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Periodic orbit theory for the quantized baker's map
ANNALS
OF PHYSICS
208. 5099510
Abstracts
Periodic Orbit Unicamp, National
(1991)
of Papers
to Appear
in Future
Theory for the Quantked Baker’s Map. A. M. OZORIO DE ALMEIDA. 13081 Campinas, Sao Paulo, Brasil; AND M. SARACENO. Departamento de Energia Atomica. 1429 Buenos Aires, Argentina.
The semiclassical limit for iteration of the baker’s map is constructed by quantizing iteration of the classical map. The resulting propagator can be expressed in terms of the function, leading to explicit expressions for the actions of all the periodic orbits. The for the smoothed density of quasi-energy levels is derived taking full account of the underlying phase space. Comparison with exact results shows excellent agreement for are much larger than the average level spacing.
Axial
Issues
Instituto de Fisicd, de Fisica, Comision
the corresponding classical generating periodic orbit sum discreteness of the smoothings which
Chunneling in Perfect Crystals, the Continuum Model, und t/7e Method of‘ Averaging. H. S. DUMAS. Institute for Mathematics and its Applications, University of Minnesota. Minneapolis, Minnesota 55455; J. A. ELLISON. Department of Mathematics, University of New Mexico, Albuquerque, New Mexico 87131; AND A. W. S.&EP;Z. Naval Research Laboratory, Washington, DC 20375 and Department of Physics, Catholic University, Washington, DC 20064.
We present a mathematically rigorous treatment of axial channeling motions of energetic. positively charged particles based on the classical relativistic perfect crystal model. More specifically, we reduce the study of motions in a six-dimensional phase space to the study of associated motions in a four-dimensional space. Our main mathematical tool is a recently improved version of the classical method of averaging for ordinary differential equations, which we discuss separately. Applying the method at orders one, two, and three. we extract successively better approximations to perfect crystal model motions (the time of validity is the same for each approximation and scales inversely with the square root of incident particle energy). We call these approximations first-, second-, and third-order continuum model solutions, respectively, and our first-order continuum model solutions are precisely those arising in Lindhard’s classical continuum model for axial channeling. The second-order solutions introduce the effects of lattice periodicity in what we believe to be the simplest possible way, e.g., in a computationally simpler way than the standard constant longitudinal momentum approximation. The third-order continuum model solutions introduce the first nontrivial relativistic corrections. and are also, we believe, the most precise approximations which may be useful in applications. After introducing these approximations and discussing their use, we conclude by briefly discussing the present state of the mathematics of channeling.
Quantum Solitons in a Hamiltonian Frumework. JOHN A. PARMENTOLA. Physics Department. West Virginia University, Morgantown, West Virginia 26506; XIN-HUA YANG, School of Physics and Astronomy. 116 Church Street. Minneapolis, Minnesota 55455; AND ISMAIL ZAHED. Department of Physics, State University of New York at Stony Brook, Stony Brook, New York 11794-3800. We to two work. Sakita
discuss the computation of the one-soliton energy for the I$” kink-tield theory in I+ 1 dimensions loops within the hamiltonian framework of Christ and Lee. We compare our calculation to other We also discuss the connection between the quantization method of Tomboulis and Gervais and and the method proposed by Christ and Lee.
509 0003-4916/91
$7.50
CopyrIght t 1997 by Academrc Press. Inc All riphtr of reproductmn m dny form reserved.
OF PHYSICS
208. 5099510
Abstracts
Periodic Orbit Unicamp, National
(1991)
of Papers
to Appear
in Future
Theory for the Quantked Baker’s Map. A. M. OZORIO DE ALMEIDA. 13081 Campinas, Sao Paulo, Brasil; AND M. SARACENO. Departamento de Energia Atomica. 1429 Buenos Aires, Argentina.
The semiclassical limit for iteration of the baker’s map is constructed by quantizing iteration of the classical map. The resulting propagator can be expressed in terms of the function, leading to explicit expressions for the actions of all the periodic orbits. The for the smoothed density of quasi-energy levels is derived taking full account of the underlying phase space. Comparison with exact results shows excellent agreement for are much larger than the average level spacing.
Axial
Issues
Instituto de Fisicd, de Fisica, Comision
the corresponding classical generating periodic orbit sum discreteness of the smoothings which
Chunneling in Perfect Crystals, the Continuum Model, und t/7e Method of‘ Averaging. H. S. DUMAS. Institute for Mathematics and its Applications, University of Minnesota. Minneapolis, Minnesota 55455; J. A. ELLISON. Department of Mathematics, University of New Mexico, Albuquerque, New Mexico 87131; AND A. W. S.&EP;Z. Naval Research Laboratory, Washington, DC 20375 and Department of Physics, Catholic University, Washington, DC 20064.
We present a mathematically rigorous treatment of axial channeling motions of energetic. positively charged particles based on the classical relativistic perfect crystal model. More specifically, we reduce the study of motions in a six-dimensional phase space to the study of associated motions in a four-dimensional space. Our main mathematical tool is a recently improved version of the classical method of averaging for ordinary differential equations, which we discuss separately. Applying the method at orders one, two, and three. we extract successively better approximations to perfect crystal model motions (the time of validity is the same for each approximation and scales inversely with the square root of incident particle energy). We call these approximations first-, second-, and third-order continuum model solutions, respectively, and our first-order continuum model solutions are precisely those arising in Lindhard’s classical continuum model for axial channeling. The second-order solutions introduce the effects of lattice periodicity in what we believe to be the simplest possible way, e.g., in a computationally simpler way than the standard constant longitudinal momentum approximation. The third-order continuum model solutions introduce the first nontrivial relativistic corrections. and are also, we believe, the most precise approximations which may be useful in applications. After introducing these approximations and discussing their use, we conclude by briefly discussing the present state of the mathematics of channeling.
Quantum Solitons in a Hamiltonian Frumework. JOHN A. PARMENTOLA. Physics Department. West Virginia University, Morgantown, West Virginia 26506; XIN-HUA YANG, School of Physics and Astronomy. 116 Church Street. Minneapolis, Minnesota 55455; AND ISMAIL ZAHED. Department of Physics, State University of New York at Stony Brook, Stony Brook, New York 11794-3800. We to two work. Sakita
discuss the computation of the one-soliton energy for the I$” kink-tield theory in I+ 1 dimensions loops within the hamiltonian framework of Christ and Lee. We compare our calculation to other We also discuss the connection between the quantization method of Tomboulis and Gervais and and the method proposed by Christ and Lee.
509 0003-4916/91
$7.50
CopyrIght t 1997 by Academrc Press. Inc All riphtr of reproductmn m dny form reserved.