The anisotropic oscillator on curved spaces: A new exactly solvable model

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• 作者：Á ; ngel Ballesteros^a ; ^{angelb@ubu.es" class="auth_mail" title="E-mail the corresponding author} ; Francisco J. Herranz^a ; ^{fjherranz@ubu.es" class="auth_mail" title="E-mail the corresponding author} ; Şengü ; l Kuru^b ; ^{kuru@science.ankara.edu.tr" class="auth_mail" title="E-mail the corresponding author} ; Javier Negro^c ; ^{jnegro@fta.uva.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Superintegrability ; Factorization ; Higher-order symmetry ; Deformation ; Curvature
• 刊名：Annals of Physics
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：373
• 期：Complete
• 页码：399-423
• 全文大小：860 K

文摘

We present a new exactly solvable (classical and quantum) model that can be interpreted as the generalization to the two-dimensional sphere and to the hyperbolic space of the two-dimensional anisotropic oscillator with any pair of frequencies formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301075&_mathId=si22.gif&_user=111111111&_pii=S0003491616301075&_rdoc=1&_issn=00034916&md5=0df38813462b5a9ef9d468ef498ceb16" title="Click to view the MathML source">ω_xflow="scroll">ωx and formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301075&_mathId=si23.gif&_user=111111111&_pii=S0003491616301075&_rdoc=1&_issn=00034916&md5=bf8052cdde241f257b53e76861847859" title="Click to view the MathML source">ω_yflow="scroll">ωy. The new curved Hamiltonian formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301075&_mathId=si5.gif&_user=111111111&_pii=S0003491616301075&_rdoc=1&_issn=00034916&md5=9d2a2264525314674085dffaa6cff9b6" title="Click to view the MathML source">H_κflow="scroll">Hκ depends on the curvature formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301075&_mathId=si25.gif&_user=111111111&_pii=S0003491616301075&_rdoc=1&_issn=00034916&md5=2a62c2b023a6bfc05ffd37a72161b8e4" title="Click to view the MathML source">κflow="scroll">κ of the underlying space as a deformation/contraction parameter, and the Liouville integrability of formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301075&_mathId=si5.gif&_user=111111111&_pii=S0003491616301075&_rdoc=1&_issn=00034916&md5=9d2a2264525314674085dffaa6cff9b6" title="Click to view the MathML source">H_κflow="scroll">Hκ relies on its separability in terms of geodesic parallel coordinates, which generalize the Cartesian coordinates of the plane. Moreover, the system is shown to be superintegrable for commensurate frequencies formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301075&_mathId=si27.gif&_user=111111111&_pii=S0003491616301075&_rdoc=1&_issn=00034916&md5=16407af728696a4f0881313664176b33" title="Click to view the MathML source">ω_x:ω_yflow="scroll">ωx:ωy, thus mimicking the behaviour of the flat Euclidean case, which is always recovered in the formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301075&_mathId=si28.gif&_user=111111111&_pii=S0003491616301075&_rdoc=1&_issn=00034916&md5=b56c47474f4eb73f48f3930d00e85f60" title="Click to view the MathML source">κ→0flow="scroll">κ→0 limit. The additional constant of motion in the commensurate case is, as expected, of higher-order in the momenta and can be explicitly deduced by performing the classical factorization of the Hamiltonian. The known formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301075&_mathId=si29.gif&_user=111111111&_pii=S0003491616301075&_rdoc=1&_issn=00034916&md5=da28fa737aa274510daa8594fc2c0cd4" title="Click to view the MathML source">1:1flow="scroll">1:1 and formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301075&_mathId=si30.gif&_user=111111111&_pii=S0003491616301075&_rdoc=1&_issn=00034916&md5=58e483f22277278eb2a2fdb12da3b6f7" title="Click to view the MathML source">2:1flow="scroll">2:1 anisotropic curved oscillators are recovered as particular cases of formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301075&_mathId=si5.gif&_user=111111111&_pii=S0003491616301075&_rdoc=1&_issn=00034916&md5=9d2a2264525314674085dffaa6cff9b6" title="Click to view the MathML source">H_κflow="scroll">Hκ, meanwhile all the remaining formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301075&_mathId=si27.gif&_user=111111111&_pii=S0003491616301075&_rdoc=1&_issn=00034916&md5=16407af728696a4f0881313664176b33" title="Click to view the MathML source">ω_x:ω_yflow="scroll">ωx:ωy curved oscillators define new superintegrable systems. Furthermore, the quantum Hamiltonian f&_user=111111111&_pii=S0003491616301075&_rdoc=1&_issn=00034916&md5=0ace1bb175577cb68c4dcaefcd80e5d8">f" data-inlimgeid="1-s2.0-S0003491616301075-si33.gif">flow="scroll">Hˆκ is fully constructed and studied by following a quantum factorization approach. In the case of commensurate frequencies, the Hamiltonian f&_user=111111111&_pii=S0003491616301075&_rdoc=1&_issn=00034916&md5=0ace1bb175577cb68c4dcaefcd80e5d8">f" data-inlimgeid="1-s2.0-S0003491616301075-si33.gif">flow="scroll">Hˆκ turns out to be quantum superintegrable and leads to a new exactly solvable quantum model. Its corresponding spectrum, that exhibits a maximal degeneracy, is explicitly given as an analytical deformation of the Euclidean eigenvalues in terms of both the curvature formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301075&_mathId=si25.gif&_user=111111111&_pii=S0003491616301075&_rdoc=1&_issn=00034916&md5=2a62c2b023a6bfc05ffd37a72161b8e4" title="Click to view the MathML source">κflow="scroll">κ and the Planck constant formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301075&_mathId=si36.gif&_user=111111111&_pii=S0003491616301075&_rdoc=1&_issn=00034916&md5=cf3128b88246ccd6a7b6fb05447fc569" title="Click to view the MathML source">ħflow="scroll">ħ. In fact, such spectrum is obtained as a composition of two one-dimensional (either trigonometric or hyperbolic) Pösch–Teller set of eigenvalues.