arXiv subjects

Xijun Hu

Publications and source records attributed to Xijun Hu.

At least 19 recordsLinked to original sources

arXiv2026

Relative Periodic Orbits in the Gutzwiller-type Anisotropic Kepler Problem and $n$-body Problem

We study the relative periodic orbits in the Gutzwiller-type anisotropic Kepler problem, which is a generalized model derived from the classical Gutzwiller anisotropic Kepler problem. By reducing the rotational symmetry of the $z$-axis, we obtain a reduced system with two degrees of freedom and its energy surface forms a compact and regular three-sphere in a certain parameter range. Combining the estimation of the Seifert rotation number of the planar Kepler orbit and the CHHL formula introduced in \cite{CHHL23}, we prove that this system admits infinitely many periodic orbits on every compact regular energy surface. This model can be applied to the $(1+2n)$-body problem to find infinitely many relative periodic orbits with hip-hop symmetry as well as relative periodic orbits in the $n$-pyramidal problem.

math.DS↗

arXiv2026

Relative Periodic Orbits in the Spatial Anisotropic Kepler Problem

The spatial anisotropic Kepler problem comes from quantum mechanics, which models electron motion in semiconductors with donor impurities and depends on the anisotropic parameter $\beta\in(-1,+\infty)$. After reducing the system modulo rotational symmetry, we investigate periodic orbits in this two-degree-of-freedom setting. Combining an index comparison in \cite{HLOQS26}, the volume formula in \cite{CHHL23} and Franks' theorem, we prove that the system possesses infinitely many periodic orbits on any fixed compact and regular energy surface for $\beta\in(-1,0]$.

math.DS↗

arXiv2026

ECH Constraints and Twist Dynamics in the Spatial Isosceles Three-Body Problem

We study dynamical constraints arising from Embedded Contact Homology (ECH) in the spatial isosceles three-body problem. For energies below the critical level, the dynamics on the energy surface is identified with a Reeb flow on the tight three-sphere. We obtain quantitative estimates for the Euler orbit, including monotonicity of its transverse rotation number and a strict inequality comparing its action with the contact volume. Combined with the ECH classification of Reeb flows on the tight three-sphere with two simple periodic orbits, these estimates rule out the two-orbit scenario, thus forcing every compact energy surface below the critical level to have infinitely many periodic orbits. The result admits a dynamical interpretation via disk-like global surfaces of section bounded by the Euler orbit. In this setting, the rotation number and the contact volume define a non-trivial twist interval which encodes the relative winding of periodic orbits. For energies above the critical level, where the energy surface is non-compact, we prove the existence of infinitely many periodic orbits and infinitely many parabolic trajectories via twist estimates near infinity.

math.SG↗

arXiv2025

Index theory for singular Lagrangian systems and Bessel-type differential operators

The aim of the present manuscript is to develop an index theory for singular Lagrangian systems, with a particular focus on the important class of singular operators given by Bessel type differential operators. The main motivation is to address several challenges posed by singular operators, which appear in a wide range of applications: celestial mechanics (for instance, perturbations in planetary motion), oscillatory systems with time dependent forcing, electromagnetism (such as wave equations in nonuniform media), and quantum mechanics (notably certain Schroedinger equations with periodic potentials). We pursue two principal objectives. First, we establish a spectral flow formula and a Morse Index Theorem for gap-continuous paths of singular Sturm Liouville operators. By means of these index formulas, we construct a Morse index theory for a broad class of Bessel type differential operators and apply it to a family of asymptotic solutions of the gravitational n body problem. Finally, our new index theory provides new insight into a phenomenon first observed by Rellich concerning the spectrum of one-parameter families of Sturm Liouville operators with varying domains.

math.DS↗

arXiv2025

Quantitative Linear Stability Analysis of Elliptic Relative Equilibria in the Planar N-Body Problem

