American Institute of Physics: Journal of Mathematical Physics: Table of Contents
Table of Contents for Journal of Mathematical Physics. List of articles from both the latest and ahead of print issues.
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American Institute of Physics: Journal of Mathematical Physics: Table of Contents
American Institute of Physics
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Journal of Mathematical Physics
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Partially regular weak solutions to the fractional Navier–Stokes equations with the critical dissipation
https://aip.scitation.org/doi/10.1063/5.0088047?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>We show that there exist partially regular weak solutions of Navier–Stokes equations with fractional dissipation [math] in the critical case of [math], which satisfy certain local energy inequalities and whose singular sets have a locally finite twodimensional parabolic Hausdorff measure. Actually, this problem had been studied by Chen and Wei [Discrete Contin. Dyn. Syst. 36(10), 5309–5322 (2016)]; in this paper, they established the partial regularity of suitable weak solutions for [math]. A point is that they admitted the existence of suitable weak solutions but did not give the proof. It should be noted that, when [math], the existence of suitable weak solutions is not trivial due to the possible lack of compactness. To overcome this difficulty, we shall use a parabolic concentrationcompactness theorem introduced by Wu [Arch. Ration. Mech. Anal. 239(3), 1771–1808 (2021)]. For the partial regularity theory, we will apply the idea of Chen and Wei.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>We show that there exist partially regular weak solutions of Navier–Stokes equations with fractional dissipation [math] in the critical case of [math], which satisfy certain local energy inequalities and whose singular sets have a locally finite twodimensional parabolic Hausdorff measure. Actually, this problem had been studied by Chen and Wei [Discrete Contin. Dyn. Syst. 36(10), 5309–5322 (2016)]; in this paper, they established the partial regularity of suitable weak solutions for [math]. A point is that they admitted the existence of suitable weak solutions but did not give the proof. It should be noted that, when [math], the existence of suitable weak solutions is not trivial due to the possible lack of compactness. To overcome this difficulty, we shall use a parabolic concentrationcompactness theorem introduced by Wu [Arch. Ration. Mech. Anal. 239(3), 1771–1808 (2021)]. For the partial regularity theory, we will apply the idea of Chen and Wei.
Partially regular weak solutions to the fractional Navier–Stokes equations with the critical dissipation
10.1063/5.0088047
Journal of Mathematical Physics
20221101T12:55:50Z
© 2022 Author(s).
Jiaqi Yang

Logarithmic Schrödinger equations in infinite dimensions
https://aip.scitation.org/doi/10.1063/5.0102156?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>We study the logarithmic Schrödinger equation with a finite range potential on [math]. Through a groundstate representation, we associate and construct a global Gibbs measure and show that it satisfies a logarithmic Sobolev inequality. We find estimates on the solutions in arbitrary dimension and prove the existence of weak solutions to the infinitedimensional Cauchy problem.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>We study the logarithmic Schrödinger equation with a finite range potential on [math]. Through a groundstate representation, we associate and construct a global Gibbs measure and show that it satisfies a logarithmic Sobolev inequality. We find estimates on the solutions in arbitrary dimension and prove the existence of weak solutions to the infinitedimensional Cauchy problem.
Logarithmic Schrödinger equations in infinite dimensions
10.1063/5.0102156
Journal of Mathematical Physics
20221101T12:55:51Z
© 2022 Author(s).
Larry Read
Bogusław Zegarliński
Mengchun Zhang

ZeroMach limit of the compressible Navier–Stokes–Korteweg equations
https://aip.scitation.org/doi/10.1063/5.0124119?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>We consider the Cauchy problem for the compressible Navier–Stokes–Korteweg system in three dimensions. Under the assumption of the global existence of strong solutions to incompressible Navier–Stokes equations, we demonstrate that the compressible Navier–Stokes–Korteweg system admits a global unique strong solution without smallness restrictions on initial data when the Mach number is sufficiently small. Furthermore, we derive the uniform convergence of strong solutions for compressible Navier–Stokes–Korteweg equations toward those for incompressible Navier–Stokes equations as long as the solution of the limiting system exists.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>We consider the Cauchy problem for the compressible Navier–Stokes–Korteweg system in three dimensions. Under the assumption of the global existence of strong solutions to incompressible Navier–Stokes equations, we demonstrate that the compressible Navier–Stokes–Korteweg system admits a global unique strong solution without smallness restrictions on initial data when the Mach number is sufficiently small. Furthermore, we derive the uniform convergence of strong solutions for compressible Navier–Stokes–Korteweg equations toward those for incompressible Navier–Stokes equations as long as the solution of the limiting system exists.
ZeroMach limit of the compressible Navier–Stokes–Korteweg equations
10.1063/5.0124119
Journal of Mathematical Physics
20221101T12:55:52Z
© 2022 Author(s).
Qiangchang Ju
Jianjun Xu

Regularity for 3D inhomogeneous incompressible MHD equations with vacuum
https://aip.scitation.org/doi/10.1063/5.0111586?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>In this paper, we consider the conditional regularity for 3D inhomogeneous incompressible magnetohydrodynamic equations in Vishik spaces and give a regularity criterion of weak solutions in terms of a gradient velocity vector.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>In this paper, we consider the conditional regularity for 3D inhomogeneous incompressible magnetohydrodynamic equations in Vishik spaces and give a regularity criterion of weak solutions in terms of a gradient velocity vector.
Regularity for 3D inhomogeneous incompressible MHD equations with vacuum
10.1063/5.0111586
Journal of Mathematical Physics
20221101T02:07:33Z
© 2022 Author(s).
JaeMyoung Kim

On a semilinear wave equation in antide Sitter spacetime: The critical case
https://aip.scitation.org/doi/10.1063/5.0086614?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>In the present paper, we prove the blowup in finite time for local solutions of a semilinear Cauchy problem associated with a wave equation in antide Sitter spacetime in the critical case. According to this purpose, we combine a result for ordinary differential inequalities with an iteration argument by using an explicit integral representation formula for the solution to a linear Cauchy problem associated with the wave equation in antide Sitter spacetime in one space dimension.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>In the present paper, we prove the blowup in finite time for local solutions of a semilinear Cauchy problem associated with a wave equation in antide Sitter spacetime in the critical case. According to this purpose, we combine a result for ordinary differential inequalities with an iteration argument by using an explicit integral representation formula for the solution to a linear Cauchy problem associated with the wave equation in antide Sitter spacetime in one space dimension.
On a semilinear wave equation in antide Sitter spacetime: The critical case
10.1063/5.0086614
Journal of Mathematical Physics
20221102T02:30:52Z
© 2022 Author(s).
Alessandro Palmieri
Hiroyuki Takamura

Global wellposedness to the Cauchy problem of 2D nonhomogeneous Bénard system with large initial data and vacuum
https://aip.scitation.org/doi/10.1063/5.0106653?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>This paper establishes the global wellposedness of strong solutions to the nonhomogeneous Bénard system with positive density at infinity in the whole space [math]. We obtain the global existence and uniqueness of strong solutions for general large initial data. Our method relies on the dedicate energy estimates and a logarithmic interpolation inequality.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>This paper establishes the global wellposedness of strong solutions to the nonhomogeneous Bénard system with positive density at infinity in the whole space [math]. We obtain the global existence and uniqueness of strong solutions for general large initial data. Our method relies on the dedicate energy estimates and a logarithmic interpolation inequality.
Global wellposedness to the Cauchy problem of 2D nonhomogeneous Bénard system with large initial data and vacuum
10.1063/5.0106653
Journal of Mathematical Physics
20221102T02:30:50Z
© 2022 Author(s).
Huanyuan Li

