The first variation of the scattering matrix

Helton, J. William; Ralston, James

doi:10.1016/0022-0396(76)90127-3

Cited by 21 publications

(6 citation statements)

References 8 publications

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“…Thus, we can sandwich any domain in an isophasal class between fixed discs D r and D R . By [7], s(λ) is monotonic in the domain, so we obtain (2.6).…”

Section: Lemma 22mentioning

confidence: 75%

“…By standard methods, this reduces to solving L k − λ 2 ũ k = g k ,ũ k = 0 on ∂Ω,ũ k outgoing at Georgetown University on July 10, 2015 http://imrn.oxfordjournals.org/ with g k → g in C ∞ and proving thatũ k →ũ, whereũ solves the limiting problem. These equations may be solved using the method of the Appendix of [7] (which in turn is based on the proof of Theorem 5.2 of [11]). Tracing through the proof, it is not hard to show that indeedũ k →ũ in L 2 loc .…”

Section: Proof Of Theorem 41mentioning

confidence: 99%

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Hassell¹,

Zelditch²

1999

Internat. Math. Res. Notices

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The formula is remarkable since the first term on the right is nonnegative and almost the square of the Sobolev 1/2 norm of φ. An analogous problem for the exterior domainIn this paper, we are interested in the exterior Laplacian. Let Ω = R 2 \ O, the exterior of an obstacle, and let ∆ Ω be the Laplacian on L 2 (Ω) with domain H 2 (Ω) ∩ H 1 0 (Ω) (i.e., the Dirichlet Laplacian). The operator ∆ Ω is selfadjoint with continuous spectrum on [0, ∞) (see [21 , Chapter 8]). We wish to formulate a problem about exterior domains that is analogous to the isospectral problem.Here, the O((ilg λ) 2 ) term is uniform over each isophasal class.

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“…Thus, we can sandwich any domain in an isophasal class between fixed discs D r and D R . By [7], s(λ) is monotonic in the domain, so we obtain (2.6).…”

Section: Lemma 22mentioning

confidence: 75%

Section: Proof Of Theorem 41mentioning

confidence: 99%

Untitled

Hassell¹,

Zelditch²

1999

Internat. Math. Res. Notices

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“…It follows from Lemma 7.1 together with the equality ψ + 0 (x, kω) = e ikωx that the operator G * 0 (k) is the farfield operator for the classical obstacle scattering problem with obstacle Ω. Moreover, S(k) := Id − 2i G * 0 (k), is the scattering matrix in the sense of [18]. It follows from [18, (2.1) and the remark after (1.9)] that all the eigenvalues λ = 1 of S(k) move in the counter-clockwise direction on the circle |z| = 1 in C continuously and with strictly positive velocities as k grows.…”

Section: Proof Of Theorem 211mentioning

confidence: 99%

“…where u(x, θ) is the solution of problem (2.11) with u 0 (x) = e −ikθx (note that [18] uses a different sign convention in the radiation condition (2.11c) resulting in a different sign of ∂β/∂k). It follows from this formula that ∂β ∂k (k) > 0: 1. the term xη x is positive by assumption, 2.…”

Section: Proof Of Theorem 211mentioning

confidence: 99%

Monochromatic Identities for the Green Function and Uniqueness Results for Passive Imaging

Agaltsov¹,

Hohage²,

Novikov³

2018

SIAM J. Appl. Math.

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For many wave propagation problems with random sources it has been demonstrated that cross correlations of wave fields are proportional to the imaginary part of the Green function of the underlying wave equation. This leads to the inverse problem to recover coefficients of a wave equation from the imaginary part of the Green function on some measurement manifold. In this paper we prove, in particular, local uniqueness results for the Schrödinger equation with one frequency and for the acoustic wave equation with unknown density and sound speed and two frequencies. As the main tool of our analysis, we establish new algebraic identities between the real and the imaginary part of Green's function, which in contrast to the well-known Kramers-Kronig relations involve only one frequency.

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“…For any potential V ≥ 0, it is not difficult to show, using the monotonicity result of [4], that the number of non-zero almost partially transparent frequencies of ∆−λV is unbounded as λ → ∞. We sketch an argument now.…”

Section: Monotonicity Of Phase Shifts and The Existence Of Almost Parmentioning

confidence: 99%

Potential scattering and the continuity of phase-shifts

Gell-Redman

Hassell

2012

Mathematical Research Letters

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Abstract. Let S(k) be the scattering matrix for a Schrödinger operator (Laplacian plus potential) on R n with compactly supported smooth potential. It is well known that S(k) is unitary and that the spectrum of S(k) accumulates on the unit circle only at 1; moreover, S(k) depends analytically on k and therefore its eigenvalues depend analytically on k provided they stay away from 1.We give examples of smooth, compactly supported potentials on R n for which (i) the scattering matrix S(k) does not have 1 as an eigenvalue for any k > 0, and (ii) there exists k 0 > 0 such that there is an analytic eigenvalue branch e 2iδ(k) of S(k) converging to 1 as k ↓ k 0 . This shows that the eigenvalues of the scattering matrix, as a function of k, do not necessarily have continuous extensions to or across the value 1. In particular this shows that a 'micro-Levinson theorem' for non-central potentials in R 3 claimed in a 1989 paper of R. Newton is incorrect.

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The first variation of the scattering matrix

Cited by 21 publications

References 8 publications

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Monochromatic Identities for the Green Function and Uniqueness Results for Passive Imaging

Potential scattering and the continuity of phase-shifts

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