Schedule and abstracts

Sunday
May 19		 Monday
May 20		 Tuesday
May 21		 Wednesday
May 22		 Thursday
May 23
10h–10h50 Alexander Fedotov: Monodromization method and Harper equation Alexey Ilyin: Spectral inequalities on the sphere Joachim Krieger: Randomization improved Strichartz estimates and applications Elena Kopylova: Global attraction to solitary waves for Dirac equations with point interactions Alexander Its: Large time asymptotics for the cylindrical Korteweg--de Vries equation. II
10h50–11h20 Coffee and registration Coffee Coffee Coffee Coffee
11h20–12h10 Tatiana Suslina: Homogenization of hyperbolic equations with periodic coefficients Piero d'Ancona: On supercritical defocusing dispersive equations outside a ball David Stuart: Solitons and Modulation Equations in Quantum Field Theory Vladimir Georgiev: On Traveling Solitary Waves and Absence of Small Data Scattering for Nonlinear Half-Wave Equations Eduard Kirr: Shadowing Solitary Waves, Time-Dependent Wave Operators and Applications to Asymptotic Stability
12h10–13h Dmitri Yafaev: Analytic scattering theory for Jacobi operators and Bernstein–Szegő asymptotics of orthogonal polynomials Scipio Cuccagna: The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states Anatoli Babin: A mathematical justification of Dirac's theory of relativistic classical electron Vladimir Sukhanov: Large time asymptotics for the cylindrical Korteweg–De Vries equation Burak Erdogan: Talbot effect for dispersive PDE on the torus
13h–15h10 Lunch Lunch Lunch Lunch Lunch
15h10–16h Ekaterina Shchetka: Semiclassical asymptotics of the spectrum of the subcritical almost Mathieu operator Nabile Boussaid: Carleman inequalities for Dirac operators Michael Sigal: Long-time behaviour of the density functional theory Hermitage Boris Vainberg: Singularities in the spectrum of exterior elliptic problems
16h–16h30 Coffee Coffee Coffee Hermitage Tea
16h30–17h20 Sergey Morozov: Self-adjoint realisations of supercritical Coulomb-Dirac operators Alexander Komech: On orbital stability of crystals in the Schrödinger–Poisson model Anton Savostianov: Smooth uniform attractors for measure driven quintic damped wave equations on 3D torus Hermitage
18h–... Welcome (PDMI) Boat trip (Fontanka, 27) Hermitage

All the talks will take place at the following location:
Marble Hall (floor 2)
PDMI ("POMI"), St.Petersburg Branch of V.A. Steklov Institute of Mathematics
of the Russian Academy of Sciences
27 Fontanka, St.Petersburg 191023, Russia

Metro station "Gostiny Dvor" (line M3, "green")

Abstracts

Anatoli Babin (Irvine): A mathematical justification of Dirac's theory of relativistic classical electron
Dirac in his seminal paper in 1938 gave a derivation of Lorentz–Abraham–Dirac (LAD) equation that describes classical radiating electron in relativistic regimes. The derivation is based on analysis of energy and momentum conservation laws for electromagnetic (EM) fields generated by a point charge. To overcome difficulties caused by the singularity of the EM field of a point charge and its infinite energy, Dirac regularized the field by an appropriate subtraction and introduced charge mass renormalization. Still, as Landau and Lifshitz wrote, "Since the subtraction of one infinity from another is not an entirely correct mathematical operation, this leads to a series of further difficulties". In my talk I present a mathematical justification and refinement of Dirac's approach. We obtain point charge dynamics as a limit of dynamics of a distributed charge as its size tends to zero. The key point is to introduce equations for a field describing the charge rather than describe the charge distribution directly as Lorentz and Abraham did. This allows one to consistently describe interaction between charge and EM field in the framework of relativistic invariant Lagrangian field theory. Then the limit point dynamics is found using asymptotically semilocalized version of Ehrenfest theorem. Einstein's mass-energy equivalence is also obtained.
Nabile Boussaid (Besançon): Carleman inequalities for Dirac operators
The aim of my talk is to present some Carleman inequalities for Dirac operators we have obtained with Andrew Comech.

Besides a sketch of the proof of such inequalities, the aim is to present various applications in the analysis of spectral problems of Dirac operators.

