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BEGIN:VEVENT
SUMMARY:Pavel Exner (Doppler Institute for Mathematical Physics and Applie
d Mathematics)
DTSTART:20201103T143000Z
DTEND:20201103T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/1
DESCRIPTION:Title: Spectral properties of spiral quantum waveguides\n
by Pavel Exner (Doppler Institute for Mathematical Physics and Applied Mat
hematics) as part of Spectral theory and related topics\n\n\nAbstract\nWe
discuss properties of a particle confined to a spiral-shaped region with D
irichlet boundary. As a case study we analyze in detail the Archimedean sp
iral for which the spectrum above the continuum threshold is absolutely co
ntinuous away from the thresholds. The subtle difference between the radia
l and perpendicular width implies\, however\, that in contrast to ‘less
curved’ waveguides\, the discrete spectrum is empty in this case. We als
o discuss modifications such a multi-arm Archimedean spirals and spiral wa
veguides with a central cavity\; in the latter case bound state already ex
ist if the cavity exceeds a critical size. For more general spiral regions
the spectral nature depends on whether they are ‘expanding’ or ‘shr
inking’. The most interesting situation occurs in the asymptotically Arc
himedean case where the existence of bound states depends on the direction
from which the asymptotics is reached.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Denis Borisov (Bashkir State Pedagogical University and Institute
o Mathematics UFRC RAS)
DTSTART:20201110T133000Z
DTEND:20201110T143000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/2
DESCRIPTION:Title: Accumulation of resonances and eigenvalues for operato
rs with distant perturbations\nby Denis Borisov (Bashkir State Pedagog
ical University and Institute o Mathematics UFRC RAS) as part of Spectral
theory and related topics\n\n\nAbstract\nWe consider a model one-dimension
al problem with distant perturbations\, for which we study a phenomenon of
emerging of infinitely many eigenvalues and resonances near the bottom of
the essential spectrum. We show that they accumulate to a certain segment
of the essential spectrum. Then we discuss possible generalization of thi
s result to multi-dimensional models and various situations of resonances
and eigenvalues distributions.\n\nZoom link: https://zoom.us/j/91097279226
\nFor password please ask the organizers: fbakharev@yandex.ru\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Pankrashkin (Carl von Ossietzky University of Oldenburg
)
DTSTART:20201117T143000Z
DTEND:20201117T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/3
DESCRIPTION:Title: Some convergence results for Dirac operators with larg
e parameters\nby Konstantin Pankrashkin (Carl von Ossietzky University
of Oldenburg) as part of Spectral theory and related topics\n\n\nAbstract
\nWe consider Euclidean Dirac operators with piecewise constant mass poten
tials and investigate their spectra in several asymptotic regimes in which
the mass becomes large in some regions. If the mass jumps along a smooth
interface\, then it appears that the (low-lying) discrete spectrum of such
an operator converges to the (low-lying) discrete spectrum of an effectiv
e operator acting either on or in the interior of the interface. The effec
tive operators admit a simple geometric interpretation in terms of the spi
n geometry\, and the results can be extended to a class of spin manifolds
as well. Most questions remain open if the jump interface is non-smooth. B
ased on joint works with Brice Flamencourt\, Markus Holzmann\, Andrei Moro
ianu\, and Thomas Ourmieres-Bonafos.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Fedotov (St. Peterburg State University)
DTSTART:20201124T143000Z
DTEND:20201124T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/4
DESCRIPTION:Title: On Hierarchical Behavior of Solutions to the Maryland
Equation in the Semiclassical Approximation\nby Alexander Fedotov (St.
Peterburg State University) as part of Spectral theory and related topics
\n\n\nAbstract\nWe describe a multiscale selfsimilar struture of solutions
to one of the most popular models of the almost periodic operator theory\
, the difference Schroedinger equation with a potential of the form a $\\c
tg(b n+c)$\, where $a$\, $b$ and $c$ are constants\, and $n$ is an integer
variable. The talk is based on a joint work with F.Klopp.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anne-Sophie Bonnet-Ben Dhia (Institut Polytechnique de Paris)
DTSTART:20201201T143000Z
DTEND:20201201T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/5
DESCRIPTION:Title: A new complex frequency spectrum for the analysis of t
ransmission efficiency in waveguide-like geometries\nby Anne-Sophie Bo
nnet-Ben Dhia (Institut Polytechnique de Paris) as part of Spectral theory
and related topics\n\n\nAbstract\nWe consider a waveguide\, with one inle
t and one outlet\, and some arbitrary perturbation in between. In general\
, an ingoing wave in the inlet will produce a reflected wave\, due to inte
raction with the perturbation. Our objective is to give an answer to the f
ollowing important questions: what are the frequencies at which the transm
ission is the best one? And in particular\, do they exist frequencies for
which the transmission is perfect\, in the sense that nothing is propagati
ng back in the inlet?\n\nOur approach relies on a simple idea\, which cons
ists in using a complex scaling in an original manner: while the same str
etching parameter is classically used in the inlet and the outlet\, here w
e take them as two complex conjugated parameters. As a result\, we select
ingoing waves in the inlet and outgoing waves in the outlet\, which is exa
ctly what arises when the transmission is perfect. This simple idea works
very well\, and provides useful information on the transmission qualities
of the system\, much faster than any traditional approach. More precisely\
, we define a new complex spectrum which contains as real eigenvalues both
the frequencies where perfect transmission occurs and the frequencies cor
responding to trapped modes (also known as bound states in the continuum).
