arXiv subjects

Yuekai Bi

Publications and source records attributed to Yuekai Bi.

4 recordsLinked to original sources

arXiv2013

New Relativistic Wave Equations for Two-Particle Systems

We seek to introduce a mathematical method to derive the relativistic wave equations for two-particle system. According to this method, if we define stationary wave functions as special solutions like $Ψ(\mathbf{r}_1,\mathbf{r}_2,t)=ψ(\mathbf{r}_1,\mathbf{r}_2)e^{-iEt/\hbar},\, ψ(\mathbf{r}_1,\mathbf{r}_2)\in\mathscr{S} (\mathbb{R}^3\times\mathbb{R}^3)$, and properly define the relativistic reduced mass $μ_0$, then some new relativistic two-body wave equations can be derived. On this basis, we obtain the two-body Sommerfeld fine-structure formula for relativistic atomic two-body systems such as the pionium and pionic hydrogen atoms bound states, using which, we discuss the pair production and annihilation of $π+$ and $π-$.

math-ph↗

arXiv2011

The Cauchy problem for higher-order linear partial differential equation

For the linear partial differential equation $P(\partial_x,\partial_t)u=f(x,t)$, where $x\in\mathbb{R}^n,\;t\in\mathbb{R}^1$, with $P(\partial_x,\partial_t)$ is $\prod^m_{i=1}(\frac{\partial}{\partial{t}}-a_iP(\partial_x))$ or $\prod^m_{i=1}(\frac{\partial^2}{\partial{t^2}}-a_i^2P(\partial_x))$, the authors give the analytic solution of the cauchy problem using the abstract operators $e^{tP(\partial_x)}$ and $\frac{\sinh(tP(\partial_x)^{1/2})}{P(\partial_x)^{1/2}}$. By representing the operators with integrals, explicit solutions are obtained with an integral form of a given function.

math.AP↗

arXiv2010

New Properties of Fourier Series and Riemann Zeta Function

We establish the mapping relations between analytic functions and periodic functions using the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, including the mapping relations between power series and trigonometric series, and by using such mapping relations we obtain a general method to find the sum function of a trigonometric series. According to this method, if each coefficient of a power series is respectively equal to that of a trigonometric series, then if we know the sum function of the power series, we can obtain that of the trigonometric series, and the non-analytical points of which are also determined at the same time, thus we obtain a general method to find the sum of the Dirichlet series of integer variables, and derive several new properties of $ζ(2n+1)$.

math.AP↗

arXiv2010

Stationary Solutions of the Klein-Gordon Equation in a Potential Field

We seek to introduce a mathematical method to derive the Klein-Gordon equation and a set of relevant laws strictly, which combines the relativistic wave functions in two inertial frames of reference. If we define the stationary state wave functions as special solutions like $Ψ(\mathbf{r},t)=ψ(\mathbf{r})e^{-iEt/\hbar}$, and define $m=E/c^2$, which is called the mass of the system, then the Klein-Gordon equation can clearly be expressed in a better form when compared with the non-relativistic limit, which not only allows us to transplant the solving approach of the Schrödinger equation into the relativistic wave equations, but also proves that the stationary solutions of the Klein-Gordon equation in a potential field have the probability significance. For comparison, we have also discussed the Dirac equation. By introducing the concept of system mass into the Klein-Gordon equation with the scalar and vector potentials, we prove that if the Schr\"{o dinger equation in a certain potential field can be solved exactly, then under the condition that the scalar and vector potentials are equal, the Klein-Gordon equation in the same potential field can also be solved exactly by using the same method.

math.AP↗