We prove the existence and nonexistence of L2(R3)-normalized solutions of two coupled semi-relativistic Hartree equations, which arise from the studies of boson stars and multi-component Bose–Einstein condensates. Under certain condition on the strength of intra-specie and inter-specie interactions, by proving some delicate energy estimates, we give a precise description on the concentration behavior of ground state solutions of the system. Furthermore, an optimal blowing up rate for the ground state solutions of the system is also proved.
{"title":"Ground states for mass critical two coupled semi-relativistic Hartree equations with attractive interactions","authors":"Thi Anh Thu Doan","doi":"10.1063/5.0178731","DOIUrl":"https://doi.org/10.1063/5.0178731","url":null,"abstract":"We prove the existence and nonexistence of L2(R3)-normalized solutions of two coupled semi-relativistic Hartree equations, which arise from the studies of boson stars and multi-component Bose–Einstein condensates. Under certain condition on the strength of intra-specie and inter-specie interactions, by proving some delicate energy estimates, we give a precise description on the concentration behavior of ground state solutions of the system. Furthermore, an optimal blowing up rate for the ground state solutions of the system is also proved.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"27 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide a new characterisation of the Standard Model gauge group GSM as a subgroup of Spin(10). The new description of GSM relies on the geometry of pure spinors. We show that GSM ⊂ Spin(10) is the group that stabilises a pure spinor Ψ1 and projectively stabilises another pure spinor Ψ2, with Ψ1,2 orthogonal and such that their arbitrary linear combination is still a pure spinor. Our characterisation of GSM relies on the facts that projective pure spinors describe complex structures on R10, and the product of two commuting complex structures is a what is known as a product structure. For the pure spinors Ψ1,2 satisfying the stated conditions the complex structures determined by Ψ1,2 commute and the arising product structure is R10=R6⊕R4, giving rise to a copy of Pati–Salam gauge group inside Spin(10). Our main statement then follows from the fact that GSM is the intersection of the Georgi–Glashow SU(5) that stabilises Ψ1, and the Pati–Salam Spin(6) × Spin(4) arising from the product structure determined by Ψ1,2. We have tried to make the paper self-contained and provided a detailed description of the creation/annihilation operator construction of the Clifford algebras Cl(2n) and the geometry of pure spinors in dimensions up to and including ten.
{"title":"Geometry of Spin(10) symmetry breaking","authors":"Kirill Krasnov","doi":"10.1063/5.0210073","DOIUrl":"https://doi.org/10.1063/5.0210073","url":null,"abstract":"We provide a new characterisation of the Standard Model gauge group GSM as a subgroup of Spin(10). The new description of GSM relies on the geometry of pure spinors. We show that GSM ⊂ Spin(10) is the group that stabilises a pure spinor Ψ1 and projectively stabilises another pure spinor Ψ2, with Ψ1,2 orthogonal and such that their arbitrary linear combination is still a pure spinor. Our characterisation of GSM relies on the facts that projective pure spinors describe complex structures on R10, and the product of two commuting complex structures is a what is known as a product structure. For the pure spinors Ψ1,2 satisfying the stated conditions the complex structures determined by Ψ1,2 commute and the arising product structure is R10=R6⊕R4, giving rise to a copy of Pati–Salam gauge group inside Spin(10). Our main statement then follows from the fact that GSM is the intersection of the Georgi–Glashow SU(5) that stabilises Ψ1, and the Pati–Salam Spin(6) × Spin(4) arising from the product structure determined by Ψ1,2. We have tried to make the paper self-contained and provided a detailed description of the creation/annihilation operator construction of the Clifford algebras Cl(2n) and the geometry of pure spinors in dimensions up to and including ten.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"71 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Despite the large number of publications on symmetry analysis of the barotropic vorticity equation on the β-plane, its group foliations have not been considered previously. The present publication aims to address this shortcoming. Group foliations are constructed for the equation, and based on them, invariant solutions are derived, some of which generalize previously known exact analytical solutions. There is also a discussion of the pros and cons of the group foliation approach including consideration of some numerical issues.
