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BEGIN:VEVENT
SUMMARY:Pavel Exner (Doppler Institute for Mathematical Physics and Applie
d Mathematics)
DTSTART;VALUE=DATE-TIME:20201103T143000Z
DTEND;VALUE=DATE-TIME:20201103T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/1
DESCRIPTION:Title: Spectral properties of spiral quantum waveguides\nby Pave
l Exner (Doppler Institute for Mathematical Physics and Applied Mathematic
s) as part of Spectral theory and related topics\n\n\nAbstract\nWe discuss
properties of a particle confined to a spiral-shaped region with Dirichle
t boundary. As a case study we analyze in detail the Archimedean spiral fo
r which the spectrum above the continuum threshold is absolutely continuou
s away from the thresholds. The subtle difference between the radial and p
erpendicular width implies\, however\, that in contrast to ‘less curved
’ waveguides\, the discrete spectrum is empty in this case. We also disc
uss modifications such a multi-arm Archimedean spirals and spiral waveguid
es with a central cavity\; in the latter case bound state already exist if
the cavity exceeds a critical size. For more general spiral regions the s
pectral nature depends on whether they are ‘expanding’ or ‘shrinking
’. The most interesting situation occurs in the asymptotically Archimede
an case where the existence of bound states depends on the direction from
which the asymptotics is reached.\n
LOCATION:https://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;VALUE=DATE-TIME:20201110T133000Z
DTEND;VALUE=DATE-TIME:20201110T143000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/2
DESCRIPTION:Title: Accumulation of resonances and eigenvalues for operators with
distant perturbations\nby Denis Borisov (Bashkir State Pedagogical Un
iversity and Institute o Mathematics UFRC RAS) as part of Spectral theory
and related topics\n\n\nAbstract\nWe consider a model one-dimensional prob
lem with distant perturbations\, for which we study a phenomenon of emergi
ng of infinitely many eigenvalues and resonances near the bottom of the es
sential spectrum. We show that they accumulate to a certain segment of the
essential spectrum. Then we discuss possible generalization of this resul
t to multi-dimensional models and various situations of resonances and eig
envalues distributions.\n\nZoom link: https://zoom.us/j/91097279226 \nFor
password please ask the organizers: fbakharev@yandex.ru\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Pankrashkin (Carl von Ossietzky University of Oldenburg
)
DTSTART;VALUE=DATE-TIME:20201117T143000Z
DTEND;VALUE=DATE-TIME:20201117T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/3
DESCRIPTION:Title: Some convergence results for Dirac operators with large param
eters\nby Konstantin Pankrashkin (Carl von Ossietzky University of Old
enburg) as part of Spectral theory and related topics\n\n\nAbstract\nWe co
nsider Euclidean Dirac operators with piecewise constant mass potentials a
nd investigate their spectra in several asymptotic regimes in which the ma
ss becomes large in some regions. If the mass jumps along a smooth interfa
ce\, then it appears that the (low-lying) discrete spectrum of such an ope
rator converges to the (low-lying) discrete spectrum of an effective opera
tor acting either on or in the interior of the interface. The effective op
erators admit a simple geometric interpretation in terms of the spin geome
try\, 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. Based on
joint works with Brice Flamencourt\, Markus Holzmann\, Andrei Moroianu\,
and Thomas Ourmieres-Bonafos.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Fedotov (St. Peterburg State University)
DTSTART;VALUE=DATE-TIME:20201124T143000Z
DTEND;VALUE=DATE-TIME:20201124T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/4
DESCRIPTION:Title: On Hierarchical Behavior of Solutions to the Maryland Equatio
n in the Semiclassical Approximation\nby Alexander Fedotov (St. Peterb
urg State University) as part of Spectral theory and related topics\n\n\nA
bstract\nWe describe a multiscale selfsimilar struture of solutions to one
of the most popular models of the almost periodic operator theory\, the d
ifference Schroedinger equation with a potential of the form a $\\ctg(b n+
c)$\, where $a$\, $b$ and $c$ are constants\, and $n$ is an integer variab
le. The talk is based on a joint work with F.Klopp.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anne-Sophie Bonnet-Ben Dhia (Institut Polytechnique de Paris)
DTSTART;VALUE=DATE-TIME:20201201T143000Z
DTEND;VALUE=DATE-TIME:20201201T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/5
DESCRIPTION:Title: A new complex frequency spectrum for the analysis of transmis
sion efficiency in waveguide-like geometries\nby Anne-Sophie Bonnet-Be
n Dhia (Institut Polytechnique de Paris) as part of Spectral theory and re
lated topics\n\n\nAbstract\nWe consider a waveguide\, with one inlet and o
ne outlet\, and some arbitrary perturbation in between. In general\, an in
going wave in the inlet will produce a reflected wave\, due to interaction
with the perturbation. Our objective is to give an answer to the followin
g important questions: what are the frequencies at which the transmission
is the best one? And in particular\, do they exist frequencies for which t
he transmission is perfect\, in the sense that nothing is propagating back
in the inlet?\n\nOur approach relies on a simple idea\, which consists in
using a complex scaling in an original manner: while the same stretching
parameter is classically used in the inlet and the outlet\, here we 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 exactly wh
at arises when the transmission is perfect. This simple idea works very we
ll\, and provides useful information on the transmission qualities of the
system\, much faster than any traditional approach. More precisely\, we de
