In this paper, we revisit the classic problem of diffraction of electromagnetic waves by an aperture in a perfectly conducting plane. We formulate the diffraction problem using a boundary integral equation that is defined on the aperture using Dyadic Green’s function. This integral equation turns out to align with the one derived by Bethe using fictitious magnetic charges and currents. We then investigate the boundary integral equation using a saddle point formulation and establish the well-posedness of the boundary integral equation, including the existence and uniqueness of the solution in an appropriately defined Sobolev space.
{"title":"A rigorous theory on electromagnetic diffraction by a planar aperture in a perfectly conducting screen","authors":"Ying Liang, Hai Zhang","doi":"10.1063/5.0179521","DOIUrl":"https://doi.org/10.1063/5.0179521","url":null,"abstract":"In this paper, we revisit the classic problem of diffraction of electromagnetic waves by an aperture in a perfectly conducting plane. We formulate the diffraction problem using a boundary integral equation that is defined on the aperture using Dyadic Green’s function. This integral equation turns out to align with the one derived by Bethe using fictitious magnetic charges and currents. We then investigate the boundary integral equation using a saddle point formulation and establish the well-posedness of the boundary integral equation, including the existence and uniqueness of the solution in an appropriately defined Sobolev space.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"2 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719687","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}
C. Langrenez, D. R. M. Arvidsson-Shukur, S. De Bièvre
The Kirkwood–Dirac (KD) quasiprobability distribution can describe any quantum state with respect to the eigenbases of two observables A and B. KD distributions behave similarly to classical joint probability distributions but can assume negative and nonreal values. In recent years, KD distributions have proven instrumental in mapping out nonclassical phenomena and quantum advantages. These quantum features have been connected to nonpositive entries of KD distributions. Consequently, it is important to understand the geometry of the KD-positive and -nonpositive states. Until now, there has been no thorough analysis of the KD positivity of mixed states. Here, we investigate the dependence of the full convex set of states with positive KD distributions on the eigenbases of A and B and on the dimension d of the Hilbert space. In particular, we identify three regimes where convex combinations of the eigenprojectors of A and B constitute the only KD-positive states: (i) any system in dimension 2; (ii) an open and dense probability one set of bases in dimension d = 3; and (iii) the discrete-Fourier-transform bases in prime dimension. Finally, we show that, if for example d = 2m, there exist, for suitable choices of A and B, mixed KD-positive states that cannot be written as convex combinations of pure KD-positive states. We further explicitly construct such states for a spin-1 system.
柯克伍德-迪拉克（Kirkwood-Dirac，KD）准概率分布可以描述与两个观测值 A 和 B 的特征基有关的任何量子态。KD 分布的行为与经典联合概率分布类似，但可以取负值和非实值。近年来，KD 分布已被证明有助于描绘非经典现象和量子优势。这些量子特征与 KD 分布的非正值条目有关。因此，了解 KD 正态和非正态的几何结构非常重要。迄今为止，还没有对混合态的 KD 正性进行过深入分析。在这里，我们研究了具有正 KD 分布的全凸状态集合对 A 和 B 的特征基以及对希尔伯特空间维数 d 的依赖性。我们特别指出了 A 和 B 的特征投影的凸组合构成唯一 KD 为正的状态的三种情况：(i) 维数为 2 的任何系统；(ii) 维数为 d = 3 的开放且密集的概率一基集；(iii) 质数维的离散傅立叶变换基。最后，我们证明，例如 d = 2m，在适当选择 A 和 B 的情况下，存在混合 KD 正态，它们不能被写成纯 KD 正态的凸组合。我们进一步明确地构建了自旋-1 系统的这种状态。
