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SUMMARY:Offline Mini-Courses in Spectral Theory and Mathematical Physics
DTSTART;VALUE=DATE-TIME:20210611T065500Z
DTEND;VALUE=DATE-TIME:20211231T163000Z
DTSTAMP;VALUE=DATE-TIME:20240725T125900Z
UID:indico-event-360@indico.eimi.ru
DESCRIPTION:Offline Mini-Courses\nin Spectral Theory and Mathematical Phys
ics\n\n11 June–31 December\, 2021\n\nThese are mini-courses for the EI
MI thematic program on Spectral Theory and Mathematical Physics (reschedu
led from 2020 due to COVID-19 pandemic). The target audience includes gra
duate\, master and senior bachelor students of any mathematical speciality
\; senior researchers are also welcome.\n\nThe meetings will be held in t
he Leonhard Euler International Mathematical Institute (Pesochnaya nab. 10
\, St. Petersburg\, Russia).\n\nPlease\, see the timetable page for the sc
hedule.\n\nVideo recordings of the talks are available on the EIMI YouTube
channel.\n\nSergey Dobrokhotov (Ishlinsky Institute for Problems in Mech
anics RAS & Moscow Institute of Physics and Technology\, Russia)\nFeynmann
–Maslov calculus of functions of noncommuting operators and applications
to adiabatic and semiclassical problems\n\n13 and 20 October\, 2021\n\nAt
the elementary level\, we discuss the basic objects and formulas in the F
eynman–Maslov operator theory about functions of non-commuting operators
. The main objects here are differential and pseudo-differential operators
with a small parameter (h-pseudo-differential operators). As a simple eff
ective application\, we consider the use of this theory in adiabatic probl
ems (in particular\, in dimension reduction problems). These include vecto
r problems (for example\, problems about wave functions in graphene)\, pro
blems about wave propagation in waveguides (for example\, waves in nanotub
es)\, problems for equations with rapidly changing coefficients (averaging
methods)\, etc. As a result of the application of the considered approach
(formulated in the form of an algorithm)\, the initial problems are reduc
ed to simpler problems described by "effective" Hamiltonians or modes\, an
d containing\, in particular\, dispersion effects. Then one can use the se
mi-classical approximation to construct asymptotic solutions of the reduce
d equations.\n\nThe first part of the lectures is devoted to elementary de
finitions and important effective formulas of operator calculus and applic
ations to vector problems (the simplest problems with an "operator-valued
symbol"). The second part is devoted to the more complicated adiabatic pro
blems mentioned above.\n\nFrédéric Klopp (IMJ-PRG\, Sorbonne Université
\, France)\nRecent results in localization\n\n15 and 29 September\, 2021\n
\nThe mini-course will be devoted to two recent results on localization. T
he first result deals with one particle localization and is joint work wit
h J.Schenker (https://arxiv.org/abs/2105.13215). It shows that\, for quite
general random models\, while localization cannot be uniform on an non em
pty open interval of energy\, a result which goes back to the 90's\, it al
most is i.e. only a small fraction of the states does not localize uniform
ly. The second result deals with many body localization. For a simple one
dimensional random Hamiltonian\, we show that\, in the thermodynamic limit
:\n\n\n at zero temperature\, for a small enough particle density\, the gr
ound state of the Hamiltonian of many fermions subjected to this one dimen
sional random Hamiltonian interacting through a compactly supported repuls
ive potential exhibits localization\;\n at positive temperature\, for a sm
all enough chemical potential\, the Gibbs state of the grand canonical ens
emble for the same Hamiltonian exhibits localization.\n\n\nHere\, in both
cases\, a state "exhibits localization" if its two particle density matrix
decays exponentially off the diagonal.\n\nVladimir Nazaikinskii (Ishlinsk
y Institute for Problems in Mechanics RAS\, Russia)\nGeometry and semiclas
sical asymptotics\n\n8 and 29 September\, 2021\n\nMaslov's canonical opera
tor is one of the most powerful tools for constructing global semiclassica
l asymptotics for linear differential equations and systems. We will outli
ne the rich geometry underlying the canonical operator (Lagrangian manifol
