We obtain asymptotic resolvent expansions at the threshold of the essential spectrum for magnetic Schrödinger and Pauli operators in dimension three. These operators are treated as perturbations of the Laplace operator in L2(R3) and L2(R3;C2), respectively. The main novelty of our approach is to show that the relative perturbations, which are first order differential operators, can be factorized in suitably chosen auxiliary spaces. This allows us to derive the desired asymptotic expansions of the resolvents around zero. We then calculate their leading and sub-leading terms explicitly. Analogous factorization schemes for more general perturbations, including e.g. finite rank perturbations, are discussed as well.
{"title":"Resolvent expansions of 3D magnetic Schrödinger operators and Pauli operators","authors":"Arne Jensen, Hynek Kovařík","doi":"10.1063/5.0211421","DOIUrl":"https://doi.org/10.1063/5.0211421","url":null,"abstract":"We obtain asymptotic resolvent expansions at the threshold of the essential spectrum for magnetic Schrödinger and Pauli operators in dimension three. These operators are treated as perturbations of the Laplace operator in L2(R3) and L2(R3;C2), respectively. The main novelty of our approach is to show that the relative perturbations, which are first order differential operators, can be factorized in suitably chosen auxiliary spaces. This allows us to derive the desired asymptotic expansions of the resolvents around zero. We then calculate their leading and sub-leading terms explicitly. Analogous factorization schemes for more general perturbations, including e.g. finite rank perturbations, are discussed as well.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"20 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770758","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}
An integrable discretization of the SIR model with vaccination is proposed. Through the discretization, the conserved quantities of the continuous model are inherited to the discrete model, since the discretization is based on the intersection structure of the non-algebraic invariant curve defined by the conserved quantities. Uniqueness of the forward/backward evolution of the discrete model is demonstrated in terms of the single-valuedness of the Lambert W function on the positive real axis. Furthermore, the exact solution to the continuous SIR model with vaccination is constructed via the integrable discretization. When applied to the original SIR model, the discretization procedure leads to two kinds of integrable discretization, and the exact solution to the continuous SIR model is also deduced. It is furthermore shown that the discrete SIR model geometrically linearizes the time evolution by using the non-autonomous parallel translation of the line intersecting the invariant curve.
本文提出了带疫苗接种的 SIR 模型的可积分离散化方法。通过离散化，连续模型的守恒量被继承到离散模型中，因为离散化是基于守恒量定义的非代数不变曲线的交集结构。离散模型的前向/后向演化的唯一性是通过正实轴上兰伯特 W 函数的单值性来证明的。此外，还通过可积分离散化构建了带有疫苗接种的连续 SIR 模型的精确解。当应用于原始 SIR 模型时，离散化过程会导致两种可积分离散化，并推导出连续 SIR 模型的精确解。此外，离散 SIR 模型还利用与不变曲线相交的直线的非自主平行平移将时间演化几何线性化。
{"title":"Exact solutions to SIR epidemic models via integrable discretization","authors":"Atsushi Nobe","doi":"10.1063/5.0152442","DOIUrl":"https://doi.org/10.1063/5.0152442","url":null,"abstract":"An integrable discretization of the SIR model with vaccination is proposed. Through the discretization, the conserved quantities of the continuous model are inherited to the discrete model, since the discretization is based on the intersection structure of the non-algebraic invariant curve defined by the conserved quantities. Uniqueness of the forward/backward evolution of the discrete model is demonstrated in terms of the single-valuedness of the Lambert W function on the positive real axis. Furthermore, the exact solution to the continuous SIR model with vaccination is constructed via the integrable discretization. When applied to the original SIR model, the discretization procedure leads to two kinds of integrable discretization, and the exact solution to the continuous SIR model is also deduced. It is furthermore shown that the discrete SIR model geometrically linearizes the time evolution by using the non-autonomous parallel translation of the line intersecting the invariant curve.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"4 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770897","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 work, we address the question of the impossibility of certain single-letter formulas by exploiting the semi-algebraic nature of various entropy-constrained sets. The focus lies on studying the properties of the level sets of relative entropy, mutual information, and Rényi entropies. We analyze the transcendental structure of the set of states in which one of the aforementioned entropy quantities is fixed. Our results rule out (semi)algebraic single-shot characterizations of these entropy measures with bounded ancilla for both the classical and quantum cases.