{"title":"Group foliations of the β-plane barotropic vorticity equation","authors":"E. I. Kaptsov","doi":"10.1063/5.0188918","DOIUrl":"https://doi.org/10.1063/5.0188918","url":null,"abstract":"Despite the large number of publications on symmetry analysis of the barotropic vorticity equation on the β-plane, its group foliations have not been considered previously. The present publication aims to address this shortcoming. Group foliations are constructed for the equation, and based on them, invariant solutions are derived, some of which generalize previously known exact analytical solutions. There is also a discussion of the pros and cons of the group foliation approach including consideration of some numerical issues.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"7 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We generalize σ-matrices to higher arities using the polyadization procedure proposed by the author. We build the nonderived n-ary version of SU2 using cyclic shift block matrices. We introduce the polyadic trace, which has an additivity property analogous to the ordinary trace for block diagonal matrices. The so called elementary Σ-matrices are ordinary matrix units, their sums are full Σ-matrices which can be treated as a polyadic analog of σ-matrices. The expression of n-ary SU2 in terms of full Σ-matrices is given using the Hadamard product. We then generalize the Pauli group in two ways: for the binary case we introduce the extended phase shifted σ-matrices with multipliers in cyclic groups of order 4q (q > 4), and for the polyadic case we construct the correspondent finite n-ary semigroup of phase-shifted elementary Σ-matrices of order 4qn−1+1, and the finite n-ary group of phase-shifted full Σ-matrices of order 4q. Finally, we introduce the finite n-ary group of heterogeneous full Σhet-matrices of order 4qn−14. Some examples of the lowest arities are presented.
{"title":"Polyadic sigma matrices","authors":"Steven Duplij","doi":"10.1063/5.0211252","DOIUrl":"https://doi.org/10.1063/5.0211252","url":null,"abstract":"We generalize σ-matrices to higher arities using the polyadization procedure proposed by the author. We build the nonderived n-ary version of SU2 using cyclic shift block matrices. We introduce the polyadic trace, which has an additivity property analogous to the ordinary trace for block diagonal matrices. The so called elementary Σ-matrices are ordinary matrix units, their sums are full Σ-matrices which can be treated as a polyadic analog of σ-matrices. The expression of n-ary SU2 in terms of full Σ-matrices is given using the Hadamard product. We then generalize the Pauli group in two ways: for the binary case we introduce the extended phase shifted σ-matrices with multipliers in cyclic groups of order 4q (q > 4), and for the polyadic case we construct the correspondent finite n-ary semigroup of phase-shifted elementary Σ-matrices of order 4qn−1+1, and the finite n-ary group of phase-shifted full Σ-matrices of order 4q. Finally, we introduce the finite n-ary group of heterogeneous full Σhet-matrices of order 4qn−14. Some examples of the lowest arities are presented.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"6 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the asymptotic behavior of solutions of the two-dimensional stochastic Navier-Stokes (SNS) equation with no-slip boundary condition in the small viscosity limit. Several equivalent dissipation conditions of the Kato type are derived to ensure that the convergence from the SNS equation to the corresponding stochastic Euler equation holds in the energy space. We do not assume any smallness on the noise of the SNS equation.
{"title":"On Kato’s conditions for the inviscid limit of the two-dimensional stochastic Navier-Stokes equation","authors":"Ya-guang Wang, Meng Zhao","doi":"10.1063/5.0175063","DOIUrl":"https://doi.org/10.1063/5.0175063","url":null,"abstract":"We study the asymptotic behavior of solutions of the two-dimensional stochastic Navier-Stokes (SNS) equation with no-slip boundary condition in the small viscosity limit. Several equivalent dissipation conditions of the Kato type are derived to ensure that the convergence from the SNS equation to the corresponding stochastic Euler equation holds in the energy space. We do not assume any smallness on the noise of the SNS equation.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"451 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nicolas Crampé, Luc Frappat, Julien Gaboriaud, Eric Ragoucy, Luc Vinet, Meri Zaimi
Bivariate Griffiths polynomials of Racah type are constructed from univariate Racah polynomials. The bispectral properties of the former are deduced from simple properties of the latter. A duality relation and the orthogonality of these polynomials are provided. The domain of validity for the indices and variables of these polynomials is also determined. Particular limits on the parameters entering the polynomials allow to define several Griffiths polynomials of other types. One special limit connects them to the original Griffiths polynomials (of Krawtchouk type). Finally, a connection with the 9j symbols is made.
