# Transition Probabilities and Almost Periodic Functions††thanks:  Mathematics
subject Classification: Primary [$30F40$, $20M25$]; Secondary [$20H10$,
$20N05$, $17D05$, $16S34$]. Keywords and phrases: generalized, probability,
transition, support, Fock space, almost periodic function.

Juriaans, S. O  Queiroz, P.C.111IME-USP and UFMA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study transition probabilities of generalized functions as introduced by
Colombeau and Gsponer. We formally introduz the study of H. Bohr almost
periodic functions in the generalized context and use them to give exact
values of transition probabilities in dimension $\ 2$. We also prove the
existence of transition probabilities, in the generalized sense, for any
moderate net $\ T=(T_{\varepsilon})$ with $\
T_{\varepsilon}:\mathbb{H}\longrightarrow\mathbb{H}$, $\mathbb{H}$ a Hilbert
space. In particular, we consider the case of selfadjoint Hilbert-Schmidt
operators.

This alludes to the possibility of existence of transition probabilities in
the Fock space, in the sense introduced by Colombeau and Gsponer, if the
spectrum of the operators involved are pure infinities or infinitesimals. This
was already indicated in two papers coauthored by J. Aragona and J.F.
Colombeau et. al.

##  1 Introduction

Schwartz’s Theory of Distributions is one of his major contribution to the
Theory of Generalized Functions. It marked major advances in the field of
P.D.E.’s. Being a linear environment, it does not deal with problems involving
products of distributions. Although there still exist the believe that
Schwartz’s Impossibility Theorem alludes to the non-existence of an
associative diferencial milieu in which these products can be dealt with, this
is certainly not the case. In fact, such environments do exists and have more
than proven their consistency and importance (see [21, 22, 23, 24, 25, 26, 27,
28, 29, 38, 39, 40, 41, 43, 49, 51, 52, 53, 54, 57, 61, 64, 65, 66, 67, 68]).
It was in the eighties that Rossinger and Colombeau undertook the challenge
and started to develop non-linear theories in which the multiplication of
distributions make sense. Important researchers such as Aragona, Biagioni,
Egorov, Grosser, Kunzinger, Oberguggenberger, Pilipovic, Scarpalézos,
Steinbauer and Vickers, to name a few, made possible the development of the
algebraic and topological theory of these new generalized environments. The
algebraic developments were key in permitting to view the real potential of
the generalized milieu. In particular, the boolean algebra $\
{\mathcal{B}}(\overline{\mathbb{K}})$ of $\ \overline{\mathbb{K}}$ is key in
this development together with the group of invertible elements $\
Inv(\overline{\mathbb{K}})$. The latter is open and dense in $\
\overline{\mathbb{K}}$ and its complement consists of zero divisors each of
which has a nontrivial element of $\ {\mathcal{B}}(\overline{\mathbb{K}})$ in
its annihilator, i.e., given a non unit $\ x\in\overline{\mathbb{K}}$ there
exists a nontrivial idempotent $\ e\in{\mathcal{B}}(\overline{\mathbb{K}})$
such that $e\cdot x=0$. It also holds that an element $\
x\in\overline{\mathbb{K}}$ is a unit if and only if there exists $\
n\in\mathbb{N}$ such that $\ x\geq\alpha^{n}$, where $\
\alpha=[\varepsilon\longrightarrow\varepsilon]$ is the natural, or standard,
gauge of $\ \overline{\mathbb{K}}$. Each idempotent is uniquely linked to a
certain kind of subset in the parameter space defining these generalized
milieu. The idempotents are crucial and important in the definition of the
notion of interleaving (see [60]), the existence of roots for generalized
polinomial functions (see [45, 81]) and the notion of the support and the
existence of an atlas for a generalized differential manifolds (see [46]) as
we shall see next. The connection with non standard Analysis (see [63]) is
clear but in a very different way than expected (see [33]).

These developments led to the introduction of a generalized milieu which
extends in a natural way the classical Newtonian Calculus. This is called
Generalized Differential Calculus (see [4, 7, 8, 9, 12, 46, 47]). Many
Classical results do hold in this generalized environment. For example, one
has a Generalized Fixed Point Theorem which is useful when proving existence
of solution for equations in the the generalized environment (see [46]).

Generalized Differential Calculus is extended in the following way. Given a
classical differential manifold $\ M$ there exist a Generalized Differential
Manifold $\ M^{*}$ into which $\ M$ embeds discretely. Problems involving
distributions on $\ M$ translates naturally into questions involving $\
C^{\infty}$ functions on $\ M^{*}$. These new environments are as natural as
the classical ones and several classical results have an analogue in the
general environment making it possible to prove most results intrinsically.
But they have a feature which exists in the classical environments:
infinitesimals and infinities coexists in these milieu and effectively effects
physical reality although going undetected by this physical reality (see
[46]).

Another important notion, introduce recently in [46], is the notion of
supports of generalized objects. For genereralized functions there is such a
notion but it refers more to the domains of the functions. Here the idea is to
look at compactly supported elements or elements which, when multiplied by an
idempotent, becomes compactly supported and are in the halo of point of
physical reality.

Colombeau and Gsponer, see [28, 31], introduced the notion of transition
probabilities in their attempt to establish a theory to deal with certain
problems in QFT. Since then, transition probabilities were studied in [3, 6,
31, 42] among to carry out the blue print layout in [28, 31]. Hopefully this
will turnout to be another major contribution of J.F. Colombeau in the
understanding of physical reality. In [5, 6] progress was made and in [6] some
questions raised remained unanswered. The existence of transition
probabilities of $\ 2\times 2$ matrices were simulated using MatLab indicating
their existence and making progress in the pursue of the ideas of Colombeau
and Gsponer. In [5] light is shed on the general case showing that it is
feasible to carry out the proposed blue print mentioned before. In this
manuscript, we first generalized the notion of transition probability
introducing the generalized transition probability and proving the existence
of the latter for any symmetric matrix in $\ M_{n}(\overline{\mathbb{C}})$. We
first prove this for elements of $\ M_{n}(\overline{\mathbb{C}}_{as})$ and
then go on to prove this in the general case. We revisit the examples given in
[6] and give explicit calculation of the values involved. We also look at
compactly supported symmetric operators acting on Hilbert spaces and establish
results analogous to the finite dimensional case.

The manuscript is organized as follows. In the next section, the basics of the
Theory of Generalized Functions and Generalized Differential Calculus is
recalled to give the reader a good overview of the status quo. Notation is
recalled and established aiming to facilitate the reading of the manuscript.
Next, the basics of almost periodic functions is collected and extended to the
generalized environment. In particular, an a.p.f. may become a periodic
function in the generalized environment. In the fourth section, the basics of
Fock Space and the generalized context constructed by Colombeau and Gsponer
are recalled. The generalized transition probability is defined having the
original definition as a particular case. In section 5, generalized transition
probabilities of finite matrices is studied. The focus is on compactly
supported matrices. It is proved that in this case the generalized transition
probability always exists and its support is calculated. The infinite
dimensional case is considered, in particular, Hilbert-Schmidt operators and
compactly supported self-adjoint operators are considered. In the last
section, the examples given in [3, 6] are revisited and the exact value of all
the transition probabilities are given. A satisfactory solution is given in
the finite dimensional case. The infinite dimensional case is revisited and
some results about the generalized transition probability are proved in this
case.

##  2 Preliminaries

We refer the reader to the following excellent texts on the Theory of
Colombeau Generalized Functions: [2, 21, 22, 23, 25, 26, 27, 28, 32, 49, 56,
57, 61, 60, 70, 77, 78, 82, 83, 84]. For the algebraic part of the theory we
refer the reader to [10, 12, 13, 45, 49, 80, 81]. The reader interested in the
foundations of the Generalized Differential Calculus we refer to [4, 7, 46].
The topological foundation of the theory can be found in [8, 9, 12, 18, 69,
70, 71]. For applications, more on the theory and physics we refer the reader
to [1, 10, 11, 14, 20, 34, 35, 38, 39, 40, 41, 43, 48, 85]. Here we shall
recall part of the theory in the form and sequel that we shall be using it in
this paper.

Given a field $\ \mathbb{K}\in\\{\mathbb{R},\mathbb{C}\\}$ consider the
product space of nets $\ \mathbb{K}^{I}$, where $\ I=]0,1]$. We call $\
\alpha=[\varepsilon\longrightarrow\varepsilon]$ the natural gauge and if
$x,y\in\ \mathbb{R}^{I}$ we write $\ x\ \leq y$ iff there exists $\ \eta>0\in
I$ such that $x(\varepsilon)\leq y(\varepsilon),\ \forall\
\varepsilon\in]0,\eta]$. The set of moderate nets of $\ \mathbb{K}^{I}$ is $\
\mathcal{E}_{M}(\mathbb{K})=\\{x\in\ \mathbb{K}^{I}\ :\ $ there exists $\
n\in\mathbb{Z}$ such that $|x|\leq\alpha^{n}\\}$, where
$|x|(\varepsilon)=|x(\varepsilon)|$. It is clear that $\
\mathcal{E}_{M}(\mathbb{K})$ is a subring of $\ \mathbb{K}^{I}$ and that
$\mathcal{N}(\mathbb{K})$ $=\\{x\in\mathcal{E}_{M}(\mathbb{K})\
:|x|\leq\alpha^{q},\forall q\in\mathbb{N}\\}$ is an ideal of this subring. The
Colombeau ring of generalized numbers over $\ \mathbb{K}$ is the quotient ring
$\ \overline{\mathbb{K}}=\mathcal{E}_{M}(\mathbb{K})/\mathcal{N}(\mathbb{K})$.
It holds that $\ \overline{\mathbb{C}}=\ \overline{\mathbb{R}}+i\
\overline{\mathbb{R}}$, $i^{2}=-1$. The ring of Colombeau Generalized real
numbers $\ \overline{\mathbb{R}}$ will be the basic under lying structure for
our generalized differential Calculus, Geometry and Analysis. It is an
ultrametric topological ring. This topology is defined in [18, 69, 70, 71] and
was coined the sharp topology. The norm of an element $\
x\in\overline{\mathbb{R}}$ is $\ \|x\|=e^{-V(x)}$, where $\ V(x)$ is the
supremum of the set $\ \\{r\in\mathbb{R}\ :\ |x|\leq\alpha^{r}\\}$. It is
interesting to note that the classical equality $\ H^{2}=H$ does not hold in
the generalised milieu, i.e., the Heaviside Function $\ H$ is not an
idempotent! The latter is a statement about the perception of our physical
reality.

