Global Uniqueness in the Inverse Scattering Problem¶for the Schrödinger Operator¶with External Yang–Mills Potentials

We consider the Schr?dinger equation in R ⁿ , n≥ 3, with external Yang–Mills potentials having compact supports. We prove the uniqueness modulo a gauge transformation of the solution of the inverse boundary value problem in a bounded convex domain. A similar uniqueness result holds for the inverse scattering problem at a fixed energy. Received: 11 August 2000 / Accepted: 24 May 2001     

Uniqueness of the Invariant Measure¶for a Stochastic PDE Driven by Degenerate Noise

We consider the stochastic Ginzburg–Landau equation in a bounded domain. We assume the stochastic forcing acts only on high spatial frequencies. The low-lying frequencies are then only connected to this forcing through the non-linear (cubic) term of the Ginzburg–Landau equation. Under these assumptions, we show that the stochastic PDE has a unique invariant measure. The techniques of proof combine a controllability argument for thelow-lying frequencies with an infinite dimensional version of the Malliavin calculus to show positivity and regularity of the invariant measure. This then implies the uniqueness of that measure. Received: 10 September 2000 / Accepted: 13 December 2000     

Spin and Abelian Electromagnetic Duality¶on Four-Manifolds

We investigate the electromagnetic duality properties of an Abelian gauge theory on a compact oriented four-manifold by analysing the behaviour of a generalised partition function under modular transformations of the dimensionless coupling constants. The true partition function is invariant under the full modular group but the generalised partition function exhibits more complicated behaviour depending on topological properties of the four-manifold concerned. It is already known that there may be “modular weights” which are linear combinations of the Euler number and Hirzebruch signature of the four-manifold. But sometimes the partition function transforms only under a subgroup of the modular group (the Hecke subgroup). In this case it is impossible to define real spinor wave-functions on the four-manifold. But complex spinors are possible provided the background magnetic fluxes are appropriately fractional rather than integral. This gives rise to a second partition function which enables the full modular group to be realised by permuting the two partition functions, together with a third. Thus the full modular group is realised in all cases. The demonstration makes use of various constructions concerning integral lattices and theta functions that seem to be of intrinsic interest. Received: 5 June 2000 / Accepted: 9 October 2000     

Absolutely Continuous Invariant Measures¶for Piecewise Real-Analytic Expanding Maps¶on the Plane

We prove the existence of absolutely continuous invariant measures for piecewise real-analytic expanding maps on bounded regions in the plane. Received: 5 June 1998 / Accepted: 11 May 1999     

Microlocal Analysis and¶Interacting Quantum Field Theories:¶Renormalization on Physical Backgrounds

We present a perturbative construction of interacting quantum field theories on smooth globally hyperbolic (curved) space-times. We develop a purely local version of the Stückelberg–Bogoliubov–Epstein–Glaser method of renormalization by using techniques from microlocal analysis. Relying on recent results of Radzikowski, K?hler and the authors about a formulation of a local spectrum condition in terms of wave front sets of correlation functions of quantum fields on curved space-times, we construct time-ordered operator-valued products of Wick polynomials of free fields. They serve as building blocks for a local (perturbative) definition of interacting fields. Renormalization in this framework amounts to extensions of expectation values of time-ordered products to all points of space-time. The extensions are classified according to a microlocal generalization of Steinmann scaling degree corresponding to the degree of divergence in other renormalization schemes. As a result, we prove that the usual perturbative classification of interacting quantum field theories holds also on curved space-times. Finite renormalizations are deferred to a subsequent paper. As byproducts, we describe a perturbative construction of local algebras of observables, present a new definition of Wick polynomials as operator-valued distributions on a natural domain, and we find a general method for the extension of distributions which were defined on the complement of some surface. Received: 31 March 1999 / Accepted: 10 June 1999     

