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MAA 2007) Dedicated to the Centenary of Mark Krein Odessa, Ukraine, April 9–14, 2007 International Conference MODERN ANALYSIS AND APPLICATIONS (MAA 2007) dedicated to the centenary of Mark Krein Odessa, Ukraine, April 9–14, 2007 Book of abstracts Kyiv - 2007 International Conference MODERN ANALYSIS AND APPLICATIONS dedicated to the centenary of Mark Krein ORGANIZERS: Ministry of Education and Science of Ukraine • National Academy of Sciences of Ukraine • Municipality of Odessa • Institute of Mathematics, National Academy of Sciences of Ukraine • Odessa National I.I. Mechnikov University • Odessa Nationa Maritime University • South Ukrainian State Pedagogical University “K.D. Ushynsky” • Institute of Mathematics, Economics and Mechanics of Odessa National • I.I. Mechnikov University Department of Mathematical Physics of Odessa National I.I. Mechnikov • University Department of Theoretical Physics of Odessa National I.I. Mechnikov University • SPONSORS: Swedish Institute • European Science Foundation • Municipality of Odessa • Manufactured Goods Market Limited • JV Dipolos Limited • Imexbank (Joint Stock commercial bank) • Pivdennyi Bank • Port-Franco Bank • c 2007 Institute of Mathematics, National Acad. Sci. of Ukraine ABSTRACTS OF TALKS Rustam Abdullaev, Bahrom Hikmetov AN ANALOGUE OF ARENS ALGEBRA CONSTRUCTED BY A STATE ON VON NEUMANN ALGEBRA Let M be a von Neumann algebra with faithful normal finite trace µ. For any faithful normal state ϕ on M exists positive operator h affiliated to M such that ϕ(x)= µ(hx) for every x M + ∈ (see[1]). On M we introduce two norms 1 x µ = (µ( x p)) p , k kp | | 1 1 1 x ϕ = (µ( h 2p xh 2p )p) p . k kp | | µ The completion of M by the norm p denote by Lp(M,µ), the completion of M by the norm ϕ k · k p denote by Lp(M, ϕ). It is easy to see, that Lp(M,µ) is a partical case of Lp(M, ϕ). For k · k 1 regular normal states (i.e when h− is local measurable ) space Lp(M, ϕ) can be described by local measurable operators affiliated to M (see[2]). Next we consider only regular states. The set Lp(M, ϕ)-is called Arens space. Consider on it the topology t, generated by p [1; ) ∈ ∞ ϕ ω system norm T p p [1; ) (see[3]). Denote it by (L (M, ϕ),t). {k · k } ∈ ∞ ω Theorem 1. The space (L (M, ϕ),tϕ) is complete metricable local convex space with dual space, which is isometrical isomorphic to Lq(M, ϕ). q (1; ] ∈S∞ Theorem 2. The space Lω(M,µ) consides with the space Lω(M, ϕ) if and only if 1 h Lq(M,µ) and h− Lq(M, ϕ). ∈ q (1; ] ∈ q (1; ] ∈S∞ ∈S∞ Theorem 3. On noncommutative algebra von Neumann exists a normal state ψ, such that Lω(M, ψ) = Lω(M,ν) for any faithful normal finite trace ν. 6 [1] Pedersen G.K., Takesaki M. The Radon Nikodym theorem for von Neumann algebras, Acta Math, (1973), v.130, no. 1-2, 53-87 [2] Trunov H.B K teorii normalnix vesov na algebrax fon Neymana.,Izv.Vuzov. Matem, (1982),no. 9, 61-70 [3] Abdullaev R.Z. Izomorfizmi algebr Arensa., Sibirskiy jurnal industrialnoy matematiki, (1988), v.1, no. 2, 3-13 Vadym Adamyan, Boris Pavlov KREIN FORMULA IN TRANSPORT PROBLEMS FOR QUANTUM NETWORKS Quantum switches, spin filters and other types of quantum nano- electronic devices are quantum networks. Mathematical design of quantum networks with prescribed transport properties is a challenging problem of modern physics. Contemporary mathematics is already able to contribute to the solution of the simplest type of transport problems- to the one-body scattering problem for networks constructed on a surface of a semiconductor of quantum dots and straight quantum wire connecting them to each other or to infinity. Construction of the network with prescribed 4 transport properties - with given scattering matrix - is a subject of the inverse scattering problem for quantum networks, which is still beyond the limits of modern mathematics. Optimization of design of the network based on straightforward computing is expensive and non-efficient because of large number of geometrical and physical parameters defining the properties of the network. One can expect that substitution of the network by the corresponding quantum graph, possibly with resonances nodes supplied with an inner structure, looks like an attractive alternative to the straightforward computing. This model is solvable, but it requires appropriate fitting. We suggest a procedure of fitting of this model based on semi-analytic approach, involving direct computing for eigenfunctions of discrete spectrum on quantum dots and subsequent rational approximation of the Dirichlet-to-Neumann maps and asymptotic calculation of the relevant scattering matrix. Attachment of the semi-infinite quantum wires to quantum dots defines the transformation of bound states in quantum dots into resonances. This is a typical difficulty for perturbation problems on the continuous spectrum. We overcome this difficulty via introduction of an inter- mediate Hamiltonian which is obtained by Glazman splitting of the Schr¨odinger operator on the quantum network into orthogonal sum of two operators with complementary