On time-dependent orthogonal polynomials on the unit circle

Two index formulas for operators defined by infinite band matrices are proved. These results may be interpreted as a generalization of a classical theorem of M. G. Krein on orthogonal polynomials. The proofs are based on dichotomy and nonstationary inertia theory.Dedicated to the memory of M. G. Krein, a mathematical giant, a great teacher and wonderful friend.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 18–36, January–February, 1994.     

On the absolute continuity of Schrödinger operators with spherically symmetric,long-range potentials,I

Let H = ?Δ + V, where the potential V is spherically symmetric and can be decomposed as a sum of a short-range and a long-range term, V(r) = V_S(r) + V_L(r). Assume that for some r₀, V_L(r) ?C^2k(r₀, ∞) and that there exist μ > 0, δ > 0, such that $\text{(}\text{d}\text{dr}\text{)}{}^{\mathrm{j}}\text{V}{}_{\mathrm{<mtext>L</mtext>}}\text{(r) = O(r}{}^{\mathrm{?\mu ?j\delta }}\text{)}\text{as}\text{r → ∞, 1 ? j ? 2k}$. Assume further that min(2kμ, (2k ? 1)δ + μ) > 1. Under this weak decay condition on V_L(r) it is shown in this paper that the positive spectrum of H is absolutely continuous and that the absolutely continuous part of H is unitarily equivalent to ?Δ, provided that the singularity of V at 0 is properly restricted. In particular, some oscillation of V_L(r) at infinity is allowed.     

Nonstationary Szegö theorem,band sequences and maximum entropy

Well known theorems concerning positive definite extensions of banded Toeplitz matrices are generalized. The proofs are based upon the theory ofm-sequences which is developed herein. The nonstationary versions of the maximum distance and maximum entropy problems are discussed.     

Approximations of Strongly Continuous Families of Unbounded Self-Adjoint Operators

In this paper we analyze supergeometric locally covariant quantum field theories. We develop suitable categories SLoc of super-Cartan supermanifolds, which generalize Lorentz manifolds in ordinary quantum field theory, and show that, starting from a few representation theoretic and geometric data, one can construct a functor \({{\mathfrak{A}}}\) : SLoc\({\to}\)S*Alg to the category of super-*-algebras, which can be interpreted as a non-interacting super-quantum field theory. This construction turns out to disregard supersymmetry transformations as the morphism sets in the above categories are too small. We then solve this problem by using techniques from enriched category theory, which allows us to replace the morphism sets by suitable morphism supersets that contain supersymmetry transformations as their higher superpoints. We construct super-quantum field theories in terms of enriched functors \({{\mathfrak{e}\mathfrak{A}}}\) : eSLoc\({\to}\)eS*Alg between the enriched categories and show that supersymmetry transformations are appropriately described within the enriched framework. As examples we analyze the superparticle in 1|1-dimensions and the free Wess–Zumino model in 3|2-dimensions.     

Dichotomy,discrete Bohl exponents,and spectrum of block weighted shifts

Supported by a Dr. Chaim Weizmann fellowship for scientific research.     

Remarks on a Nonlinear Parabolic Equation

The equation , , is studied in and in the periodic case. It is shown that the equation is well-posed in and possesses regularizing properties. For nonnegative initial data and the solution decays in as . In the periodic case it tends uniformly to a limit. A consistent difference scheme is presented and proved to be stable and convergent.

     

Time evolution of discrete fourth‐order elliptic operators

The evolution equation is considered. A discrete parabolic methodology is developed, based on a discrete elliptic (fourth‐order) calculus. The main ingredient of this calculus is a discrete biharmonic operator (DBO). In the general case, it is shown that the approximate solutions converge to the continuous one. An “almost optimal” convergence result (O(h^{4 ? ?})) is established in the case of constant coefficients, in particular in the pure biharmonic case. Several numerical test cases are presented that not only corroborate the theoretical accuracy result, but also demonstrate high‐order accuracy of the method in nonlinear cases. The nonlinear equations include the well‐studied Kuramoto–Sivashinsky equation. Numerical solutions for this equation are shown to approximate remarkably well the exact solutions. The numerical examples demonstrate the great improvement achieved by using the DBO instead of the standard (five‐point) discrete bilaplacian.     

Hyperbolic conservation laws on the sphere. A geometry-compatible finite volume scheme

We consider entropy solutions to the initial value problem associated with scalar nonlinear hyperbolic conservation laws posed on the two-dimensional sphere. We propose a finite volume scheme which relies on a web-like mesh made of segments of longitude and latitude lines. The structure of the mesh allows for a discrete version of a natural geometric compatibility condition, which arose earlier in the well-posedness theory established by Ben-Artzi and LeFloch. We study here several classes of flux vectors which define the conservation law under consideration. They are based on prescribing a suitable vector field in the Euclidean three-dimensional space and then suitably projecting it on the sphere’s tangent plane; even when the flux vector in the ambient space is constant, the corresponding flux vector is a non-trivial vector field on the sphere. In particular, we construct here “equatorial periodic solutions”, analogous to one-dimensional periodic solutions to one-dimensional conservation laws, as well as a wide variety of stationary (steady state) solutions. We also construct “confined solutions”, which are time-dependent solutions supported in an arbitrarily specified subdomain of the sphere. Finally, representative numerical examples and test cases are presented.     

Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds

Motivated by many applications (geophysical flows, general relativity), we attempt to set the foundations for a study of entropy solutions to non-linear hyperbolic conservation laws posed on a (Riemannian or Lorentzian) manifold. The flux of the conservation laws is viewed as a vector-field on the manifold and depends on the unknown function as a parameter. We introduce notions of entropy solutions in the class of bounded measurable functions and in the class of measure-valued mappings. We establish the well-posedness theory for conservation laws on a manifold, by generalizing both Kruzkov's and DiPerna's theories originally developed in the Euclidian setting. The class of geometry-compatible (as we call it) conservation laws is singled out as an important case of interest, which leads to robust L^p estimates independent of the geometry of the manifold. On the other hand, general conservation laws solely enjoy the L¹ contraction property and leads to a unique contractive semi-group of entropy solutions. Our framework allows us to construct entropy solutions on a manifold via the vanishing diffusion method or the finite volume method.