Geodesic
New method of solution of the one-dimensional Schr\"odinger equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 88 (1991) no. 3, pp. 477-480

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The potential in the Schrödinger equation is divided by gaps of infinitesimal width into individual potential barriers, the tops of which are approximated by quadratic potentials. For each barrier, the total wave function within the barrier is found, and also the reflection and transmission amplitudes. The method of recursion relations is then used to construct the reflection amplitude for the complete potential, it being expressed in terms of the amplitudes of the individual potential barriers in the form of a continued fraction. The transmission amplitude for the complete potential and the wave function at any given part of the potential are found similarly. 
@article{TMF_1991_88_3_a12,
     author = {V. K. Ignatovich},
     title = {New method of solution of the one-dimensional {Schr\"odinger} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {477--480},
     publisher = {mathdoc},
     volume = {88},
     number = {3},
     year = {1991},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1991_88_3_a12/}
}
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V. K. Ignatovich. New method of solution of the one-dimensional Schr\"odinger equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 88 (1991) no. 3, pp. 477-480. http://geodesic.mathdoc.fr/item/TMF_1991_88_3_a12/