branches of the absolutely continuous spectra, constituted by the open and closed branches of the continuous spectrum, for given Fermi level. Based on Dirichlet-to-Neumann map of the intermediate Hamil- tonian, we eliminate calculation of coefficients in front of exponentially decreasing modes in the scattered waves, reducing the problem for scattered waves to solution of a finite linear system. The rational approximation of the Dirichlet-to-Neumann map of the Intermediate Hamiltonian is interpreted as Weyl-Titchmarsh function of the corresponding solvable model on the relevant quantum graph which is automatically fitted such that the model scattering matrix serves an approximation of the scattering matrix of the full Hamiltonian on an essential spectral interval near the Fermi level. On the other hand, the scattering matrix of the model is represented by the modified Krein formula, which gives an approximation of the full scattering matrix. The solvable model may serve an intermediate step in the analytic perturbation procedure on the continuous spectrum, because spectral characteristics of the full Hamiltonian are connected to the corresponding spectral characteristics of the solvable model on the essential spectral interval by the convergent standard analytic perturbation procedure. Mikhail Agranovich SPECTRAL PROBLEMS FOR STRONGLY ELLIPTIC SYSTEMS IN LIPSCHITZ DOMAINS We will discuss the regularity of solutions to boundary value problems for strongly elliptic partial differential systems in Lipschitz domains. The results are applied to regularity properties of eigenfunctions of spectral boundary value problems. Some generalizations of the author’s results published in Functional Analysis and Its Applications, 40, 2006, No. 4, will be explained. Sergey Aizikovich BILATERAL ASYMPTOTIC SOLUTION OF ONE CLASS OF DUAL INTEGRAL EQUATIONS OF THE STATIC CONTACT PROBLEMS FOR INHOMOGENEOUS WITH DEPTH FOUNDATION 5 Sergio Albeverio, Rostyslav O. Hryniv, Yaroslav V. Mykytyuk THE KREIN EQUATION AND INVERSE SPECTRAL PROBLEMS In early 1950-ies, M.Krein suggested a method of solving the inverse spectral problems for Sturm– Liouville operators using special integral equations. We show how this method can further be developed to provide solutions to inverse spectral problems for a wide class of singular Sturm– Liouville operators and of Dirac operators with summable potentials. Daniel Alpay, I. Gohberg CONTINUOUS AND DISCRETE SYSTEMS WITH RATIONAL SPECTRAL DATA We first review the theory of canonical differential expressions in the rational case, and in particular discuss the class of potentials introduced by the authors in [1]. Then, we define and study the discrete analogue of canonical differential expressions. We focus on the rational case. Two kinds of discrete systems are to be distinguished: one–sided and two–sided. In both cases the analogue of the potential is a sequence of numbers in the open unit disk (Schur coefficients). We define the characteristic spectral functions of the discrete systems and provide exact realization formulas for them when the Schur coefficients are of a special form called strictly pseudo–exponential and introduced in [2]. The matrix-valued case, to appear in [3] will be briefly outlined. [1] D. Alpay and I. Gohberg. Inverse spectral problem for differential operators with rational scattering matrix functions. Journal of differential equations, 118:1–19, 1995. [2] D. Alpay and I. Gohberg. Connections between the Carath´eodory-Toeplitz and the Nehari extension problems: the discrete scalar case. Integral Equations Operator Theory, 37(2):125– 142, 2000. [3] D. Alpay and I. Gohberg. Discrete systems and their characteristic spectral functions. Mediterranean Journal of Mathematics, 2007 (in press). Arthur Amirshadyan BOUNDARY INDEFINITE NEVANLINNA-PICK INTERPOLATION PROBLEM The Problem ∂IPκ. Given are: real points zj and symmetric matrices Wj, Dj (j=1,..,m). Find the matrix function F (λ) N (dCn), satisfying the conditions ∈ κ lim F (λ)= W (j = 1, .., m) (1) ∧ j λ zj −→ lim F ′(λ) D (j = 1, .., m). (2) ∧ j λ zj ≤ −→ It is assumed that there exist nontangential limits in (1), (2). 6 The condition for the solvablity of this problem will be stated in terms of the Pick matrix m P = (Pjk)j,k=1, of the form [7], [1]: Wj W − k , ifj = k; zj z Pjk = − k 6 ( Dj, ifj = k. In what follows, we assume that det P = 0. 6 The matrices V = (In, .., In), W = (W1, .., Wm), Z = diag(z1In, .., zmIn) are called the data of the Problem ∂IP . Let Φ := (Z λ)P + V W . Now we define the the solution matrix κ λ − ∗ 2 W 1 1 Ω(λ)=(Ω ) =: I + (Z λ)− P− ( V ∗,W ∗). ij i,j=1 2n V − − Theorem 1. Suppose that det P = 0, sq P = κ0 κ, detΦzj = 0 (j=1,..,m). Then the 6 − ≤ 6 formula 1 F (λ)=(Ω (λ)ψ(λ) Ω (λ)φ(λ))(Ω (λ)ψ(λ) Ω (λ)φ(λ))− (3) 12 − 11 22 − 21 establishes a one-to-one correspondence between the solution set of Problem ∂IPκ and the set of Nκ κ0 – pairs φ(λ), ψ(λ) , for which − { } det(Ω (λ)ψ(λ) Ω (λ)φ(λ)) 0 (4) 22 − 21 6≡ 1 1 lim (λ z )φ(λ)(V Φ− V ∗φ(λ)+ ψ(λ))− 0 (j = 1, ..., m).
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