In addition\, we also obtain complex eigenfrequencies which can be exploi
ted to predict frequency ranges of good transmission. Let us finally menti
on that this new spectral problem is PT -symmetric for systems with mirror
symmetry.\n\nSeveral illustrations performed with finite elements in seve
ral simple 2D cases will be shown.\n\nIt is a common work with Lucas Ches
nel (INRIA) and Vincent Pagneux (CNRS).\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrii Khrabustovskyi (University of Hradec Kralove)
DTSTART:20201208T143000Z
DTEND:20201208T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/6
DESCRIPTION:Title: Homogenization of the Robin Laplacian in a domain with
small holes: operator estimates\nby Andrii Khrabustovskyi (University
of Hradec Kralove) as part of Spectral theory and related topics\n\n\nAbs
tract\nIn the talk we revisit the problem of homogenization of the Robin L
aplacian in a domain with a lot of tiny holes.\n\nLet $\\varepsilon>0$ be
a small parameter\, $\\Omega$ be an open set in $\\mathbb{R}^n$ with $n\\g
e 2$\, and $\\Omega_\\varepsilon$ be a perforated domain obtained by remov
ing from $\\Omega$ a family of tiny identical balls of the radius $d_\\var
epsilon=o(\\varepsilon)$ $(\\varepsilon\\to 0)$ distributed periodically w
ith a period \\varepsilon. We denote by $\\Delta_{\\Omega_\\varepsilon\,\\
alpha_\\varepsilon}$ the Laplacian on $\\Omega_\\varepsilon$ subject to th
e Dirichlet condition $u=0$ on the external boundary of $\\Omega_\\varepsi
lon$ and the Robin conditions on the boundary of the balls:\n\n \\[{\\par
tial u\\over\\partial \\nu}+\\alpha_\\varepsilon u=0\,\\quad \\alpha_\\var
epsilon>0\,\\]\n\nwhere $\\nu$ is an outward-facing unit normal. By $\\Del
ta_\\Omega$ we denote the Dirichlet Laplacian on $\\Omega$. It is known (K
aizu (1985\, 1989)\, Berlyand & Goncharenko (1990)\, Goncharenko (1997)\,
Shaposhnikova et al. (2018)) that $\\Delta_{\\Omega_\\varepsilon\,\\alpha_
\\varepsilon}$ converges in a strong resolvent sense either to zero (solid
ifying holes)\, to $\\Delta_\\Omega$ (fading holes) or to the operator $\\
Delta_{\\Omega}-q$ with a constant potential $q>0$ (critical case) as $\\v
arepsilon\\to 0$. The form of the limiting operator depends on certain rel
ations between$ \\varepsilon$\, $d_\\varepsilon$ and $\\alpha_\\varepsilon
$.\n\nWe will discuss our recent improvements of these results. Namely\, f
or all three cases we show the norm resolvent convergence of the above ope
rators and derive estimates in terms of operator norms. As an application
we establish the Hausdorff convergence of spectra.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frédéric Klopp (Institut de Mathématiques de Jussieu – Paris
Rive Gauche)
DTSTART:20201215T143000Z
DTEND:20201215T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/7
DESCRIPTION:Title: A new look at localization\nby Frédéric Klopp (I
nstitut de Mathématiques de Jussieu – Paris Rive Gauche) as part of Spe
ctral theory and related topics\n\n\nAbstract\nThe talk is devoted to new\
, improved bounds for the eigenfunctions of random operators in the local
ized regime. We prove that\, in the localized regime with good probabilit
y\, each eigenfunction is exponentially decaying outside a ball of a cer
tain radius\, which we call the "localization onset length." We count the
number of eigenfunctions having onset length larger than\, say\, $\\ell>
0$ and find it to be smaller than $\\exp(-c\\ell)$ times the total number
of eigenfunctions in the system (for some positive constant $c$). Thus\,
most eigenfunctions localize on finite size balls independent of the sy
stem size.\n\nWe apply our techniques to obtain decay estimates for the $
k$-particles density matrices of eigenstates of $n$ non interacting fermi
onic quantum particles subjected to the random potential $V_\\omega$ in a
large box.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Logunov (Princeton University)
DTSTART:20201221T133000Z
DTEND:20201221T143000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/8
DESCRIPTION:Title: Nodal sets\, quasiconformal mappings and how to apply
them to Landis’ conjecture\nby Alexander Logunov (Princeton Universi
ty) as part of Spectral theory and related topics\n\n\nAbstract\nA while a
go Nadirashvili proposed a beautiful idea how to attack problems on zero s
ets of Laplace eigenfunctions using quasiconformal mappings\, aiming to es
timate the length of nodal sets (zero sets of eigenfunctions) on closed tw
o-dimensional surfaces. The idea have not yet worked out as it was planned
. However it appears to be useful for Landis' Conjecture. We will explain
how to apply the combination of quasiconformal mappings and zero sets to q
uantitative properties of solutions to $\\Delta u + V u =0$ on the plane\,
where $V$ is a real\, bounded function. The method reduces some questions
about solutions to Shrodinger equation $\\Delta u + V u =0$ on the plane
to questions about harmonic functions. Based on a joint work with E.Malinn
ikova\, N.Nadirashvili and F. Nazarov.\n\nThis will be a joint session wit
h Saint Petersburg V.I. Smirnov seminar on mathematical physics. You could
connect to the session via the link: https://us02web.zoom.us/j/8214785310
2?pwd=VkFVR092dVJKMHk3VWFBU3RXcThjUT09\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Sobolev (University College London)
DTSTART:20210217T143000Z
DTEND:20210217T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/9
