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Factorization of scattering matrices due to partitioning of potentials in one-dimensional Schrödinger-type equations. (English) Zbl 0910.34069

Summary: The one-dimensional Schrödinger 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 Schrödinger equation. For such Schrödinger equations, factorization formulas are obtained that relate the scattering matrices of the fragments to the scattering matrix of the full problem.

MSC:

34L25 Scattering theory, inverse scattering involving ordinary differential operators
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.)
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