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:
 mesoscopic physics, nanostructures
 spectral and scattering properties of quantum waveguides, layers
 geometrically induced bound states and Hardytype 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 nonselfadjoint generator
 stability/instability of solutions
 quantum mechanics with nonHermitian operators
 spectral theory for nonselfadjoint operators
 PTsymmetry
 nonHermitian Robintype 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.


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 (noncompact) 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 nontrivial, i.e. curved.
Furthermore, very little is known about the structure of the spectrum
of the Laplacian in noncompact 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 Hardytype inequality for the Laplacian).
I have also been interested in
quantitative properties of the eigenvalues,
consequences of the Hardytype inequality on the transport in the waveguide,
different geometries,
other types of boundary conditions,
etc.
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 nontrivial case,
i.e. if the configuration space of the waveguide is
a tubular neighbourhood of an infinite curve
in a twodimensional surface
(not necessarily embedded in the threedimensional 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 Hardytype inequality for the Laplacian).
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 socalled
isoperimetric inequalities,
which are aimed to estimate a not easily accessible physical quantity
(e.g., principal frequency of a drum,
groundstate 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 lowlying eigenvalues of the Laplace operator,
notably on noncompact or noncomplete manifolds,
subject to various boundary conditions.
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 nodalline 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 multiplyconnected regions.
The conjecture is still open for simplyconnected regions.
Moreover, very little is known in higher dimensions.
In collaboration with P.Freitas, we have proved that
 the nodalline conjecture does not hold for unbounded regions,
 the nodalline conjecture holds for thin curved tubes
(of arbitrary crosssection and in any dimension).
It should be stressed that it is for the first time
when the conjecture has been proved for nonconvex regions
without any symmetry conditions.
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
(KleinGordon equation in a curved spacetime),
viscoelasticity, etc.
Positive damping corresponds to a dissipation,
while negative "damping" models a supply of energy into the system.

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 timestability/instability of solutions
in the difficult situation of indefinite damping,
notably in the context of spectral analysis
of the (nonselfadjoint) generator
associated with the damped wave equation in unbounded regions.
Quantum mechanics with nonHermitian operators
Although quantum mechanic is conceptually a selfadjoint theory,
there are numbers of physical problems
that require the analysis of nonselfadjoint operators
(e.g., the study of resonances via the method of complex scaling,
derivation of the famous LandauZener formula
for the adiabatic transition probability between eigenstates
of a timedependent twolevel system, etc.).
Another strong impetus to go beyond the selfadjointness
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) selfadjointness.
It is often (but not always!) the case if the system is PTsymmetric,
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 selfadjoint operators, the nonselfadjoint 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 PTsymmetric quantum mechanics),
I have recently introduced and studied a new class of nonHermitian models,
whose novelty consists in that the Hamiltonian of the system
acts simply as the free Hamiltonian and a nontrivial PTsymmetric
interaction is introduced via Robintype boundary conditions
with complex coupling.

Broken PTsymmetry in a waveguide with nonHermitian 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.)