Estimation of decay rates to largesolutions of 3D compressible magnetohydrodynamic system
https://aip.scitation.org/doi/10.1063/5.0096472?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>The aim of this paper is to get an estimation of decay rates to firstorder and secondorder derivatives of space for largesolutions to 3D compressible magnetohydrodynamic system. While the condition (σ0 − 1, u0, Q0) ∈ L1 ∩ H2 is satisfied via a classical energy method and Fourier splitting method, firstorder and secondorder derivatives of space for largesolutions tending to 0 by L2rate [math] are shown. It is a necessary supplement to the result of Gao, Wei, and Yao [Appl. Math. Lett. 102, 106100 (2020)] in which they only obtained an estimation of decay rates to magnetic fields. Meanwhile, compared with the work of Gao, Wei, and Yao [Physica D 406, 132506 (2020)], we find that the appearance of magnetic fields does not have any bad effect on the estimation of decay rates to both the velocity field and density.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>The aim of this paper is to get an estimation of decay rates to firstorder and secondorder derivatives of space for largesolutions to 3D compressible magnetohydrodynamic system. While the condition (σ0 − 1, u0, Q0) ∈ L1 ∩ H2 is satisfied via a classical energy method and Fourier splitting method, firstorder and secondorder derivatives of space for largesolutions tending to 0 by L2rate [math] are shown. It is a necessary supplement to the result of Gao, Wei, and Yao [Appl. Math. Lett. 102, 106100 (2020)] in which they only obtained an estimation of decay rates to magnetic fields. Meanwhile, compared with the work of Gao, Wei, and Yao [Physica D 406, 132506 (2020)], we find that the appearance of magnetic fields does not have any bad effect on the estimation of decay rates to both the velocity field and density.
Estimation of decay rates to largesolutions of 3D compressible magnetohydrodynamic system
10.1063/5.0096472
Journal of Mathematical Physics
20221102T02:30:51Z
© 2022 Author(s).
Shuai Wang
Fei Chen
Yongye Zhao
Chuanbao Wang

Some geometric inequalities on Riemannian manifolds associated with the generalized modified Ricci curvature
https://aip.scitation.org/doi/10.1063/5.0116994?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>In this paper, by establishing a Reillytype formula with respect to the ϕLaplacian associated with the generalized modified Ricci curvature, we first obtain sharp lower bound estimates for the first eigenvalue of the ϕLaplacian. On the other hand, some new Brascamp–Liebtype and Colesantitype inequalities under some suitable boundary conditions are achieved. As applications, we also obtain some relationships between the weighted mean curvature of a boundary submanifold and the mean curvature of submanifold x : ∂M → RN(K) into space form RN(K).
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>In this paper, by establishing a Reillytype formula with respect to the ϕLaplacian associated with the generalized modified Ricci curvature, we first obtain sharp lower bound estimates for the first eigenvalue of the ϕLaplacian. On the other hand, some new Brascamp–Liebtype and Colesantitype inequalities under some suitable boundary conditions are achieved. As applications, we also obtain some relationships between the weighted mean curvature of a boundary submanifold and the mean curvature of submanifold x : ∂M → RN(K) into space form RN(K).
Some geometric inequalities on Riemannian manifolds associated with the generalized modified Ricci curvature
10.1063/5.0116994
Journal of Mathematical Physics
20221102T02:30:57Z
© 2022 Author(s).
Guangyue Huang
Mingfang Zhu

Harmonic and anharmonic oscillators on the Heisenberg group
https://aip.scitation.org/doi/10.1063/5.0106068?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>In this article, we present a notion of the harmonic oscillator on the Heisenberg group Hn, which, under a few reasonable assumptions, forms the natural analog of a harmonic oscillator on [math]: a negative sum of squares of operators on Hn, which is essentially selfadjoint on L2(Hn) with purely discrete spectrum and whose eigenvectors are Schwartz functions forming an orthonormal basis of L2(Hn). The differential operator in question is determined by the Dynin–Folland group—a stratified nilpotent Lie group—and its generic unitary irreducible representations, which naturally act on L2(Hn). As in the Euclidean case, our notion of harmonic oscillator on Hn extends to a whole class of socalled anharmonic oscillators, which involve leftinvariant derivatives and polynomial potentials of order greater or equal 2. These operators, which enjoy similar properties as the harmonic oscillator, are in onetoone correspondence with positive Rockland operators on the Dynin–Folland group. The latter part of this article is concerned with spectral multipliers. We obtain useful LpLqestimates for a large class of spectral multipliers of the subLaplacian [math] and, in fact, of generic Rockland operators on graded groups. As a byproduct, we obtain explicit hypoelliptic heat semigroup estimates and recover the continuous Sobolev embeddings on graded groups, provided 1 < p ≤ 2 ≤ q < ∞.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>In this article, we present a notion of the harmonic oscillator on the Heisenberg group Hn, which, under a few reasonable assumptions, forms the natural analog of a harmonic oscillator on [math]: a negative sum of squares of operators on Hn, which is essentially selfadjoint on L2(Hn) with purely discrete spectrum and whose eigenvectors are Schwartz functions forming an orthonormal basis of L2(Hn). The differential operator in question is determined by the Dynin–Folland group—a stratified nilpotent Lie group—and its generic unitary irreducible representations, which naturally act on L2(Hn). As in the Euclidean case, our notion of harmonic oscillator on Hn extends to a whole class of socalled anharmonic oscillators, which involve leftinvariant derivatives and polynomial potentials of order greater or equal 2. These operators, which enjoy similar properties as the harmonic oscillator, are in onetoone correspondence with positive Rockland operators on the Dynin–Folland group. The latter part of this article is concerned with spectral multipliers. We obtain useful LpLqestimates for a large class of spectral multipliers of the subLaplacian [math] and, in fact, of generic Rockland operators on graded groups. As a byproduct, we obtain explicit hypoelliptic heat semigroup estimates and recover the continuous Sobolev embeddings on graded groups, provided 1 < p ≤ 2 ≤ q < ∞.
Harmonic and anharmonic oscillators on the Heisenberg group
10.1063/5.0106068
Journal of Mathematical Physics
20221102T02:30:56Z
© 2022 Author(s).
David Rottensteiner
Michael Ruzhansky

The asymptotic stability of solitons for the focusing mKdV equation with weak weighted Sobolev initial data
https://aip.scitation.org/doi/10.1063/5.0085253?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>In this paper, we consider the Cauchy problem for the focusing modified Korteweg–de Vries (mKdV) equation in line with the weak weighted Sobolev initial data and without the smallnorm assumption. We use the inverse scattering transform, the autoB[math]cklund transformation, and the [math]steepest descent method to obtain the asymptotic stability of the solitons of the mKdV equation.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>In this paper, we consider the Cauchy problem for the focusing modified Korteweg–de Vries (mKdV) equation in line with the weak weighted Sobolev initial data and without the smallnorm assumption. We use the inverse scattering transform, the autoB[math]cklund transformation, and the [math]steepest descent method to obtain the asymptotic stability of the solitons of the mKdV equation.
The asymptotic stability of solitons for the focusing mKdV equation with weak weighted Sobolev initial data
10.1063/5.0085253
Journal of Mathematical Physics
20221104T10:32:18Z
© 2022 Author(s).
Anran Liu
Engui Fan

Navier–Stokes–Cahn–Hilliard system of equations
https://aip.scitation.org/doi/10.1063/5.0097137?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>A growing interest in considering the “hybrid systems” of equations describing more complicated physical phenomena was observed throughout the last 10 years. We mean here, in particular, the socalled Navier–Stokes–Cahn–Hilliard equation, the Navier–Stokes–Poison equations, or the Cahn–Hilliard–Hele–Shaw equation. There are specific difficulties connected with considering such systems. Using the semigroup approach, we discuss here the existenceuniqueness of solutions to the Navier–Stokes–Cahn–Hilliard system, explaining, in particular, the limitation of maximal regularity of the local solutions imposed by the chosen boundary conditions.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>A growing interest in considering the “hybrid systems” of equations describing more complicated physical phenomena was observed throughout the last 10 years. We mean here, in particular, the socalled Navier–Stokes–Cahn–Hilliard equation, the Navier–Stokes–Poison equations, or the Cahn–Hilliard–Hele–Shaw equation. There are specific difficulties connected with considering such systems. Using the semigroup approach, we discuss here the existenceuniqueness of solutions to the Navier–Stokes–Cahn–Hilliard system, explaining, in particular, the limitation of maximal regularity of the local solutions imposed by the chosen boundary conditions.
Navier–Stokes–Cahn–Hilliard system of equations
10.1063/5.0097137
Journal of Mathematical Physics
20221104T10:32:28Z
© 2022 Author(s).
Tomasz Dlotko

