Pub Date : 2020-09-08DOI: 10.5506/APHYSPOLB.52.359
G. Rudolph, M. Schmidt
Using natural lifting operations, we give a coordinate-free proof of the fact that the connection used by Bordemann, Neumaier and Waldmann to construct the Fedosov standard ordered star product on the cotangent bundle of a Riemannian manifold is obtained by symplectification of the complete lift of the corresponding Levi-Civita connection, in the sense of Yano and Patterson. In terms of local coordinates, this has already been shown by Plebanski, Przanowski and Turrubiates.
{"title":"On a Connection Used in Deformation Quantization","authors":"G. Rudolph, M. Schmidt","doi":"10.5506/APHYSPOLB.52.359","DOIUrl":"https://doi.org/10.5506/APHYSPOLB.52.359","url":null,"abstract":"Using natural lifting operations, we give a coordinate-free proof of the fact that the connection used by Bordemann, Neumaier and Waldmann to construct the Fedosov standard ordered star product on the cotangent bundle of a Riemannian manifold is obtained by symplectification of the complete lift of the corresponding Levi-Civita connection, in the sense of Yano and Patterson. In terms of local coordinates, this has already been shown by Plebanski, Przanowski and Turrubiates.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75028740","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 : 2020-09-08DOI: 10.1142/S0129055X21500197
T. Komatsu, N. Konno, Hisashi Morioka, E. Segawa
We consider the time-independent scattering theory for time evolution operators of one-dimensional two-state quantum walks. The scattering matrix associated with the position-dependent quantum walk naturally appears in the asymptotic behavior at spatial infinity of generalized eigenfunctions. The asymptotic behavior of generalized eigenfunctions is a consequence of an explicit expression of the Green function associated with the free quantum walk. When the position-dependent quantum walk is a finite rank perturbation of the free quantum walk, we derive a kind of combinatorial constructions of the scattering matrix by counting paths of quantum walkers. We also mention some remarks on the tunneling effect.
{"title":"Generalized eigenfunctions for quantum walks via path counting approach","authors":"T. Komatsu, N. Konno, Hisashi Morioka, E. Segawa","doi":"10.1142/S0129055X21500197","DOIUrl":"https://doi.org/10.1142/S0129055X21500197","url":null,"abstract":"We consider the time-independent scattering theory for time evolution operators of one-dimensional two-state quantum walks. The scattering matrix associated with the position-dependent quantum walk naturally appears in the asymptotic behavior at spatial infinity of generalized eigenfunctions. The asymptotic behavior of generalized eigenfunctions is a consequence of an explicit expression of the Green function associated with the free quantum walk. When the position-dependent quantum walk is a finite rank perturbation of the free quantum walk, we derive a kind of combinatorial constructions of the scattering matrix by counting paths of quantum walkers. We also mention some remarks on the tunneling effect.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89552222","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 : 2020-09-05DOI: 10.1142/9789811272158_0003
R. Frank, Konstantin Merz, H. Siedentop
We consider heavy neutral atoms of atomic number $Z$ modeled with kinetic energy $(c^2p^2+c^4)^{1/2}-c^2$ used already by Chandrasekhar. We study the behavior of the one-particle ground state density on the length scale $Z^{-1}$ in the limit $Z,ctoinfty$ keeping $Z/c$ fixed. We give a short proof of a recent result by the authors and Barry Simon showing the convergence of the density to the relativistic hydrogenic density on this scale.
{"title":"Relativistic Strong Scott Conjecture: A Short Proof","authors":"R. Frank, Konstantin Merz, H. Siedentop","doi":"10.1142/9789811272158_0003","DOIUrl":"https://doi.org/10.1142/9789811272158_0003","url":null,"abstract":"We consider heavy neutral atoms of atomic number $Z$ modeled with kinetic energy $(c^2p^2+c^4)^{1/2}-c^2$ used already by Chandrasekhar. We study the behavior of the one-particle ground state density on the length scale $Z^{-1}$ in the limit $Z,ctoinfty$ keeping $Z/c$ fixed. We give a short proof of a recent result by the authors and Barry Simon showing the convergence of the density to the relativistic hydrogenic density on this scale.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76492706","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}
We consider polygon and simplex equations, of which the simplest nontrivial examples are pentagon (5-gon) and Yang--Baxter (2-simplex), respectively. We examine the general structure of (2n+1)-gon and 2n-simplex equations in direct sums of vector spaces. Then we provide a construction for their solutions, parameterized by elements of the Grassmannian Gr(n+1,2n+1).