DESCRIPTION:Title: On spectral properties of the one-particle density mat
rix\nby Alexander Sobolev (University College London) as part of Spect
ral theory and related topics\n\n\nAbstract\nThe one-particle density matr
ix $\\gamma(x\, y)$ is one of the key objects in the quantum-mechanical ap
proximation schemes. The self-adjoint operator $\\Gamma$ with the kernel $
\\gamma(x\, y)$ is trace class but a sharp estimate on the decay of its ei
genvalues was unknown. In this talk I will present a sharp bound and an as
ymptotic formula for the eigenvalues of $\\Gamma$.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christophe Hazard (Institut Polytechnique de Paris)
DTSTART:20210303T141500Z
DTEND:20210303T151500Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/10
DESCRIPTION:Title: Curiosities about the spectrum of a cavity containing
a negative material\nby Christophe Hazard (Institut Polytechnique de
Paris) as part of Spectral theory and related topics\n\n\nAbstract\nIn ele
ctromagnetism\, a negative material is a dispersive material for which the
real parts of the electric permittivity and/or the magnetic permeability
become negative in some frequency range(s). In the last decades\, the extr
aordinary properties of these materials have generated a great effervescen
ce among the communities of physicists and mathematicians. The aim of this
talk is to focus on their spectral properties. Using a simple scalar two-
dimensional model\, we will show that negative material are responsible fo
r various unusual resonance phenomena which are related to various compone
nts of an essential spectrum. This is a common work with Sandrine Paolanto
ni.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wencai Liu (Texas A&M University)
DTSTART:20210317T141500Z
DTEND:20210317T151500Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/11
DESCRIPTION:Title: Irreducibility of the Fermi variety for discrete peri
odic Schr\\"odinger operators\nby Wencai Liu (Texas A&M University) as
part of Spectral theory and related topics\n\n\nAbstract\nLet $H_0$ be a
discrete periodic Schr\\"odinger operator on $\\Z^d$:\n$$H_0=-\\Delta+V\,
$$\nwhere $\\Delta$ is the discrete Laplacian and $V:\\Z^d\\to \\R$ is per
iodic. We prove that for any $d\\geq3$\, the Fermi variety at every energy
level is irreducible (modulo periodicity). For $d=2$\, we prove that
the Fermi variety at every energy level except for the average of the po
tential is irreducible (modulo periodicity) and the Fermi variety at the
average of the potential has at most two irreducible components (modulo
periodicity). This is sharp since for $d=2$ and a constant potential $V
$\, the Fermi variety at $V$-level has exactly two irreducible componen
ts (modulo periodicity). In particular\, we show that the Bloch variety
is irreducible \n(modulo periodicity) for any $d\\geq 2$.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Yuditskii (Johannes Kepler Universität Linz)
DTSTART:20210310T141500Z
DTEND:20210310T151500Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/12
DESCRIPTION:Title: Reflectionless canonical systems: almost periodicity
and character-automorphic Fourier transforms\nby Peter Yuditskii (Joha
nnes Kepler Universität Linz) as part of Spectral theory and related topi
cs\n\n\nAbstract\nWe develop a comprehensive theory of reflectionless cano
nical systems with an arbitrary Dirichlet-regular Widom spectrum with the
Direct Cauchy Theorem property. This generalizes\, to an infinite gap sett
ing\, the constructions of finite gap quasiperiodic (algebro-geometric) so
lutions of stationary integrable hierarchies. Instead of theta functions o
n a compact Riemann surface\, the construction is based on reproducing ker
nels of character-automorphic Hardy spaces in Widom domains with respect t
o Martin measure. We also construct unitary character-automorphic Fourier
transforms which generalize the Paley-Wiener theorem. Finally\, we find th
e correct notion of almost periodicity which holds for canonical system pa
rameters in Arov gauge\, and we prove generically optimal results for almo
st periodicity for Potapov-de Branges gauge\, and Dirac operators. Based o
n joint work with Roman Bessonov and Milivoje Lukic.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Svetlana Jitomirskaya (University of California)
DTSTART:20210224T150000Z
DTEND:20210224T160000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/13
DESCRIPTION:Title: Spectral properties of the unbounded GPS model\nb
y Svetlana Jitomirskaya (University of California) as part of Spectral the
ory and related topics\n\n\nAbstract\nWe discuss spectral properties of th
e unbounded GPS model: a family of discrete 1D Schrodinger operators with
unbounded potential and exact mobility edge. Based on papers in progress j
oint with Xu\, You (Nankai) and Zhao (UCI).\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Kuchment (Texas A&M University)
DTSTART:20210407T141500Z
DTEND:20210407T151500Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/14
DESCRIPTION:Title: The nodal mysteries\nby Peter Kuchment (Texas A&M
University) as part of Spectral theory and related topics\n\n\nAbstract\n
Nodal patterns of oscillating membranes have been known for hundreds of ye
ars. Leonardo da Vinci\, Galileo Galilei\, and Robert Hooke have observed
them. By the nineteenth century they acquired the name of Chladni figures.