Dynamical stability of random delayed FitzHugh–Nagumo lattice systems driven by nonlinear Wong–Zakai noise
https://aip.scitation.org/doi/10.1063/5.0125383?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>In this paper, two problems related to FitzHugh–Nagumo lattice systems are analyzed. The first one is concerned with the asymptotic behavior of random delayed FitzHugh–Nagumo lattice systems driven by nonlinear Wong–Zakai noise. We obtain a new result ensuring that such a system approximates the corresponding deterministic system when the correlation time of Wong–Zakai noise goes to infinity rather than to zero. We first prove the existence of tempered random attractors for the random delayed lattice systems with a nonlinear drift function and a nonlinear diffusion term. The pullback asymptotic compactness of solutions is proved thanks to the Ascoli–Arzelà theorem and uniform tailestimates. We then show the upper semicontinuity of attractors as the correlation time tends to infinity. As for the second problem, we consider the corresponding deterministic version of the previous model and study the convergence of attractors when the delay approaches zero. That is, the upper semicontinuity of attractors for the delayed system to the nondelayed one is proved.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>In this paper, two problems related to FitzHugh–Nagumo lattice systems are analyzed. The first one is concerned with the asymptotic behavior of random delayed FitzHugh–Nagumo lattice systems driven by nonlinear Wong–Zakai noise. We obtain a new result ensuring that such a system approximates the corresponding deterministic system when the correlation time of Wong–Zakai noise goes to infinity rather than to zero. We first prove the existence of tempered random attractors for the random delayed lattice systems with a nonlinear drift function and a nonlinear diffusion term. The pullback asymptotic compactness of solutions is proved thanks to the Ascoli–Arzelà theorem and uniform tailestimates. We then show the upper semicontinuity of attractors as the correlation time tends to infinity. As for the second problem, we consider the corresponding deterministic version of the previous model and study the convergence of attractors when the delay approaches zero. That is, the upper semicontinuity of attractors for the delayed system to the nondelayed one is proved.
Dynamical stability of random delayed FitzHugh–Nagumo lattice systems driven by nonlinear Wong–Zakai noise
10.1063/5.0125383
Journal of Mathematical Physics
20221108T11:07:48Z
© 2022 Author(s).
Shuang Yang
Yangrong Li
Tomás Caraballo

Existence of multibump solutions for a nonlinear Kirchhofftype system
https://aip.scitation.org/doi/10.1063/5.0122696?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>In this paper, we use the Lyapunov–Schmidt reduction method to obtain the existence of multibump solutions for a nonlinear Kirchhofftype system with the parameter ɛ. As a result, when ɛ → 0, the system has more and more multibump positive solutions.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>In this paper, we use the Lyapunov–Schmidt reduction method to obtain the existence of multibump solutions for a nonlinear Kirchhofftype system with the parameter ɛ. As a result, when ɛ → 0, the system has more and more multibump positive solutions.
Existence of multibump solutions for a nonlinear Kirchhofftype system
10.1063/5.0122696
Journal of Mathematical Physics
20221109T12:40:07Z
© 2022 Author(s).
Weiming Liu

Is the continuum SSH model topological?
https://aip.scitation.org/doi/10.1063/5.0064037?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>The discrete Hamiltonian of Su, Schrieffer, and Heeger (SSH) [Phys. Rev. Lett. 42, 1698–1701 (1979)] is a wellknown onedimensional translationinvariant model in condensed matter physics. The model consists of two atoms per unit cell and describes incell and outofcell electronhopping between two sublattices. It is among the simplest models exhibiting a nontrivial topological phase; to the SSH Hamiltonian, one can associate a winding number, the Zak phase, which depends on the ratio of hopping coefficients and takes on values 0 and 1 labeling the two distinct phases. We display two homotopically equivalent continuum Hamiltonians whose tight binding limits are SSH models with different topological indices. The topological character of the SSH model is, therefore, an emergent rather than fundamental property, associated with emergent chiral or sublattice symmetry in the tightbinding limit. In order to establish that the tightbinding limit of these continuum Hamiltonians is the SSH model, we extend our recent results on the tightbinding approximation [J. Shapiro and M. I. Weinstein, Adv. Math. 403, 108343 (2022)] to lattices, which depend on the tightbinding asymptotic parameter λ.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>The discrete Hamiltonian of Su, Schrieffer, and Heeger (SSH) [Phys. Rev. Lett. 42, 1698–1701 (1979)] is a wellknown onedimensional translationinvariant model in condensed matter physics. The model consists of two atoms per unit cell and describes incell and outofcell electronhopping between two sublattices. It is among the simplest models exhibiting a nontrivial topological phase; to the SSH Hamiltonian, one can associate a winding number, the Zak phase, which depends on the ratio of hopping coefficients and takes on values 0 and 1 labeling the two distinct phases. We display two homotopically equivalent continuum Hamiltonians whose tight binding limits are SSH models with different topological indices. The topological character of the SSH model is, therefore, an emergent rather than fundamental property, associated with emergent chiral or sublattice symmetry in the tightbinding limit. In order to establish that the tightbinding limit of these continuum Hamiltonians is the SSH model, we extend our recent results on the tightbinding approximation [J. Shapiro and M. I. Weinstein, Adv. Math. 403, 108343 (2022)] to lattices, which depend on the tightbinding asymptotic parameter λ.
Is the continuum SSH model topological?
10.1063/5.0064037
Journal of Mathematical Physics
20221114T12:25:30Z
© 2022 Author(s).
Jacob Shapiro
Michael I. Weinstein

Potentials vs geometry, revisited
https://aip.scitation.org/doi/10.1063/5.0101057?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>We revisit an old subject to discuss relationships between the dynamics for particles subjected to potentials and the dynamics for particles moving freely on background geometries, in the context of nonrelativistic quantum mechanics. In particular, we illustrate how selected geometries can be used to regularize singular potentials. We also compute scattering amplitudes for quanta incident on a static nonrelativistic wormhole.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>We revisit an old subject to discuss relationships between the dynamics for particles subjected to potentials and the dynamics for particles moving freely on background geometries, in the context of nonrelativistic quantum mechanics. In particular, we illustrate how selected geometries can be used to regularize singular potentials. We also compute scattering amplitudes for quanta incident on a static nonrelativistic wormhole.
Potentials vs geometry, revisited
10.1063/5.0101057
Journal of Mathematical Physics
20221122T11:05:52Z
© 2022 Author(s).
T. Curtright
S. Subedi

The axiomatic and the operational approaches to resource theories of magic do not coincide
https://aip.scitation.org/doi/10.1063/5.0085774?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>Stabilizer operations (SO) occupy a prominent role in faulttolerant quantum computing. They are defined operationally by the use of Clifford gates, Pauli measurements, and classical control. These operations can be efficiently simulated on a classical computer, a result which is known as the Gottesman–Knill theorem. However, an additional supply of magic states is enough to promote them to a universal, faulttolerant model for quantum computing. To quantify the needed resources in terms of magic states, a resource theory of magic has been developed. SO are considered free within this theory; however, they are not the most general class of free operations. From an axiomatic point of view, these are the completely stabilizerpreserving (CSP) channels, defined as those that preserve the convex hull of stabilizer states. It has been an open problem to decide whether these two definitions lead to the same class of operations. In this work, we answer this question in the negative, by constructing an explicit counterexample. This indicates that recently proposed stabilizerbased simulation techniques of CSP maps are strictly more powerful than Gottesman–Knilllike methods. The result is analogous to a wellknown fact in entanglement theory, namely, that there is a gap between the operationally defined class of local operations and classical communication and the axiomatically defined class of separable channels.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>Stabilizer operations (SO) occupy a prominent role in faulttolerant quantum computing. They are defined operationally by the use of Clifford gates, Pauli measurements, and classical control. These operations can be efficiently simulated on a classical computer, a result which is known as the Gottesman–Knill theorem. However, an additional supply of magic states is enough to promote them to a universal, faulttolerant model for quantum computing. To quantify the needed resources in terms of magic states, a resource theory of magic has been developed. SO are considered free within this theory; however, they are not the most general class of free operations. From an axiomatic point of view, these are the completely stabilizerpreserving (CSP) channels, defined as those that preserve the convex hull of stabilizer states. It has been an open problem to decide whether these two definitions lead to the same class of operations. In this work, we answer this question in the negative, by constructing an explicit counterexample. This indicates that recently proposed stabilizerbased simulation techniques of CSP maps are strictly more powerful than Gottesman–Knilllike methods. The result is analogous to a wellknown fact in entanglement theory, namely, that there is a gap between the operationally defined class of local operations and classical communication and the axiomatically defined class of separable channels.
The axiomatic and the operational approaches to resource theories of magic do not coincide
10.1063/5.0085774
Journal of Mathematical Physics
20221101T02:07:29Z
© 2022 Author(s).
Arne Heimendahl
Markus Heinrich
David Gross

