Bundles of Pseudodifferential Operators and Modular Forms

We describe connections between pseudodifferential operators and modular forms in terms of vector bundles over a Riemann surface whose fibers are the spaces of certain pseudodifferential operators.     

Invertibility of parabolic pseudodifferential operators with rapidly increasing symbols

We consider pseudodifferential operators with rapidly increasing double symbols analytic with respect to the variable dual to the time on the lower complex half-plane. We construct invertibility theory for these operators in weighted Sobolev spaces with weights related to growths of symbols and give applications to heat equations with potentials of power, exponential, and superexponential growths. Dedicated to the memory of Professor Leonid Romanovich Volevich     

Sharp Spectral Asymptotics and Weyl Formula for Elliptic Operators with Non-smooth Coefficients

The aim of this paper is to give the Weyl formula for eigenvalues of self-adjoint elliptic operators, assuming that first-order derivatives of the coefficients are Lipschitz continuous. The approach is based on the asymptotic formula of Hörmander"s type for the spectral function of pseudodifferential operators having Lipschitz continuous Hamiltonian flow and obtained via a regularization procedure of nonsmooth coefficients.     

Pseudodifferential Operators of Several Variables and Baker Functions

The KP hierarchy consists of an infinite system of nonlinear partial differential equations and is determined by Lax equations, which can be constructed using pseudodifferential operators. The KP hierarchy and the associated Lax equations can be generalized by using pseudodifferential operators of several variables. We construct Baker functions associated to those generalized Lax equations of several variables and prove some of the properties satisfied by such functions.     

Spectral Boundary Conditions for Generalizations of Laplace and Dirac Operators

Spectral boundary conditions for Laplace-type operators on a compact manifold X with boundary are partly Dirichlet, partly (oblique) Neumann conditions, where the partitioning is provided by a pseudodifferential projection; they have an interest in string and brane theory. Relying on pseudodifferential methods, we give sufficient conditions for the existence of the associated resolvent and heat operator, and show asymptotic expansions of their traces in powers and power-log terms, allowing a smearing function . The leading log-coefficient is identified as a non-commutative residue, which vanishes when =1. The study has new consequences for well-posed (spectral) boundary problems for first-order, Dirac-like elliptic operators (generalizing the Atiyah-Patodi-Singer problem). It is found e.g. that the zeta function is always regular at zero. In the selfadjoint case, there is a stability of the zeta function value and the eta function regularity at zero, under perturbations of the boundary projection of order -dim X.     

Bounded Subquotients of Pseudodifferential Operator Modules

Recently there have been several papers on the action of the Virasoro Lie algebra on the projective decompositions of the modules of pseudodifferential operators on the circle. We use their results to prove that a wide class of the uniserial (completely indecomposable) bounded modules of the Virasoro Lie algebra may be realized as subquotients of such modules of pseudodifferential operators. This gives easy proofs of the existence of many previously known uniserial modules, and moreover yields some hitherto undiscovered.Partially supported by NSA grant MDA 904-03-1-0004.     

On Darboux–Bäcklund Transformations for the Q-Deformed Korteweg–de Vries Hierarchy

We study Darboux–Bäcklund transformations (DBTs) for the q-deformed Korteweg–de Vries hierarchy by using the q-deformed pseudodifferential operators. The elementary DBTs are triggered by the gauge operators constructed from the (adjoint) wave functions of the associated linear systems. Iterating these elementary DBTs, we obtain not only q-deformed Wronskian-type but also binary-type representations of the tau-function of the hierarchy.     

Surfaces on Lie groups,on Lie algebras,and their integrability

We study the cohomology of the Schwinger term arising in second quantization of the class of observables belonging to the restricted general linear algebra. We prove that, for all pseudodifferential operators in 3+1 dimensions of this type, the Schwinger term is equivalent to the twisted Radul cocycle, a modified version of the Radul cocycle arising in non-commutative differential geometry. In the process we also show how the ordinary Radul cocycle for any pair of pseudodifferential operators in any dimension can be written as the phase space integral of the star commutator of their symbols projected to the appropriate asymptotic component.     

THE FIRST COHOMOLOGY OF THE SUPERCONFORMAL ALGEBRA K(1|4)

The infinitesimal deformations of the embedding of the Lie superalgebra of contact vector fields on the supercircle S^1|4 into the Poisson superalgebra of symbols of pseudodifferential operators on S^1|2 are explicitly calculated.     

Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem

Fermionic quantization, or Clifford algebra, is combined with pseudodifferential operators to simplify the proof of the Atiyah-Singer index theorem for the Dirac operator on a spin manifold.     

The Atiyah-Singer index theorem as passage to the classical limit in quantum mechanics

A new approach to the Atiyah-Singer index theorem is described, using the technique of continuous fields ofC ^*-algebras. The proof is given in the case of elliptic pseudodifferential operators on ℝ ⁿ .     

High Energy Limits of Laplace-Type and Dirac-Type EigenFunctions and Frame Flows

We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators and therefore the system remains non-commutative in the high-energy limit. We discuss to what extent the space of stationary high-energy states behaves classically. The first author was supported by NSERC, FQRNT and Dawson fellowship.     

Algebraic index theorem

We prove the Atiyah-Singer index theorem where the algebra of pseudodifferential operators is replaced by an arbitrary deformation quantization of the algebra of functions on a symplectic manifold.Partially supported by NSF Grant DMS-9101817.     

守恒无伪解麦克斯韦方程间断元研究:(Ⅰ)一维和二维情况

考察了电、磁场分量分别基于不同近似函数空间展开的一维和二维Maxwell方程间断元求解方法。结合中心数值通量和电、磁场分量近似函数空间的不同组合,构造了各种间断元算子。通过用这些算子在规则和不规则网格上编码分析一维和二维金属腔的谐振模式,详细考察了算子的收敛和伪解支持性,并据此对基函数进行了优选。算子在时域和频域对谐振模式的计算结果彼此符合良好。优选的Maxwell方程间断元算子不仅同时具备能量守恒和免于伪解的特性,且无需引入辅助变量,为设计高品质Maxwell方程间断元算法和研发相关电磁场模拟软件提供了支撑。     

Pseudodifferential Symbols on Riemann Surfaces and Krichever–Novikov Algebras

We define the Krichever-Novikov-type Lie algebras of differential operators and pseudodifferential symbols on Riemann surfaces, along with their outer derivations and central extensions. We show that the corresponding algebras of meromorphic operators and symbols have many invariant traces and central extensions, given by the logarithms of meromorphic vector fields. Very few of these extensions survive after passing to the algebras of operators and symbols holomorphic away from several fixed points. We also describe the associated Manin triples and KdV-type hierarchies, emphasizing the similarities and differences with the case of smooth symbols on the circle.     

The action functional in non-commutative geometry

We establish the equality between the restriction of the Adler-Manin-Wodzicki residue or non-commutative residue to pseudodifferential operators of order –n on ann-dimensional compact manifoldM, with the trace which J. Dixmier constructed on the Macaev ideal. We then use the latter trace to recover the Yang Mills interaction in the context of non-commutative differential geometry.     

Quantum Computational Logic

A quantum computational logic is constructed by employing density operators on spaces of qubits and quantum gates represented by unitary operators. It is shown that this quantum computational logic is isomorphic to the basic sequential effect algebra [0, 1].     

The Egorov theorem for transverse Dirac-type operators on foliated manifolds

Egorov’s theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first-order transversally elliptic operator with the (classical) evolution of its symbols determined by the parallel transport along the orbits of the associated transverse bicharacteristic flow. For a particular case of a transverse Dirac operator, the transverse bicharacteristic flow is shown to be given by the transverse geodesic flow and the parallel transport by the parallel transport determined by the transverse Levi-Civita connection. These results allow us to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.     

Higher spin fields and the Gelfand-Dickey algebra

We show that in 2-dimensional field theory, higher spin algebras are contained in the algebra of formal pseudodifferential operators introduced by Gelfand and Dickey to describe integrable nonlinear differential equations in Lax form. The spin 2 and 3 algebras are discussed in detail and the generalization to all higher spins is outlined. This provides a conformal field theory approach to the representation theory of Gelfand—Dickey algebras.Supported in part by the NSF Grant PHY-84-04931     

Band asymptotics of eigenvalues for the Zoll manifold

It is well known that the spectrum of the Laplace-Beltrami operator for a Zoll metric on the sphere consists of clusters of eigenvalues. This Letter considers the asymptotic band structure of these clusters and gives a explicit formula for it on the basis of the theory of Hamiltonian dynamical systems and symbol calculations of pseudodifferential operators.