Mathematically\, they represent zero sets of eigenfunctions of the Laplac
e (or a more general) operator. In spite of such long history\, many myste
ries about these patterns (even in domains of Euclidean spaces\, and even
more on manifolds) still abound and attract recent attention of leading re
searchers working in physics\, mathematics (including PDEs\, math physics\
, and number theory) and even medical imaging. The talk will survey these
issues\, with concentration on some recent results. No prior knowledge is
assumed.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jari Taskinen (University of Helsinki)
DTSTART:20210324T141500Z
DTEND:20210324T151500Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/15
DESCRIPTION:Title: Spectra of the Robin-Laplace- and Steklov-problems in
bounded\, cuspidal domains\nby Jari Taskinen (University of Helsinki)
as part of Spectral theory and related topics\n\n\nAbstract\nIt is well-k
nown by works of several authors that the spectrum of the Neumann-Laplace
operator may be non-discrete even in bounded domains\, if the boundary
of the domain has some irregularities. In the same direction\, in a paper
in 2008 with S.A. Nazarov we considered the Steklov spectral problem in a
bounded domain $\\Omega \\subset \\mathbb{R}^n$\, $n \\geq 2$\, with a pe
ak and showed that the spectrum may be discrete or continuous depending o
n the sharpness of the peak. Later\, we proved that the spectrum of the Ro
bin Laplacian in non-Lipschitz domains may be quite pathological since\, i
n addition\nto countably many eigenvalues\, the residual spectrum may co
ver the whole complex plain. \n\nWe have recently complemented this study
in two papers\, where we consider the spectral Robin-Laplace- and Steklov
-problems in a bounded domain $\\Omega$ with a peak and also in\na family
$\\Omega_\\varepsilon$ of domains blunted at the small distance $\\varepsi
lon >0$ from the peak tip. The blunted domains are Lipschitz and the spect
ra of the corresponding problems on\n$\\Omega_\\varepsilon$ are discrete.
We study the behaviour of the discrete spectra as $\\varepsilon \\to
0$ and their relations with the spectrum of case with $\\Omega$. In par
ticular we find various subfamilies of eigenvalues which behave in differ
ent ways (e.g. "blinking" and "stable" families") and we describe a mechan
ism how the discrete spectra turn into the continuous one in this process.
\n\n The work is a co-operation with Sergei A. Nazarov (St. Petersburg)
and Nicolas Popoff (Bordeaux).\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Patrick Gerard (Université Paris-Saclay)
DTSTART:20210414T141500Z
DTEND:20210414T151500Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/16
DESCRIPTION:Title: Spectral theory of first order operators with Toeplit
z coefficients on the circle and applications to the Benjamin-Ono equation
\nby Patrick Gerard (Université Paris-Saclay) as part of Spectral the
ory and related topics\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leonid Parnovski (University College London)
DTSTART:20210331T141500Z
DTEND:20210331T151500Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/17
DESCRIPTION:Title: Floating mats and sloping beaches: spectral asymptoti
cs of the Steklov problem on polygons\nby Leonid Parnovski (University
College London) as part of Spectral theory and related topics\n\n\nAbstra
ct\nI will discuss asymptotic behavior of the eigenvalues of the Steklov p
roblem (aka Dirichlet-to-Neumann operator) on curvilinear polygons. The an
swer is completely unexpected and depends on the arithmetic properties of
the angles of the polygon.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergey Khrushchev (Satbayev University)
DTSTART:20210421T141500Z
DTEND:20210421T151500Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/18
DESCRIPTION:Title: Uniformly convergent Fourier series with universal po
wer parts on closed subsets of measure zero\nby Sergey Khrushchev (Sat
bayev University) as part of Spectral theory and related topics\n\n\nAbstr
act\nGiven a closed subset $E$ of Lebesgue measure zero on the unit circle
$\\mathbb{T}$ there is a function $f$ on $\\mathbb{T}$ with uniformly con
vergent symmetric Fourier series\n\n \\[ S_n(f\,\\zeta)=\\sum_{k=-n}^n\\h
at{f}(k)\\zeta^k\\underset{\\mathbb{T}}{\\rightrightarrows} f(\\zeta)\,\\]
\n\nsuch that for every continuous function $g$ on $E$\, there is a subseq
uence of partial power sums\n\n \\[ S^+_n(f\,\\zeta)=\\sum_{k=0}^n\\hat{f
}(k)\\zeta^k\\]\n\nof $f$\, which converges to $g$ uniformly on $E$. Here\
n\n \\[ \\hat{f}(k)=\\int_{\\mathbb{T}}\\bar{\\zeta}^kf(\\zeta)\\\, dm(\\
zeta)\,\\]\n\nand $m$ is the normalized Lebesgue measure on $\\mathbb{T}$.
