Existence of the Møller wave operators in quantum scattering theory

Existence of the wave operators of quantum mechanical scattering theory is established. The Hamiltonians are constant coefficient partial differential operators and perturbations of them by variable coefficient linear partial differential operators. The coefficients may be short range in the usual sense but they may also be oscillating. The problem of establishing existence is reduced to that of approximating solutions of a certain partial differential equations on cones in phase space. The proof is based on a refinement of Cook's argument. Research supported in part by a fellowship, awarded by the University of Toledo.     

Hecke Operators on Linear Ordinary Differential Equations

We construct Hecke operators acting on the space of certain linear ordinary differential equations, and describe a Hermitian inner product on the space of such differential equations. We also determine the adjoint of the Hecke operator with respect to this inner product, and prove that the space of ordinary differential equations associated to an automorphic form for a certain discrete subgroup of SL(2, R) has a basis consisting of common eigenvectors of a class of Hecke operators.     

Differential symmetry breaking operators: I. General theory and F-method

We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method (F-method) based on the algebraic Fourier transform for generalized Verma modules, which characterizes differential symmetry breaking operators by means of certain systems of partial differential equations. In contrast to the setting of real flag varieties, continuous symmetry breaking operators of Hermitian symmetric spaces are proved to be differential operators in the holomorphic setting. In this case, symmetry breaking operators are characterized by differential equations of second order via the F-method.     

Arithmetic partial differential equations, I

We develop an arithmetic analogue of linear partial differential equations in two independent “space-time” variables. The spatial derivative is a Fermat quotient operator, while the time derivative is the usual derivation. This allows us to “flow” integers or, more generally, points on algebraic groups with coordinates in rings with arithmetic flavor. In particular, we show that elliptic curves carry certain canonical “arithmetic flows” that are arithmetic analogues of the convection, heat, and wave equations, respectively. The same is true for the additive and the multiplicative group.     

Approximation of arithmetic sums by linear operators

We consider estimates of certain arithmetic sums by means of approximation by linear operators. These estimates are of interest in connection with a law of distribution of primes in arithmetic progressions.Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 925–936, December, 1973.     

Abstract potential operator and spectral methods for a class of degenerate evolution problems

The spectral theory of operators in Banach spaces is employed to treat a class of degenerate evolution equations. A basic role is played by the assumption that the Banach space under consideration may be expressed as a direct sum of two suitable subspaces. Two methods for solving the problem are studied. The first method is based on the expansion of the resolvent of a closed operator into Laurent series in a neighbourhood of 0. The second one makes use of the theory of abstract potential operators. In particular, an extension of the Hille-Yosida theorem on infinitesimal generators of (C₀) semigroups of linear operators is obtained. Some examples relative to operators appearing in many applications to partial differential equations are given.     

Surjectivity of differential operators and the division problem in certain function and distribution spaces

We give an overview on surjectivity conditions for partial differential operators and operators defined by multiplication with polynomials on certain function and distribution spaces of Laurent Schwartz. We complement the classical results by treating the surjectivity of operators on the space of slowly increasing functions and on the space of rapidly decreasing distributions, respectively.     

Dual Extremum Principles for Non-negative Unsymmetric Operators

Dual extremum principles are constructed for non-negative unsymmetricoperator equations. The theory is in terms of functional-valuedlinear operators defined on an infinite-dimensional space whichneed not even be normed. Saddle operators and saddle functionalare constructed as essential ingredients. An extension to affinesub-spaces required for certain partial differential equationsis included. New dual extremum principles for versions of theheat equation and the wave equation are stated in illustration.     

Evolution equations for abstract differential operators

We study in this paper the wellposedness and regularity of solutions of evolution equations associated with abstract differential operators on a Banach space. The results can be applied to many partial differential equations on different function spaces.     

On an explicit representation of the solution of linear stochastic partial differential equations with delays

Based on the analysis of a certain class of linear operators on a Banach space, we provide a closed form expression for the solutions of certain linear partial differential equations with non-autonomous input, time delays and stochastic terms, which takes the form of an infinite series expansion.     

