Microlocal properties of scattering matrices

We consider the scattering theory for a pair of operators H₀ and H = H₀ + V on L²(M, m), where M is a Riemannian manifold, H₀ is a multiplication operator on M, and V is a pseudodifferential operator of order ? μ, μ > 1. We show that a time-dependent scattering theory can be constructed, and the scattering matrix is a pseudodifferential operator on each energy surface. Moreover, the principal symbol of the scattering matrix is given by a Born approximation type function. The main motivation of the study comes from applications to discrete Schrödigner operators, and it also applies to various differential operators with constant coefficients and short-range perturbations on Euclidean spaces.     

Completeness of Averaged Scattering Solutions and Inverse Scattering at a Fixed Energy

We prove that the averaged scattering solutions to the Schrödinger equation with short-range electromagnetic potentials (V, A) where V(x) = O(|x|^?ρ), A(x) = O(|x|^?ρ), |x| → ∞, ρ > 1, are dense in the set of all solutions to the Schrödinger equation that are in L ²(K) where K is any connected bounded open set in ? ⁿ ,n ≥ 2, with smooth boundary. We use this result to prove that if two short-range electromagnetic potentials (V ₁, A ₁) and (V ₂, A ₂) in ? ⁿ , n ≥ 3, have the same scattering matrix at a fixed positive energy and if the electric potentials V _j and the magnetic fields F _j : = curl A _j , j = 1, 2, coincide outside of some ball they necessarily coincide everywhere. In a previous paper of Weder and Yafaev the case of electric potentials and magnetic fields that are asymptotic sums of homogeneous terms at infinity was studied. It was proven that all these terms can be uniquely reconstructed from the singularities in the forward direction of the scattering amplitude at a fixed positive energy. The combination of the new uniqueness result of this paper and the result of Weder and Yafaev implies that the scattering matrix at a fixed positive energy uniquely determines electric potentials and magnetic fields that are a finite sum of homogeneous terms at infinity, or more generally, that are asymptotic sums of homogeneous terms that actually converge, respectively, to the electric potential and to the magnetic field.     

Inverse Spectral Problems in Rectangular Domains

We consider the Schrödinger operator ? Δ + q in domains of the form R = {x ∈ ? ⁿ : 0 ≤ x _i  ≤ a _i , i = 1,…, n} with either Dirichlet or Neumann boundary conditions on the faces of R, and study the constraints on q imposed by fixing the spectrum of ? Δ + q with these boundary conditions. We work in the space of potentials, q, which become real-analytic on ? ⁿ when they are extended evenly across the coordinate planes and then periodically. Our results have the corollary that there are no continuous isospectral deformations for these operators within that class of potentials. This work is based on new formulas for the trace of the wave group in this setting. In addition to the inverse spectral results these formulas lead to asymptotic expansions for the traces of the wave and heat kernels on rectangular domains.     

Stability of Planar Waves in the Allen–Cahn Equation

We study the asymptotic stability of planar waves for the Allen–Cahn equation on ? ⁿ , where n ≥ 2. Our first result states that planar waves are asymptotically stable under any—possibly large—initial perturbations that decay at space infinity. Our second result states that the planar waves are asymptotically stable under almost periodic perturbations. More precisely, the perturbed solution converges to a planar wave as t → ∞. The convergence is uniform in ? ⁿ . Lastly, the existence of a solution that oscillates permanently between two planar waves is shown, which implies that planar waves are not asymptotically stable under more general perturbations.     

Generalized Skew Derivations with Nilpotent Values in Prime Rings

Let ? be a prime ring, 𝒞 the extended centroid of ?, ? a Lie ideal of ?, F be a nonzero generalized skew derivation of ? with associated automorphism α, and n ≥ 1 be a fixed integer. If (F(xy) ? yx) ⁿ  = 0 for all x, y ∈ ?, then ? is commutative and one of the following statements holds:

(1) Either ? is central;

(2) Or ? ? M ₂(𝒞), the 2 × 2 matrix ring over 𝒞, with char(𝒞) = 2.     

