# PDF A Short Introduction to Perturbation Theory for Linear Operators

Firstly, one can emphasize the Lindset series of the energy and the eigenfunctions described in Section 2. To recall Section 3. In quantum mechanics, perturbation theory relies on the Hermitian properties of the Hamiltonian, which establish the relationship between the energy states and the eigenfunctions.

However, the energy parallax theory is not based on an explicit operator, which establishes a direct relationship between the energy increments and the perturbation of the general solution. The energy spaces p and together with the Theorem 1 describe how the perturbation in the power series of the energy leads to consider additional solutions based on the higher order derivatives of the primary original solution. Technically, we showed in  the inclusion of the energy sub spaces i. Note that our formulation of the energy parallax is at the moment restricted to functions in , and thus finite energy function.

Thus, every function general solution of the PDE, additional solutions defined on the energy spaces, superposition of all the solutions should be finite energy function to guarantee that they are in .

Perturbation theory may be difficult to implement when the system is described by a set of PDEs. The term field is first coined by M. Faraday in The work of J. Maxwell leaded to the discovery of the propagation of EM waves .

## Frontiers | A Perturbative Approach to the Tunneling Phenomena | Physics

A turning point is the introduction of the special theory of relativity by. Einstein in with no longer relationship between the speed of the observer and the velocity of the waves. Field theory becomes even more important with the development of quantum mechanics in the late s and the work of P. Dirac using the emerging theory of quantum field theory to explain the energy decay of an atom between different quantum states . Let us recall an example of variation of EM energy density in the skin layer of a conductor.

The theory of energy space is now applied to the possible variations of electromagnetic energy density due to, for example, skin depth effect  inside some conductive material. Beyond this application, the interest is to give a physical meaning of taking into account those additional solutions in various energy spaces.

Thus, let us formulate the variation in time of energy density u at the second order with a Taylor series development such as:. Note that at the first order. The higher order terms are based on the assumptions that the. As discussed before, those solutions are finite energy functions and in i. To recall Section 2, the definition of the energy space , we can state in.

• !
• The NeWS Book: An Introduction to the Network/Extensible Window System.
• !
• GRE CBT?
• !
• .
• Relativity The Special and the General Theory!

Here f is either the electric or magnetic field i. With the concept of multiplicity of the solutions e.

Lecture 27: Bounded linear operators in L_p

If g is a general solution of some linear PDEs, then can be identified as a special form of the solution conditionally to its existence. Now considering the wave equation, the electric field and magnetic fields are solutions and belong to the subspace and associated with the variation of energy density. Furthermore, we can consider the solutions in associated with the variation of energy density , which can be explained with the concept of multiplicity of the solutions. The solutions of interest in are for the electric field and the magnetic field.

The Taylor Series development of the energy of for example the electric field on a nominated position in space i. Therefore, taking into account the second order term of the energy density means that additional solutions should also be considered in the EM modeling. We are taking the example of the variation of EM energy density inside a copper wall due to planar waves reflecting and refracting on it . To recall the previous example, the EM field is now including , and , , contribution of the subspaces and respectively when using the concept of multiplicity of the solutions i.

Theorem 1 for the higher order derivatives of the energy density see We call the total EM field and inside the copper plate skin layer with associated permittivity and permeability. They are solutions of the Maxwell equations:. Now, the variation of energy density 14 together with the equation of charge conservation is formulated such as:.

Now, writing , and is the first derivative in time i. The Poynting vector is defined as and its derivative. Thus, the second order term of the energy density is the contribution of the EM field generated by and is:. The last line is the contribution from only the fields and. Finally, the creation of the wave defined by the EM field , means that some material properties may allow to create two type of EM waves namely , and ,. In a broad sense, uncertainty principles are a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties describing a system, known as complimentary variables e.

So far in our comparison between the energy parallax and the perturbation theory, the development is based on the mathematical properties of the functions in i. In particular, some examples uses special type of waves i. Here, we show with the uncertainty principles, how the variables in our system i.

Let us define the electric field E function in with variables in time t and space x. Following the definition of , one can write. Associating the quantities and with and  , one can write the uncertainty principle in time and frequency,. Theorem 1 , the broadening of the frequency spectrum could be interpreted as additional waves with larger or shifted frequency bands. Finally, if we want to look at the inequality involving the position x, one needs to use the wave-particle duality and consider the wave as a photon.

In this case, we can use the Heisenberg uncertainty principle in quantum mechanics to state the relationship between x and the moment p . The Woodward effect, also referred to as a Mach effect, is part of a hypothesis proposed by James F. Woodward in . The hypothesis states that transient mass fluctuations arise in any object that absorbs internal energy while undergoing a proper acceleration.

Recently, the Woodward effect was applied to asymmetric EM cavities i. The Woodward effect is based on a formula which the author implicitly assumed that the rest mass of the piezoelectric material via the famous Einstein's relation in special relativity the rest energy associated with the rest mass m and its variation via electrostrictive effect. In order to apply this formula to an asymmetric EM cavity, the author in  formulated the hypothesis that the EM excitation on the walls creates electric charges i.

The Woodward effect can be mathematically derived in various ways   . Note that in the appendices, we also show a derivation based on the model of a point mass particle moving in a varying electric field.

If we define the mass density such as , then from  , one can write the elementary mass variation per unit of volume. Let us define the the rest energy , then. In some particular cases such as an EM cavity, we assume that the variation in time of the rest energy is equal to the variation of EM energy density u i.

It allows then to state the relationship between the Woodward effect and the EM energy density. The EM energy density u follows the general definition of the sum of energy density from the electric and magnetic fields . Note that in  , the author defines the Electro magnetic and gravitational coupling using equation Discussion 2 : The above equation shows that the variation of mass density is a linear relationship with the first and second derivative of the EM energy density.

To recall Example 2 in Section 3. As we are dealing with evanescent waves functions in with , with 3D space and time in the skin layer of the EM cavity, we can apply the results of Example 2 with the multiplicity of the solutions i. The interpretation of the Woodward effect using the energy parallax is that the solutions are in k in using the same definitions as in Section 2. In other words, we need to take into account the evanescent waves associated with the electric and magnetic fields and their first and second derivative in time.

This work is a discussion on the energy parallax and the comparison with the perturbation theory. One of the motivation is that the energy parallax is based on the multiplicity of the solutions i. Theorem 1 developed by  for the functions in the , i. Note that we give some meaning to the variation of energy via the uncertainty inequality time, frequency based on the superposition of waves using the energy parallax.

The perturbation theory is well defined when the system can be described with an operator e. However, complex systems using multiple operators or various PDEs may be best described in terms of the variation of the total energy. In this way, the energy parallax can be seen as an alternative. In the first example, the energy parallax is applied to the evanescent waves in the skin layer of a dielectric material i.