The magnetic Laplacian with a step magnetic field has been intensively studied during the last years. We adapt the construction introduced by Bonnaillie-Noël et al. [Bull. London Math. Soc. 56, 2529 (2024)] to prove the existence of bound states of a new effective operator involving a magnetic step field on a domain with an almost flat magnetic barrier. This result emphasizes the fact that even a small non-smoothness of the discontinuity region can cause the appearance of eigenvalues below the essential spectrum. We also give an example where this effective operator arises.
在过去几年里，人们一直在深入研究带有阶跃磁场的磁拉普拉斯。我们改编了 Bonnaillie-Noël 等人[Bull. London Math. Soc. 56, 2529 (2024)]引入的构造，证明了在几乎平坦的磁屏障域上涉及磁阶场的新有效算子的边界态存在。这一结果强调了这样一个事实，即即使不连续区域很小的不光滑度也会导致出现低于本征谱的特征值。我们还给出了出现这种有效算子的一个例子。
{"title":"Discrete spectrum of the magnetic Laplacian on almost flat magnetic barriers","authors":"Germán Miranda","doi":"10.1063/5.0208990","DOIUrl":"https://doi.org/10.1063/5.0208990","url":null,"abstract":"The magnetic Laplacian with a step magnetic field has been intensively studied during the last years. We adapt the construction introduced by Bonnaillie-Noël et al. [Bull. London Math. Soc. 56, 2529 (2024)] to prove the existence of bound states of a new effective operator involving a magnetic step field on a domain with an almost flat magnetic barrier. This result emphasizes the fact that even a small non-smoothness of the discontinuity region can cause the appearance of eigenvalues below the essential spectrum. We also give an example where this effective operator arises.","PeriodicalId":508452,"journal":{"name":"Journal of Mathematical Physics","volume":"44 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141688695","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}
Magnetohydrodynamic rotating shallow water system (MRSW) is a proposed model for a thin layer of electrically conducting fluid, which plays an important role in astrophysical plasma studies. For the spatial periodic domain, a mathematically rigorous framework is developed for deriving reduced systems for MRSW equations with general unbalanced initial data. It is shown that the reduced slow dynamics are the magnetohydrodynamic quasi-geostrophic equations.
{"title":"The QG limit of magnetohydrodynamic rotating shallow water system","authors":"Yue Fang, Jiawei Wang, Xiao Wang, Xin Xu","doi":"10.1063/5.0197052","DOIUrl":"https://doi.org/10.1063/5.0197052","url":null,"abstract":"Magnetohydrodynamic rotating shallow water system (MRSW) is a proposed model for a thin layer of electrically conducting fluid, which plays an important role in astrophysical plasma studies. For the spatial periodic domain, a mathematically rigorous framework is developed for deriving reduced systems for MRSW equations with general unbalanced initial data. It is shown that the reduced slow dynamics are the magnetohydrodynamic quasi-geostrophic equations.","PeriodicalId":508452,"journal":{"name":"Journal of Mathematical Physics","volume":"4 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141415561","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 an inverse scattering problems for the biharmonic Schrödinger operator Δ2 + A · ∇ + V in three dimensions. By the Helmholtz decomposition, we take A = ∇p + ∇ ×ψ. The main contributions of this work are twofold. First, we derive a stability estimate of determining the divergence-free part ∇ ×ψ of A by far-field data at multiple wavenumbers. As a consequence, we further derive a quantitative stability estimate of determining −12∇⋅A+V. Both the stability estimates improve as the upper bound of the wavenumber increases, which exhibit the phenomenon of increased stability. Second, we obtain the uniqueness of recovering both A and V by partial far-field data. The analysis employs scattering theory to obtain an analytic domain and an upper bound for the resolvent of the fourth order elliptic operator. Notice that due to an obstruction to uniqueness, the corresponding results do not hold in general for the Laplacian, i.e., Δ + A · ∇ + V. This can be explained by the fact that the resolvent of the biharmonic operator enjoys a faster decay estimate with respect to the wavenumber compared with the Laplacian.
