On the Riemann–Hilbert problem for the one-dimensional Schrödinger equation. (English) Zbl 0777.34056

Summary: A matrix Riemann-Hilbert problem associated with the one-dimensional Schrödinger equation is considered, and the existence and uniqueness of its solutions are studied. The solution of this Riemann-Hilbert problem yields the solution of the inverse scattering problem for a larger class of potentials than the usual Faddeev class. Some examples of explicit solutions of the Riemann-Hilbert problem are given, and the connection with ambiguities in the inverse scattering problem is established.


34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81Q40 Bethe-Salpeter and other integral equations arising in quantum theory
34L25 Scattering theory, inverse scattering involving ordinary differential operators
81U40 Inverse scattering problems in quantum theory
Full Text: DOI


[1] DOI: 10.1063/1.524447 · Zbl 0446.34029
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[4] DOI: 10.1088/0266-5611/3/4/006 · Zbl 0641.35068
[5] DOI: 10.1088/0266-5611/4/2/013 · Zbl 0669.34030
[6] DOI: 10.1063/1.526014 · Zbl 0557.35112
[7] DOI: 10.1088/0266-5611/1/4/003 · Zbl 0602.47036
[8] DOI: 10.1088/0266-5611/3/1/012 · Zbl 0617.35106
[9] DOI: 10.1103/PhysRevLett.58.2159
[10] DOI: 10.1002/cpa.3160320202 · Zbl 0388.34005
[11] DOI: 10.1007/BF01342848 · Zbl 0044.31201
[12] DOI: 10.1007/BF01203119 · Zbl 0790.47012
[13] DOI: 10.1090/trans2/014/09 · Zbl 0098.07501
[14] DOI: 10.1090/trans2/014/09 · Zbl 0098.07501
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