On the Schrödinger equation with steplike potentials. (English) Zbl 0968.81020

Summary: The one-dimensional Schrödinger equation is considered when the potential is asymptotic to a positive constant on the right half line in a certain sense. The zero-energy limits of the scattering coefficients are obtained under weaker assumptions than used elsewhere, and the continuity of the scattering coefficients from the left are established. The scattering coefficients for the potential are expressed in terms of the corresponding coefficients for the pieces of the potential on the positive and negative half lines. The number of bound states for the whole potential is related to the number of bound states for the two pieces. Finally, an improved result is given on the small-energy asymptotics of reflection coefficients for potentials supported on a half line.


81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81U15 Exactly and quasi-solvable systems arising in quantum theory
34L25 Scattering theory, inverse scattering involving ordinary differential operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81U20 \(S\)-matrix theory, etc. in quantum theory
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[1] Buslaev V, Vestn. Leningr. Univ., Ser. 4: Fiz., Khim. 17 pp 56– (1962)
[2] DOI: 10.1512/iumj.1985.34.34008 · Zbl 0553.34015
[3] DOI: 10.1016/0370-1573(94)00110-O
[4] DOI: 10.1080/10448639408217666
[5] DOI: 10.1103/PhysRevB.52.10831
[6] DOI: 10.1103/PhysRevB.52.10827
[7] DOI: 10.1016/0921-4526(95)00974-4
[8] DOI: 10.1016/0921-4526(95)00975-2
[9] DOI: 10.1063/1.531338 · Zbl 0843.34080
[10] DOI: 10.1088/0266-5611/4/2/013 · Zbl 0669.34030
[11] DOI: 10.1063/1.532271 · Zbl 1001.34074
[12] DOI: 10.1063/1.529883 · Zbl 0762.35075
[13] DOI: 10.1063/1.531754 · Zbl 0910.34069
[14] DOI: 10.1090/trans2/065/04 · Zbl 0181.56704
[15] DOI: 10.1002/cpa.3160320202 · Zbl 0388.34005
[16] DOI: 10.1063/1.532510 · Zbl 0931.34069
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