Noncommutative Geometry¶and Gauge Theory on Fuzzy Sphere

The differential algebra on the fuzzy sphere is constructed by applying Connes' scheme. The U(1) gauge theory on the fuzzy sphere based on this differential algebra is defined. The local U(1) gauge transformation on the fuzzy sphere is identified with the left U(N+1) transformation of the field, where a field is a bimodule over the quantized algebra . The interaction with a complex scalar field is also given. Received: 21 January 1998 / Accepted: 4 February 2000     

Superconformal Invariance and the Geography¶of Four-Manifolds

The correlation functions of supersymmetric gauge theories on a four-manifold X can sometimes be expressed in terms of topological invariants of X. We show how the existence of superconformal fixed points in the gauge theory can provide nontrivial information about four-manifold topology. In particular, in the example of gauge group SU(2) with one doublet hypermultiplet, we derive a theorem relating classical topological invariants such as the Euler character and signature to sum rules for Seiberg–Witten invariants. A short account of this paper can be found in [1]. Received: 19 December 1998 / Accepted: 7 March 1999     

Effect of the free surface and the rigid plane on structural vibration and acoustic radiation

The coupled fluid-structure interaction equation is estabhshed for bodies in the half-space fluid domain, especially sitting on the infinite plane, based on the BEM (Boundary Element Method) theory. Then, the natural frequencies, vibration responses and the acoustic radiation for a box are calculated, and the effect of the free surface and the rigid plane is discussed. Finally, several relative factors including the plate thickness, the structure damping and the distance between the body and the infinite plane are studied. The results show that the effect of the free surface and the rigid plane on the structural natural frequencies, vibration responses and the acoustic radiation cannot be ignored.     

Two remarks on the painlevé test

Using the example of two previously discussed autonomous differential equations it is pointed out that the Painlevé test with an ansatz w = Σ_n=p^∞w_n(z − z₀)ⁿ, p < 0, integer, will not reveal movable branch points if all leading order contribution arise from terms with equal degree of nonlinearity. Just contrary to this case, for the reduced damped and driven φ⁴ equation with p = −1 two resonance conditions arise at order n = 3 which cannot be satisfied simultaneously.     

Eigenvalues of the Dirac Operator on Manifolds¶with Boundary

Under standard local boundary conditions or certain global APS boundary conditions, we get lower bounds for the eigenvalues of the Dirac operator on compact spin manifolds with boundary. For the local boundary conditions, limiting cases are characterized by the existence of real Killing spinors and the minimality of the boundary. Received: 22 August 2000 / Accepted: 15 March 2001     

Szegő Difference Equations, Transfer Matrices¶and Orthogonal Polynomials on the Unit Circle

We develop the theory of orthogonal polynomials on the unit circle based on the Szegő recurrence relations written in matrix form. The orthogonality measure and C-function arise in exactly the same way as Weyl's function in the Weyl approach to second order linear differential equations on the half-line. The main object under consideration is the transfer matrix which is a key ingredient in the modern theory of one-dimensional Schr?dinger operators (discrete and continuous), and the notion of subordinacy from the Gilbert–Pearson theory. We study the relations between transfer matrices and the structure of orthogonality measures. The theory is illustrated by the Szegő equations with reflection coefficients having bounded variation. Received: 26 February 2001 / Accepted: 28 May 2001     

Renormalization Group¶and the Melnikov Problem for PDE's

We give a new proof of persistence of quasi-periodic, low dimensional elliptic tori in infinite dimensional systems. The proof is based on a renormalization group iteration that was developed recently in [BGK] to address the standard KAM problem, namely, persistence of invariant tori of maximal dimension in finite dimensional, near integrable systems. Our result covers situations in which the so called normal frequencies are multiple. In particular, it provides a new proof of the existence of small-amplitude, quasi-periodic solutions of nonlinear wave equations with periodic boundary conditions. Received: 29 January 2001 / Accepted: 8 March 2001     

Symplectic Structures of Moduli Space¶of Higgs Bundles over a Curve and Hilbert Scheme¶of Points on the Canonical Bundle