Which bath Hamiltonians matter for thermal operations?
https://aip.scitation.org/doi/10.1063/5.0117534?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>In this article, we explore the set of thermal operations from a mathematical and topological point of view. First, we introduce the concept of Hamiltonians with a resonant spectrum with respect to some reference Hamiltonian, followed by proving that when defining thermal operations, it suffices to only consider bath Hamiltonians, which satisfy this resonance property. Next, we investigate the continuity of the set of thermal operations in certain parameters, such as energies of the system and temperature of the bath. We will see that the set of thermal operations changes discontinuously with respect to the Hausdorff metric at any Hamiltonian, which has the socalled degenerate Bohr spectrum, regardless of the temperature. Finally, we find a semigroup representation of (enhanced) thermal operations in two dimensions by characterizing any such operation via three real parameters, thus allowing for a visualization of this set. Using this, in the qubit case, we show commutativity of (enhanced) thermal operations and convexity of thermal operations without the closure. The latter is done by specifying the elements of this set exactly.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>In this article, we explore the set of thermal operations from a mathematical and topological point of view. First, we introduce the concept of Hamiltonians with a resonant spectrum with respect to some reference Hamiltonian, followed by proving that when defining thermal operations, it suffices to only consider bath Hamiltonians, which satisfy this resonance property. Next, we investigate the continuity of the set of thermal operations in certain parameters, such as energies of the system and temperature of the bath. We will see that the set of thermal operations changes discontinuously with respect to the Hausdorff metric at any Hamiltonian, which has the socalled degenerate Bohr spectrum, regardless of the temperature. Finally, we find a semigroup representation of (enhanced) thermal operations in two dimensions by characterizing any such operation via three real parameters, thus allowing for a visualization of this set. Using this, in the qubit case, we show commutativity of (enhanced) thermal operations and convexity of thermal operations without the closure. The latter is done by specifying the elements of this set exactly.
Which bath Hamiltonians matter for thermal operations?
10.1063/5.0117534
Journal of Mathematical Physics
20221104T10:32:20Z
© 2022 Author(s).
Frederik vom Ende

Optimal form of the Kretschmann–Schlingemann–Werner theorem for energyconstrained quantum channels and operations
https://aip.scitation.org/doi/10.1063/5.0102141?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>It is proved that the energyconstrained Bures distance between arbitrary infinitedimensional quantum channels is equal to the operator Enorm distance from any given Stinespring isometry of one channel to the set of all Stinespring isometries of another channel with the same environment. The same result is shown to be valid for arbitrary quantum operations.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>It is proved that the energyconstrained Bures distance between arbitrary infinitedimensional quantum channels is equal to the operator Enorm distance from any given Stinespring isometry of one channel to the set of all Stinespring isometries of another channel with the same environment. The same result is shown to be valid for arbitrary quantum operations.
Optimal form of the Kretschmann–Schlingemann–Werner theorem for energyconstrained quantum channels and operations
10.1063/5.0102141
Journal of Mathematical Physics
20221108T11:07:47Z
© 2022 Author(s).
M. E. Shirokov

The quantumtoclassical graph homomorphism game
https://aip.scitation.org/doi/10.1063/5.0072288?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>Motivated by nonlocal games and quantum coloring problems, we introduce a graph homomorphism game between quantum graphs and classical graphs. This game is naturally cast as a “quantum–classical game,” that is, a nonlocal game of two players involving quantum questions and classical answers. This game generalizes the graph homomorphism game between classical graphs. We show that winning strategies in the various quantum models for the game is an analog of the notion of noncommutative graph homomorphisms due to Stahlke [IEEE Trans. Inf. Theory 62(1), 554–577 (2016)]. Moreover, we present a game algebra in this context that generalizes the game algebra for graph homomorphisms given by Helton et al. [New York J. Math. 25, 328–361 (2019)]. We also demonstrate explicit quantum colorings of all quantum complete graphs, yielding the surprising fact that the algebra of the four coloring game for a quantum graph is always nontrivial, extending a result of Helton et al. [New York J. Math. 25, 328–361 (2019)].
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>Motivated by nonlocal games and quantum coloring problems, we introduce a graph homomorphism game between quantum graphs and classical graphs. This game is naturally cast as a “quantum–classical game,” that is, a nonlocal game of two players involving quantum questions and classical answers. This game generalizes the graph homomorphism game between classical graphs. We show that winning strategies in the various quantum models for the game is an analog of the notion of noncommutative graph homomorphisms due to Stahlke [IEEE Trans. Inf. Theory 62(1), 554–577 (2016)]. Moreover, we present a game algebra in this context that generalizes the game algebra for graph homomorphisms given by Helton et al. [New York J. Math. 25, 328–361 (2019)]. We also demonstrate explicit quantum colorings of all quantum complete graphs, yielding the surprising fact that the algebra of the four coloring game for a quantum graph is always nontrivial, extending a result of Helton et al. [New York J. Math. 25, 328–361 (2019)].
The quantumtoclassical graph homomorphism game
10.1063/5.0072288
Journal of Mathematical Physics
20221108T11:07:43Z
© 2022 Author(s).
Michael Brannan
Priyanga Ganesan
Samuel J. Harris

SICPOVMs from Stark units: Prime dimensions n2 + 3
https://aip.scitation.org/doi/10.1063/5.0083520?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>We propose a recipe for constructing a fiducial vector for a symmetric informationally complete positive operator valued measure (SICPOVM) in a complex Hilbert space of dimension of the form d = n2 + 3, focusing on prime dimensions d = p. Such structures are shown to exist in 13 prime dimensions of this kind, the highest being p = 19 603. The real quadratic base field K (in the standard SICPOVM terminology) attached to such dimensions has fundamental units uK of norm −1. Let [math] denote the ring of integers of K; then, [math] splits into two ideals: [math] and [math]. The initial entry of the fiducial is the square ξ2 of a geometric scaling factor ξ, which lies in one of the fields [math]. Strikingly, each of the other p − 1 entries of the fiducial vector is a product of ξ and the square root of a Stark unit. These Stark units are obtained via the Stark conjectures from the value at s = 0 of the first derivatives of partial Lfunctions attached to the characters of the ray class group of [math] with modulus [math], where ∞1 is one of the real places of K.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>We propose a recipe for constructing a fiducial vector for a symmetric informationally complete positive operator valued measure (SICPOVM) in a complex Hilbert space of dimension of the form d = n2 + 3, focusing on prime dimensions d = p. Such structures are shown to exist in 13 prime dimensions of this kind, the highest being p = 19 603. The real quadratic base field K (in the standard SICPOVM terminology) attached to such dimensions has fundamental units uK of norm −1. Let [math] denote the ring of integers of K; then, [math] splits into two ideals: [math] and [math]. The initial entry of the fiducial is the square ξ2 of a geometric scaling factor ξ, which lies in one of the fields [math]. Strikingly, each of the other p − 1 entries of the fiducial vector is a product of ξ and the square root of a Stark unit. These Stark units are obtained via the Stark conjectures from the value at s = 0 of the first derivatives of partial Lfunctions attached to the characters of the ray class group of [math] with modulus [math], where ∞1 is one of the real places of K.
SICPOVMs from Stark units: Prime dimensions n2 + 3
10.1063/5.0083520
Journal of Mathematical Physics
20221108T11:07:46Z
© 2022 Author(s).
Marcus Appleby
Ingemar Bengtsson
Markus Grassl
Michael Harrison
Gary McConnell

Stability and convergence of dynamical decoupling with finite amplitude controls
https://aip.scitation.org/doi/10.1063/5.0101259?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>Dynamical decoupling is a key method to mitigate errors in a quantum mechanical system, and we studied it in a series of papers dealing, in particular, with the problems arising from unbounded Hamiltonians. The standard bangbang model of dynamical decoupling, which we also used in those papers, requires decoupling operations with infinite amplitude, which is, strictly speaking, unrealistic from a physical point of view. In this paper, we look at decoupling operations of finite amplitude, discuss under what assumptions dynamical decoupling works with such finite amplitude operations, and show how the bangbang description arises as a limit, hence justifying it as a reasonable approximation.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>Dynamical decoupling is a key method to mitigate errors in a quantum mechanical system, and we studied it in a series of papers dealing, in particular, with the problems arising from unbounded Hamiltonians. The standard bangbang model of dynamical decoupling, which we also used in those papers, requires decoupling operations with infinite amplitude, which is, strictly speaking, unrealistic from a physical point of view. In this paper, we look at decoupling operations of finite amplitude, discuss under what assumptions dynamical decoupling works with such finite amplitude operations, and show how the bangbang description arises as a limit, hence justifying it as a reasonable approximation.
Stability and convergence of dynamical decoupling with finite amplitude controls
10.1063/5.0101259
Journal of Mathematical Physics
20221111T01:58:24Z
© 2022 Author(s).
Daniel Burgarth
Paolo Facchi
Robin Hillier

