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Asymptotics of the scattering coefficients for a generalized Schrödinger equation. (English) Zbl 0953.34071

In this article, the asymptotics of the reflection and transmission coefficients of the generalized 1-D Schrödinger equation \[ \psi''(k,x)+F(k)\psi(k,x)=[ikP(x)+Q(x)]\psi(k,x),\qquad x\in{\mathbb{R}}, \] where \(P\) and \(Q\) are real potentials that are integrable on the real line after having been multiplied by \((1+|x|)\), as \(k\) approaches one of the (real or complex) zeros of \(F(k)\) are studied in detail. Special attention is paid to the cases \(F(k)=k^2\) (the usual Schrödinger equation if \(P(x)\equiv 0\); the so-called Jaulent equation if \(P(x)\not\equiv 0\)) and \(F(k)=k^2+m^2\) with \(m>0\) (the so-called Kaup equation if \(P(x)\not\equiv 0\)). The proofs are based on ideas first expounded by M. Klaus [Inverse Probl. 4, No. 2, 505-512 (1988; Zbl 0669.34030)].

MSC:

34L25 Scattering theory, inverse scattering involving ordinary differential operators
81U20 \(S\)-matrix theory, etc. in quantum theory
47A40 Scattering theory of linear operators

Citations:

Zbl 0669.34030
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References:

[1] DOI: 10.1007/BF01645775
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[4] DOI: 10.1063/1.523064
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[6] DOI: 10.1063/1.532271 · Zbl 1001.34074
[7] DOI: 10.1143/PTP.54.396 · Zbl 1079.37514
[8] DOI: 10.1016/0022-247X(81)90266-3 · Zbl 0477.35074
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[11] DOI: 10.1002/cpa.3160320202 · Zbl 0388.34005
[12] DOI: 10.1088/0266-5611/4/2/013 · Zbl 0669.34030
[13] DOI: 10.1063/1.530089 · Zbl 0777.34056
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