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Representationtheoretic aspects of twodimensional quantum systems in singular vector potentials canonical commutation relations, quantum algebras, and reduction to lattice quantum systems
Title:  Representationtheoretic aspects of twodimensional quantum systems in singular vector potentials canonical commutation relations, quantum algebras, and reduction to lattice quantum systems 
Authors:  Arai, A. Browse this author 
Issue Date:  1Jun1997 
Journal Title:  Hokkaido University Preprint Series in Mathematics 
Volume:  385 
Start Page:  1 
End Page:  32 
Abstract:  Investigated are some representationtheoretic aspects of a twodimensional quantum system of a charged particle in a vector potential A which may be singular on an infinite discrete subset D of R2 . For each vector v in a set V(D) C R2 \ {O}, the projection Pv of the physical momentum operator P := p  aA to the direction of vis defined by Pv := v · P as an operator acting in L2 (R2 ), where p = (iDx, iDy ) [(x, y) E R2] with Dx (resp. Dy ) being the generalized partial differential operator in the variable x (resp. y) and a E R is a parameter denoting the charge of the particle. It is proven that Pv is essentially selfadjoint and an explicit formula is derived for the strongly continuous oneparameter unitary group { eitPv heR generated by the selfadjoint operator E'v ( the closure of Pv ), i.e., the magnetic translation to the direction of the vector v. The magnetic translations along curves in R2 \ D are also considered. Conjugately to Pv and Pw [w E V(D)], a selfadjoint multiplication operator Qv,w is introduced, which is a linear combination of the position operators x and y, such that, if A is flat on R2 \ D, then 1r¢".w := {Qv,w,Qw,v,Pv,Pw} gives a representation of the canonical commutation relations (CCR) with two degrees of freedom. Properties of the representation 1r¢".w are analyzed. In particular, established is a necessary and sufficient condition for 1r¢".w to be unitarily equivalent ( or inequivalent) to the Schrodinger representation of CCR. The case where rr w is inequivalent to the Schrodinger representation corresponds to the AharonvBohm effect. Quantum algebraic structures [quantum plane and the quantum group Uq(sl2)] associated with the pair {Pv,.Pw} are also discussed. Moreover, for every A in a class of vector potentials having singularities on the infinite lattice L(w1,w2) := {mw1 + nw2 lm, n E Z} [the case D = L(w1,w2)], where w E R2 and w2 E R2 are linearly independent, it is shown that the magnetic translations eiPwi, j = 1, 2, with A replaced by a modified vector potential are reduced by the Hilbert space £2(L(w1,w2)) identified with a closed subspace of L2(R2 ). This result, which may be regarded as one of the most important novel results of the present paper, establishes a connection of continuous quantum systems in vector potentials to lattice ones. 
Type:  bulletin (article) 
URI:  http://hdl.handle.net/2115/69135 
Appears in Collections:  理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics

Submitter: 数学紀要登録作業用