Internal quantum reference frames for finite Abelian groups
https://aip.scitation.org/doi/10.1063/5.0088485?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>Employing internal quantum systems as reference frames is a crucial concept in quantum gravity, gauge theories, and quantum foundations whenever external relata are unavailable. In this work, we give a comprehensive and selfcontained treatment of such quantum reference frames (QRFs) for the case when the underlying configuration space is a finite Abelian group, significantly extending our previous work [M. Krumm, P. A. Höhn, and M. P. Müller, Quantum 5, 530 (2021)]. The simplicity of this setup admits a fully rigorous quantum information–theoretic analysis, while maintaining sufficient structure for exploring many of the conceptual and structural questions also pertinent to more complicated setups. We exploit this to derive several important structures of constraint quantization with quantum information–theoretic methods and to reveal the relation between different approaches to QRF covariance. In particular, we characterize the “physical Hilbert space”—the arena of the “perspectiveneutral” approach—as the maximal subspace that admits frameindependent descriptions of purifications of states. We then demonstrate the kinematical equivalence and, surprising, dynamical inequivalence of the “perspectiveneutral” and the “alignability” approach to QRFs. While the former admits unitaries generating transitions between arbitrary subsystem relations, the latter, remarkably, admits no such dynamics when requiring symmetrypreservation. We illustrate these findings by example of interacting discrete particles, including how dynamics can be described “relative to one of the subystems.”
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>Employing internal quantum systems as reference frames is a crucial concept in quantum gravity, gauge theories, and quantum foundations whenever external relata are unavailable. In this work, we give a comprehensive and selfcontained treatment of such quantum reference frames (QRFs) for the case when the underlying configuration space is a finite Abelian group, significantly extending our previous work [M. Krumm, P. A. Höhn, and M. P. Müller, Quantum 5, 530 (2021)]. The simplicity of this setup admits a fully rigorous quantum information–theoretic analysis, while maintaining sufficient structure for exploring many of the conceptual and structural questions also pertinent to more complicated setups. We exploit this to derive several important structures of constraint quantization with quantum information–theoretic methods and to reveal the relation between different approaches to QRF covariance. In particular, we characterize the “physical Hilbert space”—the arena of the “perspectiveneutral” approach—as the maximal subspace that admits frameindependent descriptions of purifications of states. We then demonstrate the kinematical equivalence and, surprising, dynamical inequivalence of the “perspectiveneutral” and the “alignability” approach to QRFs. While the former admits unitaries generating transitions between arbitrary subsystem relations, the latter, remarkably, admits no such dynamics when requiring symmetrypreservation. We illustrate these findings by example of interacting discrete particles, including how dynamics can be described “relative to one of the subystems.”
Internal quantum reference frames for finite Abelian groups
10.1063/5.0088485
Journal of Mathematical Physics
20221115T10:21:14Z
© 2022 Author(s).
Philipp A. Höhn
Marius Krumm
Markus P. Müller

On the discrete Dirac spectrum of a point electron in the zerogravity Kerr–Newman spacetime
https://aip.scitation.org/doi/10.1063/5.0084471?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>The discrete spectrum of the Dirac operator for a point electron in the maximal analytically extended Kerr–Newman spacetime is determined in the zeroG limit (zGKN), under some restrictions on the electrical coupling constant and on the radius of the ringsingularity of the zGKN spacetime. The spectrum is characterized by a triplet of integers, associated with winding numbers of orbits of dynamical systems on cylinders. A dictionary is established that relates the spectrum with the known hydrogenic Dirac spectrum. Numerical illustrations are presented. Open problems are listed.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>The discrete spectrum of the Dirac operator for a point electron in the maximal analytically extended Kerr–Newman spacetime is determined in the zeroG limit (zGKN), under some restrictions on the electrical coupling constant and on the radius of the ringsingularity of the zGKN spacetime. The spectrum is characterized by a triplet of integers, associated with winding numbers of orbits of dynamical systems on cylinders. A dictionary is established that relates the spectrum with the known hydrogenic Dirac spectrum. Numerical illustrations are presented. Open problems are listed.
On the discrete Dirac spectrum of a point electron in the zerogravity Kerr–Newman spacetime
10.1063/5.0084471
Journal of Mathematical Physics
20221116T10:32:13Z
© 2022 Author(s).
Michael K.H. Kiessling
Eric Ling
A. Shadi TahvildarZadeh

Kundt threedimensional left invariant spacetimes
https://aip.scitation.org/doi/10.1063/5.0091202?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>Kundt spacetimes are of great importance to general relativity. We show that a Kundt spacetime is a Lorentz manifold with a nonsingular isotropic geodesic vector field having its orthogonal distribution integrable and determining a totally geodesic foliation. We give the local structure of Kundt spacetimes and some properties of left invariant Kundt structures on Lie groups. Finally, we classify all left invariant Kundt structures on threedimensional simply connected unimodular Lie groups.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>Kundt spacetimes are of great importance to general relativity. We show that a Kundt spacetime is a Lorentz manifold with a nonsingular isotropic geodesic vector field having its orthogonal distribution integrable and determining a totally geodesic foliation. We give the local structure of Kundt spacetimes and some properties of left invariant Kundt structures on Lie groups. Finally, we classify all left invariant Kundt structures on threedimensional simply connected unimodular Lie groups.
Kundt threedimensional left invariant spacetimes
10.1063/5.0091202
Journal of Mathematical Physics
20221101T02:07:32Z
© 2022 Author(s).
Mohamed Boucetta
Aissa Meliani
Abdelghani Zeghib

Twodimensional Lifshitzlike AdS black holes in F(R) gravity
https://aip.scitation.org/doi/10.1063/5.0104272?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>Twodimensional (2D) Lifshitzlike black holes in special F(R) gravity cases are extracted. We indicate an essential singularity at r = 0, covered by an event horizon. Then, conserved and thermodynamic quantities, such as temperature, mass, entropy, and the heat capacity of 2D Lifshitzlike black holes in F(R) gravity, are evaluated. Our analysis shows that 2D Lifshitzlike black hole solutions can be physical solutions, provided that the cosmological constant is negative (Λ < 0). Indeed, there is a phase transition between stable and unstable cases by increasing the radius of AdS black holes. In other words, the 2D Lifshitzlike AdS black holes with large radii are physical and enjoy thermal stability. The obtained 2D Lifshitzlike AdSblack holes in F(R) gravity turn into the wellknown 2D Schwarzschild AdSblack holes when the Lifshitzlike parameter is zero (s = 0). Moreover, correspondence between these black hole solutions and the 2D rotating black hole solutions is found by adjusting the Lifshitzlike parameter.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>Twodimensional (2D) Lifshitzlike black holes in special F(R) gravity cases are extracted. We indicate an essential singularity at r = 0, covered by an event horizon. Then, conserved and thermodynamic quantities, such as temperature, mass, entropy, and the heat capacity of 2D Lifshitzlike black holes in F(R) gravity, are evaluated. Our analysis shows that 2D Lifshitzlike black hole solutions can be physical solutions, provided that the cosmological constant is negative (Λ < 0). Indeed, there is a phase transition between stable and unstable cases by increasing the radius of AdS black holes. In other words, the 2D Lifshitzlike AdS black holes with large radii are physical and enjoy thermal stability. The obtained 2D Lifshitzlike AdSblack holes in F(R) gravity turn into the wellknown 2D Schwarzschild AdSblack holes when the Lifshitzlike parameter is zero (s = 0). Moreover, correspondence between these black hole solutions and the 2D rotating black hole solutions is found by adjusting the Lifshitzlike parameter.
Twodimensional Lifshitzlike AdS black holes in F(R) gravity
10.1063/5.0104272
Journal of Mathematical Physics
20221102T02:30:51Z
© 2022 Author(s).
B. Eslam Panah

