#### Journal of Mathematical Physics

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ISSN / EISSN
:
0022-2488 / 1089-7658

Published by: AIP Publishing
(10.1063)

Total articles ≅ 24,610

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#### Latest articles in this journal

Journal of Mathematical Physics, Volume 62; https://doi.org/10.1063/5.0061763

**Abstract:**

We study the statistics and the arithmetic properties of the Robin spectrum of a rectangle. A number of results are obtained for the multiplicities in the spectrum depending on the Diophantine nature of the aspect ratio. In particular, it is shown that for the square, unlike the case of Neumann eigenvalues where there are unbounded multiplicities of arithmetic origin, there are no multiplicities in the Robin spectrum for a sufficiently small (but nonzero) Robin parameter except a systematic symmetry. In addition, uniform lower and upper bounds are established for the Robin–Neumann gaps in terms of their limiting mean spacing. Finally, the pair correlation function of the Robin spectrum on a Diophantine rectangle is shown to be Poissonian.

Journal of Mathematical Physics, Volume 62; https://doi.org/10.1063/5.0054823

**Abstract:**

We gain tight rigorous bounds on the renormalization fixed point for period doubling in families of unimodal maps with degree 4 critical point. We use a contraction mapping argument to bound essential eigenfunctions and eigenvalues for the linearization of the operator and for the operator controlling the scaling of added noise. Multi-precision arithmetic with rigorous directed rounding is used to bound operations in a space of analytic functions yielding tight bounds on power series coefficients and universal constants to over 320 significant figures.

Journal of Mathematical Physics, Volume 62; https://doi.org/10.1063/5.0057987

**Abstract:**

We study certain non-tracial von Neumann algebras generated by some self-adjoint operators satisfying mixed

*q*-commutation relations. Such algebras are discussed in the work of Bikram*et al.*[“Mixed q-deformed Araki-Woods von Neumann algebras,” (submitted)]. We prove the analog of Nelson’s hypercontractivity inequality for the mixed*q*-Ornstein–Uhlenbeck semigroup. We also show that the mixed*q*-Ornstein–Uhlenbeck semigroup is ultracontractive.
Journal of Mathematical Physics, Volume 62; https://doi.org/10.1063/5.0055911

**Abstract:**

We consider the long time dynamics of nonlinear Schrödinger equations with an external potential. More precisely, we look at Hartree type equations in three or higher dimensions with small initial data. We prove an optimal decay estimate, which is comparable to the decay of free solutions. Our proof relies on good control on a high Sobolev norm of the solution to estimate the terms in Duhamel’s formula.

Journal of Mathematical Physics, Volume 62; https://doi.org/10.1063/5.0054450

**Abstract:**

Whether the three dimensional isentropic compressible Navier–Stokes equations admit weak solutions for arbitrary initial data with adiabatic exponent

*γ*> 1 remains a challenging problem. The only available results under*γ*> 1 were achieved by either assuming the initial data with small energy due to Hoff [J. Differ. Equations 120(1), 215–254 (1995)] or under the spherically symmetric condition by Jiang and Zhang [Commun. Math. Phys. 215, 559–581 (2001)] and Huang [J. Differ. Equations 262, 1341–1358 (2017)]. In this paper, we establish the existence of weak solutions with higher regularity of the three-dimensional periodic compressible isentropic Navier–Stokes equations in small time for the adiabatic exponent*γ*> 1 in the presence of vacuum. It can be viewed as a local version of Hoff’s work and also extends the result of Desjardins [Commun. Partial Differ. Equations 22(5–6), 977–1008 (1997)] by removing the assumption of*γ*> 3.
Journal of Mathematical Physics, Volume 62; https://doi.org/10.1063/5.0061119

**Abstract:**

The Struckmeier and Riedel (SR) approach is extended in real space to isolate dynamical invariants for one- and two-dimensional time-dependent Hamiltonian systems. We further develop the SR-formalism in $z\stackrel{\u0304}{z}$ complex phase space characterized by

*z*=*x*+*iy*and $\stackrel{\u0304}{z}=x-iy$ and construct invariants for some physical systems. The obtained quadratic invariants contain a function*f*_{2}(*t*), which is a solution of a linear third-order differential equation. We further explore this approach into extended complex phase space defined by*x*=*x*_{1}+*ip*_{2}and*p*=*p*_{1}+*ix*_{2}to construct a quadratic invariant for a time-dependent quadratic potential. The derived invariants may be of interest in the realm of numerical simulations of explicitly time-dependent Hamiltonian systems.
Journal of Mathematical Physics, Volume 62; https://doi.org/10.1063/5.0056957

**Abstract:**

Isochrone potentials are spherically symmetric potentials within which a particle orbits with a radial period that is independent of its angular momentum. Whereas all previous results on isochrone mechanics have been established using classical analysis and geometry, in this article, we revisit the isochrone problem of motion using tools from Hamiltonian dynamical systems. In particular, we (1) solve the problem of motion using a well-adapted set of angle-action coordinates and generalize the notion of eccentric anomaly to all isochrone orbits, and (2) we construct the Birkhoff normal form for a particle orbiting a generic radial potential and examine its Birkhoff invariants to prove that the class of isochrone potentials is in correspondence with parabolas in the plane. Along the way, several fundamental results of celestial mechanics, such as the Bertrand theorem or the Kepler equation and laws, are obtained as special cases of more general properties characterizing isochrone mechanics.

Journal of Mathematical Physics, Volume 62; https://doi.org/10.1063/5.0055250

**Abstract:**

We consider the Dirac equation, written in polar formalism, in the presence of general Coulomb-like potentials, that is, potentials arising from the time component of the vector potential and depending only on the radial coordinate, in order to study the conditions of integrability, given as some specific form for the solution: we find that the angular dependence can always be integrated, while the radial dependence is reduced to finding the solution of a Riccati equation so that it is always possible, at least in principle. We exhibit the known case of the Coulomb potential and one special generalization as examples to show the versatility of the method.

Journal of Mathematical Physics, Volume 62; https://doi.org/10.1063/5.0058652

**Abstract:**

In this paper, we study the asymptotic behavior of solutions to the incompressible magnetohydrodynamic (MHD) equations in an exterior domain. We will show that, under some assumption, any nontrivial velocity field u and magnetic field h obey a minimal decaying rate exp(−

*C*|*x*|^{2}log |*x*|) at infinity. Our proof is based on appropriate Carleman estimates. As a consequence, we establish a Liouville-type result for the three dimensional incompressible MHDs in an exterior domain.
Journal of Mathematical Physics, Volume 62; https://doi.org/10.1063/5.0055683

**Abstract:**

In this paper, we use the generalized Prüfer variables to study the spectral type of a class of random Jacobi operators $({H}_{\tau ,\omega}^{\lambda}u)(n)={\tau}_{n}u(n+1)+{\tau}_{n-1}u(n-1)+\lambda {a}_{n}{\omega}_{n}u(n),$ in which the decay speed of the parameters

*a*_{n}is*n*^{−α}for some*α*> 0. We will show that the operator has an absolutely continuous spectrum for $\alpha >\frac{1}{2}$ , a pure point spectrum for $0<\alpha <\frac{1}{2}$ , and a transition from a singular continuous spectrum to a pure point spectrum in $\alpha =\frac{1}{2}$ .