Integrable Structure of Conformal Field Theory II. Q-operator and DDV equation

This paper is a direct continuation of [1] where we began the study of the integrable structures in Conformal Field Theory. We show here how to construct the operators ${\bf Q}_{\pm}(\lambda)$ which act in the highest weight Virasoro module and commute for different values of the parameter λ. These operators appear to be the CFT analogs of the Q - matrix of Baxter [2], in particular they satisfy Baxter's famous T- Q equation. We also show that under natural assumptions about analytic properties of the operators as the functions of λ the Baxter's relation allows one to derive the nonlinear integral equations of Destri-de Vega (DDV) [3] for the eigenvalues of the Q-operators. We then use the DDV equation to obtain the asymptotic expansions of the Q - operators at large λ; it is remarkable that unlike the expansions of the T operators of [1], the asymptotic series for Q(λ) contains the “dual” nonlocal Integrals of Motion along with the local ones. We also discuss an intriguing relation between the vacuum eigenvalues of the Q - operators and the stationary transport properties in the boundary sine-Gordon model. On this basis we propose a number of new exact results about finite voltage charge transport through the point contact in the quantum Hall system. Received: 2 December 1996 / Accepted: 11 March 1997     

Quadratic transformations of the sixth Painlevé equation with application to algebraic solutions

In 1991, one of the authors showed the existence of quadratic transformations between the Painlevé VI equations with local monodromy differences (1/2, a, b, ±1/2) and (a, a, b, b). In the present paper we give concise forms of these transformations. They are related to the quadratic transformations obtained by Manin and Ramani–Grammaticos–Tamizhmani via Okamoto transformations. To avoid cumbersome expressions with differentiation, we use contiguous relations instead of the Okamoto transformations. The 1991 transformation is particularly important as it can be realized as a quadratic‐pull back transformation of isomonodromic Fuchsian equations. The new formulas are illustrated by derivation of explicit expressions for several complicated algebraic Painlevé VI functions. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Luminescent copolymers for applications in multicolor‐light‐emitting devices

Light‐emitting diodes based on organic materials [organic light‐emitting diodes (OLEDs)] have attracted much interest over the past decade. Several different attempts have been made to realize multicolor OLEDs. This article describes a new approach based on energy transfer in a donor/acceptor system. A copolymer containing both donor and acceptor compounds as comonomer units is prepared. The polymer consists of a derivative of a luminescent dye [4‐dicyanmethylene‐2‐methyl‐6‐4H‐pyran (DCM); acceptor compound], which is copolymerized with fluorene (donor compound) to combine the properties of an electroactive polymer with a highly luminescent dye. Photochemical processing is achieved by UV irradiation of this copolymer in the presence of gaseous trialkylsilanes. This reagent selectively saturates the C?C bonds in the DCM comonomer units while leaving the fluorene units essentially unaffected. As a result of the photochemical process, the red electroluminescence of the acceptor compound vanishes, and the blue‐green electroluminescence from the polyfluorene units is recovered. Compared with previous approaches based on polymer blends, this copolymer approach avoids problems associated with phase‐separation phenomena in the active layer of OLEDs. © 2006Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 4317–4327, 2006     

Sobolev Spaces with only Trivial Isometries

We will give some conditions for Sobolev spaces on bounded Lipschitz domains to admit only trivial isometries.     

The maximum principle for the Bergman space and the Möbius pseudodistance for the annulus

It is shown that the formula for the Möbius pseudodistance for the annulus yields better estimates than previously known for the constant in the Bergman space maximum principle.

     

Rectification of sets in Euclidean spaces

We criticize traditional definitions of the arc length which require semi-continuity from below. Symmetric definitions of lower and uppern-lengths (n-dimensional volumes) are introduced for a wide class of sets in Euclidean spaces, and the additivity of both functionals is proved.     