fine a new complex spectrum which contains as real eigenvalues both the fr
equencies where perfect transmission occurs and the frequencies correspond
ing to trapped modes (also known as bound states in the continuum). In add
ition\, we also obtain complex eigenfrequencies which can be exploited to
predict frequency ranges of good transmission. Let us finally mention that
this new spectral problem is PT -symmetric for systems with mirror symmet
ry.\n\nSeveral illustrations performed with finite elements in several si
mple 2D cases will be shown.\n\nIt is a common work with Lucas Chesnel (IN
RIA) and Vincent Pagneux (CNRS).\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrii Khrabustovskyi (University of Hradec Kralove)
DTSTART;VALUE=DATE-TIME:20201208T143000Z
DTEND;VALUE=DATE-TIME:20201208T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
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 Hra
dec Kralove) as part of Spectral theory and related topics\n\n\nAbstract\n
In the talk we revisit the problem of homogenization of the Robin Laplacia
n 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 $n\\ge 2$\,
and $\\Omega_\\varepsilon$ be a perforated domain obtained by removing fro
m $\\Omega$ a family of tiny identical balls of the radius $d_\\varepsilon
=o(\\varepsilon)$ $(\\varepsilon\\to 0)$ distributed periodically with a p
eriod \\varepsilon. We denote by $\\Delta_{\\Omega_\\varepsilon\,\\alpha_\
\varepsilon}$ the Laplacian on $\\Omega_\\varepsilon$ subject to the Diric
hlet condition $u=0$ on the external boundary of $\\Omega_\\varepsilon$ an
d the Robin conditions on the boundary of the balls:\n\n \\[{\\partial u\
\over\\partial \\nu}+\\alpha_\\varepsilon u=0\,\\quad \\alpha_\\varepsilon
>0\,\\]\n\nwhere $\\nu$ is an outward-facing unit normal. By $\\Delta_\\Om
ega$ we denote the Dirichlet Laplacian on $\\Omega$. It is known (Kaizu (1
985\, 1989)\, Berlyand & Goncharenko (1990)\, Goncharenko (1997)\, Shaposh
nikova et al. (2018)) that $\\Delta_{\\Omega_\\varepsilon\,\\alpha_\\varep
silon}$ converges in a strong resolvent sense either to zero (solidifying
holes)\, to $\\Delta_\\Omega$ (fading holes) or to the operator $\\Delta_{
\\Omega}-q$ with a constant potential $q>0$ (critical case) as $\\varepsil
on\\to 0$. The form of the limiting operator depends on certain relations
between$ \\varepsilon$\, $d_\\varepsilon$ and $\\alpha_\\varepsilon$.\n\nW
e will discuss our recent improvements of these results. Namely\, for all
three cases we show the norm resolvent convergence of the above operators
and derive estimates in terms of operator norms. As an application we esta
blish the Hausdorff convergence of spectra.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frédéric Klopp (Institut de Mathématiques de Jussieu – Paris
Rive Gauche)
DTSTART;VALUE=DATE-TIME:20201215T143000Z
DTEND;VALUE=DATE-TIME:20201215T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/7
DESCRIPTION:Title: A new look at localization\nby Frédéric Klopp (Institut
de Mathématiques de Jussieu – Paris Rive Gauche) as part of Spectral t
heory and related topics\n\n\nAbstract\nThe talk is devoted to new\, impro
ved bounds for the eigenfunctions of random operators in the localized re
gime. We prove that\, in the localized regime with good probability\, ea
ch eigenfunction is exponentially decaying outside a ball of a certain ra
dius\, 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 eig
enfunctions in the system (for some positive constant $c$). Thus\, most
eigenfunctions localize on finite size balls independent of the system si
ze.\n\nWe apply our techniques to obtain decay estimates for the $k$-part
icles density matrices of eigenstates of $n$ non interacting fermionic qu
antum particles subjected to the random potential $V_\\omega$ in a large
box.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Logunov (Princeton University)
DTSTART;VALUE=DATE-TIME:20201221T133000Z
DTEND;VALUE=DATE-TIME:20201221T143000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/8
DESCRIPTION:Title: Nodal sets\, quasiconformal mappings and how to apply them to
Landis’ conjecture\nby Alexander Logunov (Princeton University) as
part of Spectral theory and related topics\n\n\nAbstract\nA while ago Nadi
rashvili proposed a beautiful idea how to attack problems on zero sets of
Laplace eigenfunctions using quasiconformal mappings\, aiming to estimate
the length of nodal sets (zero sets of eigenfunctions) on closed two-dimen
sional surfaces. The idea have not yet worked out as it was planned. Howev
er it appears to be useful for Landis' Conjecture. We will explain how to
apply the combination of quasiconformal mappings and zero sets to quantita
tive 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 ques
tions about harmonic functions. Based on a joint work with E.Malinnikova\,
N.Nadirashvili and F. Nazarov.\n\nThis will be a joint session with Saint
Petersburg V.I. Smirnov seminar on mathematical physics. You could connec
t to the session via the link: https://us02web.zoom.us/j/82147853102?pwd=V
kFVR092dVJKMHk3VWFBU3RXcThjUT09\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Sobolev (University College London)
DTSTART;VALUE=DATE-TIME:20210217T143000Z
DTEND;VALUE=DATE-TIME:20210217T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/9
DESCRIPTION:Title: On spectral properties of the one-particle density matrix
\nby Alexander Sobolev (University College London) as part of Spectral the
ory and related topics\n\n\nAbstract\nThe one-particle density matrix $\\g
amma(x\, y)$ is one of the key objects in the quantum-mechanical approxima
tion schemes. The self-adjoint operator $\\Gamma$ with the kernel $\\gamma
(x\, y)$ is trace class but a sharp estimate on the decay of its eigenvalu
es was unknown. In this talk I will present a sharp bound and an asymptoti
