**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 [1–3]. 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 [4–7]. 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 [8–10] 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):

$$\begin{aligned} u^{\mathfrak{m}}(\mathbf{r}, \boldsymbol{\omega}) &= u^{\mathfrak{m}(\circ)}(\mathbf{r}, \boldsymbol{\omega}) \\ &+ \lambda(\boldsymbol{\omega}) \int\_0^\infty \mathbf{K}\_n^{\mathfrak{m}(\circ)}(\boldsymbol{\omega}; \mathbf{r}, \mathbf{r}') u^n(\mathbf{r}', \boldsymbol{\omega}) d\boldsymbol{r}' \end{aligned} \tag{1}$$

**3. Connection between the eigenvalues** *η<sup>e</sup>* **and the function***λ*

our previous results [1], the solutions of Eq. (3) that is all the *y<sup>m</sup>*

functions appearing in Eqs. (1) and (3) as follows:

*On the Zap Integral Operators over Fourier Transforms DOI: http://dx.doi.org/10.5772/intechopen.94573*

And also

The orthogonality relation is

**G***<sup>n</sup>*ð Þ<sup>∘</sup>

*um*ð Þ¼ **<sup>r</sup>**, *<sup>ω</sup> um*ð Þ<sup>∘</sup> ð Þ **<sup>r</sup>**, *<sup>ω</sup>*

�X<sup>∞</sup> *e*¼1

∞ð

0 *ym*

which must comply with an integral equation:

Two other conditions must be satisfied: The first is that Fredholm determinant is zero

functions (and operators):

**3**

*<sup>M</sup><sup>m</sup>*ð Þ¼ **<sup>r</sup>**, **<sup>r</sup>**0; *<sup>ω</sup> η ω*ð ÞΔð Þ *<sup>η</sup>*,*<sup>ω</sup>*

The second that the Fredholm eigenvalue equals to one:

<sup>þ</sup> *η ω*ð Þ <sup>ð</sup>

∞

0

But thanks to our second order approximation Eq. (2) we can show that other

<sup>Ψ</sup>ð Þ� **<sup>r</sup>**;*<sup>ω</sup> <sup>M</sup><sup>m</sup>*ð Þ� **<sup>r</sup>**, **<sup>r</sup>**0; *<sup>ω</sup>* <sup>Δ</sup>ð Þ *<sup>η</sup>*, *<sup>ω</sup> um*ð Þ **<sup>r</sup>**;*<sup>ω</sup>* (10)

interesting conditions are satisfied, for example if we define some particular

ð ∞

*ym <sup>e</sup>* ð Þ **<sup>r</sup>**,*<sup>ω</sup> <sup>y</sup><sup>n</sup>*

0

*<sup>e</sup>* ð Þ **<sup>r</sup>**; *<sup>ω</sup> Amnyn*

**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,

We know that because of the Hilbert-Schmidt theory [2, 3] and more recently by

orthogonal functions and then a set of eigenvalues *ηe*ð Þ *ω* **.** Thus we can relate the

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

*e Ce ym <sup>e</sup>* ð Þ**<sup>r</sup>** *<sup>y</sup><sup>n</sup> <sup>e</sup>* ð Þ**s** *λ* � *η<sup>e</sup>*

*<sup>e</sup>* **r**<sup>0</sup> ð Þ ,*ω λ ω*ð Þ� *<sup>η</sup>e*ð Þ *<sup>ω</sup> <sup>u</sup><sup>m</sup>*ð Þ<sup>∘</sup> **<sup>r</sup>**<sup>0</sup> ð Þ ;*<sup>ω</sup> dr*<sup>0</sup>

*<sup>i</sup>* ð Þ **r**;*ω dr* ¼ 0 if*i* 6¼ *e* (6)

**<sup>K</sup>***<sup>m</sup>*ð Þ<sup>∘</sup> *<sup>n</sup>* ð Þ *<sup>ω</sup>*; **<sup>r</sup>**, **<sup>s</sup>** *<sup>M</sup><sup>n</sup>*ð Þ **<sup>s</sup>**, **<sup>r</sup>**0;*<sup>ω</sup> ds*

Δð Þ¼ *η*, *ω* 0 (8)

*η ω*ð Þ¼ 1 (9)

*<sup>m</sup>* ð Þ¼ *<sup>ω</sup>*; **<sup>r</sup>**, **<sup>s</sup>** <sup>X</sup>

*<sup>e</sup>* ð Þ **r**; *ω* , form a set of

(4)

(5)

(7)

Where the kernel **<sup>K</sup>***<sup>m</sup>*ð Þ<sup>∘</sup> *<sup>n</sup> <sup>ω</sup>*; **<sup>r</sup>**, **<sup>r</sup>**<sup>0</sup> ð Þ, is the product of the interaction *Am <sup>t</sup>* ð Þ *ω*; **r**, **s** (may be a non-local potential) with the free Green function **G***<sup>t</sup>*ð Þ<sup>∘</sup> *<sup>n</sup>* ð Þ *ω*; **r**, **s** .

And we make the ansatz of two successive approximations (a second order approach) [9], by the consideration that *λ ω*ð Þ is a number with a very small absolute value (j j *λ ω*ð Þ ≪ 1), we arrive to the integral equation we named the GIFE:

$$\begin{split} \mathbf{s}^{m}(\mathbf{r}, \boldsymbol{\omega}) &= \mathbf{s}^{m(\circ)}(\mathbf{r}, \boldsymbol{\omega}) + \boldsymbol{\Theta}^{m}(\mathbf{r}, \boldsymbol{\omega}) \\ &+ \nu(\boldsymbol{\omega}) \int\_{0}^{\infty} \mathbf{K}\_{n}^{m(\circ)}(\boldsymbol{\omega}; \mathbf{r}, \mathbf{r}') \mathbf{s}^{n}(\mathbf{r}', \boldsymbol{\omega}) d\boldsymbol{r}' \end{split} \tag{2}$$

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]:

$$\chi\_{\epsilon}^{m}(\mathbf{r};\alpha) = \eta\_{\epsilon}(\alpha) \int\_{0}^{\infty} \mathbf{K}\_{n}^{m(\circ)}(\alpha; \mathbf{r}, \mathbf{r}') \chi\_{\epsilon}^{n} dr' \tag{3}$$

Eq. (3) has a special index *e* that mean a specific resonance [1, 4, 5, 8, 9]. Among the three Eqs. (1), (2), and (3) there are a common ingredient, for each equation we have used different names: *λ*, *ν* and *η* [1–3] but any of them can be incorporated to the kernel or used as an independent function or even an eigenvalue. In order to connect the homogeneous and inhomogeneous equation we must define some functions as we will see in the next sections.