{"title":"Characterizing the geometry of the Kirkwood–Dirac-positive states","authors":"C. Langrenez, D. R. M. Arvidsson-Shukur, S. De Bièvre","doi":"10.1063/5.0164672","DOIUrl":"https://doi.org/10.1063/5.0164672","url":null,"abstract":"The Kirkwood–Dirac (KD) quasiprobability distribution can describe any quantum state with respect to the eigenbases of two observables A and B. KD distributions behave similarly to classical joint probability distributions but can assume negative and nonreal values. In recent years, KD distributions have proven instrumental in mapping out nonclassical phenomena and quantum advantages. These quantum features have been connected to nonpositive entries of KD distributions. Consequently, it is important to understand the geometry of the KD-positive and -nonpositive states. Until now, there has been no thorough analysis of the KD positivity of mixed states. Here, we investigate the dependence of the full convex set of states with positive KD distributions on the eigenbases of A and B and on the dimension d of the Hilbert space. In particular, we identify three regimes where convex combinations of the eigenprojectors of A and B constitute the only KD-positive states: (i) any system in dimension 2; (ii) an open and dense probability one set of bases in dimension d = 3; and (iii) the discrete-Fourier-transform bases in prime dimension. Finally, we show that, if for example d = 2m, there exist, for suitable choices of A and B, mixed KD-positive states that cannot be written as convex combinations of pure KD-positive states. We further explicitly construct such states for a spin-1 system.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"38 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141588322","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 continue the study on the validity of the Prandtl boundary layer expansions in [Gao et al., Sci. China Math. 66, 679–722 (2023)], whereby estimating the stream-function of the remainder, we proved the case when the Euler flow is the perturbation of shear flow in a narrow domain. In this paper, we obtain a new derivatives estimate of stream-function away from the boundary layer and then prove the validity of expansions for any non-shear Euler flow, provided that the width of the domain is small.
{"title":"The steady Prandtl boundary layer expansions for non-shear Euler flow","authors":"Chen Gao, Liqun Zhang","doi":"10.1063/5.0192671","DOIUrl":"https://doi.org/10.1063/5.0192671","url":null,"abstract":"We continue the study on the validity of the Prandtl boundary layer expansions in [Gao et al., Sci. China Math. 66, 679–722 (2023)], whereby estimating the stream-function of the remainder, we proved the case when the Euler flow is the perturbation of shear flow in a narrow domain. In this paper, we obtain a new derivatives estimate of stream-function away from the boundary layer and then prove the validity of expansions for any non-shear Euler flow, provided that the width of the domain is small.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"39 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141586151","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 consider the Bogoliubov energy functional proposed by Napiórkowski, Reuvers and Solovej and analize it in the high density regime. We derive a two term asymptotic expansion of the ground state energy.
{"title":"Ground state energy of Bogoliubov energy functional in the high density limit","authors":"Norbert Mokrzański, Bartosz Pałuba","doi":"10.1063/5.0206192","DOIUrl":"https://doi.org/10.1063/5.0206192","url":null,"abstract":"We consider the Bogoliubov energy functional proposed by Napiórkowski, Reuvers and Solovej and analize it in the high density regime. We derive a two term asymptotic expansion of the ground state energy.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"11 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141575227","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}
As a direct continuation of Zwart [J. Math. Phys. 64(10), 101701 (2023)], which is built on the work of Müger and Tuset [Indagationes Math. 35(1), 114 (2024)], we reduce the Mathieu conjecture, formulated by Mathieu [Algèbra Non Commutative, Groupes Quantiques et Invariants, edited by Alex, J. and Cauchon, G. (Société Mathématique de France, Reims, 1997), Vol. 2, pp. 263–279], for Sp(N) and G2 to a conjecture involving functions over Rn×(S1)m with n,m∈N0. The proofs rely on Euler-style parametrizations of these groups, a specific version of the KAK decomposition, which we discuss and prove.