The algebraic properties of this ring were first investigated in [12]. For
examples, all maximal ideals of $\ \overline{\mathbb{R}}$ where described, the
boolean algebra $\mathcal{B}(\overline{\mathbb{R}})$ was described and proved
that its group of invertible elements $\ Inv(\overline{\mathbb{R}})$ is open
and dense in the sharp topology. The elements of
$\mathcal{B}(\overline{\mathbb{R}})$ are elegantly linked to subsets of the
parameter space $\ I$ (see [12, 33]). Moreover, if $\
0\not=x\in\overline{\mathbb{K}}$ then there exists $\
e\in\mathcal{B}(\overline{\mathbb{R}})$ such that $\ ex\in
Inv(e\overline{\mathbb{R}})$. Note that if $\
e,f\in\mathcal{B}(\overline{\mathbb{R}})$ are such that $f<e$ then $\ ef=f$.
An excellent reference is [80] (see also [13, 33]).

In [7, 8, 9] a basis for this topology is given that is compatible with the
algebraic structure. A basic neighbourhood is $\
V_{r}(0)=\\{x\in\overline{\mathbb{R}}\ :\ |x|<\alpha^{r}\\}$, $r>0$. An $\
x\in V_{1}(0)$ is such that $\ |x|<s,\forall\ s\in\mathbb{R}^{+}$. On the
other hand, $\alpha^{-1}>s,\forall\ s\in\mathbb{R}$. Consequently, in this new
milieu infinitesimals and infinities coexist and can cancel out when
multiplied.

Given $\Omega\subset\mathbb{K}^{n}$, let
$\widetilde{\Omega}_{c}=\\{x=(x_{1},\cdots,x_{n})\in\overline{\mathbb{K}}^{n}\
:\ \exists\ \eta>0,\\{\hat{x}(\epsilon)\ :\
\epsilon<\eta\\}\subset\subset\Omega\\}$, where $\ \subset\subset$ means that
the set has compact topological closure. One can prove that $\ \Omega$ is
discretely embedded in $\ \widetilde{\Omega}_{c}$. This is one of the most
important definitions in the recent developments of the Theory and was given
by Kuzinger-Oberguggenberger (see [49, 50]). The importance of this definition
is the following results proved by them: A Colombeau Generalized Function is
zero if and only if its point values on $\ \widetilde{\Omega}_{c}$ are all
zero in $\ \overline{\mathbb{R}}$ (see [49, 50]). This actually translate into
having a natural candidate for a domain for a generalized functions defined on
$\ \Omega$. Together with the algebraic developments, this is one of the
stepping stones for our Generalized Differential Calculus (see [4]).

A function $f\in\mathcal{F}(\widetilde{\Omega}_{c},\overline{\mathbb{K}})$ is
said to be differentiable if there exists $z_{0}\in\overline{\mathbb{K}}$ such
that

$\lim_{x\to
x_{0}}\frac{f(x)-f(x_{0})-z_{0}(x-x_{0})}{\alpha_{-\log(\|x-x_{0}\|)}}=0$

Denoting this limit by $f^{\prime}(x_{0})$, one shows that this is a
derivation on $\mathcal{F}(\widetilde{\Omega}_{c},\overline{\mathbb{K}})$
which satisfies almost all properties of classical differential calculus and
we have the following embedding theorem. This theorem is a complete solution
to the apparent Schwartz Impossibility Paradox in the sense of Newtonian
Calculus.

######  Theorem 2.1 (Aragona-Fernadez-Juriaans)

There exists a continuous $\ \mathbb{K}-$ linear injection
$\kappa:D^{\prime}\to
C^{\infty}(\widetilde{\Omega}_{c},\overline{\mathbb{K}})$ such that
$\kappa\Big{(}\frac{\partial f}{\partial
x_{i}}\Big{)}=\frac{\partial(\kappa(f))}{\partial x_{i}},~{}\forall~{}f\in
D^{\prime},\forall i$.

Generalized Calculus is further developed in [7] and is the basis for the
Generalized Differential Geometry and a Fixed Point Theorem in this
generalized milieu (see [46]).

The subset $\ {\overline{{\mathbb{K}}}}_{0}:=\\{x\in{\overline{{\mathbb{K}}}}\
|\ x\approx 0\\}$ is a subring of ${\overline{{\mathbb{K}}}}$ containing the
open ball $B_{1}(0)$ properly (see [12, Proposition 2.1]). More generally,
${\overline{{\mathbb{K}}}}_{as}:=\\{x\in{\overline{{\mathbb{K}}}}\ |\ x\approx
x_{0}\in\mathbb{R}\\}$ is a subring of ${\overline{{\mathbb{K}}}}$ containing
the open ball $B_{1}(0)$. In this paper we consider matrices over
${\overline{{\mathbb{K}}}}_{as}$ and consider also the case when $\
\lim\limits_{\varepsilon\longrightarrow 0}\hat{x}(\varepsilon)=\infty$,
coining the name pure infinities to such elements. Our standard reference for
matrices over ${\overline{{\mathbb{K}}}}$ will be [45, 81]. It should be noted
however that this subject was first studied in [55, 81]. The notion of
eigenvalues and eigenvectors is as defined in [45]. Examples in [45] show that
one must be careful in the generalized environment when defining these
notions. For the readers sake, we recall some basics needed in the sequel.

A vector $(v_{1},\cdots,v_{n})=v\in{\overline{{\mathbb{K}}}}^{n}$ is a free-
vector if $\ \
span_{{\overline{{\mathbb{K}}}}}[v_{1},\cdots,v_{n}]={\overline{{\mathbb{K}}}}\
$ and a generalized number $\lambda\in{\overline{{\mathbb{K}}}}$ is an
eigenvalue of an $n\times n$ matrix $A\in M_{n}({\overline{{\mathbb{K}}}})$ if
there exists a free-vector $v\in{\overline{{\mathbb{K}}}}^{n}$ such that
$A\cdot v=\lambda v$. In this case one has that $det(\lambda
I-A)=0\in{\overline{{\mathbb{K}}}}$, $Ker(\lambda I-A)$ is non-trivial and
contains a free vector. Note that $v\in{\overline{{\mathbb{K}}}}^{n}$ is a
free-vector if and only if $\|v\|\in Inv({\overline{{\mathbb{R}}}})$.

We now recall the definition of the support of a generalized object given in
[46] (see [46] for more details). It can be defined in any generalized mulieu,
in special if the nets involved are membrane like (see [7] for the definition
of a membrane). For $p=[(p_{\varepsilon})]\in\overline{\mathbb{C}}^{n}$,
consider the set $\\{q\in\mathbb{C}^{n}\ :\exists\ \varepsilon_{n}\rightarrow
0,\ p_{\varepsilon_{n}}\rightarrow q\\}$. Algebraically this can be written
as: Given $q_{0}\in\mathbb{C}^{n}$, we have that $\
q_{0}\in\\{q\in\mathbb{C}^{n}\ :\exists\ \varepsilon_{n}\rightarrow 0,\
p_{\varepsilon_{n}}\rightarrow q\\}$ if and only if there exists
$e\in{\cal{B}}(\overline{\mathbb{R}})$ such that $e\cdot p\approx e\cdot
q_{0}$ (extending the notion of association to $\overline{\mathbb{K}}^{n}$ in
the obvious way). This is a compact subset of $\ \mathbb{C}^{n}$ to which we
shall refer to as the support of the point $p$ and denote it by $supp(p)$. It
follows that there exists a complete set of orthogonal idempotents
$(e_{\lambda})$ such that

$p\ =\sum\limits_{x_{\lambda}\in supp(p)}e_{\lambda}\cdot p$
$(e_{\lambda}\cdot p\approx e_{\lambda}\cdot x_{\lambda})$

This definition can be extended to any generalized milieu and we shall do so
here in this manuscript. It can obviously be extended to include $\ \infty$ in
the case of generalized numbers. A generalized number $\ \lambda$ is an
infinity if $\ \infty\in supp(\lambda)$ and is a pure infinity if $\
supp(\lambda)=\\{\infty\\}$.

The notion of interleaving given by Oberguggenberg-Vernaeve (see [60]) once
again shows the importance of the elements of $\
\mathcal{B}(\overline{\mathbb{K}})$. The interested reader can read more about
this in several sections of [46]. This notion plays an important role in
defining a generalized manifold starting with a classical manifold and
embedding the latter discretely into the former. For the notion of membrane
and its generalization, internal sets, we refer the reader to [7, 60].

##  3 Almost Periodic Functions

In this section we shall prove the existence of mean values for some
functions. It is here also that we link the transition probabilities with the
mean value of a almost periodic function. We start recalling the concept of
almost periodic function. The basic references we used are [15, 19]. This
remarkable theory seems to be have conceived to merge naturally into the
Theory of Generalized Differential Calculus. One of the main things of this
section is the fact that an almost periodic function becomes a periodic
function in the generalized context. After recalling the definition of an
almost periodic function we prove some useful ways to calcule their mean value
and then show their link to transition probabilities.