\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexei Poltoratski (University of Wisconsin)
DTSTART:20210428T161500Z
DTEND:20210428T171500Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/19
DESCRIPTION:Title: Pointwise convergence of scattering data\nby Alex
ei Poltoratski (University of Wisconsin) as part of Spectral theory and re
lated topics\n\n\nAbstract\nt is widely understood that the scattering tra
nsform can be viewed as an analog of the Fourier transform in non-linear s
ettings. This connection brings up numerous questions on finding non-linea
r analogs of classical results of Fourier analysis. One of the fundamental
results of classical harmonic analysis is a theorem by L. Carleson on poi
ntwise convergence of the Fourier series. In this talk I will discuss conv
ergence for the scattering data of a real Dirac system on the half-line an
d present an analog of Carleson's theorem for the non-linear Fourier trans
form.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Silva (Universidad Nacional Autónoma de México)
DTSTART:20210505T141500Z
DTEND:20210505T151500Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/20
DESCRIPTION:Title: A functional model for symmetric operators and its ap
plications to spectral theory\nby Luis Silva (Universidad Nacional Aut
ónoma de México) as part of Spectral theory and related topics\n\n\nAbst
ract\nA functional model for symmetric operators\, based on the representa
tion theory developed by Krein and Straus\, is introduced for studying the
spectral properties of the corresponding selfadjoint extensions. By this
approach\, one makes use of results and techniques in de Branges space and
the moment problem theories for spectral characterization of singular dif
ferential operators.\n\nThe results presented in this talk were obtained i
n collaboration with Rafael del Rio\, G. Teschl\, and J. H. Toloza.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Zinchenko (University of New Mexico)
DTSTART:20210519T161500Z
DTEND:20210519T171500Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/21
DESCRIPTION:Title: Bounds and asymptotics for Chebyshev polynomials\
nby Maxim Zinchenko (University of New Mexico) as part of Spectral theory
and related topics\n\n\nAbstract\nThis year marks 200 birthday of P.L.Cheb
yshev. In this talk I will give an overview of some classical as well as r
ecent results on general Chebyshev-type polynomials (i.e.\, polynomials th
at minimize sup norm over a given compact set). In particular\, I will dis
cuss bounds and large degree asymptotics for such polynomials.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Comech (Texas A&M University and IITP RAS)
DTSTART:20210512T141500Z
DTEND:20210512T151500Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/22
DESCRIPTION:Title: Virtual levels and virtual states of Schrodinger oper
ators\nby Andrew Comech (Texas A&M University and IITP RAS) as part of
Spectral theory and related topics\n\n\nAbstract\nVirtual levels admit se
veral equivalent characterizations:\n\n(1) there are corresponding eigenst
ates from $L^2$ or a space “slightly weaker” than $L^2$\;\n\n(2) there
is no limiting absorption principle in the vicinity of a virtual level (e
.g. no weights such that the “sandwiched” resolvent remains uniformly
bounded)\;\n\n(3) an arbitrarily small perturbation can produce an eigenva
lue.\n\nWe study virtual levels in the context of Schrodinger operators\,
with nonselfadjoint potentials and in all dimensions. In particular\, we d
erive the “missing” limiting absorption principle — the estimates on
the resolvent — near the threshold in two dimensions in the case when t
he threshold is not a virtual level.\n\nThis is a joint work with Nabile B
oussaid based on the preprint arXiv:2101.11979\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Damanik (Rice University)
DTSTART:20210526T141500Z
DTEND:20210526T151500Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/23
DESCRIPTION:Title: Zero measure spectrum for multi-frequency Schrödinge
r operators\nby David Damanik (Rice University) as part of Spectral th
eory and related topics\n\n\nAbstract\nBuilding on works of Berthé-Steine
r-Thuswaldner and Fogg-Nous we show that on the two-dimensional torus\, Le
besgue almost every translation admits a natural coding such that the asso
ciated subshift satisfies the Boshernitzan criterion. As a consequence we
show that for these torus translations\, every quasi-periodic potential ca
n be approximated uniformly by one for which the associated Schrödinger o
perator has Cantor spectrum of zero Lebesgue measure. Joint work with Jon
Chaika\, Jake Fillman\, Philipp Gohlke.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karl-Mikael Perfekt (University of Reading)
DTSTART:20210602T141500Z
DTEND:20210602T151500Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/24
DESCRIPTION:Title: Infinitely many embedded eigenvalues for the Neumann-
Poincaré operator in 3D\nby Karl-Mikael Perfekt (University of Readin
g) as part of Spectral theory and related topics\n\n\nAbstract\nI will dis
cuss the spectral theory of the Neumann-Poincaré operator for 3D domains
with rotationally symmetric singularities\, which is directly related to t
he plasmonic eigenvalue problem for such domains. I will then describe the
construction of some special domains for which the problem features infin
itely many eigenvalues embedded in the essential/continuous spectrum. Seve
ral questions and open problems will be stated.\n\nBased on joint papers w
ith Johan Helsing and with Wei Li and Stephen Shipman.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Simanek (Baylor University)
DTSTART:20210916T143000Z
DTEND:20210916T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/25
DESCRIPTION:Title: Universality Limits for Orthogonal Polynomials\nb
y Brian Simanek (Baylor University) as part of Spectral theory and related
topics\n\n\nAbstract\nWe will consider the scaling limits of polynomial r
eproducing kernels for measures on the real line. For many years there ha
s been considerable research to find the weakest assumptions that one can
place on a measure that allows one to prove that these rescaled kernels co
nverge to the sinc kernel. Our main result will provide the weakest condi
tions that have yet been found. In particular\, it will demonstrate that
one only needs local conditions on the measure. We will also settle a con