ds in the phase space\, focal points\, caustics\, Maslov index\, etc.) and
explain its up-to-date construction suitable not only for theoretical res
earch but also for the efficient analysis of specific problems using the c
apabilities of technical computation systems such as Wolfram Mathematica.\
n\nAlexander Pushnitski (King's College London\, UK)\nAdditive and multipl
icative Hankel and Toeplitz operators\n\n8 and 15 September\, 2021\n\nIn t
he first part\, I will discuss the classical (additive) Toeplitz and Hanke
l operators. These are operators whose matrix representations have the for
m {t(j-k)} for Toeplitz and {h(j+k)} for Hankel (here j\,k are non-negativ
e integers). In the second part\, I will discuss the multiplicative Toepli
tz and Hankel operators\; these are operators represented by infinite matr
ices of the form {t(j/k)} and {h(jk)}\, where j\,k are natural numbers. It
is well known that additive Toeplitz and Hankel operators can be naturall
y realised as operators on the Hardy space. It turns out that in a similar
way the multiplicative Toeplitz and Hankel operators can be realised as o
perators acting on a certain Hilbert space built from Dirichlet series. I
will discuss the general set-up for the theory of these classes of operato
rs and mention some key questions: boundedness\, compactness\, finite rank
property\, etc.\n\nIlya Kachkovskiy (Michigan State University\, USA)\nBo
urgain’s method of proving Anderson localization for quasiperiodic opera
tors\n\n28–30 August\, 2021\n\nAnderson localization for Schrödinger op
erators can be described as the property of having purely point spectrum w
ith exponentially decaying eigenfunctions. For random operators\, it was d
iscovered in 1958 by P.Anderson and proved rigorously in many different mo
dels. For quasiperiodic operators\, perturbative methods of proving Anders
on localization based on KAM theory were developed since 1980s. A common p
roperty of these methods is that the coupling constant at the potential ne
eds to be large enough\, depending on the Diophantine properties of the fr
equency. In particular\, one cannot find a lower bound on the coupling con
stant that would guarantee Anderson localization for a full measure set of
frequencies.\n\nIn 1998\, S.Jitomirskaya obtained optimal (in measure-the
oretical setting) non-perturbative localization results for the special ca
se of the almost Mathieu potential. Shortly afterwards\, J.Bourgain and hi
s collaborators developed a non-perturbative approach that allows to treat
one-dimensional quasiperiodic operators with arbitrary real analytic pote
ntials. This approach combines several ideas from complex analysis\, harmo
nic analysis\, Diophantine approximation and semi-algebraic geometry. Modi
fications of this approach can also be applied to multi-dimensional models
.\n\nIn this mini-course\, we will consider the simplest non-trivial case
of a quasiperiodic operator with arbitrary real analytic potential (one-di
mensional and one-frequency)\, and establish non-perturbative Anderson loc
alization for this operator using Bourgain’s method.\n\nOur goal is to b
e as self-contained as reasonably possible. Spectral theorem for bounded s
elf-adjoint operators and Schnol’s theorem will be stated without proofs
. Knowledge of basic complex analysis and basic Fourier analysis is expect
ed. No knowledge of semi-algebraic geometry will be assumed\, but several
facts will be stated without proofs.\n\nLecture notes and literature are a
vailable here: Notes and Literature.\n\nMarcin Moszyński (Uniwersytet War
szawski\, Poland)\nSpectral theory for self-adjoint cyclic operators & its
analog for "finitely cyclic" ones with rigorous introduction to matrix me
asure L² spaces\n\n19–22 July\, 2021\n\nI would like to dedicate this m
ini course to the memory of my friend Sergei Naboko.\n\nThis mini-course i
s devoted to а rigorous proof of the result\, called here "x-Multiplicati
on Unitary Equivalence" (abbr.: xMUE) Тheorem\, for finitely cyclic self-
adjoint operators\, which is an analog of the "xMUE" Theorem for self-adjo
int cyclic operators (most probably*\, it is one of the Stone's results fo
r cyclic – i.e.\,"simple spectrum" – operators).\n\nA finitely cycli
c operator is a generalisation of a cyclic operator: the cyclic ("generati