{"title":"Transcendental properties of entropy-constrained sets II","authors":"Vjosa Blakaj, Chokri Manai","doi":"10.1063/5.0182728","DOIUrl":"https://doi.org/10.1063/5.0182728","url":null,"abstract":"In this work, we address the question of the impossibility of certain single-letter formulas by exploiting the semi-algebraic nature of various entropy-constrained sets. The focus lies on studying the properties of the level sets of relative entropy, mutual information, and Rényi entropies. We analyze the transcendental structure of the set of states in which one of the aforementioned entropy quantities is fixed. Our results rule out (semi)algebraic single-shot characterizations of these entropy measures with bounded ancilla for both the classical and quantum cases.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"20 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770896","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 discuss here the electron dynamics in crystals in rather strong magnetic fields. The most non-trivial part of this topic is the Novikov problem, namely, the problem of classifying non-closed trajectories for particles with an arbitrary dispersion relation in presence of a magnetic field. Here we will try to review both the classical results obtained in the study of this problem and the most recent results related to its most difficult part. At the same time, the study of electron dynamics on the Fermi surface can also be carried out in a more general setting, namely, as the study of the topology of the corresponding dynamical system and the physical phenomena associated with it. This approach is actually related to the Novikov problem, but includes the study of a wider range of issues, as well as a wider range of experimentally observed phenomena. Here we will try to describe a number of general principles relating changes in the topological pattern of trajectories on the Fermi surface to the observed phenomena in strong magnetic fields. We also note here that the study of the described effects can be quite informative for the experimental study of dispersion relations in conductors.
{"title":"Topology of dynamical systems on the Fermi surface and galvanomagnetic phenomena in normal metals","authors":"A. Ya. Maltsev, S. P. Novikov","doi":"10.1063/5.0151512","DOIUrl":"https://doi.org/10.1063/5.0151512","url":null,"abstract":"We discuss here the electron dynamics in crystals in rather strong magnetic fields. The most non-trivial part of this topic is the Novikov problem, namely, the problem of classifying non-closed trajectories for particles with an arbitrary dispersion relation in presence of a magnetic field. Here we will try to review both the classical results obtained in the study of this problem and the most recent results related to its most difficult part. At the same time, the study of electron dynamics on the Fermi surface can also be carried out in a more general setting, namely, as the study of the topology of the corresponding dynamical system and the physical phenomena associated with it. This approach is actually related to the Novikov problem, but includes the study of a wider range of issues, as well as a wider range of experimentally observed phenomena. Here we will try to describe a number of general principles relating changes in the topological pattern of trajectories on the Fermi surface to the observed phenomena in strong magnetic fields. We also note here that the study of the described effects can be quite informative for the experimental study of dispersion relations in conductors.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"64 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141742901","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}
Main purpose of this paper is to formulate canonical quantum continuity equations for white noise operators and to find their explicit unique solutions. By applying the continuity equations for white noise functionals and canonical topological isomorphisms between the spaces of white noise operators and the spaces of two-variable white noise functionals, we formulate canonical quantum continuity equations for white noise operators, and then we induce time dependent quantum evolution systems which are equivalent to the quantum continuity equations. Then we find explicit forms, in terms of the quantum analogues of generalized Fourier-Mehler transforms, of the unique solutions of the quantum continuity equations by solving the quantum evolution systems for white noise operators.