c formula for the eigenvalues of $\\Gamma$.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christophe Hazard (Institut Polytechnique de Paris)
DTSTART;VALUE=DATE-TIME:20210303T141500Z
DTEND;VALUE=DATE-TIME:20210303T151500Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/10
DESCRIPTION:Title: Curiosities about the spectrum of a cavity containing a nega
tive material\nby Christophe Hazard (Institut Polytechnique de Paris)
as part of Spectral theory and related topics\n\n\nAbstract\nIn electromag
netism\, a negative material is a dispersive material for which the real p
arts of the electric permittivity and/or the magnetic permeability become
negative in some frequency range(s). In the last decades\, the extraordina
ry properties of these materials have generated a great effervescence amon
g the communities of physicists and mathematicians. The aim of this talk i
s to focus on their spectral properties. Using a simple scalar two-dimensi
onal model\, we will show that negative material are responsible for vario
us unusual resonance phenomena which are related to various components of
an essential spectrum. This is a common work with Sandrine Paolantoni.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wencai Liu (Texas A&M University)
DTSTART;VALUE=DATE-TIME:20210317T141500Z
DTEND;VALUE=DATE-TIME:20210317T151500Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/11
DESCRIPTION:Title: Irreducibility of the Fermi variety for discrete periodic Sc
hr\\"odinger operators\nby Wencai Liu (Texas A&M University) as part o
f Spectral theory and related topics\n\n\nAbstract\nLet $H_0$ be a discret
e periodic Schr\\"odinger operator on $\\Z^d$:\n$$H_0=-\\Delta+V\,$$\nwhe
re $\\Delta$ is the discrete Laplacian and $V:\\Z^d\\to \\R$ is periodic.
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 Fe
rmi variety at every energy level except for the average of the potential
is irreducible (modulo periodicity) and the Fermi variety at the averag
e of the potential has at most two irreducible components (modulo period
icity). This is sharp since for $d=2$ and a constant potential $V$\, the
Fermi variety at $V$-level has exactly two irreducible components (mod
ulo periodicity). In particular\, we show that the Bloch variety is irr
educible \n(modulo periodicity) for any $d\\geq 2$.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Yuditskii (Johannes Kepler Universität Linz)
DTSTART;VALUE=DATE-TIME:20210310T141500Z
DTEND;VALUE=DATE-TIME:20210310T151500Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/12
DESCRIPTION:Title: Reflectionless canonical systems: almost periodicity and cha
racter-automorphic Fourier transforms\nby Peter Yuditskii (Johannes Ke
pler Universität Linz) as part of Spectral theory and related topics\n\n\
nAbstract\nWe develop a comprehensive theory of reflectionless canonical s
ystems with an arbitrary Dirichlet-regular Widom spectrum with the Direct
Cauchy Theorem property. This generalizes\, to an infinite gap setting\, t
he constructions of finite gap quasiperiodic (algebro-geometric) solutions
of stationary integrable hierarchies. Instead of theta functions on a com
pact Riemann surface\, the construction is based on reproducing kernels of
character-automorphic Hardy spaces in Widom domains with respect to Marti
n measure. We also construct unitary character-automorphic Fourier transfo
rms which generalize the Paley-Wiener theorem. Finally\, we find the corre
ct notion of almost periodicity which holds for canonical system parameter
s in Arov gauge\, and we prove generically optimal results for almost peri
odicity for Potapov-de Branges gauge\, and Dirac operators. Based on joint
work with Roman Bessonov and Milivoje Lukic.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Svetlana Jitomirskaya (University of California)
DTSTART;VALUE=DATE-TIME:20210224T150000Z
DTEND;VALUE=DATE-TIME:20210224T160000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/13
DESCRIPTION:Title: Spectral properties of the unbounded GPS model\nby Svetl
ana Jitomirskaya (University of California) as part of Spectral theory and
related topics\n\n\nAbstract\nWe discuss spectral properties of the unbou
nded GPS model: a family of discrete 1D Schrodinger operators with unbound
ed potential and exact mobility edge. Based on papers in progress joint wi
th Xu\, You (Nankai) and Zhao (UCI).\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Kuchment (Texas A&M University)
DTSTART;VALUE=DATE-TIME:20210407T141500Z
DTEND;VALUE=DATE-TIME:20210407T151500Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/14
DESCRIPTION:Title: The nodal mysteries\nby Peter Kuchment (Texas A&M Univer
sity) as part of Spectral theory and related topics\n\n\nAbstract\nNodal p
atterns of oscillating membranes have been known for hundreds of years. Le
onardo da Vinci\, Galileo Galilei\, and Robert Hooke have observed them. B
y the nineteenth century they acquired the name of Chladni figures. Mathem
atically\, they represent zero sets of eigenfunctions of the Laplace (or a
more general) operator. In spite of such long history\, many mysteries ab
out these patterns (even in domains of Euclidean spaces\, and even more on
manifolds) still abound and attract recent attention of leading researche
rs working in physics\, mathematics (including PDEs\, math physics\, and n
umber 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://researchseminars.org/talk/eimi_spectral_theory/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jari Taskinen (University of Helsinki)
DTSTART;VALUE=DATE-TIME:20210324T141500Z
DTEND;VALUE=DATE-TIME:20210324T151500Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/15
DESCRIPTION:Title: Spectra of the Robin-Laplace- and Steklov-problems in bounde
d\, cuspidal domains\nby Jari Taskinen (University of Helsinki) as par
t of Spectral theory and related topics\n\n\nAbstract\nIt is well-known by