ng") vector is replaced here by a finite system of vectors (the name an "o
perator with finite multiplicity of the spectrum" is also used). To do thi
s\, we introduce and rigorously develop some foundations of the theory of
matrix measure L² spaces\, and we study spectral properties of multiplica
tion operators (by scalar functions) in such spaces.\n\nBecause of the "mi
ni" character of the course\, some proofs will be omitted during the lectu
res\, but all the details can be found in the manuscript: Part 1 and Part
2.\n\nHere you can find an extra closing recording :)) Google Drive.\n
\nLecture 1:\n\n\n Introduction\n A detailed proof of the "xMUE" Theorem f
or self-adjoint cyclic operators\n\n\nLecture 2:\n\n\n Matrix measure and
its trace measure\, and trace density\n ℂ^d – vectors function space
L^(M) with semi-scalar product\, and its "zero-layer"\n\n\nLecture 3:\n\n\
n The L²(M) Hilbert space (completeness)\n The subspaces of simple functi
ons and of "smooth" functions (density)\n\n\nLecture 4:\n\n\n Multiplicati
on operators in L²(M)\n Self-adjoint finitely cyclic operators\n\n\nLectu
re 5:\n\n\n The spectral matrix measure for a self-adjoint operator and a
system of vectors\n The canonical spectral transformation\n\n\nLecture 6:\
n\n\n Vector polynomials and "xMUE" Theorem for self-adjoint finitely cycl
ic operators\n\n\n____\n\n* According to the opinion of Sergei N.\n\nAlexa
nder Its (IUPUI\, USA & St. Petersburg University\, Russia)\nNonlinear sad
dle point method for the cylindrical KdF equation\n\n16–18 June\, 2021\n
\nWe discuss long time asymptotics of the solution to a Cauchy problem for
the nonlinear cylindrical KdF equation. Our method is based on an asympto
tic analysis of a matrix Riemann-Hilbert problem. We use a well-known vers
ion of the nonlinear steepest descent method.\n\nElena Zhizhina (Institute
for Information Transmission Problems\, Russia)\nOn periodic homogenizati
on of non-local convolution type operators\n\n11–18 June\, 2021\n\nThe c
ourse will focus on periodic homogenization of parabolic and elliptic prob
lems for integral convolution type operators\, it is based on recent resul
ts obtained in our joint works with A.Piatnitski.\n\nFirst\, we will consi
der some models of population dynamics\, which can be described in terms o
f non-local convolution type operators. In particular\, we will introduce
evolution equations for the dynamics of the first correlation function (th
e so-called density of population). We will also discuss a number of probl
ems arising in the theory of non-local convolution type operators.\n\nThen
we will turn to homogenization of elliptic equations for non-local operat
ors with a symmetric kernel. We will show that in the topology of resolven
t convergence the family of rescaled operators converges to a second order
elliptic operator with constant coefficients. The proof of convergence in
cludes the following steps:\n\n\n construction of anzats and deriving the
equation on the first corrector\;\n construction of the limit diffusive op
erator\;\n justification of the convergence.\n\n\nOur next goal is homogen
ization of parabolic problems for operators with a non-symmetric kernel. I
t will be shown that the homogenization result holds in moving coordinates
. We will find the corresponding effective velocity and obtain the limit o
perator. In the case of small antisymmetric perturbations of a symmetric k
ernel we will show that the so-called Einstein relation holds.\n\nReferenc
es:\n\n\n A.Piatnitski\, E.Zhizhina\, Periodic homogenization of non-local
operators with a convolution type kernel\, SIAM J. Math. Anal. Vol. 49\,
No. 1\, p. 64-81\, 2017.\n A.Piatnitski\, E.Zhizhina\, Homogenization of
biased convolution type operators\, Asymptotic Analysis\, 2019\, Vol. 115
\, No. 3-4\, p. 241-262\, doi:10.3233/ASY-191533\; Arxiv: 1812.00027.\n Yu
.Kondratiev\, O.Kutoviy\, S.Pirogov\, E.Zhizhina\, Invariant measures for
spatial contact model in small dimensions\, Arxiv: 1812.00795\, 29 Novembe
r 2018\, Markov Proc. Rel. Fields (2021)\, to appear.\n\n\nhttps://indico.
eimi.ru/event/360/
LOCATION:Leonhard Euler International Mathematical Institute in St. Peters
burg
URL:https://indico.eimi.ru/event/360/
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