Gravitational constant model and correction
https://aip.scitation.org/doi/10.1063/5.0095583?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>We construct a model for considering the quantum correction of the gravitational constant. In the model, the gravitational constant originates from a coupling between the gravitational field and a scalar field. If the scalar field, as it should be in the real physical world, is a quantum field, the gravitational constant will have a quantum correction. The quantum correction, generally speaking, varies with spacetime coordinates. Therefore, the gravitational constant is no longer a constant. In different spacetime, the quantum correction is different, for the coupling in different spacetime is different. As a result, the gravitational constant in different spacetime is different, though the difference is only at the quantum level. We calculate the quantum correction of the gravitational constant in the Schwarzschild spacetime, the H3 (Euclidean AdS3) spacetime, the H3/Z spacetime, the universe model, the de Sitter spacetime, and the Rindler spacetime.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>We construct a model for considering the quantum correction of the gravitational constant. In the model, the gravitational constant originates from a coupling between the gravitational field and a scalar field. If the scalar field, as it should be in the real physical world, is a quantum field, the gravitational constant will have a quantum correction. The quantum correction, generally speaking, varies with spacetime coordinates. Therefore, the gravitational constant is no longer a constant. In different spacetime, the quantum correction is different, for the coupling in different spacetime is different. As a result, the gravitational constant in different spacetime is different, though the difference is only at the quantum level. We calculate the quantum correction of the gravitational constant in the Schwarzschild spacetime, the H3 (Euclidean AdS3) spacetime, the H3/Z spacetime, the universe model, the de Sitter spacetime, and the Rindler spacetime.
Gravitational constant model and correction
10.1063/5.0095583
Journal of Mathematical Physics
20221107T11:37:04Z
© 2022 Author(s).
YuJie Chen
ShiLin Li
YuZhu Chen
WenDu Li
WuSheng Dai

Linear quaternionvalued difference equations: Representation of solutions, controllability, and observability
https://aip.scitation.org/doi/10.1063/5.0100608?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>In this paper, we present the fundamental theory of linear quaternionvalued difference equations. Firstly, we derive general solutions for linear homogeneous equations and give the algorithm for calculating the fundamental matrix in the case of the diagonalizable form and Jordan form. Secondly, we apply the variation of the constant formula and Z transformation to study general solutions of linear nonhomogeneous equations. We obtain the representation of solutions in the case of quaternion and complex numbers. Thirdly, we adopt the ideas from the Gram matrix and the rank of the criteria to establish sufficient and necessary conditions to guarantee that linear quaternionvalued difference equations are controllable and observable in the sense of quaternionvalued and complex numbers, respectively. In addition, a direct method to solve the control function and duality is also given. Finally, we illustrate our theoretical results with some examples.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>In this paper, we present the fundamental theory of linear quaternionvalued difference equations. Firstly, we derive general solutions for linear homogeneous equations and give the algorithm for calculating the fundamental matrix in the case of the diagonalizable form and Jordan form. Secondly, we apply the variation of the constant formula and Z transformation to study general solutions of linear nonhomogeneous equations. We obtain the representation of solutions in the case of quaternion and complex numbers. Thirdly, we adopt the ideas from the Gram matrix and the rank of the criteria to establish sufficient and necessary conditions to guarantee that linear quaternionvalued difference equations are controllable and observable in the sense of quaternionvalued and complex numbers, respectively. In addition, a direct method to solve the control function and duality is also given. Finally, we illustrate our theoretical results with some examples.
Linear quaternionvalued difference equations: Representation of solutions, controllability, and observability
10.1063/5.0100608
Journal of Mathematical Physics
20221101T02:07:30Z
© 2022 Author(s).
Dan Chen
Michal Fečkan
JinRong Wang

Instability of smallamplitude periodic waves from foldHopf bifurcation
https://aip.scitation.org/doi/10.1063/5.0106152?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>We study the existence and stability of smallamplitude periodic waves emerging from foldHopf equilibria in a system of one reaction–diffusion equation coupled with one ordinary differential equation. This coupled system includes the FitzHugh–Nagumo system, caricature calcium models, and other models in realworld applications. Based on the recent results on the averaging theory, we solve periodic solutions in related threedimensional systems and then prove the existence of periodic waves arising from foldHopf bifurcations. Numerical computation by Tsai et al. [SIAM J. Appl. Dyn. Syst. 11, 1149–1199 (2012)] once suggested that the periodic waves from foldHopf bifurcations in a caricature calcium model are spectrally unstable, yet without a proof. After analyzing the linearization about periodic waves by the relatively bounded perturbation, we prove the instability of smallamplitude periodic waves through a perturbation of the unstable spectra for the linearization about the foldHopf equilibria. As an application, we prove the existence and stability of smallamplitude periodic waves from foldHopf bifurcations in the FitzHugh–Nagumo system with an applied current.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>We study the existence and stability of smallamplitude periodic waves emerging from foldHopf equilibria in a system of one reaction–diffusion equation coupled with one ordinary differential equation. This coupled system includes the FitzHugh–Nagumo system, caricature calcium models, and other models in realworld applications. Based on the recent results on the averaging theory, we solve periodic solutions in related threedimensional systems and then prove the existence of periodic waves arising from foldHopf bifurcations. Numerical computation by Tsai et al. [SIAM J. Appl. Dyn. Syst. 11, 1149–1199 (2012)] once suggested that the periodic waves from foldHopf bifurcations in a caricature calcium model are spectrally unstable, yet without a proof. After analyzing the linearization about periodic waves by the relatively bounded perturbation, we prove the instability of smallamplitude periodic waves through a perturbation of the unstable spectra for the linearization about the foldHopf equilibria. As an application, we prove the existence and stability of smallamplitude periodic waves from foldHopf bifurcations in the FitzHugh–Nagumo system with an applied current.
Instability of smallamplitude periodic waves from foldHopf bifurcation
10.1063/5.0106152
Journal of Mathematical Physics
20221102T02:30:50Z
© 2022 Author(s).
Shuang Chen
Jinqiao Duan

Charged particle motion in spherically symmetric distributions of magnetic monopoles
https://aip.scitation.org/doi/10.1063/5.0105653?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>The classical equations of motion of a charged particle in a spherically symmetric distribution of magnetic monopoles can be transformed into a system of linear equations, thereby providing a type of integrability. In the case of a single monopole, the solution was given long ago by Poincaré. In the case of a uniform distribution of monopoles, the solution can be expressed in terms of parabolic cylinder functions (essentially the eigenfunctions of an inverted harmonic oscillator). This solution is relevant to recent studies of nonassociative star products, symplectic lifts of twisted Poisson structures, and fluids and plasmas of electric and magnetic charges.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>The classical equations of motion of a charged particle in a spherically symmetric distribution of magnetic monopoles can be transformed into a system of linear equations, thereby providing a type of integrability. In the case of a single monopole, the solution was given long ago by Poincaré. In the case of a uniform distribution of monopoles, the solution can be expressed in terms of parabolic cylinder functions (essentially the eigenfunctions of an inverted harmonic oscillator). This solution is relevant to recent studies of nonassociative star products, symplectic lifts of twisted Poisson structures, and fluids and plasmas of electric and magnetic charges.
Charged particle motion in spherically symmetric distributions of magnetic monopoles
10.1063/5.0105653
Journal of Mathematical Physics
20221104T11:08:50Z
© 2022 Author(s).
Robert Littlejohn
Philip Morrison
Jeffrey Heninger

Gaussian (p, q)Jacobsthal and Gaussian (p, q)Jacobsthal Lucas numbers and their some interesting properties
https://aip.scitation.org/doi/10.1063/5.0096935?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>In the present article, we introduce two new notions, which are called Gaussian (p, q)Jacobsthal numbers sequence [math] and Gaussian (p, q)Jacobsthal Lucas numbers sequence [math], and we present and prove our exciting properties and results, which relate these sequences. We first give recurrence relations, Binet’s formulas, explicit formulas, and negative extensions of them. We then obtain some important identities for Gaussian (p, q)Jacobsthal and Gaussian (p, q)Jacobsthal Lucas numbers and some connection formulas between these Gaussian numbers. After that, we give some summation formulas and the symmetric functions of Gaussian (p, q)Jacobsthal and Gaussian (p, q)Jacobsthal Lucas numbers. In addition, by using the symmetric functions, we derive a new class of generating functions for Gaussian (p, q)Jacobsthal and Gaussian (p, q)Jacobsthal Lucas numbers.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>In the present article, we introduce two new notions, which are called Gaussian (p, q)Jacobsthal numbers sequence [math] and Gaussian (p, q)Jacobsthal Lucas numbers sequence [math], and we present and prove our exciting properties and results, which relate these sequences. We first give recurrence relations, Binet’s formulas, explicit formulas, and negative extensions of them. We then obtain some important identities for Gaussian (p, q)Jacobsthal and Gaussian (p, q)Jacobsthal Lucas numbers and some connection formulas between these Gaussian numbers. After that, we give some summation formulas and the symmetric functions of Gaussian (p, q)Jacobsthal and Gaussian (p, q)Jacobsthal Lucas numbers. In addition, by using the symmetric functions, we derive a new class of generating functions for Gaussian (p, q)Jacobsthal and Gaussian (p, q)Jacobsthal Lucas numbers.
Gaussian (p, q)Jacobsthal and Gaussian (p, q)Jacobsthal Lucas numbers and their some interesting properties
10.1063/5.0096935
Journal of Mathematical Physics
20221104T11:08:49Z
© 2022 Author(s).
N. Saba
A. Boussayoud