works of several authors that the spectrum of the Neumann-Laplace operat
or 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 peak and
showed that the spectrum may be discrete or continuous depending on the s
harpness of the peak. Later\, we proved that the spectrum of the Robin Lap
lacian in non-Lipschitz domains may be quite pathological since\, in addit
ion\nto countably many eigenvalues\, the residual spectrum may cover the
whole complex plain. \n\nWe have recently complemented this study in two
papers\, where we consider the spectral Robin-Laplace- and Steklov-proble
ms in a bounded domain $\\Omega$ with a peak and also in\na family $\\Omeg
a_\\varepsilon$ of domains blunted at the small distance $\\varepsilon >0$
from the peak tip. The blunted domains are Lipschitz and the spectra of t
he corresponding problems on\n$\\Omega_\\varepsilon$ are discrete. We st
udy the behaviour of the discrete spectra as $\\varepsilon \\to 0$ an
d their relations with the spectrum of case with $\\Omega$. In particular
we find various subfamilies of eigenvalues which behave in different way
s (e.g. "blinking" and "stable" families") and we describe a mechanism 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://researchseminars.org/talk/eimi_spectral_theory/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Patrick Gerard (Université Paris-Saclay)
DTSTART;VALUE=DATE-TIME:20210414T141500Z
DTEND;VALUE=DATE-TIME:20210414T151500Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/16
DESCRIPTION:Title: Spectral theory of first order operators with Toeplitz coeff
icients on the circle and applications to the Benjamin-Ono equation\nb
y Patrick Gerard (Université Paris-Saclay) as part of Spectral theory and
related topics\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leonid Parnovski (University College London)
DTSTART;VALUE=DATE-TIME:20210331T141500Z
DTEND;VALUE=DATE-TIME:20210331T151500Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/17
DESCRIPTION:Title: Floating mats and sloping beaches: spectral asymptotics of t
he Steklov problem on polygons\nby Leonid Parnovski (University Colleg
e London) as part of Spectral theory and related topics\n\n\nAbstract\nI w
ill discuss asymptotic behavior of the eigenvalues of the Steklov problem
(aka Dirichlet-to-Neumann operator) on curvilinear polygons. The answer is
completely unexpected and depends on the arithmetic properties of the ang
les of the polygon.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergey Khrushchev (Satbayev University)
DTSTART;VALUE=DATE-TIME:20210421T141500Z
DTEND;VALUE=DATE-TIME:20210421T151500Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/18
DESCRIPTION:Title: Uniformly convergent Fourier series with universal power par
ts on closed subsets of measure zero\nby Sergey Khrushchev (Satbayev U
niversity) as part of Spectral theory and related topics\n\n\nAbstract\nGi
ven a closed subset $E$ of Lebesgue measure zero on the unit circle $\\mat
hbb{T}$ there is a function $f$ on $\\mathbb{T}$ with uniformly convergent
symmetric Fourier series\n\n \\[ S_n(f\,\\zeta)=\\sum_{k=-n}^n\\hat{f}(k
)\\zeta^k\\underset{\\mathbb{T}}{\\rightrightarrows} f(\\zeta)\,\\]\n\nsuc
h that for every continuous function $g$ on $E$\, there is a subsequence o
f partial power sums\n\n \\[ S^+_n(f\,\\zeta)=\\sum_{k=0}^n\\hat{f}(k)\\z
eta^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://researchseminars.org/talk/eimi_spectral_theory/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexei Poltoratski (University of Wisconsin)
DTSTART;VALUE=DATE-TIME:20210428T161500Z
DTEND;VALUE=DATE-TIME:20210428T171500Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/19
DESCRIPTION:Title: Pointwise convergence of scattering data\nby Alexei Polt
oratski (University of Wisconsin) as part of Spectral theory and related t
opics\n\n\nAbstract\nt is widely understood that the scattering transform
can be viewed as an analog of the Fourier transform in non-linear settings
. This connection brings up numerous questions on finding non-linear analo
gs of classical results of Fourier analysis. One of the fundamental result
s of classical harmonic analysis is a theorem by L. Carleson on pointwise
convergence of the Fourier series. In this talk I will discuss convergence
for the scattering data of a real Dirac system on the half-line and prese
nt an analog of Carleson's theorem for the non-linear Fourier transform.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Silva (Universidad Nacional Autónoma de México)
DTSTART;VALUE=DATE-TIME:20210505T141500Z
DTEND;VALUE=DATE-TIME:20210505T151500Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/20
DESCRIPTION:Title: A functional model for symmetric operators and its applicati
ons to spectral theory\nby Luis Silva (Universidad Nacional Autónoma
de México) as part of Spectral theory and related topics\n\n\nAbstract\nA
functional model for symmetric operators\, based on the representation th
eory developed by Krein and Straus\, is introduced for studying the spectr
al properties of the corresponding selfadjoint extensions. By this approac
h\, one makes use of results and techniques in de Branges space and the mo
ment problem theories for spectral characterization of singular differenti
al operators.\n\nThe results presented in this talk were obtained in colla
boration with Rafael del Rio\, G. Teschl\, and J. H. Toloza.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Zinchenko (University of New Mexico)
DTSTART;VALUE=DATE-TIME:20210519T161500Z
DTEND;VALUE=DATE-TIME:20210519T171500Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/21
DESCRIPTION:Title: Bounds and asymptotics for Chebyshev polynomials\nby Max
im Zinchenko (University of New Mexico) as part of Spectral theory and rel
ated topics\n\n\nAbstract\nThis year marks 200 birthday of P.L.Chebyshev.
