Pub Date : 2006-08-18DOI: 10.1088/0305-4470/39/33/009
S L Altmann, P. Herzig
With the basic Clifford units being identified as mirrors, it is demonstrated how proper and improper symmetry operations of point groups in spaces of arbitrary dimensions can be parametrized. In such an approach consistency with parametrizations for groups in three dimensions can be achieved even if double groups are considered. The conversion of Clifford parameters into Cartesian matrices and vice versa is discussed and, for rotations in , also the parametrization in terms of pairs of rotations in . The formalism is illustrated by a number of examples.
{"title":"Clifford algebra and the structure of point groups in higher-dimensional spaces","authors":"S L Altmann, P. Herzig","doi":"10.1088/0305-4470/39/33/009","DOIUrl":"https://doi.org/10.1088/0305-4470/39/33/009","url":null,"abstract":"With the basic Clifford units being identified as mirrors, it is demonstrated how proper and improper symmetry operations of point groups in spaces of arbitrary dimensions can be parametrized. In such an approach consistency with parametrizations for groups in three dimensions can be achieved even if double groups are considered. The conversion of Clifford parameters into Cartesian matrices and vice versa is discussed and, for rotations in , also the parametrization in terms of pairs of rotations in . The formalism is illustrated by a number of examples.","PeriodicalId":87442,"journal":{"name":"Journal of physics A: Mathematical and general","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2006-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75633318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2006-08-18DOI: 10.1088/0305-4470/39/36/005
M. Yamamura, C. Providência, J. Providência, F. Cordeiro, Y. Tsue
The two-level pairing model was investigated in the framework of the Schwinger representation. It has been shown that zero-point fluctuations, which are important in the crossover region, are well described by certain coherent states which conserve the constants of motion. These so-called conserving approximations provide excellent descriptions of the ground-state energy and of the first excitation energy over the whole range of coupling constant values. The description of the ground-state energy provided by the Glauber coherent state is also reasonable, but not so good. Although the Glauber coherent state fails to provide the description of the excitation energy around the critical point, its performance in the strong coupling limit is very good both for the description of the ground-state energy and of the first excitation energy. The present results also show that already at the mean-field level, the bosonic description in the independent particle approximation incorporates important correlation corrections to the conventional mean-field description based on the fermionic realization of the su(2) algebra (BCS theory).
{"title":"The two-level pairing model in the Schwinger representation","authors":"M. Yamamura, C. Providência, J. Providência, F. Cordeiro, Y. Tsue","doi":"10.1088/0305-4470/39/36/005","DOIUrl":"https://doi.org/10.1088/0305-4470/39/36/005","url":null,"abstract":"The two-level pairing model was investigated in the framework of the Schwinger representation. It has been shown that zero-point fluctuations, which are important in the crossover region, are well described by certain coherent states which conserve the constants of motion. These so-called conserving approximations provide excellent descriptions of the ground-state energy and of the first excitation energy over the whole range of coupling constant values. The description of the ground-state energy provided by the Glauber coherent state is also reasonable, but not so good. Although the Glauber coherent state fails to provide the description of the excitation energy around the critical point, its performance in the strong coupling limit is very good both for the description of the ground-state energy and of the first excitation energy. The present results also show that already at the mean-field level, the bosonic description in the independent particle approximation incorporates important correlation corrections to the conventional mean-field description based on the fermionic realization of the su(2) algebra (BCS theory).","PeriodicalId":87442,"journal":{"name":"Journal of physics A: Mathematical and general","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2006-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72791359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2006-08-18DOI: 10.1088/0305-4470/39/36/011
V. Hussin, Ş. Kuru, J. Negro
A generalization of the matrix Jaynes–Cummings model in the rotating wave approximation is proposed by means of the shape-invariant hierarchies of scalar factorized Hamiltonians. A class of Darboux transformations (sometimes called SUSY transformations in this context) suitable for these generalized Jaynes–Cummings models is constructed. Finally one example is worked out using the methods developed.