Degenerate response tori in Hamiltonian systems with higher zeroaverage perturbation
https://aip.scitation.org/doi/10.1063/5.0083792?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>Consider a normally degenerate Hamiltonian system with the following Hamiltonian [math] which is associated with the standard symplectic form dθ ∧ dI ∧ dx ∧ dy, where [math] and n > 1 is an integer. The existence of response tori for the degenerate Hamiltonian system has already been proved by Si and Yi [Nonlinearity 33, 6072–6098 (2020)] if [math] satisfies some nonzero conditions, see condition (H) in the work of Si and Yi [Nonlinearity 33, 6072–6098 (2020)], where [·] denotes the average value of a continuous function on [math]. However, when [math], no results were given by Si and Yi [Nonlinearity 33, 6072–6098 (2020)] for response tori of the above system. This paper attempts at carrying out this work in this direction. More precisely, with 2p < n, if P satisfies [math] for j = 1, 2, …, p and either [math] as n − p is even or [math] as n − p is odd, we obtain the following results: (1) For [math] [see [math] in (2.1)] and ϵ sufficiently small, response tori exist for each ω satisfying a Brjunotype nonresonant condition. (2) For [math] and [math] sufficiently small, there exists a Cantor set [math] with almost full Lebesgue measure such that response tori exist for each [math] if ω satisfies a Diophantine condition. In the case where [math] and n − p is even, we prove that the system admits no response tori in most regions. The present paper is regarded as a continuation of work by Si and Yi [Nonlinearity 33, 6072–6098 (2020)].
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>Consider a normally degenerate Hamiltonian system with the following Hamiltonian [math] which is associated with the standard symplectic form dθ ∧ dI ∧ dx ∧ dy, where [math] and n > 1 is an integer. The existence of response tori for the degenerate Hamiltonian system has already been proved by Si and Yi [Nonlinearity 33, 6072–6098 (2020)] if [math] satisfies some nonzero conditions, see condition (H) in the work of Si and Yi [Nonlinearity 33, 6072–6098 (2020)], where [·] denotes the average value of a continuous function on [math]. However, when [math], no results were given by Si and Yi [Nonlinearity 33, 6072–6098 (2020)] for response tori of the above system. This paper attempts at carrying out this work in this direction. More precisely, with 2p < n, if P satisfies [math] for j = 1, 2, …, p and either [math] as n − p is even or [math] as n − p is odd, we obtain the following results: (1) For [math] [see [math] in (2.1)] and ϵ sufficiently small, response tori exist for each ω satisfying a Brjunotype nonresonant condition. (2) For [math] and [math] sufficiently small, there exists a Cantor set [math] with almost full Lebesgue measure such that response tori exist for each [math] if ω satisfies a Diophantine condition. In the case where [math] and n − p is even, we prove that the system admits no response tori in most regions. The present paper is regarded as a continuation of work by Si and Yi [Nonlinearity 33, 6072–6098 (2020)].
Degenerate response tori in Hamiltonian systems with higher zeroaverage perturbation
10.1063/5.0083792
Journal of Mathematical Physics
20221107T11:37:03Z
© 2022 Author(s).
Wen Si
Xinyu Guan

Planar central configurations with five different positive masses
https://aip.scitation.org/doi/10.1063/5.0101256?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>We study planar fivebody central configurations with different positive masses. From Laura–Andoyer equations, we recover the variational formulation in terms of the potential energy and the total inertia moment, including constraints on the distances. We use coordinates to prove the permanence of the central configuration by Newton equations of motion. The masses appear in our coordinates, forming a rigid orthocentric simplex in four dimensions, with the masses at the vertices. The linear dimensions are determined by a distance. We use different angles, with one angle measuring the rotation around the center of mass. The rest of the coordinates are angles that are constants of motion for central configurations. One angle measures the ratio of the two principal moments of inertia. The rest of the coordinates determine the rotation around the center of mass of the rigid simplex of masses with respect to the plane and to the direction of projection from four to two dimensions. We transform the Laura–Andoyer equations into different expressions in 10 and 5 dimensions, including the properties of the areas formed by five point bodies in the plane. Furthermore, we presented three numerical computed configurations of planar fivebody central configurations, one convex and two concave, which give numerical confirmation of our equations with a precision better than 10−12.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>We study planar fivebody central configurations with different positive masses. From Laura–Andoyer equations, we recover the variational formulation in terms of the potential energy and the total inertia moment, including constraints on the distances. We use coordinates to prove the permanence of the central configuration by Newton equations of motion. The masses appear in our coordinates, forming a rigid orthocentric simplex in four dimensions, with the masses at the vertices. The linear dimensions are determined by a distance. We use different angles, with one angle measuring the rotation around the center of mass. The rest of the coordinates are angles that are constants of motion for central configurations. One angle measures the ratio of the two principal moments of inertia. The rest of the coordinates determine the rotation around the center of mass of the rigid simplex of masses with respect to the plane and to the direction of projection from four to two dimensions. We transform the Laura–Andoyer equations into different expressions in 10 and 5 dimensions, including the properties of the areas formed by five point bodies in the plane. Furthermore, we presented three numerical computed configurations of planar fivebody central configurations, one convex and two concave, which give numerical confirmation of our equations with a precision better than 10−12.
Planar central configurations with five different positive masses
10.1063/5.0101256
Journal of Mathematical Physics
20221102T02:30:53Z
© 2022 Author(s).
E. Piña

Sharp hierarchical upper bounds on the critical twopoint function for longrange percolation on [math]
https://aip.scitation.org/doi/10.1063/5.0088450?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>Consider longrange Bernoulli percolation on [math] in which we connect each pair of distinct points x and y by an edge with probability 1 − exp(−β‖x − y‖−d−α), where α > 0 is fixed and β ⩾ 0 is a parameter. We prove that if 0 < α < d, then the critical twopoint function satisfies [math] for every r ⩾ 1, where [math]. In other words, the critical twopoint function on [math] is always bounded above on average by the critical twopoint function on the hierarchical lattice. This upper bound is believed to be sharp for values of α strictly below the crossover value αc(d), where the values of several critical exponents for longrange percolation on [math] and the hierarchical lattice are believed to be equal.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>Consider longrange Bernoulli percolation on [math] in which we connect each pair of distinct points x and y by an edge with probability 1 − exp(−β‖x − y‖−d−α), where α > 0 is fixed and β ⩾ 0 is a parameter. We prove that if 0 < α < d, then the critical twopoint function satisfies [math] for every r ⩾ 1, where [math]. In other words, the critical twopoint function on [math] is always bounded above on average by the critical twopoint function on the hierarchical lattice. This upper bound is believed to be sharp for values of α strictly below the crossover value αc(d), where the values of several critical exponents for longrange percolation on [math] and the hierarchical lattice are believed to be equal.
Sharp hierarchical upper bounds on the critical twopoint function for longrange percolation on [math]
10.1063/5.0088450
Journal of Mathematical Physics
20221104T10:32:21Z
© 2022 Author(s).
Tom Hutchcroft