In this talk I will give an overview of some classical as well as recent r
esults on general Chebyshev-type polynomials (i.e.\, polynomials that mini
mize sup norm over a given compact set). In particular\, I will discuss bo
unds and large degree asymptotics for such polynomials.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Comech (Texas A&M University and IITP RAS)
DTSTART;VALUE=DATE-TIME:20210512T141500Z
DTEND;VALUE=DATE-TIME:20210512T151500Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/22
DESCRIPTION:Title: Virtual levels and virtual states of Schrodinger operators\nby Andrew Comech (Texas A&M University and IITP RAS) as part of Spectr
al theory and related topics\n\n\nAbstract\nVirtual levels admit several e
quivalent characterizations:\n\n(1) there are corresponding eigenstates fr
om $L^2$ 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 eigenvalue.\n\
nWe study virtual levels in the context of Schrodinger operators\, with no
nselfadjoint potentials and in all dimensions. In particular\, we derive t
he “missing” limiting absorption principle — the estimates on the re
solvent — near the threshold in two dimensions in the case when the thre
shold is not a virtual level.\n\nThis is a joint work with Nabile Boussaid
based on the preprint arXiv:2101.11979\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Damanik (Rice University)
DTSTART;VALUE=DATE-TIME:20210526T141500Z
DTEND;VALUE=DATE-TIME:20210526T151500Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/23
DESCRIPTION:Title: Zero measure spectrum for multi-frequency Schrödinger opera
tors\nby David Damanik (Rice University) as part of Spectral theory an
d related topics\n\n\nAbstract\nBuilding on works of Berthé-Steiner-Thusw
aldner and Fogg-Nous we show that on the two-dimensional torus\, Lebesgue
almost every translation admits a natural coding such that the associated
subshift satisfies the Boshernitzan criterion. As a consequence we show th
at for these torus translations\, every quasi-periodic potential can be ap
proximated uniformly by one for which the associated Schrödinger operator
has Cantor spectrum of zero Lebesgue measure. Joint work with Jon Chaika\
, Jake Fillman\, Philipp Gohlke.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karl-Mikael Perfekt (University of Reading)
DTSTART;VALUE=DATE-TIME:20210602T141500Z
DTEND;VALUE=DATE-TIME:20210602T151500Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/24
DESCRIPTION:Title: Infinitely many embedded eigenvalues for the Neumann-Poincar
é operator in 3D\nby Karl-Mikael Perfekt (University of Reading) as p
art of Spectral theory and related topics\n\n\nAbstract\nI will discuss th
e spectral theory of the Neumann-Poincaré operator for 3D domains with ro
tationally symmetric singularities\, which is directly related to the plas
monic eigenvalue problem for such domains. I will then describe the constr
uction of some special domains for which the problem features infinitely m
any eigenvalues embedded in the essential/continuous spectrum. Several que
stions and open problems will be stated.\n\nBased on joint papers with Joh
an Helsing and with Wei Li and Stephen Shipman.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Simanek (Baylor University)
DTSTART;VALUE=DATE-TIME:20210916T143000Z
DTEND;VALUE=DATE-TIME:20210916T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/25
DESCRIPTION:Title: Universality Limits for Orthogonal Polynomials\nby Brian
Simanek (Baylor University) as part of Spectral theory and related topics
\n\n\nAbstract\nWe will consider the scaling limits of polynomial reproduc
ing kernels for measures on the real line. For many years there has been
considerable research to find the weakest assumptions that one can place o
n a measure that allows one to prove that these rescaled kernels converge
to the sinc kernel. Our main result will provide the weakest conditions t
hat have yet been found. In particular\, it will demonstrate that one onl
y needs local conditions on the measure. We will also settle a conjecture
of Avila\, Last\, and Simon by showing that convergence holds at almost e
very point in the essential support of the absolutely continuous part of t
he measure.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laurent Baratchart (INRIA)
DTSTART;VALUE=DATE-TIME:20210923T143000Z
DTEND;VALUE=DATE-TIME:20210923T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/26
DESCRIPTION:Title: Stability of periodic delay systems and harmonic transfer fu
nction\nby Laurent Baratchart (INRIA) as part of Spectral theory and r
elated topics\n\n\nAbstract\nThe Henry-Hale theorem says that a delay sys
tem with constant coefficients of the form $y(t)=\\sum_{j=1}^N a_j y(t-\\t
au_j)$ is exponentially stable if and only if $(I-\\sum_{j=1}^N e^{-z\\tau
_j})^{-1}$ is analytic in $|z|>-\\varepsilon$ for some $\\varepsilon>0$. W
e discuss an analog of this result when the $a_j$ are periodic with Hölde
r-continuous derivative\, saying that in this case exponential stability i
s equivalent to the analyticity of the so called harmonic transfer functio
n for $|z|>-\\varepsilon$\, as a function valued in operators on $L^2(T)
$ with $T$ the unit circle.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergey Denisov (University of Wisconsin–Madison)
DTSTART;VALUE=DATE-TIME:20211007T143000Z
DTEND;VALUE=DATE-TIME:20211007T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/27
DESCRIPTION:Title: Spectral theory of Jacobi matrices on trees whose coefficien
ts are generated by multiple orthogonality\nby Sergey Denisov (Univers
ity of Wisconsin–Madison) as part of Spectral theory and related topics\
n\n\nAbstract\nThe connection between Jacobi matrices and polynomials orth
ogonal on the real line is well-known. I will discuss Jacobi matrices on t
rees whose coefficients are generated by multiple orthogonal polynomials.
