichlet boundary conditions.
In my paper with Martin Kolb
[ Journal of Spectral Theory (2014)],
we argue (based on the largetime 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 Hardytype inequalities
or discrete spectra, respectively).
Nonselfadjoint spectral theory
Studying nonselfadjoint 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 nonselfadjoint 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 selfadjoint paradise
to nonselfadjoint torments
was like opening Pandora's box...
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 nonsemiclassical
pseudomodes for Schrodinger and Dirac operators.
We cover discontinuous potentials,
solving thus an open problem from a
2015 AIM workshop.
Quasiselfadjoint quantum mechanics
In my (most cited) paper with Petr Siegl
[Physical Review D (2012)],
we disprove quasiselfadjointness 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 Rieszbasis property).
In another highly cited paper
[Journal of Physics A (2006),
with Hynek Bila and Miloslav Znojil],
I introduce a new class of quasiselfadjoint toy models:
The operator acts as
the onedimensional Laplacian and the nonselfadjointness
is introduced via complex Robin boundary conditions only.
The simplicity allows for closed formulae of the metric operator
as well as of the selfadjoint Hamiltonian.
At the same time, the multidimensional generalisations
lead to peculiar spectral properties.

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.)