A set of real numbers $E\subset{\mathbb{R}}$ is said to be relatively dense in
$\ \mathbb{R}$ if there exists $L>0$ such that for each $a\in{\mathbb{R}}$ one
has that $]a,a+L[\cap E\neq\emptyset$. Roughly speaking, this means that there
are no arbitrary large gaps among the elements of $\ E$.

###### Definition 3.1

Given a complex valued function $f:{\mathbb{R}}\longrightarrow{\mathbb{C}}$
and $\varepsilon>0$, the real number
$\tau=\tau(\varepsilon)=\tau_{f}(\varepsilon)>0$ is called a translation
number of $f$ corresponding to $\varepsilon$ if $\
|f(x+\tau)-f(x)|\leq\varepsilon,\forall x\in{\mathbb{R}}$. A continuous
function $f:{\mathbb{R}}\longrightarrow{\mathbb{C}}$ is said to be almost
periodic ( an a.p.f.) if for each $\varepsilon>0$ the set
$\\{\tau\in{\mathbb{R}}\ |\tau=\tau_{f}(\varepsilon)\\}$ is relatively dense.

Consider the ${\mathbb{C}}$-vector space $V=Span_{{\mathbb{C}}}[e^{i\lambda
x}\ |\lambda\in{\mathbb{R}}]$ of trigonometric polynomials or pure vibrations
which include the harmonic vibrations which are the generating set of the
Wiener algebra. The elements of $V$ are bounded functions and thus we may
consider the normed space $(V,||\ ||_{\infty})$. Let $\overline{V}$ be its
closure in $L_{\infty}({\mathbb{R}})$. Thus $\overline{V}$ consists of uniform
limits of elements of $\ V$. Note that the elements of $\ V$ are periodic
functions and thus the set of periodic functions is dense in $\overline{V}$.

With an elementary step function we mean the characteristic function of an
interval $I\subset{\mathbb{R}}$. A step function is a finite
${\mathbb{R}}$-linear combination of elementary step functions. Since one
clearly has that $M(f(x+c))=M(f)$, for any constant $c\in{\mathbb{R}}$ and
$M(f(\lambda x))=M(f)$, for any $\lambda>0$, we way consider elementary step
functions defined on an interval of the form $[0,\beta]$ with $\beta<1$. We
say that $f$ is an elementary periodic step function if $f$ has period $1$ and
on $[0,1]$ $f$ is the characteristic function of an interval $[0,\beta]$.

If $\ f$ is periodic of period $\ L$ then we may approximate $\ f$ uniformly
by a sequence of step functions on $\ [0,L]$. Hence $\ f$ can be approximated
uniformly by periodic step functions on $\ {\mathbb{R}}$. It follows that an
a.p.f. is a uniform limit of step functions on ${\mathbb{R}}$. So if we denote
by $S$ the set of periodic step functions then we have that
$\overline{V}\subset\overline{S}$. If $\ A\in M_{n}({\overline{\mathbb{C}}})$
is a hermitian matrix then $\ exp(iA)$ is a unitary matrix and hence its
eigenvalues are of the form $\ exp(i\theta),\theta\in
Spec(A)\subset{\overline{\mathbb{R}}}$. Since we are interested in transition
probabilities, which include such expressions, we look at the general
situation. Let $[f_{\varepsilon}]=f\in\overline{\mathbb{R}}$ be integrable
when seen as a function $\ f\ :\ I\longrightarrow\mathbb{R}$. Define the
generalized transition probability of $\ f$ to be

$\nu(f)=\frac{1}{\alpha}\int\limits_{0}^{\alpha}f(t)dt=\bigg{[}\varepsilon\longrightarrow\frac{1}{\varepsilon}\int\limits_{0}^{\varepsilon}f(t)dt\bigg{]}$

In particular, if$\ f\approx f_{0}\in\mathbb{R}$ then $\nu(f)=f_{0}$, as can
be easily seen. It will also follow from a result of this section. This
particular case explains our interest in almost periodic functions and their
mean value and thus linking all sections of this manuscript with the concept
of ıtransition probability defined by Colombeau-Gsponer in [28, 31]. The
results from this section show that this definition makes sense. When a matrix
or operator $\ A$ on a Hilbert space is involved, we shall write $\
\nu(A)=\nu(A,u,v)=\nu(|\langle u\mid Av\rangle|)$ without specifying the
dependence of the pair of fixed orthogonal vectors involved in the definition.
The same will be done when transition probabilities of fixed orthogonal pairs
are involved (see the next section). This definition should be compared with
the definition of $\ \mu(e),e\in\mathcal{B}(\overline{\mathbb{K}})$ given in
[46]. Since a natural way of approximating discrete data is the use of
elementary step functions, it is natural to think of idempotents stemming from
physical reality as resulting from such elementary step functions. Hence their
probability is well defined. See [46] for more information with regard to $\
\mu(e),e\in\mathcal{B}(\overline{\mathbb{K}})$.

We recall the following result fundamental results concerning Bohr’s a.p.f
(see [15, 19]).

######  Theorem 3.2 (Fundamental Theorem of A.P.F.)

A complex valued function $\ f\ $ is almost periodic if and only if $\
f\in\overline{V}$. Moreover, $\overline{V}$ is a ${\mathbb{C}}$-algebra and
its elements are bounded uniformly continuous functions.

If $f$ is a periodic function with period $L$, then it is easy to prove that

$M(f):=\lim\limits_{T\longrightarrow\infty}\frac{1}{T}\int\limits_{0}^{T}f(x)dx=\frac{1}{L}\int_{0}^{L}f(x)dx$

In particular, $M(|sin(Ax)|)=M(|cos(Ax)|)=\frac{2}{\pi}$, $\forall\ A>0$. We
shall use this in the section of examples.

######  Theorem 3.3 (Mean Value Theorem)

For every a.p.f. the mean value

$M(f):=\lim\limits_{T\longrightarrow\infty}\frac{1}{T}\int\limits_{0}^{T}f(x)dx$

exist.

Hence, an a.p.f. may not be periodic but it still has a mean value. Theorem
3.3 is the foundation for a theory of Fourier series of almost periodic
functions. In fact, given an a.p.f. $\ f$ there exist an at most an enumerable
set of vibrations $\ e^{-i\lambda x}$ such that $\ M(f(x)e^{-i\lambda x})\neq
0$ and $\ f\equiv 0$ if and only if $\ M(|f|^{2})=0$. Parseval’s Equation also
holds:

$M(|f(x)|^{2})=\sum\limits_{M(f(x)e^{-i\lambda x})\neq 0}|M(f(x)e^{-i\lambda
x})|^{2}$

We proceed to prove alternative formulas for the mean value of an a.p.f. and
by composing them with pure infinities, we introduce their generalized
transition probability int the generalized environment.

###### Lemma 3.4

Let $f$ be an elementary periodic step function and $\gamma>0$. Then

$M(f)=\lim\limits_{T\longrightarrow\infty}\gamma\cdot
T^{\gamma}\cdot\int\limits_{T}^{\infty}\frac{f(x)}{x^{1+\gamma}}dx$

Proof. Case $\gamma=1$: We first suppose that $T=n$ is an integer and hence we
may calculate its limit as
$\lim\limits_{n\longrightarrow\infty}n\cdot\int\limits_{n}^{\infty}\frac{f(x)}{x^{2}}dx$
$=\lim\limits_{n\longrightarrow\infty}n\cdot(\sum\limits_{k=n}^{\infty}[\frac{1}{k}-\frac{1}{k+\beta}])$
$=\beta\cdot(\lim\limits_{n\longrightarrow\infty}n\cdot(\sum\limits_{k=n}^{\infty}\frac{1}{k(k+\beta)}))$.
Note that
$\int\limits_{n}^{\infty}\frac{1}{u(u+\beta)}du<\sum\limits_{k=n}^{\infty}\frac{1}{k(k+\beta)}<\int\limits_{n-1}^{\infty}\frac{1}{u(u+\beta)}du$
$=\int\limits_{n-1}^{n}\frac{1}{u(u+\beta)}du+\int\limits_{n}^{\infty}\frac{1}{u(u+\beta)}du$.
Applying L’Hospital’s rule from elementary calculus we get that
$\lim\limits_{n\longrightarrow\infty}n\cdot(\int\limits_{n-1}^{n}\frac{1}{u(u+\beta)}du)=0$.
From this it follows that
$\lim\limits_{n\longrightarrow\infty}n\cdot(\sum\limits_{k=n}^{\infty}\frac{1}{k(k+\beta)})=\lim\limits_{n\longrightarrow\infty}n\cdot(\int\limits_{n}^{\infty}\frac{1}{u(u+\beta)}du)=\lim\limits_{x\longrightarrow\infty}x\cdot(\int\limits_{x}^{\infty}\frac{1}{u(u+\beta)}du)$
$=\lim\limits_{x\longrightarrow\infty}\frac{x}{\beta}\cdot\ln(\frac{x+\beta}{x})=1$,
where we once again applied L’Hospital’s rule to get the last equality.

Now if $T\in{\mathbb{R}}$ then $T=n+r$, with $0\leq r\leq 1$. We have that
$\lim\limits_{T\longrightarrow\infty}T\cdot\int\limits_{T}^{\infty}\frac{f(x)}{x^{2}}dx$
$=\lim\limits_{T\longrightarrow\infty}T\cdot\int\limits_{n+r}^{\infty}\frac{f(x)}{x^{2}}dx=\lim\limits_{T\longrightarrow\infty}n\cdot\int\limits_{n}^{\infty}\frac{f(x)}{x^{2}}dx+\lim\limits_{T\longrightarrow\infty}r\cdot\int\limits_{n}^{\infty}\frac{f(x)}{x^{2}}dx+\lim\limits_{T\longrightarrow\infty}(n+r)\cdot\int\limits_{n}^{n+r}\frac{f(x)}{x^{2}}dx$.
The integral in the middle goes to $0$, as $T$ goes to infinity (independ of
the value of $r$), and the last integral is
$\leq(n+r)[\frac{1}{n}-\frac{1}{n+r}]=\frac{r}{n}$. From this the result
follows.