The spectral theory of such operators can be thoroughly studied and many s
harp asymptotical results can be obtained by employing the complex analysi
s methods (matrix Riemann-Hilbert approach). Based on join work with A. Ap
tekarev and M. Yattselev.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jake Fillman (Texas State University)
DTSTART;VALUE=DATE-TIME:20211014T143000Z
DTEND;VALUE=DATE-TIME:20211014T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/28
DESCRIPTION:Title: Almost-Periodic Schr\\"odinger Operators with Thin Spectra\nby Jake Fillman (Texas State University) as part of Spectral theory an
d related topics\n\n\nAbstract\nThe determination of the spectrum of a Sch
r\\"odinger operator is a fundamental problem in mathematical quantum mech
anics. We will discuss a series of results showing that almost-periodic Sc
hr\\"odinger operators can exhibit spectra that are remarkably thin in the
sense of Lebesgue measure and fractal dimensions. [joint work with D. Dam
anik\, A. Gorodetski\, and M. Lukic]\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roman Bessonov (St.Petersburg State University & PDMI)
DTSTART;VALUE=DATE-TIME:20210930T143000Z
DTEND;VALUE=DATE-TIME:20210930T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
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 Spectral
theory and related topics\n\n\nAbstract\nI’m going to discuss a necessa
ry and sufficient condition for the existence of wave operators of past an
d future for the unitary group generated by a one-dimensional Dirac operat
or on the positive half line. The criterion could be formulated both in te
rms of the operator potential and in terms of its spectral measure. In the
second case\, a necessary and sufficient condition for scattering coincid
es 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 spectral
measure. The proof essentially uses ideas from the theory of orthogonal p
olynomials on the unit circle\, in particular\, a formula discovered by S.
Khrushchev. \n\nPartially based on joint works with S. Denisov.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yurii Belov (St. Petersburg State University)
DTSTART;VALUE=DATE-TIME:20211021T143000Z
DTEND;VALUE=DATE-TIME:20211021T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/30
DESCRIPTION:Title: On the chain structure of de Branges spaces\nby Yurii Be
lov (St. Petersburg State University) as part of Spectral theory and relat
ed topics\n\n\nAbstract\nIt is well known that any measure \\mu (with \\in
t(1+x^2)^{-1}d\\mu(x)<\\infty) on the real line generates a chain of Hilbe
rt spaces of entire functions (de Branges spaces). These spaces are isomet
rically embedded in L^2(\\mu). We study the indivisible intervals and the
stability of exponential type in the chains of de Branges subspaces in ter
ms of the spectral measure.\nThe report is based on joint work with A. Bor
ichev (Aix-Marseille University).\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ole Brevig (University of Oslo)
DTSTART;VALUE=DATE-TIME:20211118T143000Z
DTEND;VALUE=DATE-TIME:20211118T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/31
DESCRIPTION:Title: Idempotent Fourier multipliers acting contractively on $L^p$
and $H^p$\nby Ole Brevig (University of Oslo) as part of Spectral the
ory and related topics\n\n\nAbstract\nWe describe the idempotent Fourier m
ultipliers on the $d$-dimensional torus $\\mathbb{T}^d$ which act contract
ively on $L^p$ and $H^p$. This topic constitutes a part of a larger progra
m designed to look systematically at contractive inequalities for Hardy sp
aces in one and several variables\, and is perhaps our only true success s
tory (so far). The presentation is based on joint work with Joaquim Ortega
-Cerd\\`{a} and Kristian Seip.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ari Laptev (Imperial College London)
DTSTART;VALUE=DATE-TIME:20211028T143000Z
DTEND;VALUE=DATE-TIME:20211028T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
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 topics\n\
n\nAbstract\nThe main result of this paper is a complete proof of a new Li
eb-Thirring type inequality for Jacobi matrices originally conjectured by
Hundertmark and Simon. In particular it is proved that the estimate on th
e sum of eigenvalues does not depend on the off-diagonal terms as long as
they are smaller than their asymptotic value. An interesting feature of th
e proof is that it employs a technique originally used by Hundertmark-Lapt
ev-Weidl concerning sums of singular values for compact operators. This te
chnique seems to be novel in the context of Jacobi matrices.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Rohleder (Stockholm University)
DTSTART;VALUE=DATE-TIME:20211216T143000Z
DTEND;VALUE=DATE-TIME:20211216T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/33
DESCRIPTION:Title: Eigenvalue inequalities for Laplace and Schrödinger operato
rs\nby Jonathan Rohleder (Stockholm University) as part of Spectral th
eory and related topics\n\n\nAbstract\nEigenvalues of elliptic differentia
l operators play a natural\nrole in many classical problems in physics and
they have been\ninvestigated mathematically in depth. For instance\, for
the Laplacian on\na bounded domain it is well-known that its eigenvalues c
orresponding to\na Neumann boundary condition lie below those that corresp
ond to a\nDirichlet condition. In the course of time nontrivial improvemen
