On the transformation operator for the Schrödinger equation with an additional linear potential. (English. Russian original) Zbl 1447.81110
Funct. Anal. Appl. 54, No. 1, 73-76 (2020); translation from Funkts. Anal. Prilozh. 54, No. 1, 93-96 (2020).
Summary: The paper considers the Schrödinger equation with an additional linear potential on the entire real line. A transformation operator with a condition at \(- \infty\) is constructed. The Gelfand-Levitan integral equation is obtained on the half-line \((- \infty, x)\).
MSC:
81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
81U40 | Inverse scattering problems in quantum theory |
34A55 | Inverse problems involving ordinary differential equations |
45A05 | Linear integral equations |
33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
Keywords:
Schrödinger equation; additional linear potential; Airy function; transformation operator; inverse problemReferences:
[1] | Calogero, F.; Degasperis, A., Lett. Nuovo Cimento, 23, 4, 143-149 (1978) · doi:10.1007/BF02763080 |
[2] | Li, Y., Chin. Ann. of Math., 2, 2, 147-155 (1981) |
[3] | Kachalov, A. P.; Kurylev, Ya V., Zap. Nauchn. Sem. LOMI, 179, 73-87 (1989) · Zbl 0715.34139 |
[4] | Its, A.; Sukhanov, V., Inverse Problems, 32, 5, 1-28 (2016) · Zbl 1381.34109 · doi:10.1088/0266-5611/32/5/055003 |
[5] | Lin, Y.; Qian, M.; Zhang, Q., Acta Math. Appl. Sinica, 5, 2, 116-136 (1989) · Zbl 0699.35200 · doi:10.1007/BF02009745 |
[6] | Graffi, S.; Harrell, E., Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A, 36, 1, 41-58 (1982) · Zbl 0506.35079 |
[7] | Chelkak, D.; Kargaev, P.; Korotyaev, E., Comm. Math. Phys., 249, 1, 133-196 (2004) · Zbl 1085.81055 · doi:10.1007/s00220-004-1105-8 |
[8] | Korotyaev, E. L., J. Spectral Theory, 7, 3, 699-732 (2017) · Zbl 1460.34071 · doi:10.4171/JST/175 |
[9] | Guseinov, I. M.; Khanmamedov, A. Kh; Mamedova, A. F., Teoret. Mat. Fiz., 195, 1, 54-63 (2018) · Zbl 1401.81047 · doi:10.4213/tmf9423 |
[10] | Bagirova, S. M.; Khanmamedov, A. Kh, Proc. Inst. Math. Mech. NAS of Azerbaijan, 44, 2, 1-10 (2018) |
[11] | Avron, J.; Herbst, I., Comm. Math. Phys., 52, 3, 239-254 (1977) · Zbl 0351.47007 · doi:10.1007/BF01609485 |
[12] | Savchuk, A. M.; Shkalikov, A. A., Funkts. Anal. Prilozhen., 51, 1, 82-98 (2017) · Zbl 1372.34131 · doi:10.4213/faa3264 |
[13] | Korotyaev, E. L., Lett. Math. Phys., 108, 5, 1307-1322 (2018) · Zbl 1397.34152 · doi:10.1007/s11005-017-1033-0 |
[14] | Abramowitz, M.; Stegun, I., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1964), Washington, D.C.: National Bureau of Standards, Washington, D.C. · Zbl 0171.38503 |
[15] | Firsova, N. E., Mat. Sb., 130, 172, 349-385 (1986) · Zbl 0616.34017 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.