Scipio Cuccagna (Trieste): The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states
In this talk we discuss a topic related to research V. Buslaev did in the 90s, see for example [1,2,3], related to the stabilization to ground states in the the Nonlinear Schrödinger Equation. We will discuss the question of the decay of the internal modes of the ground states and the so-called Fermi Golden Rule.
[1] V.S. Buslaev and G.S. Perelman: Scattering for the nonlinear Schrödinger equation: states close to a soliton. St.Petersburg Math. J. 4 (1993), 1111–1142.
[2] V.S. Buslaev and G.S. Perelman: On the stability of solitary waves for nonlinear Schrödinger equations. Nonlinear evolution equations, Amer. Math. Soc. Transl. Ser. 2 164 (1995), 75–98.
[3] V.S. Buslaev, A.I. Komech, E.A. Kopylova, and D. Stuart: On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator. Comm. Partial Differential Equations 33 (2008), 669–705.
Piero d'Ancona (Rome): On supercritical defocusing dispersive equations outside a ball
In this work I consider the defocusing semilinear wave and Schröodinger equations, with a power nonlinearity, defined on the outside of the unit ball of ℝn, and with Dirichlet conditions at the boundary. The power is assumed to be sufficiently large, p>O(n), and the space dimension is 3 or larger. Even in the radial case, the corresponding problem on ℝn is completely open. Here I construct a family of large global solutions, whose data are small perturbations of radial initial data in suitable weighted Sobolev norms of higher order. These solutions are unique in the family of energy class solutions satisfying an energy inequality.
Burak Erdogan (Urbana–Champaign): Talbot effect for dispersive PDE on the torus
In this talk we discuss qualitative behavior of certain solutions to linear and nonlinear dispersive partial differential equations such as Schrödinger and Korteweg–de Vries equations. In particular, we will present results on the fractal dimension of the solution graph and the dependence of solution profile on the algebraic properties of time.
Alexander Fedotov (St.Petersburg): Monodromization method and Harper equation
To study the cantorian geometry of the spectrum of the almost Mathieu equation, a one-dimensional quasiperiodic difference Schrödinger equation, V. Buslaev and A. Fedotov suggested the monodromization method, a renormalization method based on ideas of the Bloch–Floquet theory used to study differential equations with periodic coefficients. In this talk we briefly describe ideas of the monodromization method and some results obtained with its help.
Vladimir Georgiev (Pisa): On Traveling Solitary Waves and Absence of Small Data Scattering for Nonlinear Half-Wave Equations
We consider nonlinear half-wave equations with focusing power-type nonlinearity

i∂tu = (-Δ)1/2u - |u|p-1u, with (t,x) ∈ ℝ×ℝd,

with exponents 1 < p < ∞ for d=1 and 1 < p < (d+1)/(d-1) for d ≥ 2. We study traveling solitary waves of the form

u(t,x) = eiωtQv(x-vt)

with frequency ω ∈ ℝ, velocity v ∈ ℝd, and some finite-energy profile Qv ∈ H1/2(ℝd), Qv ≢ 0. We prove that traveling solitary waves for speeds |v| ≥ 1 do not exist. Furthermore, we generalize the non-existence result to the square root Klein–Gordon operator (-Δ+m2)1/2 and other nonlinearities. As a second main result, we show that small data scattering fails to hold for the focusing half-wave equation in any space dimension. The proof is based on the existence and properties of traveling solitary waves for speeds |v| < 1. Finally, we discuss the energy-critical case when p=(d+1)/(d-1) in dimensions d ≥ 2.