Case $\gamma\neq 1$: We will suppose first that $\gamma\in{\mathbb{Q}}$ and
$T=n\in{\mathbb{N}}$. As before, we have
$\lim\limits_{n\longrightarrow\infty}\gamma\cdot
n^{\gamma}\cdot\int\limits_{n}^{\infty}\frac{f(x)}{x^{1+\gamma}}dx$
$=\lim\limits_{n\longrightarrow\infty}n^{\gamma}\cdot(\sum\limits_{k=n}^{\infty}[\frac{1}{k^{\gamma}}-\frac{1}{(k+\beta)^{\gamma}}])$
$=\lim\limits_{x\longrightarrow\infty}x^{\gamma}\cdot(\int\limits_{x}^{\infty}[\frac{1}{u^{\gamma}}-\frac{1}{(u+\beta)^{\gamma}}]du)$
$=\frac{1}{1-\gamma}\lim\limits_{x\longrightarrow\infty}x^{\gamma}\cdot((u+\beta)^{1-\gamma}-u^{1-\gamma})$.
If $1-\gamma>0$ we set $1-\gamma=\frac{p}{q}$ and otherwise we set
$\gamma-1=\frac{p}{q}$. In the first case the limit becomes
$\frac{1}{1-\gamma}\lim\limits_{x\longrightarrow\infty}x^{\gamma}\cdot(\sqrt[q]{(u+\beta)^{p}}-\sqrt[q]{u^{p}})$
and in the latter case it becomes
$\frac{1}{1-\gamma}\lim\limits_{x\longrightarrow\infty}x^{\gamma}\cdot(\frac{1}{\sqrt[q]{(u+\beta)^{p}}}-\frac{1}{\sqrt[q]{u^{p}}})$.
Rationalizing this expression in the usual way, and calculating its limit we
obtain that it equals $\ \beta$. By continuity, the result holds for all $\
\gamma>0$.

###### Corollary 3.5

Let $f$ be a uniform limit of periodic step functions and $c>0,\gamma>0$. Then

$M(f)=\lim\limits_{T\longrightarrow\infty}\gamma\cdot
T^{\gamma}\cdot\int\limits_{T}^{\infty}\frac{f(x)}{x^{1+\gamma}}dx=\lim\limits_{\eta\longrightarrow
0}\frac{1}{\eta}\int\limits_{0}^{\eta}f\bigg{(}\frac{1}{\epsilon^{c}}\bigg{)}d\epsilon$

In particular, $\nu(f\circ\alpha^{-c})=M(f)$.

Proof. If $f$ is the uniform limit of the sequence $(f_{n})$ then clearly we
have that $\lim\limits_{n\longrightarrow\infty}M(f_{n})=M(f)$. To prove the
second part, make the chance of variable $x=\frac{1}{\varepsilon^{c}}$
obtaining that $\lim\limits_{\eta\longrightarrow
0}\frac{1}{\eta}\int\limits_{0}^{\eta}f(\frac{1}{\epsilon^{c}})d\epsilon=\lim\limits_{T\longrightarrow\infty}\gamma\cdot
T^{\gamma}\cdot\int\limits_{T}^{\infty}\frac{f(x)}{x^{1+\gamma}}dx$, where
$\gamma=\frac{1}{c}$. The result now follows from Lemma 3.4.

###### Theorem 3.6

Let $\ f$ be an a.p.f. $\gamma>0$ and $\lambda:[0,1]\longrightarrow[0,\infty]\
$ be continuos with $\lambda(0)=0$ an let $\nu$ be its inverse. Suppose that
$\nu\in\mathcal{C}^{k}([0,\lambda(1)])$, with $k\geq 2$ , is not a null
function. Then

$M(f)=\lim\limits_{T\longrightarrow\infty}\gamma\cdot
T^{\gamma}\cdot\int\limits_{T}^{\infty}\frac{f(x)}{x^{1+\gamma}}dx=\lim\limits_{\eta\longrightarrow
0}\frac{1}{\eta}\int\limits_{0}^{\eta}f\bigg{(}\frac{1}{\lambda(\epsilon)}\bigg{)}d\epsilon$

In particular, $\nu(f\circ\frac{1}{\lambda})=M(f)$.

Proof. As seen above, $\ f$ can be uniformly approximated by periodic steps
functions, and hence the first equality follows from Corollary 3.5. To prove
the second equality, note that since $\nu$ is not a null function there exists
$q\in{\mathbb{N}}^{*}$ such that $b:=\nu^{(q)}(0)\neq 0$ and $\nu^{(j)}(0)=0$,
$\forall\ 0<j<q$. Making the chance of variable
$x=\frac{1}{\lambda(\epsilon)}$ one obtains

$\lim\limits_{\eta\longrightarrow
0}\frac{1}{\eta}\int\limits_{0}^{\eta}f\bigg{(}\frac{1}{\lambda(\epsilon)}\bigg{)}d\epsilon=\lim\limits_{T\longrightarrow\infty}\frac{1}{\nu(\frac{1}{T})}\int\limits_{T}^{\infty}\frac{f(x)\nu^{\prime}(\frac{1}{x})dx}{x^{2}}$

From hypothesis we have that
$\lim\limits_{T\longrightarrow\infty}T^{q}\nu(\frac{1}{T})=\frac{b}{q!}$ and
$\ \nu^{\prime}(\frac{1}{x})=\frac{b}{q!x^{q-1}}+g(x)$,
$|g(x)|\leq\frac{C}{x^{q}}$, where $C>0$ is a constant. Hence we have that

$\lim\limits_{T\longrightarrow\infty}\frac{1}{\nu(\frac{1}{T})}\int\limits_{T}^{\infty}\frac{f(x)\nu^{\prime}(\frac{1}{x})dx}{x^{2}}=\lim\limits_{T\longrightarrow\infty}\frac{T^{q}}{T^{q}\nu(\frac{1}{T})}\int\limits_{T}^{\infty}\frac{f(x)[\frac{b}{q!x^{q-1}}+g(x)]}{x^{2}}dx=$
$\lim\limits_{T\longrightarrow\infty}\frac{T^{q}}{b/q!}\int\limits_{T}^{\infty}\frac{f(x)[\frac{b}{(q-1)!x^{q-1}}+g(x)]}{x^{2}}dx=\lim\limits_{T\longrightarrow\infty}q\cdot
T^{q}\cdot\int\limits_{T}^{\infty}\frac{f(x)dx}{x^{q+1}}+\lim\limits_{T\longrightarrow\infty}T^{q}\cdot\int\limits_{T}^{\infty}\frac{q!f(x)g(x)dx}{bx^{2}}$

Since $\ f$ is bounded we have, from our hypothesis, that

$\Big{|}\lim\limits_{T\longrightarrow\infty}T^{q}\cdot\int\limits_{T}^{\infty}\frac{q!f(x)g(x)dx}{bx^{2}}\Big{|}\leq\frac{q!C\|f\|_{\infty}}{b}\lim\limits_{T\longrightarrow\infty}T^{q}\cdot\int\limits_{T}^{\infty}\frac{dx}{x^{q+2}}=0$

It follows that
$\lim\limits_{T\longrightarrow\infty}\frac{1}{\nu(\frac{1}{T})}\int\limits_{T}^{\infty}\frac{f(x)\nu^{\prime}(\frac{1}{x})dx}{x^{2}}=\lim\limits_{T\longrightarrow\infty}q\cdot
T^{q}\cdot\int\limits_{T}^{\infty}\frac{f(x)dx}{x^{q+1}}=M(f)$.

Note that if $\nu$ is a null function then $\ \lim\limits_{\eta\longrightarrow
0}\frac{1}{\eta}\int\limits_{0}^{\eta}f(\frac{1}{\lambda(\epsilon)})d\epsilon=0$.
This is easily seen since, in this case, we have that $|\nu^{\prime}(x)|\leq
C(q)\frac{1}{x^{q}}\ $, where $C(q)$ is a constant, for all $\ q>0$.

$\nu(f\circ\frac{1}{\lambda})=M(f)$

###### Lemma 3.7

  1. 1.

Let $\ f$ be a.p.f. and $\ g$ such that
$\lim\limits_{x\rightarrow\infty}g(x)=L$. Then

$M(gf)=L\cdot M(f)$

  2. 2.

Let $\ f$ and $\ \lambda$ be as in the previous theorem and let $\
g=[g(\varepsilon)]\in\overline{\mathbb{R}}$ be such that $g\approx
g_{0}\in\mathbb{R}$. Then

$\nu\bigg{(}g\cdot\bigg{(}f\circ\frac{1}{\lambda}\bigg{)}\bigg{)}=\lim\limits_{\eta\longrightarrow
0}\frac{1}{\eta}\int\limits_{0}^{\eta}g(\varepsilon)f\bigg{(}\frac{1}{\lambda(\epsilon)}\bigg{)}d\epsilon=g_{0}\cdot
M(f)$

  3. 3.

