We define and study an algebra Ψ ∞ 1,0,V (M 0 ) of pseudodifferential operators canonically associated to a noncompact, Riemannian manifold M whose geometry at infinity is described by Lie vector fields V on compactification compact with corners.We show that the basic properties usual extend ).We also consider Diff * differential generated C ), microlocalization ).Our construction solves problem posed Melrose in 1990.Finally, we introduce semi-classical "suspended" versions ).

We study a generalization of the geodesic spray and give conditions for noncomapct manifolds with Lie structure at infinity to have positive injectivity radius. also prove that geometric operators are generated by given algebra vector fields. This is first one in series papers devoted analysis differential on infinity.

We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a algebra of vector fields compactification. extend several classical results Sobolev spaces, elliptic regularity, and mapping properties pseudodifferential operators. A tubular neighborhood theorem for submanifolds allows us also regular open subsets the traces functions in suitable spaces. Our main application is regularity result...

Let D ⊂ ℝ d , = 2, 3, be a bounded domain with piecewise smooth boundary, Y ℓ ∞ (ℕ) and U B 1 (Y), the open unit ball of Y. We consider parametric family (P y ) y∈U uniformly strongly elliptic, second-order partial differential operators P on D. Under suitable assumptions coefficients, we establish regularity result for solution u boundary value problem u(x, y) f(x, y), x ∈ D, U, mixed Dirichlet–Neumann conditions ∂ and, respectively, n Our well-posedness results are formulated in scale...

We show that a Lie algebroid on stratified manifold is integrable if, and only its restriction to each strata integrable. These results allow us construct large class of algebras pseudodifferential operators. They are also relevant for the definition graph certain singular foliations manifolds with corners construction natural operators given complex algebraic variety.

We construct algebras of pseudodifferential operators on a continuous family groupoid , this was shown not to be possible by the first author in an earlier paper. AMS 2000 Mathematics subject classification: Primary 35S05. Secondary 35J15; 47G30; 58J40; 46L87

Abstract We study the complex powers A z of an elliptic, strictly positive pseudodifferential operator using axiomatic method that combines approaches Guillemin and Seeley. In particular, we introduce a class algebras, called “Guillemin ” whose definition was inspired by [Guillemin, V. (1985). new proof Weyl's formula on asymptotic distribution eigenvalues. Adv. in Math. 55:131–160]. algebra can be thought as “abstract operators.” Most algebras operators belong to this class. Several results...

We construct closed-form asymptotic formulas for the Green's function of parabolic equations (e.g., Fokker–Planck equations) with variable coefficients in one space dimension. More precisely, let $u(t,x)=\int\mathcal{G}_t(x,y)f(y)dy$ be solution $\partial_tu-(au"+bu'+cu)=0$ $t>0$, $u(0,x)=f(x)$. Then we find computable approximations $\mathcal{G}_t^{[n]}$ $\mathcal{G}_t$. The approximate kernels are derived by applying Dyson–Taylor commutator method that have recently developed short-time...

Abstract Let M be a Riemannian manifold with smooth boundary. The main question we address in this article is: “When is the Laplace–Beltrami operator , invertible?” We consider also case of mixed boundary conditions. study leads us to class manifolds and bounded geometry introduced by Schick (Math. Nachr. 223 (2001), 103–120). thus begin some needed results on geometry. an open closed subset . say that has finite width if, definition, such distance from point uniformly x (and hence,...

Let V be a real valued potential that is smooth everywhere on R3, except at periodic, discrete set S of points, where it has singularities the Coulomb-type Z/r. We assume periodic with period lattice L. study spectrum Schrödinger operator H=−Δ+V acting space Bloch waves arbitrary, but fixed, wavevector k. T≔R3/L. u an eigenfunction H eigenvalue λ and let ϵ>0 arbitrarily small. show classical regularity u∊H5/2−ϵ(T) in usual Sobolev spaces, u∊K3/2−ϵm(T\S) weighted spaces. The index m...

Let [Formula: see text] be a second order, uniformly elliptic, positive semi-definite differential operator on complete Riemannian manifold of bounded geometry M, acting between sections vector bundle with E over M. We assume that the coefficients L are bounded. Using finite speed propagation for L, we investigate properties operators form text]. In particular, establish results distribution kernels and mapping e -tL (μ + L) s . show generates holomorphic semigroup has usual W s,p -Sobolev...

We determine the essential spectrum of N-body Hamiltonians with 2-body (or, more generally, k-body) potentials that have radial limits at infinity. The classical appearing in well known HVZ-theorem are a particular case this type corresponding to zero Our result thus extends describes usual Hamiltonians. More precisely, if configuration space system is finite dimensional real vector X, then let e(X) be C *-algebra functions on X generated by algebras C(X/Y), where Y runs over set all linear...