An impossibility theorem on approximately simulating quantum non-integrable Hamiltonian systems is presented here. This result shows that there is a tradeo between the unitary property and the energy expectation conservation law in time-descretization of quantum non-integrable systems, whose classical counterpart is Ge-Marsden's impossibility result about simulating classically non-integrable H...

We investigate dispersive estimates for the two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies a t−1 decay rate as an operator from the Hardy space H to BMO, the space of functions of bounded mean oscillation. This estimate, along with the L conservation law allows one to deduce a family of Strichartz estimates. We classify the structure o...

Many physical processes are described by conservative quasilinear systems of equations that admit a supplementary independent conservation law. A typical example is the equations of incompressible fluid dynamics. If for basic equations the conservation laws of the momentum and the volume are chosen then the conservation law of the energy is an additional conservation law. In the present paper a...

Denoting by v ̄ the velocity flow of a classical particle that is subject to a potential V , we demonstrate that the path-integral formalism of non-relativistic quantum mechanics can be obtained by superimposing wave functions that are solutions of a wave equation which in turn directly corresponds to the probability conservation equation div(ψ∗ψv ̄ ) = 0.

A group invariant solution for a steady two-dimensional jet is derived by considering a linear combination of the Lie point symmetries of Prandtl’s boundary layer equations for the jet. Only two Lie point symmetries contribute to the solution and the ratio of the constants in the linear combination is determined from conservation of total momentum flux in the downstream direction. A conservatio...

A logical error in the usual derivation of the energy conservation law is analyzed, and a way to avoid the error is presented. In earlier papers [1], [2] we identified a logical error adopted by repetition in textbooks on classical electrodynamics when the laws of conservation of energy and momentum are derived for a system consisting of electromagnetic fields and charged particles. In this pap...