Let $\ f$ and $\ g$ be as in the previous item then

$\nu(f\circ g)=\lim\limits_{\eta\longrightarrow
0}\frac{1}{\eta}\int\limits_{0}^{\eta}f\bigg{(}g(\varepsilon)\bigg{)}d\epsilon=f(g_{0})$

Proof. We have that
$\bigg{|}\frac{1}{T}\int\limits_{0}^{T}g(x)f(x)dx-\frac{1}{T}\int\limits_{0}^{T}Lf(x)dx\bigg{|}\leq\|g-L\|_{[0,T],\infty}\cdot\frac{1}{T}\int\limits_{0}^{T}|f(x)|dx$
$\leq\|g-L\|_{[0,T],\infty}\cdot(2M(|f(x)|)+1)$, for $\ T$ large, proving the
first item. The second item follows by the the previous theorem and the proof
of the previous item. For the last item, note that $\ f\circ g\approx
f(g_{0})$ because $\ f$ is continuous. Hence the transition probability is the
mean value of any sequence converging to $\ f(g_{0})$.

From the previous lemma, we get the following corollary.

###### Corollary 3.8

Let $\ f$ be an a.p.f., $\ \lambda$ a pure infinity and $\
g\in\overline{\mathbb{R}}$ such that $\ g\approx g_{0}\in\mathbb{R}$. Then $\
M(g\cdot f\circ\lambda)=g_{0}\cdot M(f)$ and, in case $\ f\circ g$ exists,
then $\ M(f\circ g)=f(g_{0})$.

The last result of this section proves that a positive function which is
bounded by an a.p.f. has generalized mean values.

###### Theorem 3.9

Let $\ 0\leq|g(x)|\leq f(x)$, $\ f$ is an a.p.f and $\ \lambda$ a pure
infinity. Then both $\ \nu(g\circ\lambda)$ and $\ \nu(|g\circ\lambda|)$ have
non-empty supports.

Proof. We may suppose that $\ g$ is a positive function. For each $0<\eta<1$,
sufficiently small, we have that
$0\leq\frac{1}{\eta}\int\limits_{0}^{\eta}g(\lambda(\varepsilon))d\epsilon\leq\frac{1}{\eta}\int\limits_{0}^{\eta}f(\lambda(\varepsilon))d\epsilon\leq
2M(f)+1$. Hence, in the generalized environment, we have that
$0\leq\nu(g\circ\lambda)\leq(2M(f)+1)\ $ and from this the result follows.

Let $\ f$ be a $\ C^{\infty}$ a.p.f. and $\nu\in\overline{\mathbb{R}}$ a
positive infinitesimal. For each $\ \varepsilon\in I$, define the set $\
M^{f,\nu}_{\varepsilon}=\\{\tau_{\varepsilon}\in\mathbb{R}\ :\
\|f(x+\tau_{\varepsilon})-f(x)\|_{\infty}\leq\nu(\varepsilon)\\}$, a set of
translation numbers of $\ f$, and consider the internal set $\
M=[M_{\varepsilon}^{f,\nu}]$. Given $\ \tau\in M$, we have that
$supp(\tau)\cap\mathbb{R}^{*}\neq\emptyset$ if and only if $\ f$ is a periodic
function. Hence if $\ f$ is non periodic then $\ M$ consists of infinitesimals
or infinities. If $\ f$ is $\ C^{\infty}$, $\ \nu$ is a null and $\
M\neq\emptyset$ then the canonical image of $\ f$ in $\
\mathcal{G}({\mathbb{R}})$ is a generalized periodic function and each non-
zero element of $\ M$ is a generalized period of $\ f$. Such a periods of $\
f$ need not be an invertible element, but if $\ \tau\in M$ is an invertible
pure infinity then $\ \frac{1}{\tau}\int\limits_{0}^{\tau}f(t)dt\approx M(f)$.
One could start with any moderate net of a.p. functions and not just with a
constant net. A generalized function which has such nets as representatives we
shall call a g.a.p.f., short for generalized a.p.f. It would be interesting to
know if there are non periodic a.p. functions which are generalized periodic.
The same question can be posed for g.a.p.f.

##  4 Fock Space

In [30, 31] Colombeau and Gsponer propose a setting for QFT aiming to deal
with the renormalization theory. We collect some fact sabout this setting. We
will be using their notation and thus it will be useful for the reader to
survey their paper for a more precise understanding of their setting.

The Fock space, introduced by the Soviet physicist Vladimir Aleksandrovich
Fock, who did foundational work on quantum mechanics and quantum
electrodynamics, is the Hilbertian direct sum

$\mathbb{F}=\bigoplus_{n=0}^{+\infty}L^{2}_{s}((\mathbb{R}^{3})^{n}),$ (1)

where $L^{2}_{s}((\mathbb{R}^{3})^{0}=\mathbb{C},\
L^{2}_{s}((\mathbb{R}^{3})^{1}=L^{2}(\mathbb{R}^{3})$ and for $n\geq 2\
L^{2}_{s}((\mathbb{R}^{3})^{n})$ is the closed subspace of the space
$L^{2}((\mathbb{R}^{3})^{n})$ made of those (complex valued) functions which
are symmetric in their $n$ arguments in $\mathbb{R}^{3}.$ That is, an element
$F$ of $\mathbb{F}$ is an infinite sequence
$F=(f_{0},f_{1},\dots,f_{n},\dots)$ such that

$\|F\|^{2}:=|f_{0}|^{2}+\sum_{n=1}^{+\infty}\|f_{n}\|^{2}_{L^{2}(\mathbb{R}^{3n})}<+\infty.$
(2)

They introduce the generalized functions over the Fock Space and prove that
$S_{\tau}(t)$, the scattering operator, is a unitary operator in this
generalized environment. According to them, this proof is actually due to
Perron (see also [5]). We recall the following results of [42].

###### Proposition 4.1

The interacting field operator satisfies the formula.

$\Phi(x,t)=(S_{\tau}(t))^{-1}\Phi_{0}(x,t)S_{\tau}(t)$

###### Proposition 4.2

The scattering operator is solution of the ODE.

$\partial_{t}S_{\tau}(t)=-i\frac{g}{N+1}\int_{x\in\mathbb{R}^{3}}(\Phi_{0}(x,t))^{N+1}dx\
S_{\tau}(t),\ \ S_{\tau}(\tau)=id$

The detailed formal calculations for the proofs of propositions 4.1 and 4.2
are carried out in [30]. Using this generalized setting over the Fock Space,
Colombeau and Gsponer (see [30, 42]) propose how to obtain numerical
predictions of the transition probabilities, i.e., the probability that a
state $F$ before interaction produces a state $G$ after interaction.
Considering an observable $O(x,t)$ as a product of field operators, then,
after interaction, the formula one gets is

$O(x,t)=S_{\tau}(t)^{-1}O_{0}(x,t)S_{\tau}(t).$ (3)

The formula

$|<F_{1}|O(x,t)F_{2}>|=|<F_{1}|S_{\tau}(t)^{-1}O_{0}(x,t)S_{\tau}(t)F_{2}>|=|<S_{\tau}(t)F_{1},O_{0}(x,t)S_{\tau}(t)F_{2}>|$

shows that $S_{\tau}(t)F_{i}$ are the states that give the results from
experiments in the new free field theory after the interaction has ceased
($\tau$ in the past before interaction, $t$ in the future after interaction).
Therefore the probability that a state $F$ before interaction would be
transformed after interaction into a state $G$ is $|<G,S_{\tau}(t)F>|$, which
is therefore interpreted as the transition probability of $F$ into $G$ due to
the interaction. A way to view this transition probability is the following:
Let $\ u,v$ be orthogonal vectors of norm one of a Hilbert space $\
{\mathbb{H}}$ and let $\ S$ be a unitary operator on $\ {\mathbb{H}}$. Then
one sees $|<\vec{u},S\vec{v}>|$ as probability in the following way: as a
probability by considering an average of the values
$|<\vec{u},S(\epsilon)\vec{v}>|$ on a large number of small values of
$\epsilon$. For instance according to a classical interpretation of
renormalization in the case of the anomalous magnetic moment of the electron
various values of $\epsilon$ could represent various spontaneous creations of
virtual electron-positron pairs in the vacuum [62] section 26.9. In short, the
transition probability should be

$\mathcal{P}_{0}(v\rightarrow u)=\lim_{\eta\rightarrow 0,\
N\rightarrow+\infty}\frac{1}{N}\sum_{i=1}^{N}|<\vec{u},S(\epsilon_{i})\vec{v}>|,random\
\epsilon_{i}\in]0,\eta[$ (4)

provided this limit exists. Note that this average is naturally related to our
notion of support of a generalized object. A non-discrete way to express this
average is

$\mathcal{P}_{0}(v\rightarrow u)=\lim_{\eta\rightarrow
0}\frac{1}{\eta}\int_{0}^{\eta}|\langle u,S_{\epsilon}v\rangle|d\epsilon.$ (5)

This leads naturally to the definition of the generalized transition
probability as

$\mathcal{P}(v\rightarrow
u)=\bigg{[}\varepsilon\longrightarrow\frac{1}{\varepsilon}\int_{0}^{\varepsilon}|\langle
u,S_{t}v\rangle|dt\bigg{]}$

Hence the existence of $\ \mathcal{P}_{0}(v\rightarrow u)$ is equivalent to

$\mathcal{P}(v\rightarrow u)\approx\mathcal{P}_{0}(v\rightarrow u)$

or, equivalently,

$supp(\mathcal{P}(v\rightarrow u))=\\{\mathcal{P}_{0}(v\rightarrow u)\\}$

Since, in general, this obviously need not be the case, this leads us to
define any element of $\ supp(\mathcal{P}(v\rightarrow u))$ as a transition
probability. If we suppose that $\ S_{t}$ depends continuously on $\ t$ and
that $\ supp(\mathcal{P}(v\rightarrow u))=\\{\mathcal{P}_{0}(v\rightarrow
u)\\}$ then we have that

$\mathcal{P}(v\rightarrow u)\approx\bigg{|}\lim_{\varepsilon\rightarrow
0}\frac{1}{\varepsilon}\int_{0}^{\varepsilon}\langle u,S_{t}v\rangle
dt\bigg{|}=\mathcal{P}_{0}(v\rightarrow u)$

Henceforth, in this paper, the existence of a transition probability will be
equal to the the support of the generalized transition probability being not
empty. Colombeau and Gsponer require that the support of the generalized
transition probability has a unique element.

