## 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

Zbl 0669.34030
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### References:

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