{"title":"Generalized Jaynes–Cummings Hamiltonians by shape-invariant hierarchies and their SUSY partners","authors":"V. Hussin, Ş. Kuru, J. Negro","doi":"10.1088/0305-4470/39/36/011","DOIUrl":"https://doi.org/10.1088/0305-4470/39/36/011","url":null,"abstract":"A generalization of the matrix Jaynes–Cummings model in the rotating wave approximation is proposed by means of the shape-invariant hierarchies of scalar factorized Hamiltonians. A class of Darboux transformations (sometimes called SUSY transformations in this context) suitable for these generalized Jaynes–Cummings models is constructed. Finally one example is worked out using the methods developed.","PeriodicalId":87442,"journal":{"name":"Journal of physics A: Mathematical and general","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2006-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82629425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2006-08-18DOI: 10.1088/0305-4470/39/33/L01
M. Calixto, V. Aldaya
We explore an intriguing connection between the Fermi–Dirac and Bose–Einstein statistics and the thermal baths obtained from a vacuum radiation of coherent states of zero modes in a second quantized (many-particle) theory on the compact O(3) and noncompact O(2, 1) isometry subgroups of the de Sitter and anti-de Sitter spaces, respectively. The high frequency limit is retrieved as a (zero-curvature) group contraction to the Newton–Hooke (harmonic oscillator) group. We also make some comments on the vacuum energy density and the cosmological constant problem.
{"title":"Curvature, zero modes and quantum statistics","authors":"M. Calixto, V. Aldaya","doi":"10.1088/0305-4470/39/33/L01","DOIUrl":"https://doi.org/10.1088/0305-4470/39/33/L01","url":null,"abstract":"We explore an intriguing connection between the Fermi–Dirac and Bose–Einstein statistics and the thermal baths obtained from a vacuum radiation of coherent states of zero modes in a second quantized (many-particle) theory on the compact O(3) and noncompact O(2, 1) isometry subgroups of the de Sitter and anti-de Sitter spaces, respectively. The high frequency limit is retrieved as a (zero-curvature) group contraction to the Newton–Hooke (harmonic oscillator) group. We also make some comments on the vacuum energy density and the cosmological constant problem.","PeriodicalId":87442,"journal":{"name":"Journal of physics A: Mathematical and general","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2006-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79920309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2006-08-18DOI: 10.1088/0305-4470/39/33/008
O. Agrawal
This paper presents the Euler–Lagrange equations and the transversality conditions for fractional variational problems. The fractional derivatives are defined in the sense of Riemann–Liouville and Caputo. The connection between the transversality conditions and the natural boundary conditions necessary to solve a fractional differential equation is examined. It is demonstrated that fractional boundary conditions may be necessary even when the problem is defined in terms of the Caputo derivative. Furthermore, both fractional derivatives (the Riemann–Liouville and the Caputo) arise in the formulations, even when the fractional variational problem is defined in terms of one fractional derivative only. Examples are presented to demonstrate the applications of the formulations.
{"title":"Fractional variational calculus and the transversality conditions","authors":"O. Agrawal","doi":"10.1088/0305-4470/39/33/008","DOIUrl":"https://doi.org/10.1088/0305-4470/39/33/008","url":null,"abstract":"This paper presents the Euler–Lagrange equations and the transversality conditions for fractional variational problems. The fractional derivatives are defined in the sense of Riemann–Liouville and Caputo. The connection between the transversality conditions and the natural boundary conditions necessary to solve a fractional differential equation is examined. It is demonstrated that fractional boundary conditions may be necessary even when the problem is defined in terms of the Caputo derivative. Furthermore, both fractional derivatives (the Riemann–Liouville and the Caputo) arise in the formulations, even when the fractional variational problem is defined in terms of one fractional derivative only. Examples are presented to demonstrate the applications of the formulations.","PeriodicalId":87442,"journal":{"name":"Journal of physics A: Mathematical and general","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2006-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85616871","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2006-08-18DOI: 10.1088/0305-4470/39/36/012
I. Mendaš
The properties and structure of the eight-dimensional parameter space of three-state density matrices are examined using the SU(3) generator expansion. Three-dimensional cross sections of the space of generalized Bloch vectors are determined by the Monte Carlo sampling method and an appropriate classification of 56 different three-parameter density matrices into ten types, according to the shape of the corresponding region of allowed values of three relevant parameters, is introduced. Positions of the representative points corresponding to the pure states are determined for all ten types.
