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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:20260510T174207Z
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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