23 research outputs found
Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle
Bohr-Sommerfeld type quantization conditions of semiclassical eigenvalues for
the non-selfadjoint Zakharov-Shabat operator on the circle are derived using an
exact WKB method. The conditions are given in terms of the action associated
with the unit circle or the action associated with turning points following the
absence or presence of real turning points.Comment: 3 figures. Included additional references in the introductio
Generalized axially symmetric potentials with distributional boundary values
We study a counterpart of the classical Poisson integral for a family of
weighted Laplace differential equations in Euclidean half space, solutions of
which are known as generalized axially symmetric potentials. These potentials
appear naturally in the study of hyperbolic Brownian motion with drift. We
determine the optimal class of tempered distributions which by means of the
so-called -convolution can be extended to generalized axially
symmetric potentials. In the process, the associated Dirichlet boundary value
problem is solved, and we obtain sharp order relations for the asymptotic
growth of these extensions.Comment: 25 pages. Added references, fixed typo
Perfect partial reconstructions for multiple simultaneous sources
A major focus of research in the seismic industry of the past two decades has
been the acquisition and subsequent separation of seismic data using multiple
sources fired simultaneously. The recently introduced method of {\it signal
apparition} provides a new take on the problem by replacing the random
time-shifts usually employed to encode the different sources by fully
deterministic periodic time-shifts. In this paper we give a mathematical proof
showing that the signal apparition method results in optimally large regions in
the frequency-wavenumber space where exact separation of sources is achieved.
These regions are diamond-shaped and we prove that using any other method of
source encoding results in strictly smaller regions of exact separation. The
results are valid for arbitrary number of sources. Numerical examples for
different number of sources (three resp.~four sources) demonstrate the exact
recovery of these diamond-shaped regions. The theoretical proofs'
implementation in the field is illustrated by the results of a conducted field
test.Comment: 16 pages, 5 figures. Expanded Section 3 with an additional numerical
experiment and the results of a field test. Added reference
Hofstadter butterflies and metal/insulator transitions for moir\'e heterostructures
We consider a tight-binding model recently introduced by Timmel and Mele for
strained moir\'e heterostructures. We consider two honeycomb lattices to which
layer antisymmetric shear strain is applied to periodically modulate the
tunneling between the lattices in one distinguished direction. This effectively
reduces the model to one spatial dimension and makes it amenable to the theory
of matrix-valued quasi-periodic operators. We then study the transport and
spectral properties of this system, explaining the appearance of a
Hofstadter-type butterfly. For sufficiently incommensurable moir\'e length and
strong coupling between the lattices this leads to the occurrence of
localization phenomena
Mathematics of magic angles in a model of twisted bilayer graphene
We provide a mathematical account of the recent Physical Reviews Letter by
Tarnopolsky--Kruchkov--Vishwanath. The new contributions are a spectral
characterization of magic angles, its accurate numerical implementation and an
exponential estimate on the squeezing of all bands as the angle decreases.
Pseudospectral phenomena due to the non-hermitian nature of operators appearing
in the model play a crucial role in our analysis.Comment: We added a trace formula summing up powers of magic angle
Quenched limit theorems for random U(1) extensions of expanding maps
The Lyapunov spectra of random U(1) extensions of expanding maps on the torus
were investigated in our previous work [NW2015]. Using the result, we extend
the recent spectral approach for quenched limit theorems for expanding maps
[DFGV2018] and hyperbolic maps [DFGV2019] to our partially hyperbolic dynamics.
Quenched central limit theorems, large deviations principles and local central
limit theorems for random U(1) extensions of expanding maps on the torus are
proved via corresponding theorems for abstract random dynamical systems.Comment: 39 page
Chiral limit of twisted trilayer graphene
We initiate the mathematical study of the Bistritzer-MacDonald Hamiltonian
for twisted trilayer graphene in the chiral limit (and beyond). We develop a
spectral theoretic approach to investigate the presence of flat bands under
specific magic parameters. This allows us to derive trace formulae that show
that the tunnelling parameters that lead to flat bands are nowhere continuous
as functions of the twisting angles
On the microlocal properties of the range of systems of principal type
The purpose of this paper is to study microlocal conditions for inclusion
relations between the ranges of square systems of pseudodifferential operators
which fail to be locally solvable. The work is an extension of earlier results
for the scalar case in this direction, where analogues of results by L.
H\"ormander about inclusion relations between the ranges of first order
differential operators with coefficients in which fail to be locally
solvable were obtained. We shall study the properties of the range of systems
of principal type with constant characteristics for which condition (\Psi) is
known to be equivalent to microlocal solvability.Comment: Added Theorem 4.7, Corollary 4.8 and Lemma A.4, corrected misprints.
The paper has 40 page