The spectral density of the scattering matrix of the magnetic Schrödinger operator for high energies

Bulger, Daniel; Pushnitski, Alexander

doi:10.4171/jst/54

Cited by 3 publications

(3 citation statements)

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“…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%

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Microlocal properties of scattering matrices

Nakamura

2016

Communications in Partial Differential Equations

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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.

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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

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“…Relation to other works The distribution of the eigenvalues of the scattering matrix has been studied since the eighties ([BY82], [BY84], [SY85]). More recently, in the (non-semiclassical) highenergy limit, it was studied in [BP12], and extended to more general Hamiltonians in [BP13] and [Nak14]. For related topics in the physics literature for obstacle scattering, see [DS92].…”

Section: Statement Of the Resultsmentioning

confidence: 99%

Equidistribution of phase shifts in trapped scattering

Ingremeau

2018

J. Spectr. Theory

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We consider semiclassical scattering for compactly supported perturbations of the Laplacian and show equidistribution of eigenvalues of the scattering matrix at (classically) non-degenerate energy levels. The only requirement is that sets of fixed points of certain natural scattering relations have measure zero. This extends the result of [GRHZ15], where the authors proved the equidistribution result under a similar assumption on fixed points but with the condition that there is no trapping. IntroductionConsider a Riemannian manifold (X, g) which is Euclidean near infinity, in the sense that there exist compact sets X 0 ⊂ X and K 0 ⊂ R d such that (X\X 0 , g) and (R d \K 0 , g eucl ) are isometric.Let us consider an operatorWe define the scattering matrix 1 S h : C ∞ (S d−1 ) −→ C ∞ (S d−1 ), which depends on h, by S h (φ in ) := e iπ(d−1)/2 φ out .The factor e iπ(d−1)/2 is taken so that the scattering matrix is the identity operator when (X, g) = (R d , g Eucl ) and V ≡ 0.For each h ∈ (0, 1], S h can be extended by density to a unitary operator acting on L 2 (S d−1 ). S h − Id is then a trace class operator. Therefore, S h admits a sequence of eigenvalues of modulus 1, which converge to 1, and which we denote by (e iβ h,n ) n∈N .Our aim in this paper will be to study the behaviour of (e iβ h,n ) in the limit where h → 0. To do this, we define a measure µ h on S 1 byfor any continuous f : S 1 −→ C. This measure is not finite, but µ h , f is finite as soon as 1 is not in the support of f .Let us now state the assumptions we make on the manifold X and on the potential V .

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