{"title":"Quantum continuity equations and quantum evolution systems for white noise operators","authors":"Un Cig Ji","doi":"10.1063/5.0176568","DOIUrl":"https://doi.org/10.1063/5.0176568","url":null,"abstract":"Main purpose of this paper is to formulate canonical quantum continuity equations for white noise operators and to find their explicit unique solutions. By applying the continuity equations for white noise functionals and canonical topological isomorphisms between the spaces of white noise operators and the spaces of two-variable white noise functionals, we formulate canonical quantum continuity equations for white noise operators, and then we induce time dependent quantum evolution systems which are equivalent to the quantum continuity equations. Then we find explicit forms, in terms of the quantum analogues of generalized Fourier-Mehler transforms, of the unique solutions of the quantum continuity equations by solving the quantum evolution systems for white noise operators.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"30 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743151","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 non-smoothness of the fundamental solution for the Schrödinger equation with a spherically symmetric and super-quadratic potential in the sense that V(x) ≥ C|x|2+ɛ at infinity with constants C > 0 and ɛ > 0. More precisely, we show the fundamental solution E(t, x, y) does not belong to C1 as a function of (t, x, y), which partially solves Yajima’s conjecture.
{"title":"Non-smoothness of the fundamental solutions for Schrödinger equations with super-quadratic and spherically symmetric potential","authors":"Keiichi Kato, Wataru Nakahashi, Yukihide Tadano","doi":"10.1063/5.0184443","DOIUrl":"https://doi.org/10.1063/5.0184443","url":null,"abstract":"We study non-smoothness of the fundamental solution for the Schrödinger equation with a spherically symmetric and super-quadratic potential in the sense that V(x) ≥ C|x|2+ɛ at infinity with constants C > 0 and ɛ > 0. More precisely, we show the fundamental solution E(t, x, y) does not belong to C1 as a function of (t, x, y), which partially solves Yajima’s conjecture.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"36 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141742900","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 deals with the model of compressible viscous and barotropic fluids coupled with the Poisson equation in a bounded domain Ω⊂R3 with C2+α (0 < α < 1) boundary ∂Ω. We prove the existence and weak-strong uniqueness of dissipative solutions when the adiabatic exponent γ > 1. We find that the Poisson term ρ∇Φ is not integrable when γ∈(1,32). We will make full use of the Poisson equation and energy inequality to overcome this difficulty. Finally, we obtain that ρ∇Φ leads to the decrease of Reynolds stress R and the increase of the energy dissipation defect E.
{"title":"Generalized solutions to the model of compressible viscous fluids coupled with the Poisson equation","authors":"Zhong Tan, Hui Yang","doi":"10.1063/5.0190282","DOIUrl":"https://doi.org/10.1063/5.0190282","url":null,"abstract":"This paper deals with the model of compressible viscous and barotropic fluids coupled with the Poisson equation in a bounded domain Ω⊂R3 with C2+α (0 &lt; α &lt; 1) boundary ∂Ω. We prove the existence and weak-strong uniqueness of dissipative solutions when the adiabatic exponent γ &gt; 1. We find that the Poisson term ρ∇Φ is not integrable when γ∈(1,32). We will make full use of the Poisson equation and energy inequality to overcome this difficulty. Finally, we obtain that ρ∇Φ leads to the decrease of Reynolds stress R and the increase of the energy dissipation defect E.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"26 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743148","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 will prove that Anderson localization takes place for quasiperiodic Schrödinger operators with even C2 cos-type potentials, weak Liouvillean frequencies and Diophantine phases when the coupling is sufficiently large.
在本文中，我们将证明当耦合足够大时，对于具有偶 C2 cos 型势能、弱刘维尔频率和 Diophantine 相的准周期薛定谔算子，会发生安德森局域化。
{"title":"Arithmetic version of Anderson localization for a class of C2 quasiperiodic Schrödinger operators","authors":"Zhen Zhang, Xiong Li","doi":"10.1063/5.0157371","DOIUrl":"https://doi.org/10.1063/5.0157371","url":null,"abstract":"In this paper, we will prove that Anderson localization takes place for quasiperiodic Schrödinger operators with even C2 cos-type potentials, weak Liouvillean frequencies and Diophantine phases when the coupling is sufficiently large.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"63 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141742902","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}
Prem Nigam Kar, Jitendra Prakash, David E. Roberson
We study a class of nonlocal games, called transitive games, for which the set of perfect strategies forms a semigroup. We establish several interesting correspondences of bisynchronous transitive games with the theory of compact quantum groups. In particular, we associate a quantum permutation group with each bisynchronous transitive game and vice versa. We prove that the existence of a C*-strategy, the existence of a quantum commuting strategy, and the existence of a classical strategy are all equivalent for bisynchronous transitive games. We then use some of these correspondences to establish necessary and sufficient conditions for some classes of correlations, that arise as perfect strategies of transitive games, to be nonlocal.