{"title":"Griffiths polynomials of Racah type","authors":"Nicolas Crampé, Luc Frappat, Julien Gaboriaud, Eric Ragoucy, Luc Vinet, Meri Zaimi","doi":"10.1063/5.0209006","DOIUrl":"https://doi.org/10.1063/5.0209006","url":null,"abstract":"Bivariate Griffiths polynomials of Racah type are constructed from univariate Racah polynomials. The bispectral properties of the former are deduced from simple properties of the latter. A duality relation and the orthogonality of these polynomials are provided. The domain of validity for the indices and variables of these polynomials is also determined. Particular limits on the parameters entering the polynomials allow to define several Griffiths polynomials of other types. One special limit connects them to the original Griffiths polynomials (of Krawtchouk type). Finally, a connection with the 9j symbols is made.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"11 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Understanding the dispersive properties of photonic crystals is a fundamental and well-studied problem. However, the introduction of singular permittivities and damping complicates the otherwise straightforward theory. In this paper, we study photonic crystals with a Drude–Lorentz model for the permittivity, motivated by halide perovskites. We demonstrate how the introduction of singularities and damping affects the spectral band structure and show how to interpret the notion of a “band gap” in this setting. We study a one-dimensional model for which we present explicit solutions.
{"title":"The effect of singularities and damping on the spectra of photonic crystals","authors":"Konstantinos Alexopoulos, Bryn Davies","doi":"10.1063/5.0164213","DOIUrl":"https://doi.org/10.1063/5.0164213","url":null,"abstract":"Understanding the dispersive properties of photonic crystals is a fundamental and well-studied problem. However, the introduction of singular permittivities and damping complicates the otherwise straightforward theory. In this paper, we study photonic crystals with a Drude–Lorentz model for the permittivity, motivated by halide perovskites. We demonstrate how the introduction of singularities and damping affects the spectral band structure and show how to interpret the notion of a “band gap” in this setting. We study a one-dimensional model for which we present explicit solutions.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"2 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present work we present a general framework which guarantees the existence of optimal domains for isoperimetric problems within the class of C1,1-regular domains satisfying a uniform ball condition as long as the desired objective function satisfies certain properties. We then verify that the helicity isoperimetric problem studied in [Cantarella et al., J. Math. Phys. 41, 5615 (2000)] satisfies the conditions of our framework and hence establish the existence of optimal domains within the given class of domains. We additionally use the same framework to prove the existence of optimal domains among uniform C1,1-domains for a first curl eigenvalue problem which has been studied recently for other classes of domains in [Enciso et al., Trans. Am. Math. Soc. 377, 4519–4540 (2024)].
{"title":"Existence of optimal domains for the helicity maximisation problem among domains satisfying a uniform ball condition","authors":"Wadim Gerner","doi":"10.1063/5.0163183","DOIUrl":"https://doi.org/10.1063/5.0163183","url":null,"abstract":"In the present work we present a general framework which guarantees the existence of optimal domains for isoperimetric problems within the class of C1,1-regular domains satisfying a uniform ball condition as long as the desired objective function satisfies certain properties. We then verify that the helicity isoperimetric problem studied in [Cantarella et al., J. Math. Phys. 41, 5615 (2000)] satisfies the conditions of our framework and hence establish the existence of optimal domains within the given class of domains. We additionally use the same framework to prove the existence of optimal domains among uniform C1,1-domains for a first curl eigenvalue problem which has been studied recently for other classes of domains in [Enciso et al., Trans. Am. Math. Soc. 377, 4519–4540 (2024)].","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"46 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper focuses on the global well-posedness of the Oberbeck–Boussinesq approximation for the unsteady motion of a drop in another bounded fluid separated by a closed interface with surface tension. We assume that the initial state of the drop is close to a ball BR with the same volume as the drop, and that the boundary of the drop is a small perturbation of the boundary of BR. To begin, we introduce the Hanzawa transformation with an added barycenter point to obtain the linearized Oberbeck–Boussinesq approximation in a fixed domain. From there, we establish time-weighted estimates of solutions for the shifted equation using maximal Lp–Lq regularities for the two-phase fluid motion of the linearized system, as obtained by Hao and Zhang [J. Differ. Equations 322, 101–134 (2022)]. Using time decay estimates of the semigroup, we then obtain decay time-weighted estimates of solutions for the linearized problem. Additionally, we prove that these estimates are less than the sum of the initial value and its own square and cube by estimating the corresponding non-linear terms. Finally, the existence and uniqueness of solutions in the finite time interval (0, T) was proven by Hao and Zhang [Commun. Pure Appl. Anal. 22(7), 2099–2131 (2023)]. After that, we demonstrate that the solutions can be extended beyond T by analyzing the properties of the roots of algebraic equations.