Uniformly Conditioned Bases of Spectral Subspaces

A condition number of an ordered basis of a finite-dimensional normed space is defined in an intrinsic manner. This concept is extended to a sequence of bases of finite-dimensional normed spaces, and is used to determine uniform conditioning of such a sequence. We address the problem of finding a sequence of uniformly conditioned bases of spectral subspaces of operators of the form T _n  = S _n  + U _n , where S _n is a finite-rank operator on a Banach space and U _n is an operator which satisfies an invariance condition with respect to S _n . This problem is reduced to constructing a sequence of uniformly conditioned bases of spectral subspaces of operators on ? ^n×1. The applicability of these considerations in practical as well as theoretical aspects of spectral approximation is pointed out.     

Upper and Lower Bounds on Resonances for Manifolds Hyperbolic Near Infinity

For a conformally compact manifold that is hyperbolic near infinity and of dimension n + 1, we complete the proof of the optimal O(r ⁿ⁺¹) upper bound on the resonance counting function, correcting a mistake in the existing literature. In the case of a compactly supported perturbation of a hyperbolic manifold, we establish a Poisson formula expressing the regularized wave trace as a sum over scattering resonances. This leads to an r ⁿ⁺¹ lower bound on the counting function for scattering poles.     

On the Semigroup Algebra of Binary Relations

The semigroup of binary relations on {1,…, n} with the relative product is isomorphic to the semigroup B _n of n × n zero-one matrices with the Boolean matrix product. Over any field F, we prove that the semigroup algebra FB _n contains an ideal K _n of dimension (2 ⁿ  ? 1)², and we construct an explicit isomorphism of K _n with the matrix algebra M _{2 ⁿ ?1}(F).     

Modified Wave Operators for the Hartree Equation with Data,Image and Convergence in the Same Space,II

We study modified wave operators for the Hartree equation with a long-range potential |x|^-n |x|^{-\nu} , extending the result in [12] to the whole range of the Dollard type 1/2 < n \nu < 1. We construct the modified wave operators in the whole space of (1 + |x|)^-sL² (1 + |x|)^{-s}L^2 . We also have the image, strong continuity and strong asymptotic approximation in the same space. The lower bound $ s > 1 - \nu / 2 $ s > 1 - \nu / 2 of the weight is sharp from the scaling argument. Those maps are homeomorphic onto open subsets, which implies in particular asymptotic completeness for small data.     

The Semilinear Wave Equation on Asymptotically Euclidean Manifolds

We consider the quadratically semilinear wave equation on (? ^d , 𝔤), d ≥ 3. The metric 𝔤 is non-trapping and approaches the Euclidean metric like ?x?^?ρ. Using Mourre estimates and the Kato theory of smoothness, we obtain, for ρ > 0, a Keel–Smith–Sogge type inequality for the linear equation. Thanks to this estimate, we prove long time existence for the nonlinear problem with small initial data for ρ ≥ 1. Long time existence means that, for all n > 0, the life time of the solution is a least δ^?n , where δ is the size of the initial data in some appropriate Sobolev space. Moreover, for d ≥ 4 and ρ > 1, we obtain global existence for small data.     

Fourier-Integral-Operator Approximation of Solutions to First-Order Hyperbolic Pseudodifferential Equations I: Convergence in Sobolev Spaces

An approximation Ansatz for the operator solution, U(z′,z), of a hyperbolic first-order pseudodifferential equation, ? _z  + a(z,x,D _x ) with Re (a) ≥ 0, is constructed as the composition of global Fourier integral operators with complex phases. An estimate of the operator norm in L(H ^(s),H ^(s)) of these operators is provided, which yields a convergence result for the Ansatz to U(z′,z) in some Sobolev space as the number of operators in the composition goes to ∞.     