{"title":"The classification of three-parameter density matrices for a qutrit","authors":"I. Mendaš","doi":"10.1088/0305-4470/39/36/012","DOIUrl":"https://doi.org/10.1088/0305-4470/39/36/012","url":null,"abstract":"The properties and structure of the eight-dimensional parameter space of three-state density matrices are examined using the SU(3) generator expansion. Three-dimensional cross sections of the space of generalized Bloch vectors are determined by the Monte Carlo sampling method and an appropriate classification of 56 different three-parameter density matrices into ten types, according to the shape of the corresponding region of allowed values of three relevant parameters, is introduced. Positions of the representative points corresponding to the pure states are determined for all ten types.","PeriodicalId":87442,"journal":{"name":"Journal of physics A: Mathematical and general","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2006-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87830796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2006-08-11DOI: 10.1088/0305-4470/39/35/005
O. O. Vakhnenko
A new spectral problem on one-dimensional lattices is found allowing consistently to support the zero-curvature representation for a wide class of integrable nonlinear ladder systems. The modified recurrence technique for obtaining an infinite set of conservation laws is developed and some basic conserved quantities are explicitly derived. The eigenvalue problems associated with the limiting spectral operator for the special case of rapidly vanishing boundary conditions on Schrödinger-type fields and finite background condition on a concomitant field are solved and the domains of analyticity of Jost functions are presented both analytically and graphically. This particular example shows that the original auxiliary spectral problem is basically of fourth order and must generate a set of four distinct Jost functions that have to be involved in the procedure of inverse scattering transform. Moreover, there exists a critical background value of accompanying field which separates two principally different possibilities in the organization of analyticity domains of Jost functions. This crossover should inevitably lead to qualitative rearrangements in the structure of model solutions. Thus already in the limit of low-amplitude excitations we strictly observe the loss of stability regarding the linear spectrum of Schrödinger subsystem just above the critical background value of practically unexcited concomitant field, whereas in the stability region the structure of linear spectrum is essentially controlled by the magnitude of background level via effective modification of both intersite resonant coupling and self-site coupling.
{"title":"Integrable nonlinear ladder system with background-controlled intersite resonant coupling","authors":"O. O. Vakhnenko","doi":"10.1088/0305-4470/39/35/005","DOIUrl":"https://doi.org/10.1088/0305-4470/39/35/005","url":null,"abstract":"A new spectral problem on one-dimensional lattices is found allowing consistently to support the zero-curvature representation for a wide class of integrable nonlinear ladder systems. The modified recurrence technique for obtaining an infinite set of conservation laws is developed and some basic conserved quantities are explicitly derived. The eigenvalue problems associated with the limiting spectral operator for the special case of rapidly vanishing boundary conditions on Schrödinger-type fields and finite background condition on a concomitant field are solved and the domains of analyticity of Jost functions are presented both analytically and graphically. This particular example shows that the original auxiliary spectral problem is basically of fourth order and must generate a set of four distinct Jost functions that have to be involved in the procedure of inverse scattering transform. Moreover, there exists a critical background value of accompanying field which separates two principally different possibilities in the organization of analyticity domains of Jost functions. This crossover should inevitably lead to qualitative rearrangements in the structure of model solutions. Thus already in the limit of low-amplitude excitations we strictly observe the loss of stability regarding the linear spectrum of Schrödinger subsystem just above the critical background value of practically unexcited concomitant field, whereas in the stability region the structure of linear spectrum is essentially controlled by the magnitude of background level via effective modification of both intersite resonant coupling and self-site coupling.","PeriodicalId":87442,"journal":{"name":"Journal of physics A: Mathematical and general","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2006-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78825825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2006-08-11DOI: 10.1088/0305-4470/39/32/S17
G. Lévai
The physically admissible solutions of the -symmetric Scarf I potential are identified in the domain of real and complex energies. It is found that generally there are no admissible complex-energy solutions, and there is one with real energy. In a limited range of the parameters there are two series of seemingly admissible solutions both with the real and complex energies belonging to quasi-parity q = ±; however, the two sets are not -orthogonal in the domain of real energies. The sign of the pseudo-norm of states with real energy is found to oscillate as (−1)n, in accordance with the example of other -symmetric potentials possessing an infinite number of discrete levels. It is argued that the spontaneous breakdown of -symmetry cannot be defined for the Scarf I potential. A comparison with some -symmetric extensions of the infinite square well is presented.