The uniform measure for quantum walk on hypercube: A quantum Bernoulli noises approach
https://aip.scitation.org/doi/10.1063/5.0070451?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>In this paper, we present a quantum Bernoulli noises approach to quantum walks on hypercubes. We first obtain an alternative description of a general hypercube, and then, based on the alternative description, we find that the operators [math] behave actually as the shift operators, where ∂k and [math] are the annihilation and creation operators acting on Bernoulli functionals, respectively. With the abovementioned operators as the shift operators on the position space, we introduce a discretetime quantum walk model on a general hypercube and obtain an explicit formula for calculating its probability distribution at any time. We also establish two limit theorems showing that the averaged probability distribution of the walk even converges to the uniform probability distribution. Finally, we show that the walk produces the uniform measure as its stationary measure on the hypercube provided its initial state satisfies some mild conditions. Some other results are also proven.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>In this paper, we present a quantum Bernoulli noises approach to quantum walks on hypercubes. We first obtain an alternative description of a general hypercube, and then, based on the alternative description, we find that the operators [math] behave actually as the shift operators, where ∂k and [math] are the annihilation and creation operators acting on Bernoulli functionals, respectively. With the abovementioned operators as the shift operators on the position space, we introduce a discretetime quantum walk model on a general hypercube and obtain an explicit formula for calculating its probability distribution at any time. We also establish two limit theorems showing that the averaged probability distribution of the walk even converges to the uniform probability distribution. Finally, we show that the walk produces the uniform measure as its stationary measure on the hypercube provided its initial state satisfies some mild conditions. Some other results are also proven.
The uniform measure for quantum walk on hypercube: A quantum Bernoulli noises approach
10.1063/5.0070451
Journal of Mathematical Physics
20221101T02:07:34Z
© 2022 Author(s).
Ce Wang

Commutators of random matrices from the unitary and orthogonal groups
https://aip.scitation.org/doi/10.1063/5.0041240?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>We investigate the statistical properties of C = uvu−1v−1, when u and v are independent random matrices, uniformly distributed with respect to the Haar measure of the groups U(N) and O(N). An exact formula is derived for the average value of power sum symmetric functions of C, and also for products of the matrix elements of C, similar to Weingarten functions. The density of eigenvalues of C is shown to become constant in the largeN limit, and the first N−1 correction is found.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>We investigate the statistical properties of C = uvu−1v−1, when u and v are independent random matrices, uniformly distributed with respect to the Haar measure of the groups U(N) and O(N). An exact formula is derived for the average value of power sum symmetric functions of C, and also for products of the matrix elements of C, similar to Weingarten functions. The density of eigenvalues of C is shown to become constant in the largeN limit, and the first N−1 correction is found.
Commutators of random matrices from the unitary and orthogonal groups
10.1063/5.0041240
Journal of Mathematical Physics
20221102T02:30:54Z
© 2022 Author(s).
Pedro H. S. Palheta
Marcelo R. Barbosa
Marcel Novaes

Multiparameter universal characters of Btype and integrable hierarchy
https://aip.scitation.org/doi/10.1063/5.0102146?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>We introduce the multiparameter universal characters of Btype (Btype universal character), which contain special factorial Schur Qfunctions, classical Schur Qfunctions, and classical Btype universal characters. Then, we prove that multiparameter Btype universal characters are solutions of the universal character hierarchy of Btype.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>We introduce the multiparameter universal characters of Btype (Btype universal character), which contain special factorial Schur Qfunctions, classical Schur Qfunctions, and classical Btype universal characters. Then, we prove that multiparameter Btype universal characters are solutions of the universal character hierarchy of Btype.
Multiparameter universal characters of Btype and integrable hierarchy
10.1063/5.0102146
Journal of Mathematical Physics
20221104T10:32:25Z
© 2022 Author(s).
Qianqian Yang
Chuanzhong Li

Longtime asymptotic behavior of the coupled dispersive AB system in low regularity spaces
https://aip.scitation.org/doi/10.1063/5.0102264?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>In this paper, we mainly investigate the longtime asymptotic behavior of the solution for coupled dispersive AB systems with weighted Sobolev initial data, which allows soliton solutions via the Dbar steepest descent method. Based on the spectral analysis of Lax pairs, the Cauchy problem of coupled dispersive AB systems is transformed into a Riemann–Hilbert problem, and the existence and uniqueness of its solution is proved by the vanishing lemma. The stationary phase points play an important role in determining the longtime asymptotic behavior of these solutions. We demonstrate that in any fixed time cone [math], the longtime asymptotic behavior of the solution for coupled dispersive AB systems can be expressed by [math] solitons on the discrete spectrum, the leading order term [math] on the continuous spectrum, and the allowable residual [math].
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>In this paper, we mainly investigate the longtime asymptotic behavior of the solution for coupled dispersive AB systems with weighted Sobolev initial data, which allows soliton solutions via the Dbar steepest descent method. Based on the spectral analysis of Lax pairs, the Cauchy problem of coupled dispersive AB systems is transformed into a Riemann–Hilbert problem, and the existence and uniqueness of its solution is proved by the vanishing lemma. The stationary phase points play an important role in determining the longtime asymptotic behavior of these solutions. We demonstrate that in any fixed time cone [math], the longtime asymptotic behavior of the solution for coupled dispersive AB systems can be expressed by [math] solitons on the discrete spectrum, the leading order term [math] on the continuous spectrum, and the allowable residual [math].
Longtime asymptotic behavior of the coupled dispersive AB system in low regularity spaces
10.1063/5.0102264
Journal of Mathematical Physics
20221104T10:32:23Z
© 2022 Author(s).
JinYan Zhu
Yong Chen

Integrable delaydifference and delaydifferential analogs of the KdV, Boussinesq, and KP equations
https://aip.scitation.org/doi/10.1063/5.0125308?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>Delaydifference and delaydifferential analogs of the KdV and Boussinesq (BSQ) equations are presented. Each of them has the Nsoliton solution and reduces to an already known soliton equation as the delay parameter approaches 0. In addition, a delaydifferential analog of the KP equation is proposed. We discuss its Nsoliton solution and the limit as the delay parameter approaches 0. Finally, the relationship between the delaydifferential analogs of the KdV, BSQ, and KP equations is clarified. Namely, reductions of the delay KP equation yield the delay KdV and delay BSQ equations.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>Delaydifference and delaydifferential analogs of the KdV and Boussinesq (BSQ) equations are presented. Each of them has the Nsoliton solution and reduces to an already known soliton equation as the delay parameter approaches 0. In addition, a delaydifferential analog of the KP equation is proposed. We discuss its Nsoliton solution and the limit as the delay parameter approaches 0. Finally, the relationship between the delaydifferential analogs of the KdV, BSQ, and KP equations is clarified. Namely, reductions of the delay KP equation yield the delay KdV and delay BSQ equations.
Integrable delaydifference and delaydifferential analogs of the KdV, Boussinesq, and KP equations
10.1063/5.0125308
Journal of Mathematical Physics
20221104T10:32:26Z
© 2022 Author(s).
Kenta Nakata

An approach to the Gaussian RBF kernels via Fock spaces
https://aip.scitation.org/doi/10.1063/5.0060342?af=R&feed=mostrecent
Journal of Mathematical Physics, <a href="https://aip.scitation.org/toc/jmp/63/11">Volume 63, Issue 11</a>, November 2022. <br/>We use methods from the Fock space and Segal–Bargmann theories to prove several results on the Gaussian RBF kernel in complex analysis. The latter is one of the most used kernels in modern machine learning kernel methods and in support vector machine classification algorithms. Complex analysis techniques allow us to consider several notions linked to the radial basis function (RBF) kernels, such as the feature space and the feature map, using the socalled Segal–Bargmann transform. We also show how the RBF kernels can be related to some of the most used operators in quantum mechanics and time frequency analysis; specifically, we prove the connections of such kernels with creation, annihilation, Fourier, translation, modulation, and Weyl operators. For the Weyl operators, we also study a semigroup property in this case.
Journal of Mathematical Physics, Volume 63, Issue 11, November 2022. <br/>We use methods from the Fock space and Segal–Bargmann theories to prove several results on the Gaussian RBF kernel in complex analysis. The latter is one of the most used kernels in modern machine learning kernel methods and in support vector machine classification algorithms. Complex analysis techniques allow us to consider several notions linked to the radial basis function (RBF) kernels, such as the feature space and the feature map, using the socalled Segal–Bargmann transform. We also show how the RBF kernels can be related to some of the most used operators in quantum mechanics and time frequency analysis; specifically, we prove the connections of such kernels with creation, annihilation, Fourier, translation, modulation, and Weyl operators. For the Weyl operators, we also study a semigroup property in this case.
An approach to the Gaussian RBF kernels via Fock spaces
10.1063/5.0060342
Journal of Mathematical Physics
20221110T12:54:23Z
© 2022 Author(s).
Daniel Alpay
Fabrizio Colombo
Kamal Diki
Irene Sabadini