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Scattering for Fractional Schr(?)dinger Operators with Short Range Potentials

Author: ZhaoZhen
Tutor: ZhengQuan
School: Huazhong University of Science and Technology
Course: Applied Mathematics
Keywords: Schr (o ¨) dinger operator Scattering Short domain Wave operator
CLC: O177
Type: Master's thesis
Year: 2011
Downloads: 12
Quote: 0
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Abstract


Scattering collisions are effective method used to study microscopic particles in nature there are many scattering phenomenon , such as light scattering . For quantum scattering has been a lot of research . Classical scattering theory is divided into short potential and long geopotential both cases , the existence of potential wave operator called short potential of the modified wave operator exists potential called long geopotential . With short potential has been relatively mature , with a long potential is not perfect . Compared with the non-classical integer -order high-level Schr (o | ¨) dinger operator scattering theory is not perfect . H (o | ¨) rmander for the main operator of linear partial differential operators , potential V variable coefficient partial differential operator midnight , full of short potential , and long potential short introduction. Recently, people started to pay attention Fractional Schr (o | ¨) dinger the operator physically fractional Schr (o | ¨) dinger equation for (- △) α / 2 , the main consideration lt ; α ≤ 2 situation , mathematically , consider a wider range of alpha gt ; Fractional Schr ( o | ¨ ) dinger operator , there are a small number of results , such as the nature of semigroup super compressibility equation smooth estimates , existence and uniqueness of solutions in the nonlinear case . Fractional Schr (o | ¨) dinger operators , mainly to study the scattering problem . We study the scattering problem for (- △) α / 2 and ( - △ ) α / 2 < / sup > V these two systems . The main arrangements are as follows : in the introductory section , we introduce some basic concepts and major work , and then we study the operator ( - △ ) the α / 2 < / sup> resolvent R 0 (z), the appropriate V makes VR 0 < / sub > ( z ) is in the appropriate space of a compact operator , pursuant to which we have defined for V concept of short Disturbed ; Finally, similar H (o | ¨) rmander using the stationary phase method and the compactness of the operator V to prove the existence of a short Disturbed next wave operator . After completing the main proof , we also give the the fractional Schr ( o | ¨ ) dinger equations of two asymptotic behavior of . After given the paper can be improved in some places .

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