我们考虑三维双谐薛定谔算子 Δ2 + A -∇ + V 的反散射问题。根据亥姆霍兹分解，我们取 A =∇p +∇ ×ψ。这项工作的主要贡献有两个方面。首先，我们通过多个波数的远场数据，得出了确定 A 的无发散部分 ∇ ×ψ 的稳定性估计值。因此，我们进一步得出了确定 -12∇⋅A+V 的定量稳定性估计值。这两个稳定性估计值都随着波数上限的增加而提高，表现出稳定性增强的现象。其次，我们获得了通过部分远场数据恢复 A 和 V 的唯一性。分析运用了散射理论，得到了四阶椭圆算子的解析域和解析量上界。请注意，由于唯一性的障碍，拉普拉斯算子的相应结果一般不成立，即 Δ + A -∇ + V。
{"title":"Inverse scattering problems of the biharmonic Schrödinger operator with a first order perturbation","authors":"Xiang Xu, Yue Zhao","doi":"10.1063/5.0202903","DOIUrl":"https://doi.org/10.1063/5.0202903","url":null,"abstract":"We consider an inverse scattering problems for the biharmonic Schrödinger operator Δ2 + A · ∇ + V in three dimensions. By the Helmholtz decomposition, we take A = ∇p + ∇ ×ψ. The main contributions of this work are twofold. First, we derive a stability estimate of determining the divergence-free part ∇ ×ψ of A by far-field data at multiple wavenumbers. As a consequence, we further derive a quantitative stability estimate of determining −12∇⋅A+V. Both the stability estimates improve as the upper bound of the wavenumber increases, which exhibit the phenomenon of increased stability. Second, we obtain the uniqueness of recovering both A and V by partial far-field data. The analysis employs scattering theory to obtain an analytic domain and an upper bound for the resolvent of the fourth order elliptic operator. Notice that due to an obstruction to uniqueness, the corresponding results do not hold in general for the Laplacian, i.e., Δ + A · ∇ + V. This can be explained by the fact that the resolvent of the biharmonic operator enjoys a faster decay estimate with respect to the wavenumber compared with the Laplacian.","PeriodicalId":508452,"journal":{"name":"Journal of Mathematical Physics","volume":"26 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141410380","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}
A common choice for the evolution of the knotted graphs in loop quantum gravity is to use the Pachner moves, adapted to graphs from their dual triangulations. Here, we show that the natural way to consistently use these moves is on framed graphs with edge twists, where the Pachner moves can only be performed when the twists, and the vertices the edges are incident on, meet certain criteria. For other twists, one can introduce an algebraic object, which allow any knotted graph with framed edges to be written in terms of a generalized braid group.
{"title":"Knotted 4-regular graphs. II. Consistent application of the Pachner moves","authors":"Daniel Cartin","doi":"10.1063/5.0191415","DOIUrl":"https://doi.org/10.1063/5.0191415","url":null,"abstract":"A common choice for the evolution of the knotted graphs in loop quantum gravity is to use the Pachner moves, adapted to graphs from their dual triangulations. Here, we show that the natural way to consistently use these moves is on framed graphs with edge twists, where the Pachner moves can only be performed when the twists, and the vertices the edges are incident on, meet certain criteria. For other twists, one can introduce an algebraic object, which allow any knotted graph with framed edges to be written in terms of a generalized braid group.","PeriodicalId":508452,"journal":{"name":"Journal of Mathematical Physics","volume":"60 25","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141277097","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}
The equilibrium and phase properties of the Ising model with three-spin interaction and an external field are studied within the framework of mean-field approximation. The thermodynamic properties of the model reveals two coexistence curves, signifying two distinct second-order phase transitions, dependent on the domain of the interaction parameter. The critical exponents of the magnetic order parameter are calculated in all directions of the phase space and show their agreement with the mean-field universality class.