An elliptic relative equilibrium (ERE) is a special solution of the planar $N$-body problem generated by a central configuration. Its linear stability depends on the eccentricity $e$ and the masses of the bodies. However, for $e>0$, the variational equations become non-autonomous and highly complex, particularly near $e=1$, where the system exhibits a singularity. This complicates the stability analysis as $e$ approaches one, making it challenging to derive a rigorous quantitative estimate for the stable region across $e\in[0,1)$. In this work, we address this problem. Using trace formulas for the non-degenerate Hamiltonian system of EREs, we establish an upper bound ensuring non-degeneracy for all $e\in[0,1)$. As key applications, we provide explicit stability estimates for the Lagrange, Euler, and regular $(1+n)$-gon EREs over the full range of eccentricity.

math.DS↗

arXiv2025

Relative periodic solutions in spatial Kepler problem with symmetric perturbation

The spatial Kepler problem with a perturbation satisfying the rotational symmetry w.r.t. the $z$-axis and the reflection symmetry w.r.t. the $(x, y)$-plane, can be reduced to an Hamiltonian system with 2 degrees of freedom after fixing the angular momentum. For small enough perturbations, we show that for certain choices of energy and angular momentum, the corresponding energy surface is compact and diffeomorphic to $\mathbb{S}^3$, and on each compact energy surface there is a unique $z$-symmetric brake orbit, which forms a Hopf link with a planar relative periodic orbit. Moreover under some additional technical assumptions, by applying recent results from symplectic dynamics (\cite{CHHL23}) and Franks' Theorem, we prove there are infinitely many relative periodic orbits on each compact energy surface. These results can be applied to the motion of a satellite around a uniformly mass-distributed ellipsoid and the $n$-pyramidal problem, where one point mass moves along the $z$-axis and $n$ other equal point masses form a regular $n$-gon perpendicular to the $z$-axis.

math.DS↗

arXiv2025

Tuning Charge Density Wave in the Transition from Magnetically Frustrated Conductor to Ferrimagnetic Insulator in Carbon Nanowire within Boron Nitride Nanotube

The emergence of exotic charge density wave (CDW) alongside ferrimagnetism materials opens exciting new possibilities for quantum switching, particularly in field-tuning CDW electronics. However, these two phenomena often compete and rely heavily on strong electronic correlations. While carbon nanowire arrays have been experimentally shown to exhibit ferromagnetism above 400 K, our research shows that encapsulating a linear carbon chain (LCC) within zigzag boron nitride nanotubes (BNT) induces a short-range CDW state under a competing effect of ferrimagnetism and magnetic frustrations. However, for this exotic feature to occur, the LCC needs to break the symmetry along the circular plane of the BNT. Then we utilize a Monte Carlo model to identify the optimal length of LCC@BNT to tackle its size effect, while also comparing the stability of chains provided by carbon nanotubes. The shorter LCC@BNT displays a more prominent long-range CDW pattern with a tunneling barrier of 2.3 eV on the Fermi surface, transitioning into an unconventional insulator. Meanwhile, magnetic frustrations disappear, and ferrimagnetism remains stable up to 280 K. Our discovery of ferrimagnetic CDW carbyne insulators, which function without conventional periodic lattice distortion, spin-orbit coupling, or complex d and f hybridization represents a groundbreaking shift in thinking, which demonstrates that such exotic properties are not exclusive to transition metal elements. We anticipate that spin fluctuations in LCC@BNT could enable fine-tuning of the CDW pattern, and applying an electric excitation of 2.3 eV triggers an abrupt insulator-to-conductor transition for quantum switching applications.

cond-mat.mes-hall↗

arXiv2024

Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions and Markov chain

In this paper, we study the eigenvalue problem of stochastic Hamiltonian system driven by Brownian motion and Markov chain with boundary conditions and time-dependent coefficients. For any dimensional case, the existence of the first eigenvalue is proven and the corresponding eigenfunctions are constructed by virtue of dual transformation and generalized Riccati equation system. Furthermore, we have more finely characterized the existence of all eigenvalues and constructed the related eigenfunctions for one-dimensional Hamiltonian system. Moreover, the increasing order of these eigenvalues have also been given.

math.PR↗

arXiv2023

Global Surfaces of Section and Periodic Orbits in The Spatial Isosceles Three Body Problem