jecture of Avila\, Last\, and Simon by showing that convergence holds at a
lmost every point in the essential support of the absolutely continuous pa
rt of the measure.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laurent Baratchart (INRIA)
DTSTART:20210923T143000Z
DTEND:20210923T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/26
DESCRIPTION:Title: Stability of periodic delay systems and harmonic tran
sfer function\nby Laurent Baratchart (INRIA) as part of Spectral theor
y and related topics\n\n\nAbstract\nThe Henry-Hale theorem says that a de
lay system with constant coefficients of the form $y(t)=\\sum_{j=1}^N a_j
y(t-\\tau_j)$ is exponentially stable if and only if $(I-\\sum_{j=1}^N e^{
-z\\tau_j})^{-1}$ is analytic in $|z|>-\\varepsilon$ for some $\\varepsilo
n>0$. We discuss an analog of this result when the $a_j$ are periodic with
Hölder-continuous derivative\, saying that in this case exponential stab
ility is equivalent to the analyticity of the so called harmonic transfer
function for $|z|>-\\varepsilon$\, as a function valued in operators on
$L^2(T)$ with $T$ the unit circle.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergey Denisov (University of Wisconsin–Madison)
DTSTART:20211007T143000Z
DTEND:20211007T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/27
DESCRIPTION:Title: Spectral theory of Jacobi matrices on trees whose coe
fficients are generated by multiple orthogonality\nby Sergey Denisov (
University of Wisconsin–Madison) as part of Spectral theory and related
topics\n\n\nAbstract\nThe connection between Jacobi matrices and polynomia
ls orthogonal on the real line is well-known. I will discuss Jacobi matric
es on trees whose coefficients are generated by multiple orthogonal polyno
mials. The spectral theory of such operators can be thoroughly studied and
many sharp asymptotical results can be obtained by employing the complex
analysis methods (matrix Riemann-Hilbert approach). Based on join work wit
h A. Aptekarev and M. Yattselev.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jake Fillman (Texas State University)
DTSTART:20211014T143000Z
DTEND:20211014T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/28
DESCRIPTION:Title: Almost-Periodic Schr\\"odinger Operators with Thin Sp
ectra\nby Jake Fillman (Texas State University) as part of Spectral th
eory and related topics\n\n\nAbstract\nThe determination of the spectrum o
f a Schr\\"odinger operator is a fundamental problem in mathematical quant
um mechanics. We will discuss a series of results showing that almost-peri
odic Schr\\"odinger operators can exhibit spectra that are remarkably thin
in the sense of Lebesgue measure and fractal dimensions. [joint work with
D. Damanik\, A. Gorodetski\, and M. Lukic]\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roman Bessonov (St.Petersburg State University & PDMI)
DTSTART:20210930T143000Z
DTEND:20210930T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/29
DESCRIPTION:Title: The logarithmic integral and Möller wave operators\nby Roman Bessonov (St.Petersburg State University & PDMI) as part of S
pectral theory and related topics\n\n\nAbstract\nI’m going to discuss a
necessary and sufficient condition for the existence of wave operators of
past and future for the unitary group generated by a one-dimensional Dirac
operator on the positive half line. The criterion could be formulated bot
h in terms of the operator potential and in terms of its spectral measure.
In the second case\, a necessary and sufficient condition for scattering
coincides with the finiteness of the Szegő logarithmic integral\n$$\n \
\int_{R} \\frac{\\log w}{1+x^2}dx > - \\infty\n$$\nof the density of the s
pectral measure. The proof essentially uses ideas from the theory of ortho
gonal polynomials on the unit circle\, in particular\, a formula discovere
d by S. Khrushchev. \n\nPartially based on joint works with S. Denisov.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yurii Belov (St. Petersburg State University)
DTSTART:20211021T143000Z
DTEND:20211021T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/30
DESCRIPTION:Title: On the chain structure of de Branges spaces\nby Y
urii Belov (St. Petersburg State University) as part of Spectral theory an
d related topics\n\n\nAbstract\nIt is well known that any measure \\mu (wi
th \\int(1+x^2)^{-1}d\\mu(x)<\\infty) on the real line generates a chain o
f Hilbert spaces of entire functions (de Branges spaces). These spaces are
isometrically embedded in L^2(\\mu). We study the indivisible intervals a
nd the stability of exponential type in the chains of de Branges subspaces
in terms of the spectral measure.\nThe report is based on joint work with
A. Borichev (Aix-Marseille University).\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ole Brevig (University of Oslo)
DTSTART:20211118T143000Z
DTEND:20211118T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/31
DESCRIPTION:Title: Idempotent Fourier multipliers acting contractively o
n $L^p$ and $H^p$\nby Ole Brevig (University of Oslo) as part of Spect
ral theory and related topics\n\n\nAbstract\nWe describe the idempotent Fo
urier multipliers on the $d$-dimensional torus $\\mathbb{T}^d$ which act c
ontractively on $L^p$ and $H^p$. This topic constitutes a part of a larger
program designed to look systematically at contractive inequalities for H
ardy spaces in one and several variables\, and is perhaps our only true su
ccess story (so far). The presentation is based on joint work with Joaquim
Ortega-Cerd\\`{a} and Kristian Seip.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ari Laptev (Imperial College London)
DTSTART:20211028T143000Z
DTEND:20211028T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/32
DESCRIPTION:Title: On a conjecture by Hundertmark and Simon\nby Ari
Laptev (Imperial College London) as part of Spectral theory and related to
pics\n\n\nAbstract\nThe main result of this paper is a complete proof of a
new Lieb-Thirring type inequality for Jacobi matrices originally conjectu
red by Hundertmark and Simon. In particular it is proved that the estimat
e on the sum of eigenvalues does not depend on the off-diagonal terms as l
ong as they are smaller than their asymptotic value. An interesting featur
e of the proof is that it employs a technique originally used by Hundertma
rk-Laptev-Weidl concerning sums of singular values for compact operators.