On Copure Projective Modules and Copure Projective Dimensions

Let R be any ring. A right R-module M is called n-copure projective if Ext¹(M, N) = 0 for any right R-module N with fd(N) ≤ n, and M is said to be strongly copure projective if Ext ⁱ (M, F) = 0 for all flat right R-modules F and all i ≥ 1. In this article, firstly, we present some general properties of n-copure projective modules and strongly copure projective modules. Then we define and investigate copure projective dimensions of modules and rings. Finally, more properties and applications of n-copure projective modules, strongly copure projective modules and copure projective dimensions are given over coherent rings with finite self-FP-injective dimension.     

Subsets with Generalized Derivations Having Nilpotent Values on Lie Ideals

Let R be a prime ring, with no nonzero nil right ideal, Q the two-sided Martindale quotient ring of R, F a generalized derivation of R, L a noncommutative Lie ideal of R, and b ∈ Q. If, for any u, w ∈ L, there exists n = n(u, w) ≥1 such that (F(uw) ? bwu)ⁿ = 0, then one of the following statements holds:

1. F = 0 and b = 0;

2. R ? M₂(K), the ring of 2 × 2 matrices over a field K, b² = 0, and F(x) = ?bx, for all x ∈ R.

     

Small Data Scattering for the One-Dimensional Nonlinear Dirac Equation with Power Nonlinearity

We study scattering problems for the one-dimensional nonlinear Dirac equation (?_t + α?_x + iβ)Φ = λ|Φ|^p?1Φ. We prove that if p > 3 (resp. p > 3 + 1/6), then the wave operator (resp. the scattering operator) is well-defined on some 0-neighborhood of a weighted Sobolev space. In order to prove these results, we use linear operators D(t)xD(?t) and t?_x + x?_t ? α/2, where {D(t)}_t∈? is the free Dirac evolution group. For the reader's convenience, in an appendix we list and prove fundamental properties of D(t)xD(?t) and t?_x + x?_t ? α/2.     

Annihilator Properties of Skew Monoid Rings

Let R be a ring with an endomorphism σ and F ∪ {0} the free monoid generated by U = {u ₁,…, u _t } with 0 added, and M a factor of F setting certain monomial in U to 0, enough so that, for some n, M ⁿ  = 0. In this article, we study various annihilator properties and a variety of conditions and related properties that the skew monoid ring R[M; σ] is inherited from R.     

Characterization of Multidimensional Stable Random Measures by Means of Vector Measures

Abstract

In connection with a symmetric α stable random measure Φ on a measurable space (F, ?) with values in R ^d , a complete metric space of symmetric finite measures on S _d?1 is constructed, and is employed to characterize the law of Φ by a unique positive measure on ? and a unique function on F × R ^d . The stochastic integral ∫ _F f d Φ is also defined for certain d × d matrix valued functions f, which for α = 2 reduces to the Wiener–Masani integral.     

Rate of Convergence of Chlodowsky-Type Bernstein Operators for Functions of Bounded Variation

In the current paper, we estimate the rate of pointwise convergence of the Chlodowsky-type Bernstein operators C _n (f,x) for functions defined on the interval [0,b _n ], for b _n  → ∞ as n → ∞, which are of bounded variation on [0, ∞). To prove our main result, we have used some methods and techniques from probability theory.     

Analysis of Symmetric Matrix Valued Functions

For any symmetric function f: ? ⁿ  → ? ⁿ , one can define a corresponding function on the space of n × n real symmetric matrices by applying f to the eigenvalues of the spectral decomposition. We show that this matrix valued function inherits from f the properties of continuity, Lipschitz continuity, strict continuity, directional differentiability, Fréchet differentiability, and continuous differentiability.     

Dispersive Estimates of Solutions to the Wave Equation with a Potential in Dimensions n ≥ 4

We prove dispersive estimates for solutions to the wave equation with a real-valued potential V ∈ L ^∞(R ⁿ ), n ≥ 4, satisfying V(x) = O(?x?^?(n+1)/2?ε), ε > 0.