作为 Zwart [J. Math. Phys. 64(10), 101701 (2023)]在 Müger 和 Tuset [Indagationes Math. 35(1), 114 (2024)]工作基础上的直接延续，我们将 Mathieu [Algèbra Non Commutative, Groupes Quantiques et Invariants, edited by Alex, J. and Cauchon, G. (Société Mathématique de France, Reims, 1997, Vol. 2, pp.and Cauchon, G. (Société Mathématique de France, Reims, 1997), Vol. 2, pp.证明依赖于这些群的欧拉式参数化，即 KAK 分解的一个特定版本，我们对其进行了讨论和证明。
{"title":"On the Mathieu conjecture for Sp(N) and G2","authors":"Kevin Zwart","doi":"10.1063/5.0206983","DOIUrl":"https://doi.org/10.1063/5.0206983","url":null,"abstract":"As a direct continuation of Zwart [J. Math. Phys. 64(10), 101701 (2023)], which is built on the work of Müger and Tuset [Indagationes Math. 35(1), 114 (2024)], we reduce the Mathieu conjecture, formulated by Mathieu [Algèbra Non Commutative, Groupes Quantiques et Invariants, edited by Alex, J. and Cauchon, G. (Société Mathématique de France, Reims, 1997), Vol. 2, pp. 263–279], for Sp(N) and G2 to a conjecture involving functions over Rn×(S1)m with n,m∈N0. The proofs rely on Euler-style parametrizations of these groups, a specific version of the KAK decomposition, which we discuss and prove.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"43 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141575228","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 this paper we establish spectral comparison results for Schrödinger operators on a certain class of infinite quantum graphs, using recent results obtained in the finite setting. We also show that new features do appear on infinite quantum graphs such as a modified local Weyl law. In this sense, we regard this paper as a starting point for a more thorough investigation of spectral comparison results on more general infinite metric graphs.
{"title":"Some spectral comparison results on infinite quantum graphs","authors":"P. Bifulco, J. Kerner","doi":"10.1063/5.0178226","DOIUrl":"https://doi.org/10.1063/5.0178226","url":null,"abstract":"In this paper we establish spectral comparison results for Schrödinger operators on a certain class of infinite quantum graphs, using recent results obtained in the finite setting. We also show that new features do appear on infinite quantum graphs such as a modified local Weyl law. In this sense, we regard this paper as a starting point for a more thorough investigation of spectral comparison results on more general infinite metric graphs.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"35 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528538","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 cavitation and concentration phenomena of the Riemann solutions for a reduced two-phase mixtures model with non-isentropic gas state in vanishing pressure limit. We solve the Riemann problem by constructing the regions in (p, u, s) coordinate system. Then we obtain the limiting behaviors of the Riemann solutions and the formation of δ-shock waves and vacuum as pressure vanishes. We conclude that, as pressure vanishes, the limit of Riemann solutions is the Riemann solutions of the reduced 2-dimensional pressureless gas dynamics model. Finally, we present numerical simulations which are consistent with our theoretical analysis.
我们研究了非各向同性气体状态的两相混合物模型在压力消失极限下的黎曼解的空化和浓缩现象。我们通过在 (p, u, s) 坐标系中构建区域来求解黎曼问题。然后，我们得到了黎曼解的极限行为，以及压力消失时 δ 震荡波和真空的形成。我们的结论是，当压力消失时，黎曼解的极限是缩小的二维无压气体动力学模型的黎曼解。最后，我们介绍了与我们的理论分析相一致的数值模拟。
{"title":"The singular limits of the Riemann solutions as pressure vanishes for a reduced two-phase mixtures model with non-isentropic gas state","authors":"W. Jiang, D. Jin, T. Li, T. Chen","doi":"10.1063/5.0191801","DOIUrl":"https://doi.org/10.1063/5.0191801","url":null,"abstract":"We study the cavitation and concentration phenomena of the Riemann solutions for a reduced two-phase mixtures model with non-isentropic gas state in vanishing pressure limit. We solve the Riemann problem by constructing the regions in (p, u, s) coordinate system. Then we obtain the limiting behaviors of the Riemann solutions and the formation of δ-shock waves and vacuum as pressure vanishes. We conclude that, as pressure vanishes, the limit of Riemann solutions is the Riemann solutions of the reduced 2-dimensional pressureless gas dynamics model. Finally, we present numerical simulations which are consistent with our theoretical analysis.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"8 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528539","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 this paper, Herglotz-type vakonomic dynamics and Noether theory of nonholonomic systems are studied. Firstly, Herglotz-type vakonomic dynamical equations for nonholonomic systems are derived on the premise of Herglotz variational principle. Secondly, in terms of the Herglotz-type vakonomic dynamical equations, the Noether symmetry of Herglotz-type vakonomic dynamics is explored, and the Herglotz-type vakonomic dynamical Noether theorems and their inverse theorems are deduced. Finally, the conservation laws of Appell–Hamel case with non-conservative forces are analyzed to show the validity of our results.