Here we deal with the Colombeau algebra defined over $L(\mathbb{D})$ (see [31,
Section 1.5.2]) and thus all operators involved are, by definition, bounded
operators. This can be justified since there does not exist a structure
containing the unbounded operators in which one can calculate their norm which
is necessary in the construction of a Colombeau algebra. The construction of
$L(\mathbb{D})$ probably contains most important operators used in Physics.

##  5 Transition Probabilities

In this section we study self adjoint operators over
${\overline{{\mathbb{K}}}}^{n}$ and infinite dimensional Hilbert spaces. We
start looking at the finite dimensional case. In this case, a self adjoint
operator corresponds to a hermintian matrix $A\in
M_{n}({\overline{{\mathbb{K}}}})$. Hence we are interested in matrices with
coefficients in ${\overline{{\mathbb{K}}}}_{as}$ or $\tilde{\mathbb{K}}_{c}$
and the existence of the transition probability $\
\nu(A)=|<\vec{u},exp(iA(\epsilon))\vec{v}>|$ where $u,v$ are unitary vectors.
As notice before, this need not be the case and thus we will study the support
of the generalized transition probability. Our first result gives a condition
for the existence of the transition probability.

If $\ A=(a_{ij})$ is a matrix, we denote by $\|A\|$ its euclidean norm which
is also known as its Frobenius norm. The Frobenius norm can be extended to the
generalized environment in a natural way. We shall use the same notation for
this extended norm (see [4, 13]). We denote by $\ S_{n}$ the permutation group
on $\ n$ symbols. We now state a result due to Hoffman-Wielandt fundamental
for what comes in the sequel. We refere the reader to [16, 44, 45] for more
information.

######  Theorem 5.1 (Hoffman-Wielandt)

Let ${\mathcal{A}}$ and $B$ be two normal complex matrices with spectra $\
\\{\alpha_{1},\dots,\alpha_{n}\\}$ and $\ \\{\beta_{1},\dots,\beta_{n}\\}$,
respectively. Then $\min\limits_{\sigma\in S_{n}}\
\Big{(}\sum\limits_{1}^{n}|\alpha_{i}-\beta_{\sigma(i)}|^{2}\Big{)}\leq\|A-B\|^{2}$.

Given a generalized function $\ f$ , we denote by $R_{f}$ the set of solutions
of the equation $\ f(x)=0$ in $\ \overline{\mathbb{C}}$. In case $\ f(x)$ is a
generalized polinomial, we have the following result from [81].

###### Lemma 5.2

Let $f(x)\in\overline{\mathbb{C}}[x]$ be monic. There exist $\
\lambda_{1},\dots,\lambda_{n}\in\overline{\mathbb{C}}$ such that
$f(x)=(\lambda-\lambda_{1})\cdots(\lambda-\lambda_{n})$ and
$R_{f}=interl(\\{\lambda_{1},\dots,\lambda_{n}\\})$.

If$\ A$ is a square matrix, we denote by $\ \chi_{A}(\lambda)$ its
characteristic polinomial over $\ \overline{\mathbb{C}}$. The spectrum of $\
A$ is the set of eigenvalues of $\ A$, i.e., $\ R_{\chi_{A}}$. We refer the
reader to [45, 81] for excellent explanations with respect to the subject of
eigenvalues. For our purpose, we recall that $\lambda$ is an eigenvalue of $\
A$ if and only if $\ \lambda\in Spec(A)$ (see [45, Proposition 3.20]). This is
equivalent to the following: there exists $\ v=(v_{1},\cdots
v_{n})\in\overline{\mathbb{C}}^{n}$ such that $\ Av=\lambda v$ and
$Span_{\overline{\mathbb{C}}}[v_{1},\cdots v_{n}]=\overline{\mathbb{C}}$. Such
a vector is called a free vector. We can now state two consequences of the
previous lemma which can be found in [45, 81].

###### Theorem 5.3

Let ${\mathcal{A}}$ be an $n\times n$-matrix with coefficients in $\
\overline{\mathbb{C}}$. Then theres exists $\
\lambda_{1},\dots,\lambda_{n}\in\overline{\mathbb{C}}$, eigenvalues of $\ A$
such that $\
\chi_{A}(\lambda)=(\lambda-\lambda_{1})\cdots(\lambda-\lambda_{n})$ is the
characteristic polinomial of $\ A$ and the spectrum of $\ A$ is equal to $\
interl(\\{\lambda_{1},\dots,\lambda_{n}\\})$.

###### Theorem 5.4

Let ${\mathcal{A}}$ be an $n\times n$-hermitian matrix with coefficients in $\
\overline{\mathbb{C}}$. Then the spectrum of $\ A$ is contained in $\
\overline{\mathbb{R}}$. Moreover, if there exist eigenvalues $\
\lambda_{1}\geq\cdots\geq\lambda_{n}$ of $\ A$, then they satisfy the
conclusion of the previous theorem and $\ A$ is unintary equivalent to the
diagonal matrix $\ diag(\lambda_{1},\dots,\lambda_{n})$.

If$\ A$ is an $n\times n$-matrix with coefficients in $\
\overline{\mathbb{C}}$ and $\
\\{\lambda_{1},\dots,\lambda_{n}\\}\subset\overline{\mathbb{C}}$ is contained
in the $\ Spec(A)$ is as in one of the previous theorems than we say that $\
\\{\lambda_{1},\dots,\lambda_{n}\in\overline{\mathbb{C}}\\}$ is a generating
set for $\ Spec(A)$.

Let$\ A$ is an $n\times n$-matrix with coefficients in $\
\overline{\mathbb{C}}$ and $\
\\{\lambda_{1},\dots,\lambda_{n}\\}\subset\overline{\mathbb{C}}$ a generating
set. A permutation net is a map $\ \sigma:I\longrightarrow S_{n}$. Define the
element $\ \lambda_{i}^{\sigma}$ as the element whose representative is $\
\lambda_{i}^{\sigma}(\varepsilon):=\lambda_{\sigma_{\varepsilon}(i)}(\varepsilon)$.
For each $1\leq j\leq n$ let $\ S_{ij}=\\{\varepsilon\in
I:\sigma_{\varepsilon}(i)=j\\}$ and let $\ e_{ij}$ be the idempotent generated
by $\ S_{ij}$. Then $\
\lambda_{i}^{\sigma}=\sum\limits_{j}e_{ij}\cdot\lambda_{j}\in Spec(A)$ and
$\\{e_{i1},\cdots e_{in}\\}$ is a complete set of ortogonal idempotents. This
proves that $\ \lambda_{i}^{\sigma}$ is a well defined element. Moreover, if
the matrix $(e_{ij})$ is invertable then $\
\\{\lambda_{1}^{\sigma},\cdots\lambda_{n}^{\sigma}\\}$ is a generating set for
$\ Spec(A)$.

###### Theorem 5.5

Let $A\in M_{n}({\overline{\mathbb{K}}}_{as})$ be a hermitian matrix with
$supp(A)=\\{A_{0}\\}$ and let $\\{\lambda_{1},\dots,\lambda_{n}\\}$ be a
generating set for $\ Spec(A)$. Then there exists a permutation net $\ \sigma$
such that $\\{\lambda_{1}^{\sigma},\cdots,\lambda_{n}^{\sigma}\\}$ is a
generating set for $\ Spec(A)$,
$\lambda_{i}^{\sigma}\in{\overline{\mathbb{K}}}_{as}\ \forall i\ $ and
$\\{supp(\lambda_{i}^{\sigma}),\cdots,supp(\lambda_{n}^{\sigma})\\}=Spec(A_{0})$.

Proof. Let $\ supp(A_{0})=\\{\mu_{1},\dots,\mu_{n}\\}$ as an ordered set.
Since $A$ is hermitian and $A\approx A_{0}$ it follows that $\ A_{0}$ is
hermitian and we may choose a representative for $\ A$ such that $\
A_{\varepsilon}$ is hermitian for all $\ \varepsilon\in I$. Applying the
Hoffman-Wielandt’s Theorem, there exists a permutation net $\ \sigma$ such
that

$|\lambda_{i}^{\sigma}-\mu_{i}|\leq\|A-A_{0}\|\approx 0$ (6)

from which it follows that $supp(\lambda_{i}^{\sigma})=\mu_{i}$. Since
$\lambda_{i}^{\sigma}\in Spec(A)$, and $\
\prod\limits_{i}(\lambda-\lambda_{i}^{\sigma}(\varepsilon))=\prod\limits_{i}(\lambda-\lambda_{i}(\varepsilon))$
for all $\ \varepsilon$, we have that $\
\prod\limits_{i}(\lambda-\lambda_{i}^{\sigma})=\chi_{A}(\lambda)$ and hence
the theorem is proved.

###### Corollary 5.6

Let $\ A\approx A_{0}\in M_{n}({\mathbb{C}})$ and $\lambda\in Spec(A)$. Then
$supp(\lambda)\subseteq Spec(A_{0})$ with equality if
$\lambda=\sum\limits_{i}e_{i}\cdot\lambda_{i}^{\sigma}$, where
$\\{e_{1},\cdots,e_{n}\\}$ is a complete set of non-trivial orthogonal
idempotents and the $\ \lambda_{i}^{\sigma}$ are given by the theorem.

