## Abstract

We devote the current chapter to describe a class of integral operators with properties equivalent to a killer operator of the quantum mechanics theory acting over a determined state, literally killing the state but now operating over some kind of Fourier integral transforms that satisfies a certain Fredholm integral equation, we call this operators Zap Integral Operators (ZIO). The result of this action is to eliminate the inhomogeneous term and recover a homogeneous integral equation. We show that thanks to this class of operators we can explain the presence of two extremely different solutions of the same Generalized Inhomogeneous Fredholm equation. So we can regard the Generalized Inhomogeneous Fredholm Equation as a Super-Equation with two kinds of solutions, the resonant and the conventional but coexisting simultaneously. Also, we remember the generalized projection operators and we show they are the precursors of the ZIO. We present simultaneous academic examples for both kinds of solutions.

### Keywords

- integral operators
- generalized inhomogeneous Fredholm equations
- killer operators
- evanescent waves
- electromagnetic resonances

## 1. Introduction

Recently a new question about the solutions of integral Fredholm emerges, that is the question about the type of equation each of them solve. If we follow the steps or the clue marked by the linear second order differential equations the solutions of the inhomogeneous equation do not solve de homogeneous equation. But we have shown in a recent paper that both kind of solutions of the homogeneous and also the inhomogeneous Fredholm equations satisfy a third class of integral equation we named the Generalized Inhomogeneous Fredholm Equation (GIFE) which is only a bit different for the traditional inhomogeneous [1, 2, 3]. Even more, we can transform his appearance in a continuous form from homogeneous to inhomogeneous, but preserving his very extraordinary property: the two kinds of solutions are simultaneous solutions. This situation is quite different from differential equations but not the connection between eigenfunctions and solutions of inhomogeneous equations through the Green function [4, 5, 6, 7]. And if we want to explain this behavior we find a founder: an integral operator which is hidden in the structure of the GIFE. There is no surprise in the fact that the new operator treats in different manner both kinds of solutions. Indeed, it seems to be natural that the new operators include the Green function and are close to the Fredholm operator [2, 3]. Before we define the ZIO operators we must underline the fact that in a broadcasting situation [8, 9, 10] we must take into accounts not only one kind of traveling waves but all the known ones because the complete description of the phenomena comes from the GIFE. Another important goal of this paper is to give an explanation of the simultaneous validity of two sets of boundary conditions that are very apart one to the other and the fact that there is a connection with other projection operators, the generalized projection operators (GPO) [11] that separates the constituents of a signal in orthogonal parts.

## 2. Remembering the GIFE

We remember that if we take the inhomogeneous vector integral Fredholm Eq. (1):

Where the kernel

And we make the ansatz of two successive approximations (a second order approach) [9], by the consideration that

This last equation is the one that have the property of represent a complete panorama in a broadcasting problem, that is describes both the resonant and the conventional behavior of the electromagnetic field [12].

As we have commented, Eq. (2) carries a mechanism that allows simultaneously consider both types of solution. The so called generalized source is indeed a blend of integral operators as we will see with properties we want to visualize. But first we must present the Generalized Homogeneous Fredholm Equation (GHFE) [1]:

Eq. (3) has a special index

## 3. Connection between the eigenvalues η e and the functionλ

We know that because of the Hilbert-Schmidt theory [2, 3] and more recently by our previous results [1], the solutions of Eq. (3) that is all the

By means of the spectral representation of Green function, [2, 3] we have:

And also

The orthogonality relation is

## 4. Conditions imposed over the homogeneous Fredholm equations

In accordance with the theory of homogeneous Fredholm integral equations [1, 2, 13], the first Fredholm minor is a two point function, like a Green function, which must comply with an integral equation:

Two other conditions must be satisfied:

The first is that Fredholm determinant is zero

The second that the Fredholm eigenvalue equals to one:

