The scattering problem for the Schr�dinger equation with a potential linear in time and in space. II. Correctness, smoothness, behavior of the solution at infinity

Babich, V. M.; Smyshlyaev, V. P.

doi:10.1007/bf01100135

Cited by 8 publications

(27 citation statements)

References 6 publications

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“…Although we have restricted our study to improving Popov's result, our analysis is a starting point for constructing a local paramatrix which describes the gliding wave generated by an incident wave grazing at an inflection point at the boundary,, Recently, we have noticed that Babich and Smyshlyaev [2] obtained the same result for the special case it-I. However, their method is quite different from ours.…”

Section: § 1 Introductionmentioning

confidence: 83%

Whispering Gallery Waves in a Neighborhood of a Higher Order Zero of the Curvature of the Boundary

Nakamura¹,

Shibata²,

Tanuma³

1989

Publ. Res. Inst. Math. Sci.

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Section: § 1 Introductionmentioning

confidence: 83%

Whispering Gallery Waves in a Neighborhood of a Higher Order Zero of the Curvature of the Boundary

Nakamura¹,

Shibata²,

Tanuma³

1989

Publ. Res. Inst. Math. Sci.

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“…In [4], V.M. Babich and the first author have proved that the solution of (2)-( 5) is in fact classical, using bootstrap-type techniques based on the asymptotic expansion (9), which has allowed to justify (9) with proved error estimates.…”

Section: Formulation Background and The Main Resultsmentioning

confidence: 99%

Searchlight Asymptotics for High-Frequency Scattering by Boundary Inflection

Smyshlyaev¹,

Kamotski²

2021

Preprint

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We consider an inner problem for whispering gallery highfrequency asymptotic mode's scattering by a boundary inflection. The related boundary-value problem for a Schrödinger equation on a halfline with a potential linear in both space and time appears fundamental for describing transitions from modal to scattered asymptotic patterns, and despite having been intensively studied over several decades remains largely unsolved. We prove that the solution past the inflection point has a "searchlight" asymptotics corresponding to a beam concentrated near the limit ray, and establish certain decay and smoothness properties of the related searchlight amplitude. We also discuss further interpretations of the above result: the existence of associated generalised wave operator, and of a version of a unitary scattering operator connecting the modal and scattered asymptotic regimes.

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“…in unscaled coordinates (s, n). Also the well posedness of the problem has been proved in [19] and [4]. All this research has strongly suggested that the phase in any superposition of plane waves of the form (1.4) should include a fifth power of λ, as discussed in detail in [10,8].…”

Section: Diffractionmentioning

confidence: 88%

“…in unscaled coordinates (s, n). Also the well-posedness of the problem has been proved in [14] and [13].…”

Section: Diffractionmentioning

confidence: 99%

Thin-layer solutions of the Helmholtz equation

Ockendon

Tew

2020

Eur. J. Appl. Math

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The scattering problem for the Schr�dinger equation with a potential linear in time and in space. II. Correctness, smoothness, behavior of the solution at infinity

Cited by 8 publications

References 6 publications

Whispering Gallery Waves in a Neighborhood of a Higher Order Zero of the Curvature of the Boundary

Whispering Gallery Waves in a Neighborhood of a Higher Order Zero of the Curvature of the Boundary

Searchlight Asymptotics for High-Frequency Scattering by Boundary Inflection

Thin-layer solutions of the Helmholtz equation

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