Semjon Vugalter, Timo Weidl
We prove Cwikel-Lieb-Rosenbljum and Lieb-Thirring type bounds on the discrete spectrum of a two-body pair operator and calculate spectral asymptotics for the eigenvalue moments and the local spectral density in the pseudo-relativistic limit.
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We present a Kato-type inequality for bounded domain Omega \subset R^n, n>1.
In this article we present Sobolev-type inequalities for the localization of pseudo-relativistic energy.
In this paper we use different techniques from the fractional and pseudo-operators calculus to solve partial differential equations involving operators with non integer exponents. We apply the method to equations resembling generalizations of the heat equations and discuss the possibility of extending the procedure to the relativistic Schrödinger and Dirac equations.
We consider the Dirac equation in 1+1 space-time dimension with vector, scalar and pseudo-scalar coupling. In the traditional spin (or pseudo-spin) symmetry, the difference between (or sum of) the scalar and vector potentials is a constant. Here, however, we introduce an alternative symmetry where the scalar or pseudo-scalar potential is proportional to the vector potential. This leads to a model with significant extensions to supersymmetric quantum mechanics. We present a formal solution of the problem but give explicit analytic results for specific examples.
We prove within the Hartree-Fock theory of pseudo-relativistic atoms that the maximal negative ionization charge and the ionization energy of an atom remain bounded independently of the nuclear charge Z and the fine structure constant αas long as Zαis bounded.
It is known that ground states of the pseudo-relativistic Boson stars exist if and only if the stellar mass $N>0$ satisfies $N<N^*$, where the finite constant $N^*$ is called the critical stellar mass. Lieb and Yau conjecture in [Comm. Math. Phys., 1987] that ground states of the pseudo-relativistic Boson stars are unique for each $N<N^*$. In this paper, we prove that the above uniqueness conjecture holds for the particular case where $N>0$ is small enough.
Arrest of blowup for the 3-D semi-relativistic Schrödinger-Poisson system with pseudo-relativistic diffusion
研究了三维半相对论Schrödinger-POISSON系统在伪相对论扩散下的能量范数全局适定性，讨论了在L^2范数下的适定条件，并指出无耗散时负能量初始条件导致解在有限时间内爆破。
The pseudo-Finsleroid relativistic metric was constructed upon assuming that the involved vector field $b_i$ is time-like. In the present paper it is shown that the metric admits just the alternative counterpart in which the field is space-like. The entailed pseudo-Finsleroid-spatial framework is systematically described. We face on various remarkable properties, including the constant curvature of the associated indicatrix, the explicit Hamiltonian function, transparent presentations for the angle and scalar product. The spray coefficients are found to be of a rather simple structure. The Ber…
For a pseudo-relativistic model of matter, based on the no-pair Hamiltonian, we prove that the inclusion of the interaction with the self-generated magnetic field leads to instability for all positive values of the fine structure constant. This is true no matter whether this interaction is accounted for by the Breit potential, by an external magnetic field which is chosen to minimize the energy, or by the quantized radiation field.
Pseudo-relativistic fermionic systems with attractive Yukawa potential
本文研究了具有吸引性Yukawa势的费米子系统，证明了系统稳定性与总质量临界值的关系，并探讨了亚临界和临界质量下的基态存在性与性质。
The central role played by pseudodifferential operators in relativistic dynamics is very well know. In this work, operators as the Schrodinger one (e.g: square root) are treated from the point of view of the non-local pseudodifferential Green functions. Starting from the explicit construction of the Green (semigroup) theoretical kernel, a theorem linking the integrability conditions and their dependence on the spacetime dimensions is given. Relativistic wave equations with arbitrary spin and the causality problem are discussed with the algebraic interpretation of the radical operator and their…
In this paper we study the large-Z behaviour of the ground state energy of atoms with electrons having relativistic kinetic energy sqrt(p^2c^2+m^2c^4)-mc^2. We prove that to leading order in Z the energy is the same as in the non-relativistic case, given by (non-relativistic) Thomas-Fermi theory. For the problem to make sense, we keep the product Z*alpha fixed (here alpha is Sommerfeld's fine structure constant), and smaller than, or equal to, 2/pi, which means that as Z tends to infinity, alpha tends to zero.