But thanks to our second order approximation Eq. (2) we can show that other interesting conditions are satisfied, for example if we define some particular functions (and operators):

And also

We can see that the first Fredholm minor must satisfy through Ψ the inhomogeneous equation

In order to write Eq. (2) in terms of the solutions of Eq. (1), we can define the operator:

In Eq. (13) the function

Near a resonance the two small parameters

## 5. Defining a new class of integral operators

As we said in Section **1**, hidden in the structure of Eq. (2) there are some integral operators which allow the simultaneous existence of solutions with extremely different boundary conditions. So, let us define the Zap operators by the rules:

That is, the Zap operator is associated to the integral Fredholm equation satisfied by the affected solution (

The same operator (15) acting over a homogeneous equation looks like

That is

As we can see the effect of the Zap operator is to kill or eliminate the inhomogeneous term when applied to a resonant state. But this seems very artificial because we are giving indeed two parts for the complete rule. However we can build projection operators that can make the work we need.

Now, we define Zap projection operators in the next section.

## 6. The zap projection operators and their properties

On this section, we define the so named Zap projection operators (ZPO) which enable us to project a complex broadcasting system over a reduced resonant simplest one. The Zap operators acts over Fourier transforms [14, 15] related to integral operators.

hen, based on (15) and (17), we define de following operator:

In order to get a display of the properties of this operator we propose a specific set of discrete antennas in the next example:

Suppose that we have

Then, by applying the projection operator to Eq. (19) we have (remember that when

Now, because

In the last step we have used the fact that the solution of the remaining homogeneous equation is denoted by

Eq. (21) says that if we take a blend of regular and resonant solutions we have:

So taken into account from Eqs. (18) until (22), we see that we have projected the original problem into a resonant one.

In analogy with

Let us define the complementary Zap projection operator as

Even we apply

This is because the name of the solution of the remaining inhomogeneous equation is precisely

## 7. An academic example for conventional traveling waves

In order to convince us of the utility of the

For the case of only two source points and omitting the three components of the field lifting only one, this matrix can be for example:

But first remember that

In Eq. (25)

In Eq. (26)

And then, the kernel may be

So Eq. (19) takes the form:

Where

That is

The conventional waves satisfy the scalar form of Eq. (1)

Or in accordance with Eq. (25)

Where the form of

Now we can apply the projection operator

Then, by putting

So it is irrelevant the part of the problem concerning the two sources, it is only a problem about resonances. Our problem is now to find the resonant frequencies by taking

But, what is the real advantage of the

Now the last condition over the Fredholm determinant is

The parameters

Then, Eq. (36) has two resonances:

And

## 8. Forerunners of the zap projection operators

In Section **6** we defined a new class of integral operators we named Zap projection operators that literally cleans from a broadcasting problem the inhomogeneous part and leaves a projected homogeneous version. These operators act directly over an inhomogeneous Fredholm equation and are related to the Fredholm operators. But recently, we have defined another set of operators we called generalized projection operators (GPO) which projects a complete broadcasting signal (maybe described by a GIFE) not only into several independent mutually orthogonal signals but also can reverse the time direction as we wish. These GPO may be considered as the precursors of the Zap operators and we will see why. We remember their form:

And

In Eqs. (39) and (40)

Denoting the Fourier transform like

Then, the Fourier transform of the GPO is

Where

And for the convolution we have

Then we see that the set of Fourier transforms of the GPOs behaves like a set of orthogonal basis functions for the frequency domain, that is, the resonant functions

## 9. Conclusions

We can conclude that the Zap projection operators (ZPO) can be used as an alternative approach to the generalized projection operators (GPO) that is like an alternative for clean the evanescent signals [10] from disturbances generated by the sources and at the same time to clean the source signals from resonant solutions. We can also use the two classes of projectors in a consecutively manner. The former vision suppose that the evanescent waves [10] can be considered as part of the conventional traveling waves like an everything and that we must take away the effect of the resonances with the application of the