{"title":"Phase properties of the mean-field Ising model with three-spin interaction","authors":"Godwin Osabutey","doi":"10.1063/5.0183805","DOIUrl":"https://doi.org/10.1063/5.0183805","url":null,"abstract":"The equilibrium and phase properties of the Ising model with three-spin interaction and an external field are studied within the framework of mean-field approximation. The thermodynamic properties of the model reveals two coexistence curves, signifying two distinct second-order phase transitions, dependent on the domain of the interaction parameter. The critical exponents of the magnetic order parameter are calculated in all directions of the phase space and show their agreement with the mean-field universality class.","PeriodicalId":508452,"journal":{"name":"Journal of Mathematical Physics","volume":"33 21","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141414901","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 the question of whether the high-energy eigenfunctions of certain Schrödinger operators on the d-dimensional hyperbolic space of constant curvature −κ2 are flexible enough to approximate an arbitrary solution of the Helmholtz equation Δh + h = 0 on Rd, over the natural length scale O(λ−1/2) determined by the eigenvalue λ ≫ 1. This problem is motivated by the fact that, by the asymptotics of the local Weyl law, approximate Laplace eigenfunctions do have this approximation property on any compact Riemannian manifold. In this paper we are specifically interested in the Coulomb and harmonic oscillator operators on the hyperbolic spaces Hd(κ). As the dimension of the space of bound states of these operators tends to infinity as κ ↘ 0, one can hope to approximate solutions to the Helmholtz equation by eigenfunctions for some κ > 0 that is not fixed a priori. Our main result shows that this is indeed the case, under suitable hypotheses. We also prove a global approximation theorem with decay for the Helmholtz equation on manifolds that are isometric to the hyperbolic space outside a compact set, and consider an application to the study of the heat equation on Hd(κ). Although global approximation and inverse approximation results are heuristically related in that both theorems explore flexibility properties of solutions to elliptic equations on hyperbolic spaces, we will see that the underlying ideas behind these theorems are very different.
{"title":"Inverse localization and global approximation for some Schrödinger operators on hyperbolic spaces","authors":"A. Enciso, Alba García-Ruiz, D. Peralta-Salas","doi":"10.1063/5.0156230","DOIUrl":"https://doi.org/10.1063/5.0156230","url":null,"abstract":"We consider the question of whether the high-energy eigenfunctions of certain Schrödinger operators on the d-dimensional hyperbolic space of constant curvature −κ2 are flexible enough to approximate an arbitrary solution of the Helmholtz equation Δh + h = 0 on Rd, over the natural length scale O(λ−1/2) determined by the eigenvalue λ ≫ 1. This problem is motivated by the fact that, by the asymptotics of the local Weyl law, approximate Laplace eigenfunctions do have this approximation property on any compact Riemannian manifold. In this paper we are specifically interested in the Coulomb and harmonic oscillator operators on the hyperbolic spaces Hd(κ). As the dimension of the space of bound states of these operators tends to infinity as κ ↘ 0, one can hope to approximate solutions to the Helmholtz equation by eigenfunctions for some κ > 0 that is not fixed a priori. Our main result shows that this is indeed the case, under suitable hypotheses. We also prove a global approximation theorem with decay for the Helmholtz equation on manifolds that are isometric to the hyperbolic space outside a compact set, and consider an application to the study of the heat equation on Hd(κ). Although global approximation and inverse approximation results are heuristically related in that both theorems explore flexibility properties of solutions to elliptic equations on hyperbolic spaces, we will see that the underlying ideas behind these theorems are very different.","PeriodicalId":508452,"journal":{"name":"Journal of Mathematical Physics","volume":"54 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141277840","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}
A stochastic influenza epidemic model where influenza virus can mutate into a mutant influenza virus is established to study the influence of environmental disturbance. And the transmission rate of the model is assumed to satisfy log-normal Ornstein–Uhlenbeck process. We verify that there exists a unique global positive solution to the stochastic model. By constructing proper Lyapunov functions, sufficient conditions under which the stationary distribution exists are obtained. In addition, we discuss the extinction of the disease. Furthermore, we get the accurate expression of probability density function near the endemic equilibrium of the stochastic model. Finally, several numerical simulations are carried out to verify theoretical results and examine the influence of environmental noise.