Proof. Since $\lambda\in Spec(A)$, there exists a free vector
$v\in{\overline{\mathbb{C}}}^{n}$ of norm $\ 1$ such that $Av=\lambda v$. If
$\lambda_{0}\in supp(\lambda)$, then there exists an idempotent $\ f$ such
that $f\cdot\lambda\approx f\cdot\lambda_{0}$. Since $\ v$ has norm one, there
exists an idempotent $\ e$ such that $\ ef=e$ and $\ e\cdot v\approx e\cdot
v_{0}$, with $v_{0}\in{{\mathbb{C}}}^{n}$. Using this we have,
$e\cdot\lambda_{0}v_{0}\approx e\lambda v=(Av)e\approx e\cdot A_{0}v_{0}$ and
hence, $A_{0}v_{0}=\lambda_{0}v_{0}$.

We shall use the notation of the proof of the theorem. Let $\
supp(A_{0})=\\{\mu_{1},\dots,\mu_{n}\\}$. If
$\lambda=\sum\limits_{i}e_{i}\cdot\lambda_{i}^{\sigma}$ with $\
e_{i}e_{j}=\delta_{ij}e_{i}$ and $\ e_{i}\notin\\{0,1\\}$ then $\ \lambda\cdot
e_{i}=e_{i}\cdot\lambda_{i}^{\sigma}\approx e_{i}\cdot\mu_{i}$ and hence the
result follows.

###### Corollary 5.7

$supp(Spec(A)):=\bigcup\limits_{\lambda\in Spec(A)}supp(\lambda)=Spec(A_{0}).$

###### Remark 5.8

If $\ A$ is hermitian matrix and $B$ is an arbitrary normal matrix and we
order $\ Spec(A)$ and $\ Spec(B)$ by the real part of its elements, i.e.,
$\alpha_{1}\geq\cdots\geq\alpha_{n}$ and $Re(\beta_{1})\geq\cdots\geq
Re(\beta_{n})$ then we can give a more precise description of the $\
\lambda_{i}^{\sigma}$’s. We refere the reador to [16, Observation 15.3] (see
also [45, Lemma 3.24]. In fact, in this case we have that, for each $\ i$,
$\lambda_{i}{\sigma}=\lambda_{i}$.

###### Proposition 5.9

Let $\ A$ and $\ A_{0}$ be as in the previous theorem and let $u_{0},v_{0}$ be
unitary orthogonal vectors of $\ \mathbb{C}^{n}$. Then
$supp(\nu(A))=\\{\nu(A_{0})\\}$. In particular, $\nu(A)\approx\nu(A_{0})$.

Proof. We first note that if $\ e$ is and idempotent than $\ exp(iA)\cdot
e=exp(i(eA))\approx exp(iA_{0})$. Now let $t_{0}\in supp(\nu(A))$ and let $\
e$ be an idempotent such that $e\cdot\nu(A)\approx t_{0}$. Using this, one
gets that $t_{0}\approx e\cdot\nu(A)=exp(i(eA))\approx exp(iA_{0})=\nu(A_{0})$
and hence $\ t_{0}=\nu(A_{0})$.

For each $1\leq i\leq n$, denote by $V_{\mu_{i}}(A_{0})$ the eigenspace of
$A_{0}$ corresponding to the eigenvalue $\mu_{i}\in Spec(A_{0})$ and let $\
V_{\lambda_{i}}(A)$ be the $\overline{\mathbb{K}}-$sub-modulo de
$\overline{\mathbb{K}}^{n}$ spanned by the eigenvectors of $A$ corresponding
to the eigenvalue $\ \lambda_{i}$.

Suppose that $A$, $\ A_{0}$, $\sigma$ and
$\\{\lambda_{1}^{\sigma},\cdots,\lambda_{n}^{\sigma}\\}$ are as in Theorem
5.5. Let $\ Spec(A_{0})=\\{\mu_{1},\cdots,\mu_{n}\\}$. By [45, Propositions
3.18 and 3.25], $Spec(A)\subset{\overline{\mathbb{R}}}$ and $\ A$ is unitary
equivalent to $\ diag(\lambda_{1}^{\sigma},\cdots,\lambda_{n}^{\sigma})$. It
also follows easily that $\ supp(\chi_{A})=\\{\chi_{A_{0}}\\}$ and the
algebraic multiplicity of $\ \lambda_{i}^{\sigma}$ and $\ \mu_{i}$ are equal.

###### Corollary 5.10

For each $1\leq i\leq n$, we have
$supp(V_{\lambda_{i}^{\sigma}}):=\displaystyle\bigcup_{v\in
V_{\lambda_{i}^{\sigma}}(A)}supp(v)=V_{\mu_{i}}(A_{0})$.

Proof. Let $v\in V_{\lambda_{i}^{\sigma}}(A)$ and let $v_{0}\in supp(v)$. Then
$\ e\cdot v\approx e\cdot v_{0}$, for some $\ e^{2}=e$. With the notation of
the proof of the theorem, it follows that $e\cdot(A_{0}v_{0})\approx
e\cdot(Av)=e\cdot(\lambda_{i}^{\sigma}v)\approx e\cdot\mu_{i}v_{0}$.
Consequently, $v_{0}\in V_{\mu_{i}}$. Since the algebraic multiplicity of $\
\lambda_{i}^{\sigma}$ and $\ \mu_{i}$ are equal, it follows that the
${\overline{\mathbb{C}}}-$free rank of $\ V_{\lambda_{i}^{\sigma}}$ is equal
to the $\ {\mathbb{R}}-$dimension of $V_{\mu_{i}}$ and hence, since
$supp(V_{\lambda_{i}^{\sigma}})$ is an $\ {\mathbb{R}}-$vector space, we have
that $supp(V_{\lambda_{i}^{\sigma}})=V_{\mu_{i}}(A_{0})$.

Note that if $A\in M_{n}(\tilde{\mathbb{K}}_{c})$ then we have that
$supp(A)\neq\emptyset$. Moreover, its support is compact. We are ready to
prove our main result for transition probabilities in the finite dimensional
case.

###### Theorem 5.11

Let $A=[(A_{\epsilon})_{\epsilon}]\in M_{n}(\tilde{\mathbb{K}}_{c})$ be a
hermitian matrix. Then $\ supp(\nu(A))=\\{\nu(A_{0}):A_{0}\in supp(A)\\}$.

Proof. If $A_{0}\in supp(A)$ then there exists an idempotent $\ e$ such that
$e\cdot A\approx e\cdot A_{0}$. Applying Proposition 5.9 we have that
$e\cdot\nu(A)=\nu(eA)\approx\nu(eA_{0})=e\cdot\nu(A_{0})$. Hence $\
\nu(A_{0})\in supp(\nu(A))$. Conversely, let $\ a\in supp(\nu(A))$ and $\
e=e^{2}$ such that $e\cdot a\approx e\cdot\nu(A)=\nu(e\cdot A)$. Choose $\
A_{0}\in supp(e\cdot A)\subset supp(A)$ and $\ f=f^{2}$ such that $\ ef=f$ and
$f\cdot A_{0}\approx f\cdot(eA)=f\cdot A$. Then $\ f\cdot a\approx
f\cdot\nu(A)=\nu(f\cdot A)\approx\nu(f\cdot A_{0})=f\cdot\nu(A_{0})$. Hence $\
a=\nu(A_{0})$ and the theorem is proved.

We now look at $\overline{\mathbb{K}}$-Hilbert modules, $\ {\mathcal{G}}_{H}$
where $\ H$ is a infinite dimensional Hilbert space. We refere the reader to
[36, 37] for terminology and results for these modules. We start formalizing
the notion of support in this environment. We shall also need a version of the
Hoffman-Wielandt Theorem for Hilbert spaces for which we refer the reader to
[17].

###### Definition 5.12

Let $T=(T_{\epsilon}):{\mathcal{G}}_{H}\rightarrow{\mathcal{G}}_{H}$ be a
basic map (see [36, 37]). Its support, denoted by $supp(T)$,is defined as
$supp(T):=\\{T_{0}\in{\mathcal{B}}(H):\exists\ e=e^{2}$, such that $\ e\cdot
T\approx e\cdot T_{0},i.e.,\|e\cdot(T-T_{0})\|_{\infty}\rightarrow 0\\}$.

###### Theorem 5.13

Let $\ T$ and $S$ be normal Hilbert-Schmidt operators with respective sets of
eigenvalues
$\\{\alpha_{1},\alpha_{2},\dots\\},\\{\beta_{1},\beta_{2},\dots\\}$. Then, for
each $\varepsilon>0$ there exists a permutation $\pi\in Sym({\mathbb{N}})$
such that

$\Bigg{[}\sum_{i=1}^{\infty}|\alpha_{i}-\beta_{\pi(i)}|^{2}\Bigg{]}^{1/2}\leq||T-S||+\varepsilon$
(7)

The interested reader can find the prove of this theorem in [17]. Note that in
Equation 7 we can change $\ \varepsilon$ by a null function $\
\xi(\varepsilon)$. In what follows the norm involved is the Hilbert-Schmidt
norm which we extend in a natural way to the generalized environment. Let
$\pi=(\pi_{\varepsilon})\in[Sym({\mathbb{N}})]^{]0,1]}$ be a permutation net,
$\ T=(T_{\epsilon}):{\mathcal{G}}_{H}\rightarrow{\mathcal{G}}_{H}$ a basic map
and $\ \alpha=[\varepsilon\longrightarrow\varepsilon]$ be the standard gauge.
For each $\ \varepsilon$ let $\ (t_{n}(\varepsilon))_{n\in{\mathbb{N}}}$ be an
enumeration of the eigenvalues of $\ T_{\varepsilon}$ and write $\
t(\alpha):=(t_{n}(\alpha))$ for all these enumerations. Define $\
t^{\pi}(\alpha):=(t_{\pi_{\varepsilon}(n)}(\alpha))$ and write the latter as
$\ (t_{\pi_{\alpha}(n)}(\alpha))$. Since every $\ T_{\varepsilon}$ is Hibert-
Schmidt, we have that $\ (t_{n}(\varepsilon))_{n\in{\mathbb{N}}}\in
l_{2}({\mathbb{R}})$. If we choose the $\ T_{\varepsilon}$ ’s in a ball then
we have that $t(\alpha)\in{\mathcal{G}}(l_{2}({\mathbb{R}}))$. Proceeding in
the same way as in the finite dimensional case we obtain that
$t^{\pi}(\alpha)=\sum\limits_{j=1}^{\infty}e_{nj}\cdot t_{j}(\alpha)$, where
$\ e_{nj}$ is the charateristic function of the set $\ S_{nj}:=\\{\varepsilon\
:\ \pi_{\varepsilon}(n)=j\\}$. We can now state the former theorem in the
setting of $\ \overline{\mathbb{K}}$-Hilbert modules.