We study the spatial isosceles three body problem, which is a system with two degrees of freedom after modulo the rotation symmetry. For certain choices of energy and angular momentum, we find some disk-like global surfaces of section with the Euler orbit as their common boundary, and a brake orbit passing through them. By considering the Poincaré maps of these global surfaces of section, we prove the existence of all kinds of different periodic orbits under certain assumption. Moreover, we are able to prove, for generic choices of masses, the system always has infinitely many periodic orbits. One of the key is to estimate the rotation numbers of the Euler orbit and the brake orbit with respect to the Poincaré map. For this, we establish formulas connected these numbers with the mean indices of the corresponding orbits using the Maslov-type index.

math.DS↗

arXiv2023

A symplectic dynamics approach to the spatial isosceles three-body problem

We study the spatial isosceles three-body problem from the perspective of Symplectic Dynamics. For certain choices of mass ratio, angular momentum, and energy, the dynamics on the energy surface is equivalent to a Reeb flow on the tight three-sphere. We find a Hopf link formed by the Euler orbit and a symmetric brake orbit, which spans an open book decomposition whose pages are annulus-like global surfaces of section. In the case of large mass ratios, the Hopf link is non-resonant, forcing the existence of infinitely many periodic orbits. The rotation number of the Euler orbit plays a fundamental role in the existence of periodic orbits and their symmetries. We explore such symmetries in the Hill region and show that the Euler orbit is negative hyperbolic for an open set of parameters while it can never be positive hyperbolic. Finally, we address convexity and determine for each parameter whether the energy surface is strictly convex, convex, or non-convex. Dynamical consequences of this fact are then discussed.

math.DS↗

arXiv2023

Non-minimality and instability of brake orbits for natural Lagrangians on Riemannian manifolds

We investigate minimality and stability of periodic brake orbits in natural Lagrangian systems on smooth Riemannian manifolds. We prove that every non-constant periodic brake orbit is not a minimizer of the fixed-time action, for any conormal boundary condition. Under an orbit-cylinder hypothesis, its Morse index strictly increases in the free-time setting. As a consequence, strongly nondegenerate brake orbits fail to be linearly stable under a dimensional condition; in dimension at least three, nondegenerate mountain-pass brake orbits are spectrally unstable when the monodromy is semisimple. The key ingredient is a local index contribution at each brake instant. Using Seifert collar coordinates near the Hill boundary, we reduce the normal dynamics to a one-dimensional model, exhibiting a degeneracy inherent to brake symmetry. We illustrate the results by explicit Morse index computations for the planar anisotropic oscillator, the planar pendulum, and the planar Kepler problem; in the Kepler case, the ejection--collision orbit is treated via cotangent-lift Levi--Civita--Lissajous regularization.

math.DS↗

arXiv2021

An Index Theory for Collision, Parabolic and Hyperbolic Solutions of the Newtonian $n$-body Problem

In the Newtonian $n$-body problem for solutions with arbitrary energy, which start and end either at a total collision or a parabolic/hyperbolic infinity, we prove some basic results about their Morse and Maslov indices. Moreover for homothetic solutions with arbitrary energy, we give a simple and precise formula that relates the Morse indices of these homothetic solutions to the spectra of the normalized potential at the corresponding central configurations. Potentially these results could be useful in the application of non-action minimization methods in the Newtonian $n$-body problem.

math.DS↗

arXiv2021

Schottky-diode design for the world's leading telecommunication

The Schottky diode, BN/GaN layered composite contacting to bulk aluminum, is theoretically plausible to harvest wireless energy above X-band. According to our first principle calculation, the insertion of GaN layers dramatically influences the optical properties of the layered composite. The relative dielectric constant of BN/GaN layered composite as a function of layer-to-layer separation is investigated where the optimized dielectric constant is 3.1. Furthermore, we design another Schottky diode via nanostructuring. Our first principle calculation suggests that the relative dielectric constant of boron nitride monolayer can be minimized to 1.5 only if it is deposited on aluminum monolayer. It is rare to find a semiconductor with the dielectric constant close to 1 which may push the cut-off frequency of Al/BN-based rectenna to the high-band 5G network.