ts of\nthis observation were found by Pólya\, Payne\, Levine and Weinberg
er\,\nFriedlander\, and others. In this talk we present extensions of some
of\ntheir results to further boundary conditions and to Schrödinger\nope
rators with real-valued potentials. Partially the results are joint\nworks
with Vladimir Lotoreichik and Nausica Aldeghi.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Antoine Henrot (Universite ́ de Lorraine)
DTSTART;VALUE=DATE-TIME:20211202T143000Z
DTEND;VALUE=DATE-TIME:20211202T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/34
DESCRIPTION:Title: Bounds for the first (non-trivial) Neumann eigenvalue and pa
rtial results on a nice conjecture\nby Antoine Henrot (Universite ́ d
e Lorraine) as part of Spectral theory and related topics\n\n\nAbstract\nL
et $\\mu_1(\\Omega)$ be the first non-trivial eigenvalue of the Laplace op
erator with Neumann boundary conditions. It is a classical task to look fo
r estimates of the eigenvalues involving geometric quantities like the are
a\, the perimeter\, the diameter… In this talk\, we will recall the clas
sical inequalities known for $\\mu_1$. Then we will focus on the following
conjecture: prove that $P^2(\\Omega) \\mu_1(\\Omega) \\leq 16 \\pi^2$ fo
r all plane convex domains\, the equality being achieved by the square AND
the equilateral triangle. We will prove this conjecture assuming that $\\
Omega$ has two axis of symmetry.\n\nThis is a joint work with Antoine Leme
nant and Ilaria Lucardesi (Nancy)\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Iosif Polterovich (Université de Montréal)
DTSTART;VALUE=DATE-TIME:20211209T143000Z
DTEND;VALUE=DATE-TIME:20211209T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/35
DESCRIPTION:Title: Eigenvalue inequalities on surfaces: from sharpness to stab
ility\nby Iosif Polterovich (Université de Montréal) as part of Spec
tral theory and related topics\n\n\nAbstract\nIsoperimetric inequalities f
or Laplace eigenvalues have a long history in geometric spectral theory\,
going back to the celebrated Faber-Krahn inequality for the fundamental
tone of a drum. Still\, many questions in the subject remain open\, parti
cularly in the Riemannian setting\, \nwhere interesting connections to min
imal surface theory and harmonic maps have been discovered. I will discus
s some recent advances on this topic\, including sharp bounds for highe
r eigenvalues on the 2-sphere\, as well as stability estimates for isope
rimetric eigenvalue inequalities on surfaces. The talk is based on join
t works with M. Karpukhin\, N. Nadirashvili\, M. Nahon\, A. Penskoi\, a
nd D. Stern.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rupert Frank (LMU Munich)
DTSTART;VALUE=DATE-TIME:20211125T140000Z
DTEND;VALUE=DATE-TIME:20211125T150000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/36
DESCRIPTION:Title: Eigenvalue bounds for Schrodinger operators with complex pot
entials\nby Rupert Frank (LMU Munich) as part of Spectral theory and r
elated topics\n\n\nAbstract\nWe discuss open problems and recent progress
concerning eigenvalues of Schrodinger operators with complex potentials. W
e seek bounds for individual eigenvalues or sums of them which depend on t
he potential only through some $L^p$ norm. While the analogues of these qu
estions are (almost) completely understood for real potentials\, the compl
ex case leads to completely new phenomena\, which are related to interesti
ng questions in harmonic and complex analysis.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Semyon Dyatlov (MIT)
DTSTART;VALUE=DATE-TIME:20211111T143000Z
DTEND;VALUE=DATE-TIME:20211111T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/37
DESCRIPTION:Title: What is quantum chaos?\nby Semyon Dyatlov (MIT) as part
of Spectral theory and related topics\n\n\nAbstract\nWhere do eigenfunctio
ns of the Laplacian concentrate as eigenvalues go to infinity? Do they equ
idistribute or do they concentrate in an uneven way? It turns out that 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 Ergodicity theo
rem of Shnirelman\, Colin de Verdi\\`ere\, and Zelditch\, the Quantum Uniq
ue Ergodicity conjecture of Rudnick--Sarnak\, the progress on it by Linde
nstrauss and Soundararajan\, and the entropy bounds of Anantharaman--Nonne
nmacher. I will conclude with a more recent lower bound on the mass of eig
enfunctions obtained with Jin and Nonnenmacher. It relies on a new tool ca
lled "fractal uncertainty principle" developed in the works with Bourgain
and Zahl.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Vainberg (University of North Carolina at Charlotte)
DTSTART;VALUE=DATE-TIME:20211223T143000Z
DTEND;VALUE=DATE-TIME:20211223T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/38
DESCRIPTION:Title: On the Near-Critical Behavior of Continuous Polymers\nby
Boris Vainberg (University of North Carolina at Charlotte) as part of Spe
ctral theory and related topics\n\n\nAbstract\nWe will consider a mean-fie
ld model of polymers described in terms of solutions to a parabolic equati
on with a positive potential and a coupling constant proportional to the i
nverse temperature. At the critical value of the temperature\, polymers ex
hibit a transition between folded (globular) and unfolded states (for exam
ple\, denaturation of egg white when it is boiled with the transition from
a liquid to a hard state). We will study the phase transition of polymer