{"title":"Transitive nonlocal games","authors":"Prem Nigam Kar, Jitendra Prakash, David E. Roberson","doi":"10.1063/5.0199344","DOIUrl":"https://doi.org/10.1063/5.0199344","url":null,"abstract":"We study a class of nonlocal games, called transitive games, for which the set of perfect strategies forms a semigroup. We establish several interesting correspondences of bisynchronous transitive games with the theory of compact quantum groups. In particular, we associate a quantum permutation group with each bisynchronous transitive game and vice versa. We prove that the existence of a C*-strategy, the existence of a quantum commuting strategy, and the existence of a classical strategy are all equivalent for bisynchronous transitive games. We then use some of these correspondences to establish necessary and sufficient conditions for some classes of correlations, that arise as perfect strategies of transitive games, to be nonlocal.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"24 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141742903","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 present a theory to quantify the formation of spatiotemporal macrostructures (or the non-homogeneous regions of high viscosity at moderate to high fluid inertia) for viscoelastic sub-diffusive flows, by introducing a mathematically consistent decomposition of the polymer conformation tensor, into the so-called structure tensor. Our approach bypasses an inherent problem in the standard arithmetic decomposition, namely, the fluctuating conformation tensor fields may not be positive definite and hence, do not retain their physical meaning. Using well-established results in matrix analysis, the space of positive definite matrices is transformed into a Riemannian manifold by defining and constructing a geodesic via the inner product on its tangent space. This geodesic is utilized to define three scalar invariants of the structure tensor, which do not suffer from the caveats of the regular invariants (such as trace and determinant) of the polymer conformation tensor. First, we consider the problem of formulating perturbative expansions of the structure tensor using the geodesic, which is consistent with the Riemannian manifold geometry. A constraint on the maximum time, during which the evolution of the perturbative solution can be well approximated by linear theory along the Euclidean manifold, is found. A comparison between the linear and the nonlinear dynamics, identifies the role of nonlinearities in initiating the symmetry breaking of the flow variables about the centerline. Finally, fully nonlinear simulations of the viscoelastic sub-diffusive channel flows, underscore the advantage of using these invariants in effectively quantifying the macrostructures.
{"title":"Quantifying macrostructures in viscoelastic sub-diffusive flows","authors":"T. Chauhan, K. Kalyanaraman, S. Sircar","doi":"10.1063/5.0195666","DOIUrl":"https://doi.org/10.1063/5.0195666","url":null,"abstract":"We present a theory to quantify the formation of spatiotemporal macrostructures (or the non-homogeneous regions of high viscosity at moderate to high fluid inertia) for viscoelastic sub-diffusive flows, by introducing a mathematically consistent decomposition of the polymer conformation tensor, into the so-called structure tensor. Our approach bypasses an inherent problem in the standard arithmetic decomposition, namely, the fluctuating conformation tensor fields may not be positive definite and hence, do not retain their physical meaning. Using well-established results in matrix analysis, the space of positive definite matrices is transformed into a Riemannian manifold by defining and constructing a geodesic via the inner product on its tangent space. This geodesic is utilized to define three scalar invariants of the structure tensor, which do not suffer from the caveats of the regular invariants (such as trace and determinant) of the polymer conformation tensor. First, we consider the problem of formulating perturbative expansions of the structure tensor using the geodesic, which is consistent with the Riemannian manifold geometry. A constraint on the maximum time, during which the evolution of the perturbative solution can be well approximated by linear theory along the Euclidean manifold, is found. A comparison between the linear and the nonlinear dynamics, identifies the role of nonlinearities in initiating the symmetry breaking of the flow variables about the centerline. Finally, fully nonlinear simulations of the viscoelastic sub-diffusive channel flows, underscore the advantage of using these invariants in effectively quantifying the macrostructures.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"253 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743149","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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