Alexey Ilyin (Moscow): Spectral inequalities on the sphere
We prove Berezin–Li–Yau inequalities for the Dirichlet and Neumann eigenvalues on domains on the sphere 𝕊d-1. A sharp explicit bound for the sums of the Neumann eigenvalues is obtained for all dimensions d. In the case of 𝕊2 we also obtain sharp lower bounds with correction terms for the vector Laplacian and the Stokes operator [1].
On the sphere 𝕊2 we also prove by using the method of [2] the Lieb–Thirring inequalities for orthonormal families of scalar and vector functions both on the whole sphere and on proper domains on 𝕊2. By way of applications we obtain an explicit estimate for the dimension of the attractor of the Navier–Stokes system on a domain on the sphere with Dirichlet non-slip boundary conditions [3].
The talk is based on a joint work with Ari Laptev.
[1] A.A. Ilyin, A.A. Laptev: Berezin–Li–Yau inequalities on domains on the sphere. J. Math. Anal. Appl. 473:2 (2019), 1253–1269.
[2] M. Rumin: Balanced distribution-energy inequalities and related entropy bounds. Duke Math. J. 160 (2011), 567–597.
[3] A.A. Ilyin, A.A. Laptev: Lieb–Thirring inequalities on the sphere. Algebra i Analiz, accepted.
Alexander Its (Indianapolis and St.Petersburg): Large time asymptotics for the cylindrical Korteweg--de Vries equation. II
This is a sequel to Vladimir Sukhanov's talk. We present an alternative approach to the asymptotic analysis of the solutions of the cylindrical Korteweg–de Vries equation based on the Deift–Zhou nonlinear steepest descent method for oscillatory Riemann–Hilbert problems. This approach allows, in particular, to introduce and analyze other classes of the solutions which exhibit an oscillatory type of the long time asymptotic behavior absent for the solutions from Schwartz's class.
The talk is based on a joint work with Vladimir Sukhanov.
Eduard Kirr (Urbana–Champaign): Shadowing Solitary Waves, Time-Dependent Wave Operators and Applications to Asymptotic Stability
I will discuss classical and recent results regarding asymptotic stability of solitary waves i.e., localized solutions of nonlinear wave equations propagating without changing shape. Beginning with the work of Soffer–Weinstein and Buslaev–Perelman–Sulem in the 90s we know that, under certain assumptions, solutions starting close to a solitary wave shadow nearby solitary waves before collapsing on one. The mathematical analysis relies on using dispersive estimates for the linearized dynamics at a fixed (rather arbitrarily chosen) solitary wave to control the nonlinearity. I will show that the assumptions on the nonlinearity can be significantly relaxed provided one uses a time dependent linearization which follows the different solitary waves shadowed by the actual solution. Of course, this new perspective requires new dispersive estimates for time dependent wave type operators and I will discuss how they can be obtained. This is joint work with A. Zarnescu (BCAM, Bilbao, Spain), O. Mizrak (Mersin University, Turkey), and R. Skulkhu (Mahidol University, Thailand).
Alexander Komech (Moscow, Vienna): On orbital stability of crystals in the Schrödinger–Poisson model
We consider finite periodic crystals with moving ions in the Schrödinger–Poisson model. Our main results are existence of global dynamics, existence and stability of the ground state, a linear stability, the dispersion decay, and (nonlinear) orbital stability. The results are obtained for the models with one-particle Schroedinger equation and with N-particle fermionic Schrödinger equation.
Elena Kopylova (Vienna, Moscow): Global attraction to solitary waves for Dirac equations with point interactions
The long-time asymptotics is analyzed for solutions to 1D Dirac equation coupled with nonlinear oscillator and for 3D Dirac equation with concentrated nonlinearity. Our main result is the global attraction of each "finite energy solution" to the set of all solitary waves as t → ±∞. This attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersion radiation. We justify this mechanism by the following strategy based on inflation of spectrum by the nonlinearity:
[1] E. Kopylova, On global well-posedness for Klein–Gordon equation with concentrated nonlinearities, J. Math. Anal. Appl. 443 (2016).
[2] E. Kopylova: Global attraction to solitary waves for Klein–Gordon equation with concentrated nonlinearity, Nonlinearity 30 (2017).
[3] E. Kopylova: On global attraction to stationary state for wave equation with concentrated nonlinearity, J. Dynamics and Diff. Equations 30 (2018).
[4] E. Kopylova, A.I. Komech: On global attractor of 3D Klein–Gordon equation with several concentrated nonlinearities, Dynamics of PDEs 16 (2019).
[5] E. Kopylova, A.I. Komech: Global attraction for 1D Dirac field coupled to nonlinear oscillator, accepted to Comm. Math. Phys.. arXiv:1901.08963.
Joachim Krieger (Lausanne): Randomization improved Strichartz estimates and applications
I will discuss joint work with N. Burq concerning a novel class of improved Strichartz estimates for a certain data randomization, and an application to supercritical data well-posedness for certain wave equations with null-forms.
Sergey Morozov (München): Self-adjoint realisations of supercritical Coulomb-Dirac operators
We will discuss some known results and open questions concerning spectra of self-adjoint realizations of Dirac operators with electrostatic Coulomb singularities of arbitrary magnitude. We will observe that, while such models can be introduced and rigorously studied without essential difficulties, it is hard to distinguish a unique physically interesting self-adjoint realisation. For the hydrogen atom we will study asymptotics of eigenvalues for arbitrary self-adjoint realisations in the limit of coupling approaching the critical value.
The talk is based on joint work with H. Hogreve.
Anton Savostianov (Durham): Smooth uniform attractors for measure driven quintic damped wave equations on 3D torus
In this talk we discuss the existence of uniform attractors for nonautonomous damped wave equations with nonlinearities of quintic growth. We show that Strichartz estimates, which are necessary for global well-posedness of such equations, remain valid when the forcing term is a vector-valued measure with bounded total variation. This leads to new classes of admissible external forces, which guarantee existence of smooth uniform attractors for our problems.
This is joint work with Sergey Zelik.
Ekaterina Shchetka (St.Petersburg): Semiclassical asymptotics of the spectrum of the subcritical almost Mathieu operator
For the Mathieu operator, it is known that, in the semiclassical approximation, the spectrum between the minimum and maximum of the potential is located on exponentially small intervals separated by gaps of length of the order of the semiclassical parameter. On contrary, above the maximum of the potential the spectrum is located on intervals of length of the order of the semiclassical parameter, and they are separated by exponentially small gaps (see, e.g., [1]). We describe a similar effect for the subcritical almost Mathieu operator, i.e., the operator acting in l2(ℤ) by the formula

