Conference poster
This spring I had an opportunity to visit the UK and to participate in the conference “Operators, Operator Families and Asymptotics” which was held from May 16th to May 19th, 2016. The conference was hosted by the University of Bath and supported by the LMS, the EPSRC and the Bath Institute for Mathematical Innovation.
Looking at the conference title one could expect the meeting of researchers working in abstract operator theory and related “complicated” areas. In reality the aim was different – to bring together researchers applying various techniques of operator theory to the asymptotic analysis of parameter-dependent differential equations and boundary-value problems. The physical meaning of these parameters depends on the situation modelled by the equation. For example, in case of a strongly inhomogeneous media this parameter plays the role a characteristic scale of the microstructure (so, one deals with a problem belonging to homogenization theory). The meeting aimed at making an overview of the state of the art in this rapidly developing area.
During the conference I attended plenty of nice talks. About two of them (which were the most interesting for me from the point of view of my research interests) I would like to say a few words further.
Rostyslav Hryniv, who represented University of Rzeszów (Poland) and Institute for Applied Problems of Mechanics and Mathematics (Lviv, Ukraine), gave a talk concerning Schrödinger operators with δ‘ potential (the derivative of Dirac delta-function supported at 0). Indeed, assume that we are given a formal differential expression -d2/dx2+δ‘. Then the natural question arises: how to understand it? Is there a clever way to assign to this formal operation a self-adjoint operator acting in an appropriate Hilbert space? In fact, if we have a regular function and a singular distribution it is, in general, unclear how to define their product.
One of the natural ideas, how to attack this problem, is to consider a family of Schrödinger operators Aε=-d2/dx2+Vε, where the potential Vε is smooth and converges as ε→0 to δ‘ in the sense of distributions. Suppose that Aε converges (in a suitable sense) to some operator A. Then one can say that A is an operator corresponding to a formal expression -d2/dx2+δ‘.
In the talk Rostyslav Hryniv presented the results from his recent joint work with Yuri Golovaty, in which they implemented this strategy. Namely, they found the limit operator A and established the norm resolvent convergence of the operators Aε. The operator A is defined by the operation -d2/dx2 on half-axes and a certain coupling condition at 0. Generically this coupling is, actually, a decoupling 🙂 Namely, at -0 and +0 one has homogeneous Dirichlet conditions. But for some profiles Vε one can get more involved coupling (the so-called Datta – Das Sarma conditions).
Tatiana Suslina from St. Petersburg State University presented the beautiful talk (without any computer presentation, only the blackboard and a piece of chalk!) on the operator estimates in the homogenization theory.
Recall, that one of the classical problems in homogenization theory is to describe the asymptotic behavior of the following elliptic operator acting in L2(Rn):
Aε=–div(a(x/ε)grad),
where a(y) is a periodic function, which is bounded and bounded away from zero. The classical result of homogenization theory says that Aε strongly resolvent converges as ε→0 to some elliptic operator A0 with constant coefficients, which can be found explicitly by solving the so-called cell problem on a period cell. In the late 90’s M.Birman and T.Suslina made a breakthrough in this area proving the norm resolvent convergence of these operators. In a subsequent papers they extended their results to parabolic and hyperbolic scalar equations, Maxwell operators etc.
In the talk in Bath Tatiana Suslina presented new results concerning the behavior of the operator exponential exp(itAε). She proved that the operator exp(itAε) converges to exp(itA0) in the operator norm for fixed t. Moreover, sharp order error estimates are obtained. The method of proof is based on a suitable scaling transformation, the Floquet-Bloch theory and the analytic perturbation theory.
Conference photo/ (c) Alexander V. Kiselev
Also, I would like to highlight the talks given by
– Pavel Exner (Doppler Institute for Mathematical Physics and Applied Mathematics, Prague). He presented an overview of resent results on spectral properties of multidimensional Schrödinger operators with δ potential supported on a surface with co-dimensions 1 or 2.
– David Krejcirik (Nuclear Physics Institute, Czech Academy of Sciences). His talk was devoted to a study of Laplace operators on metric graphs with various classes of non-self-adjoint interface conditions at the graph vertices.
– Stefan Neukamm (TU Dresden). He spoke about estimates for the rate of convergence in stochastic homogenization (so, this talk was close to the talk of T. Suslina)
– Shane Cooper (Bath University). He concerned the asymptotic analysis of some stratified elastic media.
In fact, all talks had something interesting and new for me (except the talk of some A.K. from Karlsruhe – his results I already knew 🙂 ).
The schedule of the conference was very convenient – we had a lot of time for discussions. I came back to Karlsruhe with a lot of new ideas and plans for the future research.
Finally, I would like to thank DFG supporting my participation in this conference through CRC 1173 “Wave phenomena: analysis and numerics”.