{"title":"Global well-posedness for two-phase fluid motion in the Oberbeck–Boussinesq approximation","authors":"Wei Zhang, Jie Fu, Chengchun Hao, Siqi Yang","doi":"10.1063/5.0220764","DOIUrl":"https://doi.org/10.1063/5.0220764","url":null,"abstract":"This paper focuses on the global well-posedness of the Oberbeck–Boussinesq approximation for the unsteady motion of a drop in another bounded fluid separated by a closed interface with surface tension. We assume that the initial state of the drop is close to a ball BR with the same volume as the drop, and that the boundary of the drop is a small perturbation of the boundary of BR. To begin, we introduce the Hanzawa transformation with an added barycenter point to obtain the linearized Oberbeck–Boussinesq approximation in a fixed domain. From there, we establish time-weighted estimates of solutions for the shifted equation using maximal Lp–Lq regularities for the two-phase fluid motion of the linearized system, as obtained by Hao and Zhang [J. Differ. Equations 322, 101–134 (2022)]. Using time decay estimates of the semigroup, we then obtain decay time-weighted estimates of solutions for the linearized problem. Additionally, we prove that these estimates are less than the sum of the initial value and its own square and cube by estimating the corresponding non-linear terms. Finally, the existence and uniqueness of solutions in the finite time interval (0, T) was proven by Hao and Zhang [Commun. Pure Appl. Anal. 22(7), 2099–2131 (2023)]. After that, we demonstrate that the solutions can be extended beyond T by analyzing the properties of the roots of algebraic equations.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"242 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper studies Density Functional Theory (DFT) models for homogeneous 1D materials in the 3D space. It follows the previous work [Gontier et al., Commun. Math. Phys. 388, 1475–1505 (2021)] about DFT models for homogeneous 2D materials in 3D. We show how to reduce the problem from a 3D energy functional to a 2D energy functional. The kinetic energy is treated as in the 2D material case by diagonalizing admissible states, and writing the kinetic energy as the infimum of a modified kinetic energy functional on reduced states. Besides, we treat here the Hartree interaction term in 2D, and show how to properly define the mean-field potential, through Riesz potential. We then show the well-posedness of the reduced model and present some numerical illustrations.
{"title":"On density functional theory models for one-dimensional homogeneous materials","authors":"Bouchra Bensiali, Salma Lahbabi, Abdallah Maichine, Othmane Mirinioui","doi":"10.1063/5.0194944","DOIUrl":"https://doi.org/10.1063/5.0194944","url":null,"abstract":"This paper studies Density Functional Theory (DFT) models for homogeneous 1D materials in the 3D space. It follows the previous work [Gontier et al., Commun. Math. Phys. 388, 1475–1505 (2021)] about DFT models for homogeneous 2D materials in 3D. We show how to reduce the problem from a 3D energy functional to a 2D energy functional. The kinetic energy is treated as in the 2D material case by diagonalizing admissible states, and writing the kinetic energy as the infimum of a modified kinetic energy functional on reduced states. Besides, we treat here the Hartree interaction term in 2D, and show how to properly define the mean-field potential, through Riesz potential. We then show the well-posedness of the reduced model and present some numerical illustrations.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"3 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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