A finite-dimensional pseudo-unitary framework is set up for describing the dynamics of free elementary particles in a purely relativistic quantum mechanical way. States of any individual particles or antiparticles are defined as suitably normalized vectors belonging to the two-complex-dimensional spaces that occur in local orthogonal decompositions of isomorphic copies of Cartan's space. The corresponding dynamical variables thus show up as bounded pseudo-Hermitian operator restrictions that possess real discrete spectra. Any measurement processes have to be performed locally in orthocronous p…
Exponential Decay outside of the Light Cone for the Pseudo-Relativistic Non-Autonomous Schrödinger Equation
研究伪相对论非自治薛定谔方程中粒子在光锥外的指数衰减行为，通过建立最大速度界来分析粒子在时间依赖势中的概率分布。
An approximate quantum-mechanical two-body equation for spinless particles incorporating relativistic kinematics is derived. The derivation is based on the relativistic energy-momentum relation $mc^{2}+ε= \sqrt{m^{2}c^{4}+p^{2}c^{2}}+V$ for each single particle, where $mc^2$ is the particle rest mass energy, $p$ its linear momentum, $ε$ its dynamical energy, and $V$ being the time-like vector interaction potential. The resulting two-body equation assumes rapid wave oscillations in a single, slowly varying potential well. A Bohr-Sommerfeld-type quantization condition is obtained. The approximat…
In this paper, we are concerned with the nonrelativistic limit of the following pseudo-relativistic equation with Hartree nonlinearity or power type nonlinearity \[ \left(\sqrt{-\hbar^2c^2 Δ+m^2c^4} - mc^2 \right) u + μu = \mathcal{N}(u), \] where $c$ denotes the speed of light. We prove that the ground states of this equation converges to the ground state of its nonrelativistic counterpart \[ -\frac{\hbar^2}{2m}Δu + μu = \mathcal{N}(u) \] with an explicit convergence rate $1/c^2$ in arbitrary order as $c \to \infty$. Moreover, we show that this rate is optimal.
We consider a multiatomic system where the nuclei are assumed to be point charges at fixed positions. Particles interact via Coulomb potential and electrons have pseudo-relativistic kinetic energy. We prove the van der Waals-London law, which states that the interaction energy between neutral atoms decays as the sixth power of the distance $|D|$ between the atoms. We rigorously compute all the terms in the binding energy up to the order $|D|^{-9}$ with error term of order $\mathcal{O}(|D|^{-10})$ . As intermediate steps we prove exponential decay of eigenfunctions of multiparticle Schrödinger …
We study geometry of confocal quadrics in pseudo-Euclidean spaces of an arbitrary dimension $d$ and any signature, and related billiard dynamics. The goal is to give a complete description of periodic billiard trajectories within ellipsoids. The novelty of our approach is based on introduction of a new discrete combinatorial-geometric structure associated to a confocal pencil of quadrics, a colouring in $d$ colours, by which we decompose quadrics of $d+1$ geometric types of a pencil into new relativistic quadrics of $d$ relativistic types. Deep insight of related geometry and combinatorics com…
We show that in a relativistically covariant formulation of the two-body bound state problem, the bound state spectrum is in agreement, up to relativistic corrections, with the non-relativistic bound state spectrum. The solution is achieved by solving the problem with support of the wave function in an O(2,1) invariant submanifold of the Minkowski spacetime. The O(3,1) invariance of the differential equation requires, however, that the solutions provide a representation of O(3,1). Such solutions are obtained by means of the method of induced representations, providing a basic insight into the …
Dynamics of a relativistic discrete body: rigidity conditions, and covariant equations of motion
本文提出了一种相对离散体的刚性条件，并探讨了与之兼容的协变运动方程，该理论具有预期的六个动力学自由度，允许比Born理论更一般的运动。
The three-body problem is reexamined in the framework of general relativity. The Newtonian three-body problem admits Euler's collinear solution, where three bodies move around the common center of mass with the same orbital period and always line up. The solution is unstable. Hence it is unlikely that such a simple configuration would exist owing to general relativistic forces dependent not only on the masses but also on the velocity of each body. However, we show that the collinear solution remains true with a correction to the spatial separation between masses. Relativistic corrections to th…
We prove uniqueness of ground states $Q$ in $H^{1/2}$ for pseudo-relativistic Hartree equations in three dimensions, provided that $Q$ has sufficiently small $L^2$-mass. This result shows that a uniqueness conjecture by Lieb and Yau in [CMP 112 (1987),147--174] holds true at least under a smallness condition. Our proof combines variational arguments with a nonrelativistic limit, which leads to a certain Hartree-type equation (also known as the Choquard-Pekard or Schroedinger-Newton equation). Uniqueness of ground states for this limiting Hartree equation is well-known. Here, as a key ingredien…
One-Dimensional Relativistic Self-Gravitating Systems
本文回顾了一维相对论引力 N 体问题的研究进展，指出一维约化因无引力辐射而便于与非相对论情形比较，并系统讨论了二体至 N 体问题的精确解、混沌现象及新特征，为相对论多体系统研究开辟了新方向。
We equip the regular Fréchet Lie group of invertible, odd-class, classical pseudodifferential operators $Cl^{0,*}_{odd}(M,E)$ -- in which $M$ is a compact smooth manifold and $E$ a (complex) vector bundle over $M$ -- with pseudo-Riemannian metrics, and we use these metrics to introduce a large class of rigid body equations. We adapt to our infinite-dimensional setting Manakov's classical observation on the integrability of Euler's equations for the rigid body, and we show that our equations can be written in Lax form (with parameter) and that they admit an infinite number of integrals of motio…
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