Semiclassical localization in a one‐dimensional random analytic potential
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4.Equation (1.1) was derived previously by U. Frisch and J.‐L. Gautero, J. Math. Phys. 25, 1378 (1984) [see their Eq. (43)]. Their derivation utilizes a Fokker‐Planck equation for the accumulation of backscattering probability, and is based on the semiclassical formula for the above‐barrier reflection coefficient.
4.[See, for example, M. V. Berry, J. Phys. A 15, 3693 (1982) and Ref. 19.] While the authors acknowledge some technical difficulties associated with the application of the latter formula for multiple reflections, they do not explicitly justify its use. By effectively providing such a justification (see Sec. IV), the derivation of the present article represents both an alternate and more complete route to Eq. (1.1).
5.The present derivation follows the treatment of the close‐coupled equations for multichannel scattering given in B. C. Eu, Semiclassical Theories of Molecular Scattering (Springer, New York, 1984).
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12.Intervals on which and are energy gaps where the density of states is zero. If the potential is periodic, then all intervals on which are of this type.
13.A deterministic potential is completely determined for all by its values at where is any fixed reference point. Any generic realization of a process for which this is not so is said to be nondeterministic.
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15.The Poisson process is such that the are identically and independently distributed according to an exponential probability distribution.
16.A proven generalization of the Saxon‐Hutner conjecture (see Ref. 17) shows that energy gaps analogous to those associated with periodic potentials can exist in the spectra associated with random potentials.
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20.In Sec. IV D the rationale behind this imprecise criterion for selecting relevant turning points is made clear.
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22.Examination of Appendix B reveals that the error estimate of Eq. (4.7) is valid only if the distance from to x is as On unbounded distance scales the deviations of from the identity matrix can remain large in the semiclassical limit.
23.Equation (4.9) holds for any x. Equation (4.10) follows by choosing x to be real. In particular, we can choose x such that
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25.The overtilde notation is restricted to Appendix C to avoid undue complication in other sections. Only in Appendix C is there a real need for this notation. The appropriate interpretation of the results of Appendix B, in terms of this notation, is provided when it is required.
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