{"title":"On the pseudo-norm and admissible solutions of the -symmetric Scarf I potential","authors":"G. Lévai","doi":"10.1088/0305-4470/39/32/S17","DOIUrl":"https://doi.org/10.1088/0305-4470/39/32/S17","url":null,"abstract":"The physically admissible solutions of the -symmetric Scarf I potential are identified in the domain of real and complex energies. It is found that generally there are no admissible complex-energy solutions, and there is one with real energy. In a limited range of the parameters there are two series of seemingly admissible solutions both with the real and complex energies belonging to quasi-parity q = ±; however, the two sets are not -orthogonal in the domain of real energies. The sign of the pseudo-norm of states with real energy is found to oscillate as (−1)n, in accordance with the example of other -symmetric potentials possessing an infinite number of discrete levels. It is argued that the spontaneous breakdown of -symmetry cannot be defined for the Scarf I potential. A comparison with some -symmetric extensions of the infinite square well is presented.","PeriodicalId":87442,"journal":{"name":"Journal of physics A: Mathematical and general","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2006-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85980177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2006-08-11DOI: 10.1088/0305-4470/39/32/S14
A. Khare, U. Sukhatme
We enlarge the limited class of analytically solvable one-dimensional periodic potentials with a finite number of band gaps by considering the complex, PT-symmetric generalized associated Lamé (GAL) potential with four parameters and applying supersymmetry. We obtain three duality relations which relate the band-edge eigenstates of the twenty four GAL potentials obtained by permutation of the four parameters.
{"title":"Periodic potentials and PT symmetry","authors":"A. Khare, U. Sukhatme","doi":"10.1088/0305-4470/39/32/S14","DOIUrl":"https://doi.org/10.1088/0305-4470/39/32/S14","url":null,"abstract":"We enlarge the limited class of analytically solvable one-dimensional periodic potentials with a finite number of band gaps by considering the complex, PT-symmetric generalized associated Lamé (GAL) potential with four parameters and applying supersymmetry. We obtain three duality relations which relate the band-edge eigenstates of the twenty four GAL potentials obtained by permutation of the four parameters.","PeriodicalId":87442,"journal":{"name":"Journal of physics A: Mathematical and general","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2006-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89601809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2006-08-11DOI: 10.1088/0305-4470/39/32/S12
A. Jarosz, M. Nowak
We present some unexpected links between the spectral properties of ensembles of large Hermitian and non-Hermitian random matrices, within the formalism of free random variables. We address the task of adding non-Hermitian random matrices. To solve this problem, we present a new approach [1] based on a generalization of the notion of the matrix-valued Green's function to the quaternion level, and on an extension of free probability theory, which allows us to introduce the setting of quaternion-valued free probability theory. We use this quaternion construction to solve the problem of a non-Hermitian random matrix of the type H1 + iH2, with Hermitian H1, H2 freely independent. Finally, we mention conformal mapping relating spectral properties of Hermitian H1 + H2 to spectral properties of the non-Hermitian H1 + iH2 model.
{"title":"Random Hermitian versus random non-Hermitian operators—unexpected links","authors":"A. Jarosz, M. Nowak","doi":"10.1088/0305-4470/39/32/S12","DOIUrl":"https://doi.org/10.1088/0305-4470/39/32/S12","url":null,"abstract":"We present some unexpected links between the spectral properties of ensembles of large Hermitian and non-Hermitian random matrices, within the formalism of free random variables. We address the task of adding non-Hermitian random matrices. To solve this problem, we present a new approach [1] based on a generalization of the notion of the matrix-valued Green's function to the quaternion level, and on an extension of free probability theory, which allows us to introduce the setting of quaternion-valued free probability theory. We use this quaternion construction to solve the problem of a non-Hermitian random matrix of the type H1 + iH2, with Hermitian H1, H2 freely independent. Finally, we mention conformal mapping relating spectral properties of Hermitian H1 + H2 to spectral properties of the non-Hermitian H1 + iH2 model.","PeriodicalId":87442,"journal":{"name":"Journal of physics A: Mathematical and general","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2006-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87539988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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