The concept of multiplicity of solutions was developed in [1] which is based on the theory of energy operators in the Schwartz space S-(R) and some subspaces called energy spaces first defined in [2] and [3]. The main...The concept of multiplicity of solutions was developed in [1] which is based on the theory of energy operators in the Schwartz space S-(R) and some subspaces called energy spaces first defined in [2] and [3]. The main idea is to look for solutions of a given linear PDE in those subspaces. Here, this work extends previous developments in S-(Rm)?(m∈Z+) using the theory of Sobolev spaces. Furthermore, we also define the concept of Energy Parallax, which is the inclusion of additional solutions when varying the energy of a predefined system locally by taking into account additional smaller quantities. We show that it is equivalent to take into account solutions in other energy subspaces. To illustrate the theory, one of our examples is based on the variation of Electro Magnetic (EM) energy density within the skin depth of a conductive material, leading to take into account derivatives of EM evanescent waves, particular solutions of the wave equation. The last example is the derivation of the Woodward effect [4] with the variations of the EM energy density under strict assumptions in general relativity. It finally leads to a theoretical definition of an electromagnetic and gravitational (EMG) coupling.展开更多
This work is a discussion on the energy parallax theory developed in [1] [2] based on the multiplicity of the solutions theorem. This theory is compared with the perturbation theory in mathematical physics. The pertur...This work is a discussion on the energy parallax theory developed in [1] [2] based on the multiplicity of the solutions theorem. This theory is compared with the perturbation theory in mathematical physics. The perturbation theory uses the increment of a solution which can be formalized with a Taylor series development. With the energy parallax theory, the convergence property of the Taylor series of the energy of a system is the key to decide to include additional solutions, defined on the so-called energy spaces [2]. The development is supported using various examples in quantum mechanics (i.e. Rayleigh-Schrödinger perturbation theory) and wave theory with the Electromagnetic (EM) energy density (i.e. evanescent waves within the skin layer of a dielectric material). Finally, we discuss the Woodward effect [3] and the application of the energy parallax when assuming that the variations of EM energy density can trigger such effect within asymmetric cavities.展开更多
文摘The concept of multiplicity of solutions was developed in [1] which is based on the theory of energy operators in the Schwartz space S-(R) and some subspaces called energy spaces first defined in [2] and [3]. The main idea is to look for solutions of a given linear PDE in those subspaces. Here, this work extends previous developments in S-(Rm)?(m∈Z+) using the theory of Sobolev spaces. Furthermore, we also define the concept of Energy Parallax, which is the inclusion of additional solutions when varying the energy of a predefined system locally by taking into account additional smaller quantities. We show that it is equivalent to take into account solutions in other energy subspaces. To illustrate the theory, one of our examples is based on the variation of Electro Magnetic (EM) energy density within the skin depth of a conductive material, leading to take into account derivatives of EM evanescent waves, particular solutions of the wave equation. The last example is the derivation of the Woodward effect [4] with the variations of the EM energy density under strict assumptions in general relativity. It finally leads to a theoretical definition of an electromagnetic and gravitational (EMG) coupling.
文摘This work is a discussion on the energy parallax theory developed in [1] [2] based on the multiplicity of the solutions theorem. This theory is compared with the perturbation theory in mathematical physics. The perturbation theory uses the increment of a solution which can be formalized with a Taylor series development. With the energy parallax theory, the convergence property of the Taylor series of the energy of a system is the key to decide to include additional solutions, defined on the so-called energy spaces [2]. The development is supported using various examples in quantum mechanics (i.e. Rayleigh-Schrödinger perturbation theory) and wave theory with the Electromagnetic (EM) energy density (i.e. evanescent waves within the skin layer of a dielectric material). Finally, we discuss the Woodward effect [3] and the application of the energy parallax when assuming that the variations of EM energy density can trigger such effect within asymmetric cavities.