s when the temperature approaches to the critical value\, and\, simultaneo
usly\, the number of monomers in a molecule goes to infinity.\n\nLet $H_\\
beta=\\frac{1}{2}\\Delta+\\beta v(x)$ and $\\beta_{\\rm cr}$ is the biffur
cation value of $\\beta$ around which the first eigenvalue $\\lambda>0$ ap
pears.\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 exterior prob
lems. \n\nMost of the presented results are joint with M. Cranston (UC Irv
ine)\, L. Koralov (UMD) and S. Molchanov (UNCC).\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simon Larson (Chalmers University of Technology)
DTSTART;VALUE=DATE-TIME:20220217T143000Z
DTEND;VALUE=DATE-TIME:20220217T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/40
DESCRIPTION:Title: On the spectrum of the Kronig–Penney model in a constant e
lectric field\nby Simon Larson (Chalmers University of Technology) as
part of Spectral theory and related topics\n\n\nAbstract\nI will discuss t
he nature of the spectrum of the one-dimensional Schr\\"odinger operators\
n$$\n - \\frac{d^2}{dx^2}-Fx + \\sum_{n \\in \\mathbb{Z}}g_n \\delta(x-n)\
n$$\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 w
e prove that if $F\\in \\pi^2 \\mathbb{Q}$ the spectrum is absolutely cont
inuous away from a discrete set of points. In the second model $g_n$ are i
ndependent random variables with mean zero and variance $\\lambda^2$. Unde
r weak assumptions on the distribution of the $g_n$ we prove that in this
setting the spectrum is almost surely pure point if $F/\\lambda^2 < 1/2$ a
nd purely singular continuous if $F/\\lambda^2> 1/2$. Based on joint work
with Rupert Frank.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Protasov (University of L’Aquila\, Moscow State Univers
ity)
DTSTART;VALUE=DATE-TIME:20220224T143000Z
DTEND;VALUE=DATE-TIME:20220224T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/42
DESCRIPTION:Title: The stability of dynamical systems with switches: a geometri
c approach\nby Vladimir Protasov (University of L’Aquila\, Moscow St
ate University) as part of Spectral theory and related topics\n\n\nAbstrac
t\nLinear switching system is a systems of ODE $x'(t) = A(t)x(t)$ with 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 sy
stem is Lyapunov asymptotically stable if its trajectory tends to zero for
every switching low $A(t)$. The stability problem has been studied in gre
at details starting with pioneering works of Molchanov\, Pyatnicky\, Opoit
sev\, etc.\, due to many engineering applications. While in case of consta
nt matrix $A$\, i.e.\, when $U$ is one-element\, the stability problem is
solved by the eigenvalues of $A$\, the systems with switches are much more
complicated. Even for two-element sets $U$\, this problem is in general a
lgorithmically undecidable (Blondel\, Tsitsiclis\, 2000). It can be solved
approximately by the Lyapunov function\, which diverges along every traje
ctory. Among them\, invariant Lyapunov functions (Barabanov norms) are esp
ecially interesting. In 2017 in a joint work with N.Guglielmi we develop a
method of construction of invariant functions. Moreover\, recently it was
proved that for a generic system\, the invariant function is unique and h
as a simple structure: it is either piecewise linear or piecewise quadrati
c. This fact is rather surprising since all specialists believed that the
general Barabanov norm possesses fractal properties and can hardly be foun
d explicitly. To solve the stability problem one needs first to discretize
the system\, and the main issue is to estimate the discretization step (t
he dwell time). We derive that estimate by the sharp constant in the Marko
v-Bernstein inequality for exponential polynomials. We present new results
in this direction and formulate several open problems.\n
LOCATION:https://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;VALUE=DATE-TIME:20220210T143000Z
DTEND;VALUE=DATE-TIME:20220210T153000Z
DTSTAMP;VALUE=DATE-TIME:20231211T002255Z
UID:eimi_spectral_theory/44
DESCRIPTION:Title: Spectral properties of zero order pseudodifferential operato
rs and applications to the NP operator in 3D elasticity\nby Grigori Ro
zenblum (Chalmers University of Technology\, Sweden\; IEMI\, Sirius Univer
sity\, Russia) as part of Spectral theory and related topics\n\n\nAbstract
\nIt is known that the Neumann-Poincaré operator $K$ in 3D elasticity is
a zero order pseudodifferential operator on a closed surface. For a homoge
neous isotropic body\, it is known that the essential spectrum of $K$ cons
ists of 3 points determined by the Lamé constants $\\lambda$\, $\\mu$ of
the material. Therefore\, the eigenvalues of $K$ can converge only to thes
e three points. We discuss a new method for the study of eigenvalues of su
ch\, polynomially compact\, pseudodifferential operators and\, in particul
ar\, find their asymptotics. The formulas for the asymptotic coefficients
are rather irrational\, however for the two-sided asymptotics of eigenvalu
es these coefficients are shown to be linear combinations of the Euler cha
racteristic and the Willmore energy of the surface with coefficients deter
mined by the Lamé constants. Some results are obtained for the eigenvalue
s of the NP operator for the case when the material of the body is non-hom
ogeneous - when the essential spectrum may consist of intervals.\n
LOCATION:https://researchseminars.org/talk/eimi_spectral_theory/44/
END:VEVENT
END:VCALENDAR