Hψk=ψk+1+ψk-1+2λ cos(2π(ωk+θ))ψk

where λ, θ and ω are parameters, and 0<λ<1. The spectrum is symmetric with respect to 0. The role of the semiclassical parameter is played by ω: if it is small, the coefficient cos(2π(ωk+θ)) changes slowly with k. In the semiclassical limit, i.e., when ω is small, on the interval (0, 2-2λ), the spectrum is located on subintervals of length of the order of ω that are separated by exponentially small gaps. On the interval (2-2λ, 2+2λ), the spectrum is located on exponentially small subintervals separated by gaps of length of the order of ω. We give asymptotic estimates for the lengths of the gaps and the intervals containing the spectrum. Two different asymptotic spectral modes appear as, for different values of the spectral parameter, there are two different types of the semiclassical iso-energy curves. To get the result we use ideas of the monodromization method suggested by V.S. Buslaev and A.A. Fedotov (see [2]) and constructions of the complex WKB method for difference equations (see [3] and [4]).
The talk is based on a joint work with A.A. Fedotov.
[1] M.V. Fedoryuk: Asymptotic Methods for Linear Ordinary Differential Equations, Librokom, Moscow, 2009.
[2] V.S. Buslaev, A.A. Fedotov: The monodromization and Harper operator, Séminaire sur les Équations aux Dérivées Partielles, 1993–1994, Exp. no. XXI, 23 pp., École Polytechnique, Palaiseau, 1994.
[3] V.S. Buslaev, A.A. Fedotov: The complex WKB method for the Harper equation, St.Petersburg Math. J. 6:3 (1995), 495–517.
[4] A.A. Fedotov, E. Shchetka: Complex WKB method for a difference Schrödinger equation with the potential being a trigonometric polynomial, St.Petersburg Math. J. 29 (2018), 363–381.

