We give an explicit construction of a positive-definite invariant inner-product for the Klein-Gordon fields, thus solving the old problem of the probability interpretation of Klein-Gordon fields without having to restrict to the subspaces of the positivefrequency solutions. Our method has a much wider domain of application and may be used to obtain the most general invariant inner-product on th...

Having shown elsewhere [1] that the Klein paradox for the Klein-Gordon (KG) equation of spin-zero particle manifests exactly the same kind of wave propagation and negative refraction phenomena, which also exist in the scattering of TM (transverse-magnetic) –polarized electromagnetic (EM) wave incident on a left handed medium (LHM), we show in this paper that it is possible to simulate the Klein...

Introduction The differential equation H(y) = 0 (see below) due to Hurwitz goes back to the famous work “Ueber die Transformation siebenter Ordnung der elliptischen Functionen” of F. Klein [14]. In this work, Klein made an important contribution to the much discussed question of the solvability of algebraic equations by means of elliptic modular functions. In order to investigate the transforma...

Abstract In this paper we investigate the small data global existence and pointwise decay of solutions to two systems coupled wave-Klein–Gordon equations in spatial dimensions. particular, consider critical (in sense time decay) semilinear nonlinearities for wave equation below-critical Klein–Gordon equation, a situation that has not been studied before context equations. An interesting feature...

In this paper the thermal transport phenomena in MEMS are investigated. The thermal Klein-Gordon transport equation for nanoscale structures is formulated and solved.

ABSTRACT: We provide an outline of a college-level lecture on systems of differential equations that is intended to offer an alternative, integrative view of the subject. Particular emphasis is placed on the fact that very elementary concepts and insights can be employed to gain access to a very wide range of important applications, including the Schrödinger equation, the Klein-Gordon equation,...

Given a positive energy solution of the Klein-Gordon equation, the motion of the free, spinless, relativistic particle is described in a fixed Lorentz frame by a Markov diffusion process with non-constant diffusion coefficient. Proper time is an increasing stochastic process and we derive a probabilistic generalization of the equation (dτ)2 = − 1 c dXνdXν . A random time-change transformation p...

We construct small amplitude breathers in 1D and 2D Klein–Gordon infinite lattices. We also show that the breathers are well approximated by the ground state of the nonlinear Schrödinger equation. The result is obtained by exploiting the relation between the Klein Gordon lattice and the discrete Non Linear Schrödinger lattice. The proof is based on a Lyapunov-Schmidt decomposition and continuum...

As already known, Schrödinger equation is the most used equation to describe non-relativistic quantum systems. Its relativistic version was developed by Klein-Gordon and Dirac, but Schrödinger equation has wide applicability in particular because it resembles classical wave dynamics. For introduction to non-relativistic quantum mechanics, see [1]. Schrödinger equation begins with definition of ...