Geodesic
Quasiexact Solution of a Relativistic Finite-Difference Analogue of the Schr\"odinger Equation for a Rectangular Potential Well
Teoretičeskaâ i matematičeskaâ fizika, Tome 132 (2002) no. 3, pp. 408-433

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We consider a well-posed formulation of the spectral problem for a relativistic analogue of the one-dimensional Schrödinger equation with differential operators replaced with operators of finite purely imaginary argument shifts $\exp ({\pm i\hbar d/dx})$. We find effective solution methods that permit determining the spectrum and investigating the properties of wave functions in a wide parameter range for this problem in the case of potentials of the type of a rectangular well. We show that the properties of solutions of these equations depend essentially on the relation between $\hbar$ and the parameters of the potential and a situation in which the solution for $\hbar \ll 1$ is nevertheless fundamentally different from its Schrödinger analogue is quite possible.
Keywords: relativistic problem on bound states, field quantization in Lorentz bases, finite-difference equations with imaginary step. 
@article{TMF_2002_132_3_a5,
     author = {K. A. Sveshnikov and P. K. Silaev},
     title = {Quasiexact {Solution} of a {Relativistic} {Finite-Difference} {Analogue} of the {Schr\"odinger} {Equation} for a {Rectangular} {Potential} {Well}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {408--433},
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     volume = {132},
     number = {3},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2002_132_3_a5/}
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K. A. Sveshnikov; P. K. Silaev. Quasiexact Solution of a Relativistic Finite-Difference Analogue of the Schr\"odinger Equation for a Rectangular Potential Well. Teoretičeskaâ i matematičeskaâ fizika, Tome 132 (2002) no. 3, pp. 408-433. http://geodesic.mathdoc.fr/item/TMF_2002_132_3_a5/