Factorization of scattering matrices due to partitioning of potentials in one-dimensional Schr{umlt o}dinger-type equations

Journal Article · Sun Dec 01 04:00:00 EST 1996 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.531754· OSTI ID:397447

Aktosun, T ^[1]; Klaus, M ^[2]; van der Mee, C ^[3]

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1. Department of Mathematics, North Dakota State University, Fargo, North Dakota 58105 (United States)
2. Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 (United States)
3. Department of Mathematics, University of Cagliari, Cagliari (Italy)

The one-dimensional Schr{umlt o}dinger equation and two of its generalizations are considered, as they arise in quantum mechanics, wave propagation in a nonhomogeneous medium, and wave propagation in a nonconservative medium where energy may be absorbed or generated. Generically, the zero-energy transmission coefficient vanishes when the potential is nontrivial, but in the exceptional case this coefficient is nonzero, resulting in tunneling through the potential. It is shown that any nontrivial exceptional potential can always be fragmented into two generic pieces. Furthermore, any nontrivial potential, generic or exceptional, can be fragmented into generic pieces in infinitely many ways. The results remain valid when Dirac delta functions are included in the potential and other coefficients are added to the Schr{umlt o}dinger equation. For such Schr{umlt o}dinger equations, factorization formulas are obtained that relate the scattering matrices of the fragments to the scattering matrix of the full problem. {copyright} {ital 1996 American Institute of Physics.}

• OSTI ID: 397447
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 12 Vol. 37; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

66 PHYSICS
DELTA FUNCTION
FACTORIZATION
POTENTIAL ENERGY
SCATTERING
SCHROEDINGER EQUATION
TRANSMISSION
WAVE PROPAGATION