{"title":"Dynamics analysis of an influenza epidemic model with virus mutation incorporating log-normal Ornstein–Uhlenbeck process","authors":"Xinhong Zhang, Xiaoshan Zhang, Daqing Jiang","doi":"10.1063/5.0179818","DOIUrl":"https://doi.org/10.1063/5.0179818","url":null,"abstract":"A stochastic influenza epidemic model where influenza virus can mutate into a mutant influenza virus is established to study the influence of environmental disturbance. And the transmission rate of the model is assumed to satisfy log-normal Ornstein–Uhlenbeck process. We verify that there exists a unique global positive solution to the stochastic model. By constructing proper Lyapunov functions, sufficient conditions under which the stationary distribution exists are obtained. In addition, we discuss the extinction of the disease. Furthermore, we get the accurate expression of probability density function near the endemic equilibrium of the stochastic model. Finally, several numerical simulations are carried out to verify theoretical results and examine the influence of environmental noise.","PeriodicalId":508452,"journal":{"name":"Journal of Mathematical Physics","volume":"12 16","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141393985","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}
In [D. Feng and W. Huang, J. Math. Pures Appl. 106, 411–452 (2016)], the authors studied weighted topological pressure and established a variational principle for it. In this paper, we introduce the notion of local weighted topological pressure and generalize Feng and Huang’s main results to localized version.
在[D. Feng and W. Huang, J. Math. Pures Appl. 106, 411-452 (2016)]中，作者研究了加权拓扑压力，并为其建立了变分原理。在本文中，我们引入了局部加权拓扑压力的概念，并将冯和黄的主要结果推广到局部版本。
{"title":"Local weighted topological pressure","authors":"Fangzhou Cai","doi":"10.1063/5.0195440","DOIUrl":"https://doi.org/10.1063/5.0195440","url":null,"abstract":"In [D. Feng and W. Huang, J. Math. Pures Appl. 106, 411–452 (2016)], the authors studied weighted topological pressure and established a variational principle for it. In this paper, we introduce the notion of local weighted topological pressure and generalize Feng and Huang’s main results to localized version.","PeriodicalId":508452,"journal":{"name":"Journal of Mathematical Physics","volume":"15 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141391892","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 provide a theoretical study of a new family of orthogonal functions on the punctured complex plane solving the eigenvalue problems for some magnetic Laplacian perturbed by a singular vector potential with zero magnetic field modeling the Aharonov–Bohm effect. The functions are defined by their β-modified Rodrigues type formula and extend the polyanalytic Itô–Hermite polynomials to the polymeromorphic setting. Mainly, we derive their different operational representations and give their explicit expressions in terms of special functions. Different generating functions and integral representations are obtained.
{"title":"Polymeromorphic Itô–Hermite functions associated with a singular potential vector on the punctured complex plane","authors":"Hajar Dkhissi, A. Ghanmi","doi":"10.1063/5.0151921","DOIUrl":"https://doi.org/10.1063/5.0151921","url":null,"abstract":"We provide a theoretical study of a new family of orthogonal functions on the punctured complex plane solving the eigenvalue problems for some magnetic Laplacian perturbed by a singular vector potential with zero magnetic field modeling the Aharonov–Bohm effect. The functions are defined by their β-modified Rodrigues type formula and extend the polyanalytic Itô–Hermite polynomials to the polymeromorphic setting. Mainly, we derive their different operational representations and give their explicit expressions in terms of special functions. Different generating functions and integral representations are obtained.","PeriodicalId":508452,"journal":{"name":"Journal of Mathematical Physics","volume":"142 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141281235","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}
In this paper, we introduce an abstract KAM (Kolmogorov–Arnold–Moser) theorem. As an application, we study the two-dimensional completely resonant beam system under periodic boundary conditions. Using the KAM theorem together with partial Birkhoff normal form method, we obtain a family of Whitney smooth small–amplitude quasi–periodic solutions for the equation system.
本文介绍了一个抽象的 KAM（Kolmogorov-Arnold-Moser）定理。作为应用，我们研究了周期性边界条件下的二维完全共振梁系统。利用 KAM 定理和部分伯克霍夫正态法，我们得到了方程系统的惠特尼光滑小振幅准周期解系列。
{"title":"KAM tori for two dimensional completely resonant derivative beam system","authors":"Shuaishuai Xue, Yingnan Sun","doi":"10.1063/5.0183958","DOIUrl":"https://doi.org/10.1063/5.0183958","url":null,"abstract":"In this paper, we introduce an abstract KAM (Kolmogorov–Arnold–Moser) theorem. As an application, we study the two-dimensional completely resonant beam system under periodic boundary conditions. Using the KAM theorem together with partial Birkhoff normal form method, we obtain a family of Whitney smooth small–amplitude quasi–periodic solutions for the equation system.","PeriodicalId":508452,"journal":{"name":"Journal of Mathematical Physics","volume":"13 7","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141398942","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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