The moduli space of triples of the form (E,θ,s) are considered, where (E,θ) is a Higgs bundle on a fixed Riemann surface X, and s is a nonzero holomorphic section of E. Such a moduli space admits a natural map to the moduli space of Higgs bundles simply by forgetting s. If (Y,L) is the spectral data for the Higgs bundle (E,θ), then s defines a section of the line bundle L over Y. The divisor of this section gives a point of a Hilbert scheme, parametrizing 0-dimensional subschemes of the total space of the canonical bundle K _X , since Y is a curve on K _X . The main result says that the pullback of the symplectic form on the moduli space of Higgs bundles to the moduli space of triples coincides with the pullback of the natural symplectic form on the Hilbert scheme using the map that sends any triple (E,θ,s) to the divisor of the corresponding section of the line bundle on the spectral curve. Received: 15 January 2000 / Accepted: 25 March 2001     

Isospin Effects of the Mean Field and Two—Body Collision on the Fragmentation Process

Isospin effects of the mean field and two-body collision on the fragmentation as well as their dependences on the momentum-dependent interaction at intermediate energy heavy ion collisions are studied by using an isospindependent quantum molecular dynamics model.We find the prominent isospin effects of the multiplicity of the intermediate mass fragments Ninf,where Nimf depends sensitively on the isospin effect of the in-medium nucleonnucleon cross section and weakly on the variation of symmetry potential in the intermediate energy region.The momentum dependence interaction enhances the sensitivity of Nimf on the isospin effect of two-body collision.     

Profound Carrier-Envelope-Phase-Difference Effect on Photoionization Rate： Numerical Results on a Model Hydrogen Atom

We discuss importance of the carrier-envelope phase difference （CEPD） effect which would be encountered in real short laser pulses. By numerically simulating photoionization rate of a one-dimensional atomic hydrogen model irradiated by short light pulses, a profound CEPD effect on the photoionization rate is revealed, which might shed light on laser-atom interaction investigation. We also suggest that photoionization process might be exploited in return to measure CEPD values.     

High-Sensitivity Temperature Sensor Based on Two Parallel Fabry–Pérot Interferometers and Vernier Effect

Journal of Russian Laser Research - We propose a novel sensor with its temperature sensitivity improved by the Vernier effect. The sensor comprises two parallel Fabry–Pérot...     

Breit–Wigner Approximation and the Distribution¶of Resonances

For operators with a discrete spectrum, {λ _j ²}, the counting function of λ _j 's, N (λ), trivially satisfies N ( λ+δ ) −N ( λ−δ ) =∑ _j δ_{λ j} ((λ−δ,λ+δ]). In scattering situations the natural analogue of the discrete spectrum is given by resonances, λ _j ∈ℂ₊, and of N (λ), by the scattering phase, s(λ). The relation between the two is now non-trivial and we prove that

Square Integrability and Uniqueness of the Solutions of the Kadomtsev–Petviashvili-I Equation

We prove that the solution of the Cauchy problem for the Kadomtsev–Petviashvili-I Equation obtained by the inverse spectral method belongs to the Sobolev space H^k(R²) for k 0, under the assumption that the initial datum is a small Schwartz function. This solution is shown to be the unique solution within a class of generalized solutions of the Kadomtsev–Petviashvili-I equation.     

Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of ℝ ⁿ . They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features. The new examples include the instanton algebra and the NC-4-spheres S ⁴ _θ. We construct the noncommutative algebras ?=C ^∞ (S ⁴ _θ) of functions on NC-spheres as solutions to the vanishing, ch _j (e) = 0, j < 2, of the Chern character in the cyclic homology of ? of an idempotent e∈M ₄ (?), e ²=e, e=e ^*. We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmannian as well as the corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC-3-sphere S ³ _θ distinct from quantum group deformations SU _q (2) of SU (2). We then construct the noncommutative geometry of S _θ ⁴ as given by a spectral triple ?, ℋ, D) and check all axioms of noncommutative manifolds. In a previous paper it was shown that for any Riemannian metric g _μν on S ⁴ whose volume form is the same as the one for the round metric, the corresponding Dirac operator gives a solution to the following quartic equation,