Geodesic
Stationary Schr\"odinger equation in nonrelativistic quantum mechanics and the~functional integral
Teoretičeskaâ i matematičeskaâ fizika, Tome 171 (2012) no. 3, pp. 452-474

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We formulate a method for representing solutions of homogeneous second-order equations in the form of a functional integral or path integral. As an example, we derive solutions of second-order equations with constant coefficients and a linear potential. The method can be used to find general solutions of the stationary Schrödinger equation. We show how to find the spectrum and eigenfunctions of the quantum oscillator equation. We obtain a solution of the stationary Schrödinger equation in the semiclassical approximation, without a singularity at the turning point. In that approximation, we find the coefficient of transmission through a potential barrier. We obtain a representation for the elastic potential scattering amplitude in the form of a functional integral.
Keywords: second-order homogeneous equation, functional integral, stationary Schrödinger equation, semiclassical approximation, elastic potential scattering amplitude. 
@article{TMF_2012_171_3_a7,
     author = {G. V. Efimov},
     title = {Stationary {Schr\"odinger} equation in nonrelativistic quantum mechanics and the~functional integral},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {452--474},
     publisher = {mathdoc},
     volume = {171},
     number = {3},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2012_171_3_a7/}
}
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G. V. Efimov. Stationary Schr\"odinger equation in nonrelativistic quantum mechanics and the~functional integral. Teoretičeskaâ i matematičeskaâ fizika, Tome 171 (2012) no. 3, pp. 452-474. http://geodesic.mathdoc.fr/item/TMF_2012_171_3_a7/