This technique seems to be novel in the context of Jacobi matrices.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Rohleder (Stockholm University)
DTSTART:20211216T143000Z
DTEND:20211216T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/33
DESCRIPTION:Title: Eigenvalue inequalities for Laplace and Schrödinger
operators\nby Jonathan Rohleder (Stockholm University) as part of Spec
tral theory and related topics\n\n\nAbstract\nEigenvalues of elliptic diff
erential operators play a natural\nrole in many classical problems in phys
ics and they have been\ninvestigated mathematically in depth. For instance
\, for the Laplacian on\na bounded domain it is well-known that its eigenv
alues corresponding to\na Neumann boundary condition lie below those that
correspond to a\nDirichlet condition. In the course of time nontrivial imp
rovements of\nthis observation were found by Pólya\, Payne\, Levine and W
einberger\,\nFriedlander\, and others. In this talk we present extensions
of some of\ntheir results to further boundary conditions and to Schröding
er\noperators with real-valued potentials. Partially the results are joint
\nworks with Vladimir Lotoreichik and Nausica Aldeghi.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Antoine Henrot (Universite ́ de Lorraine)
DTSTART:20211202T143000Z
DTEND:20211202T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/34
DESCRIPTION:Title: Bounds for the first (non-trivial) Neumann eigenvalue
and partial results on a nice conjecture\nby Antoine Henrot (Universi
te ́ de Lorraine) as part of Spectral theory and related topics\n\n\nAbst
ract\nLet $\\mu_1(\\Omega)$ be the first non-trivial eigenvalue of the Lap
lace operator with Neumann boundary conditions. It is a classical task to
look for estimates of the eigenvalues involving geometric quantities like
the area\, the perimeter\, the diameter… In this talk\, we will recall t
he classical inequalities known for $\\mu_1$. Then we will focus on the fo
llowing conjecture: prove that $P^2(\\Omega) \\mu_1(\\Omega) \\leq 16 \\p
i^2$ for all plane convex domains\, the equality being achieved by the squ
are AND the equilateral triangle. We will prove this conjecture assuming t
hat $\\Omega$ has two axis of symmetry.\n\nThis is a joint work with Antoi
ne Lemenant and Ilaria Lucardesi (Nancy)\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Iosif Polterovich (Université de Montréal)
DTSTART:20211209T143000Z
DTEND:20211209T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/35
DESCRIPTION:Title: Eigenvalue inequalities on surfaces: from sharpness t
o stability\nby Iosif Polterovich (Université de Montréal) as part
of Spectral theory and related topics\n\n\nAbstract\nIsoperimetric inequal
ities for Laplace eigenvalues have a long history in geometric spectral t
heory\, going back to the celebrated Faber-Krahn inequality for the funda
mental tone of a drum. Still\, many questions in the subject remain open\
, particularly in the Riemannian setting\, \nwhere interesting connections
to minimal surface theory and harmonic maps have been discovered. I will
discuss some recent advances on this topic\, including sharp bounds fo
r higher eigenvalues on the 2-sphere\, as well as stability estimates fo
r isoperimetric eigenvalue inequalities on surfaces. The talk is based
on joint works with M. Karpukhin\, N. Nadirashvili\, M. Nahon\, A. Pens
koi\, and D. Stern.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rupert Frank (LMU Munich)
DTSTART:20211125T140000Z
DTEND:20211125T150000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/36
DESCRIPTION:Title: Eigenvalue bounds for Schrodinger operators with comp
lex potentials\nby Rupert Frank (LMU Munich) as part of Spectral theor
y and related topics\n\n\nAbstract\nWe discuss open problems and recent pr
ogress concerning eigenvalues of Schrodinger operators with complex potent
ials. We seek bounds for individual eigenvalues or sums of them which depe
nd on the potential only through some $L^p$ norm. While the analogues of t
hese questions are (almost) completely understood for real potentials\, th
e complex case leads to completely new phenomena\, which are related to in
teresting questions in harmonic and complex analysis.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Semyon Dyatlov (MIT)
DTSTART:20211111T143000Z
DTEND:20211111T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/37
DESCRIPTION:Title: What is quantum chaos?\nby Semyon Dyatlov (MIT) a
s part of Spectral theory and related topics\n\n\nAbstract\nWhere do eigen
functions of the Laplacian concentrate as eigenvalues go to infinity? Do t
hey equidistribute or do they concentrate in an uneven way? It turns out t
hat the answer depends on the nature of the geodesic flow. I will discuss
various results in the case when the flow is chaotic: the Quantum Ergodici
ty theorem of Shnirelman\, Colin de Verdi\\`ere\, and Zelditch\, the Quant
um Unique Ergodicity conjecture of Rudnick--Sarnak\, the progress on it b
y Lindenstrauss and Soundararajan\, and the entropy bounds of Anantharaman
--Nonnenmacher. I will conclude with a more recent lower bound on the mass
of eigenfunctions obtained with Jin and Nonnenmacher. It relies on a new
tool called "fractal uncertainty principle" developed in the works with Bo
urgain and Zahl.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Vainberg (University of North Carolina at Charlotte)
DTSTART:20211223T143000Z
DTEND:20211223T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/38
DESCRIPTION:Title: On the Near-Critical Behavior of Continuous Polymers<
/a>\nby Boris Vainberg (University of North Carolina at Charlotte) as part
of Spectral theory and related topics\n\n\nAbstract\nWe will consider a m
ean-field model of polymers described in terms of solutions to a parabolic
equation with a positive potential and a coupling constant proportional t
o the inverse temperature. At the critical value of the temperature\, poly