cond-mat.mtrl-sci↗

arXiv2020

Mean index for non-periodic orbits in Hamiltonian systems

In this paper, we define mean index for non-periodic orbits in Hamiltonian systems and study its properties. In general, the mean index is an interval in R which is uniformly continuous on the systems. We show that the index interval is a point for a quasi-periodic orbit. The mean index can be considered as a generalization of rotation number which defined by Johnson and Moser in the study of almost periodic Schrodinger operators. Motivated by their works, we study the relation of Fredholm property of the linear operator and the mean index at the end of the paper.

math.DS↗

arXiv2020

Morse index theorem for heteroclinic, homoclinic and halfclinic orbits of Lagrangian systems

The purpose of this paper is to prove a new, more general version of the Morse index theorem for heteroclinic, homoclinic, and half-clinic solutions in general Lagrangian systems. In the final section, we compute the Morse index for specific heteroclinic and half-clinic solutions in classical mechanical models such as the mathematical pendulum, the Nagumo equation, and a four-dimensional competition-diffusion system.

math.DS↗

arXiv2019

Use of Hydrophobic-Hydrophobic Interactions for Direct Graphene Transfer

A new large-area wet transfer method to transfer graphene directly onto UHMWPE membranes without organic-coating, thermal treatment or other mediated substrate was developed. This method completely avoids the problems of PMMA-residue and possible PMMA coating induced tear-off by harnessing hydrophobic-hydrophobic interactions between graphene and UHMWPE, which is one of most facile and cheapest transfer methods that are currently available, especially for large-area graphene preparation and transfer. Multi-layer sandwich structure composite membrane of graphene and UHMWPE with desired number of layers can also be easily prepared by repeating the transfer method. The most significant applications of this transfer method lie on that it facilitates thin polymer membranes the possibilities to be directly observed with high magnification by electron microscopes, such as SEM and TEM. Thus the crystallization and phase-behavior of polymers including UHMWPE can be investigated with real-time and in-situ morphological observations, even in the presence of their solvent and with the need of thermal treatment. Morphologies and mechanical properties of stretched graphene/UHMWPE layered composite membranes were investigated, and strengthening effect of graphene on one side and both sides was observed.

cond-mat.mtrl-sci↗

arXiv2019

Instability of semi-Riemannian closed geodesics

A celebrated result due to Poincaré affirms that a closed non-degenerate minimizing geodesic $γ$ on an oriented Riemannian surface is hyperbolic. Starting from this classical theorem, our first main result is a general instability criterion for timelike and spacelike closed semi-Riemannian geodesics on a (non)oriented manifold. A key role is played by the spectral index, a new topological invariant that we define through the spectral flow (being the Morse index truly infinite) of a path of selfadjoint Fredholm operators. A major step in the proof of this result is a em new spectral flow formula. Bott's iteration formula, introduced by author in 1956, relates in a clear way the Morse index of an iterated closed Riemannian geodesic and the so-called $ω$-Morse indices. Our second result is a semi-Riemannian generalization of the famous Bott-type iteration formula in the case of closed (resp. timelike closed) Riemannian (resp. Lorentzian) geodesics. Our last result is a strong instability result obtained by controlling the Morse index of the geodesic and of all of its iterations.

math.DG↗

arXiv2019

Morse index and stability of the planar N-vortex problem

This paper concerns the investigation of the stability properties of relative equilibria which are rigidly rotating vortex configurations sometimes called vortex crystals, in the N-vortex problem. Such a configurations can be characterized as critical point of the Hamiltonian function restricted on the constant angular impulse hypersurface in the phase space (topologically a pseudo-sphere whose coefficients are the circulation strengths of the vortices). Relative equilibria are generated by the circle action on the so-called shape pseudo-sphere (which generalize the standard shape sphere appearing in the study of the N-body problem). Inspired by the planar gravitational $N$-body problem, and after a geometrical and dynamical discussion, we investigate the relation intertwining the stability of relative equilibria and the inertia indices of the central configurations generating such equilibria. In the last section we apply our main results to some symmetric three and four vortices relative equilibria.

math.DS↗