{"title":"Grassmannian-parameterized solutions to direct-sum polygon and simplex equations","authors":"A. Dimakis, I. Korepanov","doi":"10.1063/5.0035760","DOIUrl":"https://doi.org/10.1063/5.0035760","url":null,"abstract":"We consider polygon and simplex equations, of which the simplest nontrivial examples are pentagon (5-gon) and Yang--Baxter (2-simplex), respectively. We examine the general structure of (2n+1)-gon and 2n-simplex equations in direct sums of vector spaces. Then we provide a construction for their solutions, parameterized by elements of the Grassmannian Gr(n+1,2n+1).","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80024223","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 : 2020-09-03DOI: 10.4310/CNTP.2021.v15.n3.a6
A. Alexandrov
In this paper we investigate a relation between the Givental group of rank one and Heisenberg-Virasoro symmetry group of the KP hierarchy. We prove, that only a two-parameter family of the Givental operators can be identified with elements of the Heisenberg-Virasoro symmetry group. This family describes triple Hodge integrals satisfying the Calabi-Yau condition. Using identification of the elements of two groups we prove that the generating function of triple Hodge integrals satisfying the Calabi-Yau condition and its $Theta$-version are tau-functions of the KP hierarchy. This generalizes the result of Kazarian on KP integrability in case of linear Hodge integrals.
{"title":"KP integrability of triple Hodge integrals, I. From Givental group to hierarchy symmetries","authors":"A. Alexandrov","doi":"10.4310/CNTP.2021.v15.n3.a6","DOIUrl":"https://doi.org/10.4310/CNTP.2021.v15.n3.a6","url":null,"abstract":"In this paper we investigate a relation between the Givental group of rank one and Heisenberg-Virasoro symmetry group of the KP hierarchy. We prove, that only a two-parameter family of the Givental operators can be identified with elements of the Heisenberg-Virasoro symmetry group. This family describes triple Hodge integrals satisfying the Calabi-Yau condition. Using identification of the elements of two groups we prove that the generating function of triple Hodge integrals satisfying the Calabi-Yau condition and its $Theta$-version are tau-functions of the KP hierarchy. This generalizes the result of Kazarian on KP integrability in case of linear Hodge integrals.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81112664","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}
Using the method of moving frames we analyze the algebra of differential invariants for surfaces in three-dimensional affine geometry. For elliptic, hyperbolic, and parabolic points, we show that if the algebra of differential invariants in non-trivial, then it is generically generated by a single invariant.
{"title":"Invariants of Surfaces in Three-Dimensional Affine Geometry","authors":"O. Arnaldsson, F. Valiquette","doi":"10.3842/SIGMA.2021.033","DOIUrl":"https://doi.org/10.3842/SIGMA.2021.033","url":null,"abstract":"Using the method of moving frames we analyze the algebra of differential invariants for surfaces in three-dimensional affine geometry. For elliptic, hyperbolic, and parabolic points, we show that if the algebra of differential invariants in non-trivial, then it is generically generated by a single invariant.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73361213","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 : 2020-08-30DOI: 10.1142/S0217732321501716
R. D. Mota, D. Ojeda-Guill'en, M. Salazar-Ram'irez, V. Granados
In this paper, we begin from the Klein-Gordon ($KG$) equation in $2D$ and change the standard partial derivatives by the Dunkl derivatives to obtain the Dunkl-Klein-Gordon ($DKG$) equation. We show that the generalization with Dunkl derivative of the $z$-component of the angular momentum is what allows the separation of variables of the $DKG$ equation. Then, we show that $DKG$ equations for the $2D$ Coulomb potential and the Klein-Gordon oscillator are exactly solvable. For each of the problems, we find the energy spectrum from an algebraic point of view by introducing suitable sets of operators which close the $su(1,1)$ algebra and use the unitary theory of representations. Also, we find analytically the energy spectrum and eigenfunctions of the $DKG$ equations for both problems. Finally, we show that when the parameters of the Dunkl derivative vanish, our results are suitably reduced to those reported in the literature for these $2D$ problems.
{"title":"Exact solutions of the 2D Dunkl–Klein–Gordon equation: The Coulomb potential and the Klein–Gordon oscillator","authors":"R. D. Mota, D. Ojeda-Guill'en, M. Salazar-Ram'irez, V. Granados","doi":"10.1142/S0217732321501716","DOIUrl":"https://doi.org/10.1142/S0217732321501716","url":null,"abstract":"In this paper, we begin from the Klein-Gordon ($KG$) equation in $2D$ and change the standard partial derivatives by the Dunkl derivatives to obtain the Dunkl-Klein-Gordon ($DKG$) equation. We show that the generalization with Dunkl derivative of the $z$-component of the angular momentum is what allows the separation of variables of the $DKG$ equation. Then, we show that $DKG$ equations for the $2D$ Coulomb potential and the Klein-Gordon oscillator are exactly solvable. For each of the problems, we find the energy spectrum from an algebraic point of view by introducing suitable sets of operators which close the $su(1,1)$ algebra and use the unitary theory of representations. Also, we find analytically the energy spectrum and eigenfunctions of the $DKG$ equations for both problems. Finally, we show that when the parameters of the Dunkl derivative vanish, our results are suitably reduced to those reported in the literature for these $2D$ problems.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74200228","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 : 2020-08-29DOI: 10.1016/j.matpur.2020.07.011
F. Colombo, I. Sabadini, D. Struppa, A. Yger
{"title":"Gauss sums, superoscillations and the Talbot carpet","authors":"F. Colombo, I. Sabadini, D. Struppa, A. Yger","doi":"10.1016/j.matpur.2020.07.011","DOIUrl":"https://doi.org/10.1016/j.matpur.2020.07.011","url":null,"abstract":"","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83256217","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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