mers exhibit a transition between folded (globular) and unfolded states (f
or example\, denaturation of egg white when it is boiled with the transiti
on from a liquid to a hard state). We will study the phase transition of
polymers when the temperature approaches to the critical value\, and\, sim
ultaneously\, the number of monomers in a molecule goes to infinity.\n\nLe
t $H_\\beta=\\frac{1}{2}\\Delta+\\beta v(x)$ and $\\beta_{\\rm cr}$ is the
biffurcation value of $\\beta$ around which the first eigenvalue $\\lambd
a>0$ appears.\nWe used the detailed analysis of the resolvent $(H_\\beta-\
\lambda)^{-1}$ when $\\beta \\to\\beta _{cr}$ and simultaneously $\\lambda
\\to 0$. \n\nWe also will discuss the critical value for elliptic exteri
or problems. \n\nMost of the presented results are joint with M. Cranston
(UC Irvine)\, L. Koralov (UMD) and S. Molchanov (UNCC).\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simon Larson (Chalmers University of Technology)
DTSTART:20220217T143000Z
DTEND:20220217T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/40
DESCRIPTION:Title: On the spectrum of the Kronig–Penney model in a con
stant electric field\nby Simon Larson (Chalmers University of Technolo
gy) as part of Spectral theory and related topics\n\n\nAbstract\nI will di
scuss the nature of the spectrum of the one-dimensional Schr\\"odinger ope
rators\n$$\n - \\frac{d^2}{dx^2}-Fx + \\sum_{n \\in \\mathbb{Z}}g_n \\delt
a(x-n)\n$$\nwith $F>0$ and two different choices of the coupling constants
$\\{g_n\\}_{n\\in \\mathbb{Z}}$. In the first model $g_n \\equiv \\lambda
$ and we prove that if $F\\in \\pi^2 \\mathbb{Q}$ the spectrum is absolute
ly continuous away from a discrete set of points. In the second model $g_n
$ are independent random variables with mean zero and variance $\\lambda^2
$. Under weak assumptions on the distribution of the $g_n$ we prove that i
n this setting the spectrum is almost surely pure point if $F/\\lambda^2 <
1/2$ and purely singular continuous if $F/\\lambda^2> 1/2$. Based on join
t work with Rupert Frank.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Protasov (University of L’Aquila\, Moscow State Univers
ity)
DTSTART:20220224T143000Z
DTEND:20220224T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/42
DESCRIPTION:Title: The stability of dynamical systems with switches: a g
eometric approach\nby Vladimir Protasov (University of L’Aquila\, Mo
scow State University) as part of Spectral theory and related topics\n\n\n
Abstract\nLinear switching system is a systems of ODE $x'(t) = A(t)x(t)$ w
ith the matrix $A(t)$ taken from a given control set $U$ independently for
each $t$. In other words\, this is a linear system with a matrix control.
The system is Lyapunov asymptotically stable if its trajectory tends to z
ero for every switching low $A(t)$. The stability problem has been studied
in great details starting with pioneering works of Molchanov\, Pyatnicky\
, Opoitsev\, etc.\, due to many engineering applications. While in case of
constant matrix $A$\, i.e.\, when $U$ is one-element\, the stability prob
lem is solved by the eigenvalues of $A$\, the systems with switches are mu
ch more complicated. Even for two-element sets $U$\, this problem is in ge
neral algorithmically undecidable (Blondel\, Tsitsiclis\, 2000). It can be
solved approximately by the Lyapunov function\, which diverges along ever
y trajectory. Among them\, invariant Lyapunov functions (Barabanov norms)
are especially interesting. In 2017 in a joint work with N.Guglielmi we de
velop a method of construction of invariant functions. Moreover\, recently
it was proved that for a generic system\, the invariant function is uniqu
e and has a simple structure: it is either piecewise linear or piecewise q
uadratic. This fact is rather surprising since all specialists believed th
at the general Barabanov norm possesses fractal properties and can hardly
be found explicitly. To solve the stability problem one needs first to dis
cretize the system\, and the main issue is to estimate the discretization
step (the dwell time). We derive that estimate by the sharp constant in th
e Markov-Bernstein inequality for exponential polynomials. We present new
results in this direction and formulate several open problems.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Grigori Rozenblum (Chalmers University of Technology\, Sweden\; IE
MI\, Sirius University\, Russia)
DTSTART:20220210T143000Z
DTEND:20220210T153000Z
DTSTAMP:20260404T095040Z
UID:eimi_spectral_theory/44
DESCRIPTION:Title: Spectral properties of zero order pseudodifferential
operators and applications to the NP operator in 3D elasticity\nby Gri
gori Rozenblum (Chalmers University of Technology\, Sweden\; IEMI\, Sirius
University\, Russia) as part of Spectral theory and related topics\n\n\nA
bstract\nIt is known that the Neumann-Poincaré operator $K$ in 3D elastic
ity is a zero order pseudodifferential operator on a closed surface. For a
homogeneous isotropic body\, it is known that the essential spectrum of $
K$ consists of 3 points determined by the Lamé constants $\\lambda$\, $\\
mu$ of the material. Therefore\, the eigenvalues of $K$ can converge only
to these three points. We discuss a new method for the study of eigenvalue
s of such\, polynomially compact\, pseudodifferential operators and\, in p
articular\, find their asymptotics. The formulas for the asymptotic coeffi
cients are rather irrational\, however for the two-sided asymptotics of ei
genvalues these coefficients are shown to be linear combinations of the Eu
ler characteristic and the Willmore energy of the surface with coefficient
s determined by the Lamé constants. Some results are obtained for the eig
envalues of the NP operator for the case when the material of the body is
non-homogeneous - when the essential spectrum may consist of intervals.\n
LOCATION:https://stable.researchseminars.org/talk/eimi_spectral_theory/44/
END:VEVENT
END:VCALENDAR