{"title":"Herglotz-type vakonomic dynamics and its Noether symmetry for nonholonomic constrained systems","authors":"Li-Qin Huang, Yi Zhang","doi":"10.1063/5.0157564","DOIUrl":"https://doi.org/10.1063/5.0157564","url":null,"abstract":"In this paper, Herglotz-type vakonomic dynamics and Noether theory of nonholonomic systems are studied. Firstly, Herglotz-type vakonomic dynamical equations for nonholonomic systems are derived on the premise of Herglotz variational principle. Secondly, in terms of the Herglotz-type vakonomic dynamical equations, the Noether symmetry of Herglotz-type vakonomic dynamics is explored, and the Herglotz-type vakonomic dynamical Noether theorems and their inverse theorems are deduced. Finally, the conservation laws of Appell–Hamel case with non-conservative forces are analyzed to show the validity of our results.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"49 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528540","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 prove that for α ≥ 1, among 2d unit density lattices, minL∑P∈L(|P|2−β)e−πα|P|2 is achieved at hexagonal lattice for β≤12πα and does not exist for β>12πα. Here the hexagonal lattice with unit density can be expressed by Λ1=132[Z(1,0)⊕Z(12,32)]. This leads to two applications as follows. (1) Assume that α ≥ 1. Then, among 2d unit density lattices, minL∑P∈L|P|2e−πα|P|2 is achieved at hexagonal lattice. (2) Assume that β > α ≥ 1. Then minz∈Hθ(α;z)−bθ(β;z) is achieved at z=eiπ3 (corresponding to hexagonal lattice) for b≤βα and does not exist for b>βα. Here θ(α; z) is the two-dimensional Theta function.
{"title":"On lattice hexagonal crystallization for non-monotone potentials","authors":"Senping Luo, Juncheng Wei","doi":"10.1063/5.0200485","DOIUrl":"https://doi.org/10.1063/5.0200485","url":null,"abstract":"We prove that for α ≥ 1, among 2d unit density lattices, minL∑P∈L(|P|2−β)e−πα|P|2 is achieved at hexagonal lattice for β≤12πα and does not exist for β>12πα. Here the hexagonal lattice with unit density can be expressed by Λ1=132[Z(1,0)⊕Z(12,32)]. This leads to two applications as follows. (1) Assume that α ≥ 1. Then, among 2d unit density lattices, minL∑P∈L|P|2e−πα|P|2 is achieved at hexagonal lattice. (2) Assume that β > α ≥ 1. Then minz∈Hθ(α;z)−bθ(β;z) is achieved at z=eiπ3 (corresponding to hexagonal lattice) for b≤βα and does not exist for b>βα. Here θ(α; z) is the two-dimensional Theta function.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"68 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509063","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}
Let Γ=q1Z⊕q2Z⊕⋯⊕qdZ, with qj∈Z+ for each j ∈ {1, …, d}, and denote by Δ the discrete Laplacian on ℓ2Zd. Using Macaulay2, we first numerically find complex-valued Γ-periodic potentials V:Zd→C such that the operators Δ + V and Δ are Floquet isospectral. We then use combinatorial methods to validate these numerical solutions.
{"title":"Floquet isospectrality of the zero potential for discrete periodic Schrödinger operators","authors":"Matthew Faust, Wencai Liu, Rodrigo Matos, Jenna Plute, Jonah Robinson, Yichen Tao, Ethan Tran, Cindy Zhuang","doi":"10.1063/5.0201744","DOIUrl":"https://doi.org/10.1063/5.0201744","url":null,"abstract":"Let Γ=q1Z⊕q2Z⊕⋯⊕qdZ, with qj∈Z+ for each j ∈ {1, …, d}, and denote by Δ the discrete Laplacian on ℓ2Zd. Using Macaulay2, we first numerically find complex-valued Γ-periodic potentials V:Zd→C such that the operators Δ + V and Δ are Floquet isospectral. We then use combinatorial methods to validate these numerical solutions.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"60 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528542","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：