The umbral calculus and orthogonal polynomials

We determine all orthogonal polynomials having Boas-Buck generating functions g(t)(xf(t)), where% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqqHOo% qwcaGGOaGaamiDaiaacMcacqGH9aqpruqqYLwySbacfaGaa8hiamaa% BeaaleaacaaIWaaabeaakiaadAeacaqGGaWaaSbaaSqaaiaabgdaae% qaaOGaaeikaiaadggacaGGSaGaa8hiaiaadshacaqGPaGaaeilaiaa% bccacaqGGaGaaeiiaiaadggacqGHGjsUcaaIWaGaaiilaiaa-bcacq% GHsislcaaIXaGaaiilaiaa-bcacqGHsislcaaIYaGaaiilaiablAci% ljaacUdaaeaacqqHOoqwcaGGOaGaamiDaiaacMcacqGH9aqpcaWFGa% WaaSraaSqaaiaaicdaaeqaaOGaamOraiaabccadaWgaaWcbaGaaeOm% aaqabaGccaGGOaWaaSqaaSqaaiaaigdaaeaacaaIZaaaaOGaaiilai% aa-bcadaWcbaWcbaGaaGOmaaqaaiaaiodaaaGccaGGSaGaa8hiaiaa% dshacaGGPaGaa8hiamaaBeaaleaacaaIWaaabeaakiaadAeacaqGGa% WaaSbaaSqaaiaabkdaaeqaaOGaaeikamaaleaaleaacaaIYaaabaGa% aG4maaaakiaacYcacaWFGaWaaSqaaSqaaiaaisdaaeaacaaIZaaaaO% Gaaiilaiaa-bcacaWG0bGaaiykaiaacYcacaWFGaWaaSraaSqaaiaa% icdaaeqaaOGaamOraiaabccadaWgaaWcbaGaaeOmaaqabaGccaGGOa% WaaSqaaSqaaiaaisdaaeaacaaIZaaaaOGaaiilaiaa-bcadaWcbaWc% baGaaGynaaqaaiaaiodaaaGccaGGSaGaa8hiaiaadshacaGGPaGaai% 4oaaqaaiabfI6azjaacIcacaWG0bGaaiykaiabg2da9iaa-bcadaWg% baWcbaGaaGimaaqabaGccaWGgbGaaeiiamaaBaaaleaacaqGZaaabe% aakiaacIcadaWcbaWcbaGaaGymaaqaaiaaisdaaaGccaGGSaGaa8hi% amaaleaaleaacaaIYaaabaGaaGinaaaakiaacYcacaWFGaWaaSqaaS% qaaiaaiodaaeaacaaI0aaaaOGaaiilaiaa-bcacaWG0bGaaiykaiaa% -bcadaWgbaWcbaGaaGimaaqabaGccaWGgbGaaeiiamaaBaaaleaaca% qGZaaabeaakiaabIcadaWcbaWcbaGaaGOmaaqaaiaaisdaaaGccaGG% SaGaa8hiamaaleaaleaacaaIZaaabaGaaGinaaaakiaacYcacaWFGa% WaaSqaaSqaaiaaiwdaaeaacaaI0aaaaOGaaiilaiaa-bcacaWG0bGa% aiykaiaacYcaaeaadaWgbaWcbaGaaGimaaqabaGccaWGgbGaaeiiam% aaBaaaleaacaqGZaaabeaakiaacIcadaWcbaWcbaGaaG4maaqaaiaa% isdaaaGccaGGSaGaa8hiamaaleaaleaacaaI1aaabaGaaGinaaaaki% aacYcacaWFGaWaaSqaaSqaaiaaiAdaaeaacaaI0aaaaOGaaiilaiaa% -bcacaWG0bGaaiykaiaacYcacaGGUaGaa8hiamaaBeaaleaacaaIWa% aabeaakiaadAeacaqGGaWaaSbaaSqaaiaabodaaeqaaOGaaeikamaa% leaaleaacaaI1aaabaGaaGinaaaakiaacYcacaWFGaWaaSqaaSqaai% aaiAdaaeaacaaI0aaaaOGaaiilaiaa-bcadaWcbaWcbaGaaG4naaqa% aiaaisdaaaGccaGGSaGaa8hiaiaadshacaGGPaGaaiOlaaaaaa!C1F3!\[\begin{gathered}\Psi (t) = {}_0F{\text{ }}_{\text{1}} {\text{(}}a, t{\text{), }}a \ne 0, - 1, - 2, \ldots ; \hfill \\\Psi (t) = {}_0F{\text{ }}_{\text{2}} (\tfrac{1}{3}, \tfrac{2}{3}, t) {}_0F{\text{ }}_{\text{2}} {\text{(}}\tfrac{2}{3}, \tfrac{4}{3}, t), {}_0F{\text{ }}_{\text{2}} (\tfrac{4}{3}, \tfrac{5}{3}, t); \hfill \\\Psi (t) = {}_0F{\text{ }}_{\text{3}} (\tfrac{1}{4}, \tfrac{2}{4}, \tfrac{3}{4}, t) {}_0F{\text{ }}_{\text{3}} {\text{(}}\tfrac{2}{4}, \tfrac{3}{4}, \tfrac{5}{4}, t), \hfill \\{}_0F{\text{ }}_{\text{3}} (\tfrac{3}{4}, \tfrac{5}{4}, \tfrac{6}{4}, t),. {}_0F{\text{ }}_{\text{3}} {\text{(}}\tfrac{5}{4}, \tfrac{6}{4}, \tfrac{7}{4}, t). \hfill \\\end{gathered}\]We also determine all Sheffer polynomials which are orthogonal on the unit circle. The formula for the product of polynomials of the Boas-Buck type is obtained.     

Γ andX bandgap hydrostatic deformation potentials for epitaxial In0.52Al0.48As on InP(001)

The pressure dependence of the direct and indirect bandgap of epitaxial In_0.52Al_0.48As on InP(001) substrate has been measured using photoluminescence up to 92 kbar hydrostatic pressure. The bandgap changes from Γ toX at an applied pressure of ∼ 43 kbar. Hydrostatic deformation potentials for both the Γ andX bandgaps are deduced, after correcting for the elastic constant (bulk modulus) mismatch between the epilayer and the substrate. For the epilayer we obtain and+(2.81±0.15)eV for the Γ andX bandgaps respectively. From the pressure dependence of the normalized Γ-bandgap photoluminescence intensity a Γ-X lifetime ratio, (τ_Γ/τ _X ), of 4.1×10⁻³ is deduced.