Expansion of the boundaries of the solutions of the linear Volterra integral equations with convolution kernel

We solve a certain differential equation and system of integral equations. As applications, we characterize holomorphic symbols of commuting Toeplitz operators on the pluriharmonic Bergman space. In addition, pluriharmonic symbols of normal Toeplitz operators are characterized. Also, zero semi-commutators for certain classes of Toeplitz operators are characterized.This research is partially supported by KOSEF(98-0701-03-01-5).     

Finite and Infinite Order Differential Relations for Theta Functions

We give different linear and nonlinear differential relations on Jacobi theta functions with more emphasis on the nonlinear differential equation of the third order of Jacobi. We present different points of view with a special attention to the role played by the second order linear differential equations, and their link to the Riccati equation and the Schwarzian equation. We also study an identity for theta functions resulting from the action of certain infinite order differential operators.     

Equivalence of some coercive estimates for linear partial differential operators

The equivalence of certain coercive estimates for linear partial differential operators on ℝⁿ in a Banach space X is studied. Such estimates play an important role in the theory of differential equations and operators. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

A semigroup method for delay equations with relatively bounded operators in the delay term

We consider well-posedness and stability of abstract partial differential equations with unbounded operators in their delay terms. We show that the problem is equivalent to an abstract Cauchy problem in a product Banach space, give sufficient conditions on the well-posedness, make a complete spectral analysis and give robust stability criteria. The results are applied to the problem of small delays and to damped plate equations.     

Symmetry analysis and some exact solutions of cylindrically symmetric null fields in general relativity

The symmetry reduction method based on the Fréchet derivative of the differential operators is applied to investigate symmetries of the Field equations in general relativity corresponding to cylindrically symmetric space–time, that is a coupled system of nonlinear partial differential equations of second order. More specifically, this technique yields invariant transformation that reduce the given system of partial differential equations to a system of nonlinear ordinary differential equations. Some of the reduced systems are further studied for exact solutions.     

A Semigroup Method for Delay Equations with Relatively Bounded Operators in the Delay Term

We consider well-posedness and stability of abstract partial differential equations with unbounded operators in their delay terms. We show that the problem is equivalent to an abstract Cauchy problem in a product Banach space, give sufficient conditions on the well-posedness, make a complete spectral analysis and give robust stability criteria. The results are applied to the problem of small delays and to damped plate equations. September 12, 2000     

A note on the regularity of solutions of infinite dimensional Riccati equations

This note is concerned with the regularity of solutions of algebraic Riccati equations arising from infinite dimensional LQR control problems. We show that distributed parameter systems described by certain parabolic partial differential equations often have a special structure that smooths solutions of the corresponding Riccati equation. This analysis is motivated by the need to find specific representations for Riccati operators that can be used in the development of computational schemes for problems where the input and output operators are not Hilbert-Schmidt. This situation occurs in many boundary control problems and in certain distributed control problems associated with optimal sensor/actuator placement.     

Noether, partial Noether operators and first integrals for a linear system

We obtain Noether and partial Noether operators corresponding to a Lagrangian and a partial Lagrangian for a system of two linear second-order ordinary differential equations (ODEs) with variable coefficients. The canonical form for a system of two second-order ordinary differential equations is invoked and a special case of this system is studied for both Noether and partial Noether operators. Then the first integrals with respect to Noether and partial Noether operators are obtained for the linear system under consideration. We show that the first integrals for both the Noether and partial Noether operators are the same. This can give rise to further studies on systems from a partial Lagrangian viewpoint as systems in general do not admit Lagrangians.     

Arithmetic partial differential equations, II

We continue the study of arithmetic partial differential equations initiated in [7] by classifying “arithmetic convection equations” on modular curves, and by describing their space of solutions. Certain of these solutions involve the Fourier expansions of the Eisenstein modular forms of weight 4 and 6, while others involve the Serre-Tate expansions (Mori, 1995 [13], Buium, 2003 [4]) of the same modular forms; in this sense, our arithmetic convection equations can be seen as “unifying” the two types of expansions. The theory can be generalized to one of “arithmetic heat equations” on modular curves, but we prove that they do not carry “arithmetic wave equations.” Finally, we prove an instability result for families of arithmetic heat equations converging to an arithmetic convection equation.