The Spectral Density of the Scattering Matrix for High Energies

Bulger, Daniel; Pushnitski, Alexander

doi:10.1007/s00220-012-1551-7

Cited by 9 publications

(12 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this case, v(ξ) = ξ and Σ λ = ξ ∈ R d 1 2 |ξ| 2 = λ . Then we recover the Xray transform type approximation ( [4,5]), i.e., the principal symbol of the scattering matrix is given by s 0 (λ; x, ξ) = e −iψ(λ;x,ξ) , where…”

Section: Applications To Operators On Euclidean Spacesmentioning

confidence: 99%

See 1 more Smart Citation

Microlocal properties of scattering matrices

Nakamura

2016

Communications in Partial Differential Equations

View full text Add to dashboard Cite

We consider scattering theory for a pair of operators H 0 and H = H 0 + V on L 2 (M, m), where M is a Riemannian manifold, H 0 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, but it also applies to various differential operators with constant coefficients and short-range perturbations on Euclidean spaces.Let I be a compact interval and we assumeand p −1 0 (I) is compact. We now consider the scattering theory for the pair (H, H 0 ) on the energy interval I, i.e., we study the absolutely continuous spectrum of H on I. We denote the spectral projection of an operator A on J ⊂ R by E J (A). Then the wave operatorsexist and they are complete: Ran W I ± = E I (H)H ac (H). Moreover, the point spectrum σ(H) ∩ I is finite including the multiplicities (see Section 2).We write the energy surface of H 0 with an energy λ ∈ I byΣ λ is a regular submanifold in M , and we let m λ be the smooth density on Σ λ characterized as follows: m λ = i * m λ , wherem λ ∈ d−1 (M ) such thatm λ ∧ dp 0 = m, and i : Σ λ ֒→ M is the embedding. (Note m λ is uniquely determined whereasm λ is not.) The scattering operator is defined by S I = (W I + ) * W I − , H → H, and it commutes with H 0 . Hence S I is decomposed to a family of operators {S(λ)} λ∈I , where S(λ) is a unitary operator on L 2 (Σ λ , m λ ) for a.e. λ ∈ I. S(λ) is called the scattering matrix (see Section 5 for the detail). Our main result is the following: Theorem 1.

show abstract

Section: Applications To Operators On Euclidean Spacesmentioning

confidence: 99%

“…Recently, Bulger and Pushnitski have employed a sort of hybrid of the microlocal and the functional analytic methods to obtain spectral asymptotics of the scattering matrix ( [4,5]). In this paper we obtain analogous result for fixed energies using the standard pseudodifferential operator calculus on manifolds.…”

Section: Introductionmentioning

confidence: 99%

Microlocal properties of scattering matrices

Nakamura

2016

Communications in Partial Differential Equations

View full text Add to dashboard Cite

show abstract

“…the logarithms of the eigenvalues of the scattering matrix, were analysed by Birman-Yafaev [3,4,5,6], Sobolev-Yafaev [24], Yafaev [26] and more recently Bulger-Pushnitski [7]. In [24], an asymptotic form V ∼ cr −α , α > 2 was assumed and asymptotics of the individual phase shifts as well as the scattering cross section were obtained.…”

Section: Introductionmentioning

confidence: 99%

Approximation and Equidistribution of Phase Shifts: Spherical Symmetry

Datchev

Gell-Redman

Hassell

et al. 2013

Commun. Math. Phys.

View full text Add to dashboard Cite

Abstract. Consider a semiclassical HamiltonianH V,h := h 2 ∆ + V − E where h > 0 is a semiclassical parameter, ∆ is the positive Laplacian on R d , V is a smooth, compactly supported central potential function and E > 0 is an energy level. In this setting the scattering matrix S h (E) is a unitary operator on L 2 (S d−1 ), hence with spectrum lying on the unit circle; moreover, the spectrum is discrete except at 1.We show under certain additional assumptions on the potential that the eigenvalues of S h (E) can be divided into two classes: a finite number ∼ c d (R √ E/h) d−1 , as h → 0, where B(0, R) is the convex hull of the support of the potential, that equidistribute around the unit circle, and the remainder that are all very close to 1. Semiclassically, these are related to the rays that meet the support of, and hence are scattered by, the potential, and those that do not meet the support of the potential, respectively.A similar property is shown for the obstacle problem in the case that the obstacle is the ball of radius R.

show abstract

“…[19,Chapter 8]; this makes the analysis of S(k) rather explicit. In [6], using the Born approximation, we have determined the large energy asymptotic density of the spectrum of S(k) for A ≡ 0; we will say more about this in the next subsection. When A ≡ 0, the situation is radically different: as k → ∞, the norm S(k) − I does not tend to zero and the Born approximation is no longer valid.…”

Section: Main Results and Discussionmentioning

confidence: 99%

“…This suggests the following rescaled version of the problem: for an interval δ ⊂ R \ {0} separated away from zero, set µ k (δ) = #{n ∈ N : kθ n (k) ∈ δ}. Then it turns out (see [6]) that…”

Section: Main Results and Discussionmentioning

confidence: 99%