Michael Sigal (Toronto): Long-time behaviour of the density functional theory
In this talk I will report some recent results and work in progress on the long-time behaviour of the time-dependent density functional theory. Specifically, I will outline main ingredients of the proofs of global existence, local decay and scattering for small data.
This is a joint work with Fabio Pusateri.
David Stuart (Cambridge): Solitons and Modulation Equations in Quantum Field Theory
We study the interaction of a scalar quantum field ϕ with a fixed (external) electromagnetic field. The quartic self-interaction for ϕ, a double well potential, supports the existence of solitons in the classical theory. The existence of the corresponding quantum theory can be proved by the methods of constructive quantum field theory. The aim is to analyze the dynamics of the soliton in this quantized theory as g → 0, which corresponds to a nonrelativistic limit for the soliton, which has mass which diverges as g-2 in this limit. Initially attention is focused on the case of zero electromagnetic field, and we develop the analytical framework for quantizing the theory, identifying the appropriate degrees of freedom to describe the soliton and the g → 0 limiting dynamics and proving the Dashen–Hasslacher–Neveu semiclassical mass correction formula from [1], following [4]. After this we will discuss how to extend soliton perturbation theory – a topic on which Buslaev did very important work, [2] – to the analysis of soliton dynamics in quantum field theories. We refer to [3, §23.8] for a general review of mathematical work on solitons in the context of constructive field theory.
[1] R.F. Dashen, B. Hasslacher, and A. Neveu: Nonperturbative methods and extended hadron models in field theory, parts i-iii, Phys.Rev. D10 (1974), 4114–4142.
[2] V.S. Buslaev, G.S. Perel'man: On the stability of solitary waves for nonlinear Schrödinger equations. Nonlinear evolution equations, Amer. Math. Soc. Transl. Ser. 2 164 (1995), 75–98.
[3] J. Glimm, A. Jaffe: Quantum physics. A functional integral point of view. Springer-Verlag, New York, 1987. ISBN 0-387-96476-2
[4] D.M.A. Stuart: Hamiltonian quantization of solitons in the ϕ41+1 quantum field theory. I. The semiclassical mass shift. arXiv:1904.02588.
Vladimir Sukhanov (St.Petersburg): Large time asymptotics for the cylindrical Korteweg–De Vries equation
The talk is concerned with the large time asymptotics of the solutions of the cylindrical Korteweg–de Vries equation. We consider the solutions from Schwartz's class and calculate the asymptotics using the method based on the asymptotic solution of the associated direct scattering problem first suggested in the late 70s by V.E. Zakharov and S.V. Manakov and further developed in the 80s by V.S. Buslaev, V.V. Sukhanov, and V.Yu. Novokshenov.
Tatiana Suslina (St.Petersburg): Homogenization of hyperbolic equations with periodic coefficients
We give a survey of the results on operator error estimates in homogenization of hyperbolic equations with periodic rapidly oscillating coefficients (Birman and Suslina [2008]; Meshkova [2017]; Dorodnyi and Suslina [2018,2019]). In L2(ℝd;ℂn), we consider a selfadjoint matrix strongly elliptic operator Aε, ε>0, given by the differential expression b(D)*g(x/ε)b(D). Here g(x) is a periodic bounded and positive definite matrix-valued function, and b(D) is a first order differential operator. We study the behavior of the operators cos(t Aε1/2) and Aε-1/2 sin(t Aε1/2), t∈ℝ, for small ε. It is proved that, as ε → 0, these operators converge to cos(t A01/2) and A0-1/2 sin(t A01/2), respectively, in the norm of operators acting from the Sobolev space Hs(ℝd;ℂn) (with a suitable s) to L2(ℝd;ℂn). Here A0 = b(D)* g0b(D) is the effective operator. We prove sharp-order error estimates and study the question about the sharpness of the result with respect to the norm type. The results are applied to study the behavior of the solution uε(x,t) of the Cauchy problem for hyperbolic equation (∂t)2 uε(x,t) = (Aε uε)(x,t). Applications to the nonstationary equations of acoustics and elasticity are given. The method is based on the scaling transformation, the Floquet–Bloch theory and the analytic perturbation theory.
Boris Vainberg (Charlotte): Singularities in the spectrum of exterior elliptic problems
We will discuss singularities in the continuous spectrum of the Schrödinger operators in the whole space (joint results with V. Konotop and L. Lakshtanov) and the appearance of the discrete (negative) eigenvalues for operators of the second order in exterior domains (joint results with R. Puri).
The presence of spectral singularities is related to lasing (coherent radiation by the action of a laser) or coherent perfect absorption. We will show that a family of tunable complex valued potentials can be constructed with a spectral singularity at an a priory given wavelength, as well as with double singularities; the latter can be split by an appropriate change of potential parameters.
The existence of negative eigenvalues plays a crucial role in the large time behavior of the diffusion processes. We will study the critical value of the coefficient βcr at the potential term that separates operators with negative eigenvalues and those without them. In particular, the dependence of βcr on the boundary condition and on the distance from the support of the potential to the boundary will be discussed.
Dmitri Yafaev (St.Petersburg): Analytic scattering theory for Jacobi operators and Bernstein–Szegő asymptotics of orthogonal polynomials
We study semi-infinite Jacobi matrices H=H0+V corresponding to trace class perturbations V of the "free" discrete Schrödinger operator H0 and properties of the associated orthonormal polynomials Pn(z). Our goal is to construct various spectral quantities of the operator H, such as the weight function, eigenfunctions of its continuous spectrum, the wave operators for the pair H0,H, the scattering matrix, the spectral shift function, etc. This allows us to find the asymptotic behavior of the polynomials Pn(z) as n → ∞ and gives a new look on the Bernstein–Szegő formulas. We give a proof of these formulas under essentially more general circumstances than in the original papers.