###### Corollary 5.14

Let $T=[(T_{\epsilon})]:{\mathcal{G}}_{H}\rightarrow{\mathcal{G}}_{H}$.
Suppose that $\ T$ is a self-adjoint Hilbert-Schmidt operator, i.e.
$T=(T_{\varepsilon})$ with each $\ T_{\varepsilon}$ a selfadjoint Hilbert-
Schmidt operator, $\ T_{0}\in supp(T)$ and $\ e=e^{2}$ is such that $\ e\cdot
T\approx e\cdot T_{0}$. If $\ t_{0}=\\{t_{1},t_{2},\dots\\}$ is an enumeration
of the eigenvalues of $\ T_{0}$, then there exists a permutation net $\pi$
such that

$e\cdot\Bigg{[}\sum_{n=1}^{\infty}|t_{n}-t_{\pi_{\alpha}(n)}(\alpha)|^{2}\Bigg{]}^{1/2}\leq||e\cdot(T-T_{0})||+e\cdot\alpha\approx
0$ (8)

Moreover $\ t(\alpha)\in l_{2}({\overline{\mathbb{R}}})$ and
$t^{\pi}_{n}\longrightarrow t_{0}$.

Denote by $\ {\mathcal{B}}({\mathcal{G}}_{H})$ the elements
$T=(T_{\epsilon}):{\mathcal{G}}_{H}\rightarrow{\mathcal{G}}_{H}$ for which
there exists a representative $(T_{\epsilon})_{\epsilon}$ of $\ T$ and a
compact subset $K\subset{\mathcal{B}}(H)$ such that $T_{\epsilon}\in K$ for $\
\epsilon<\eta$, for some $\eta\in I$. We call such a $\ T$ a compactly
supported element. Note that if $\ T\in{\mathcal{B}}({\mathcal{G}}_{H})$ then
$\ supp(T)\neq\emptyset$. Note that [84, Lemma 10.1] applies in this case.

###### Theorem 5.15

Let $T=[(T_{\epsilon})]\in\ {\mathcal{B}}({\mathcal{G}}_{H})$ be a selfadjoint
operator such that each $T_{\epsilon}$ is selfadjoint. Then
$supp(\nu(T))=\\{\nu(T_{0}):T_{0}\in supp(T)\\}$.

Proof. Let $T_{0}\in supp(T)$ and $\ e=e^{2}$ such that $\ e\cdot T\approx
e\cdot T_{0}$. Since all operators involved are bounded and the exponencial
map is continuous, we have that $\ e\cdot\nu(T)=e\cdot\nu(e\cdot T)\approx
e\cdot\nu(e\cdot T_{0})=e\cdot\nu(T_{0})$. Conversely, let $\ \omega\in
supp(\nu(T))$ and $\ e=e^{2}$ such that $\ e\cdot\omega\approx e\cdot\nu(T)$.
Choose $\ f$ such that $\ f\cdot e=f$ and $\ f\cdot T\approx T_{0}$, for some
$\ T_{0}$. Then we have that $\ f\cdot\omega\approx f\cdot\nu(T)=\nu(f\cdot
T)\approx\nu(f\cdot T_{0})=f\cdot\nu(T_{0})$. It follows that $\
\omega=\nu(T_{0})$, completing the proof.

##  6 Examples

In this section we considere examples some of which were already considered in
[5] without however explicit calculations. These examples are not covered by
the results of the previous sections. Before we continue we make the following
definition. An element $\lambda\in{\overline{\mathbb{R}}}$ which satisfies
$supp(\lambda)=\emptyset$ we shall call a pure infinity. Note that this is
equivalent to $\
\lim\limits_{\varepsilon\longrightarrow\infty}\lambda(\varepsilon)=\infty$.

In a 2-dimensional Hilbert space let
$H(t)=\left(\begin{array}[]{ll}\frac{g}{\epsilon}&0\\\
0&\frac{g}{\epsilon^{2}}\end{array}\right)$
$=\left(\begin{array}[]{ll}g\alpha^{-1}&0\\\
0&g\alpha^{-2}\end{array}\right)$, where $\ g\in{\mathbb{R}}$. The solution of
the ODE

$S^{\prime}(t)=iH(t)S(t)$

is

$\Bigg{[}\varepsilon\longrightarrow
S_{\varepsilon}(t)=\left(\begin{array}[]{ll}\exp\bigg{(}{i\frac{g}{\epsilon}(t-t_{0})}\bigg{)}&\
\ \ \ \ \ \ 0\\\
0&\exp\bigg{(}{i\frac{g}{\epsilon^{2}}(t-t_{0})\bigg{)}}\end{array}\right)\Bigg{]}$

Hence, for orthogonal unitary vectors $\ u=(u_{1},u_{2})$ and $\
v=(v_{1},v_{2})$ we have that

$\langle u\mid
S_{\epsilon}(t)v\rangle=u_{1}v_{1}\exp\bigg{(}{i\frac{g}{\epsilon}(t-t_{0})}\bigg{)}+u_{2}v_{2}\exp\bigg{(}{i\frac{g}{\epsilon^{2}}(t-t_{0})}\bigg{)}$

In [5] is it asserted that the support of $\langle u,S_{\epsilon}(t)v\rangle$
is not unitary, i.e., $\langle u,S_{\epsilon}(t)v\rangle$ is not associated to
any real number. But clearly $[\varepsilon\longrightarrow\langle
u,S_{\epsilon}(t)v\rangle]$ is the sum of the composition of a.p. functions
and pure infinities. Hence it does have a mean value. Hence it is most likely
that $|\langle u,S_{\epsilon}(t)v\rangle|$ also does have a mean value. We
proceed to compute this value. If write $\ a=u_{1}v_{1},b=u_{2}v_{2}$ then $\
a+b=0$ and we have that

$|\langle
u,S_{\epsilon}(t)v\rangle|=2|a|\cdot\bigg{|}sin\bigg{(}\frac{1-\varepsilon}{\varepsilon^{2}}g(t-t_{0})\bigg{)}\bigg{|}$

It follows that $\ M(|\langle u,S_{\epsilon}(t)v\rangle|)=2|a|\cdot
M\Bigg{(}\bigg{|}sin\bigg{(}\frac{1-\varepsilon}{2\varepsilon^{2}}(t-t_{0})\bigg{)}\bigg{|}\Bigg{)}=\frac{4}{\pi}\cdot|a|$,
where we used Theorem 3.6 in the last equality. We refer the reader to [30,
Section 8] (see also [5, Theorem 5]).

We look at another example from [5, 6], considering the $2\times 2$ symmetric
matrix
$A=\Bigg{[}\varepsilon\longrightarrow\left(\begin{array}[]{ll}\frac{a}{\epsilon^{n_{1}}+\epsilon^{2n_{1}}}&\frac{c}{\epsilon-2\epsilon^{2}}\\\
\frac{c}{\epsilon-2\epsilon^{2}}&\
\frac{b}{\epsilon^{n_{2}}}\end{array}\right)\Bigg{]}$
$=\left(\begin{array}[]{ll}\frac{a}{\alpha^{n_{1}}+\alpha^{2n_{1}}}&\frac{c}{\alpha-2\alpha^{2}}\\\
\frac{c}{\alpha-2\alpha^{2}}&\ \frac{b}{\alpha^{n_{2}}}\end{array}\right)$. If
$\ u=(u_{1},u_{2}),\ v=(v_{1},v_{2})$ and one computes

$\frac{1}{N}\sum_{j=1}^{N}|\langle(u_{1},u_{2})\mid\exp(-iH)(v_{1},v_{2})\rangle|$

by choosing $\epsilon$ at random between 0 and some very small value $\eta$;
$<\dots>$ is the standard Hermitian scalar product in $\mathbb{C}^{n}$ and
uses the matlab routine expm to calculate the exponential of a matrix its
seems that a transition probability should exist, for $\
a=1,b=1.3,c=0.4,n_{1}=n_{2}=1$ and is around $0.9251$. This simulation is
reported in [5, 6]. Note however that this simulation only indicates that the
support of the generalized transition probability is not empty. However, since
values of $\ \varepsilon$ were randomly chosen, it does suggest that the
support has a unique element and hence the existence of the transition
probability.

We first consider $\
u=\bigg{(}\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\bigg{)},v=\bigg{(}\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\bigg{)}$,
$\
A=\Bigg{[}\varepsilon\longrightarrow\left(\begin{array}[]{ll}\frac{1}{\epsilon+\epsilon^{2}}&\frac{1}{\epsilon-2\epsilon^{2}}\\\
\frac{1}{\epsilon-2\epsilon^{2}}&\ \ \
\frac{1}{\epsilon}\end{array}\right)\Bigg{]}$ and prove the existence of the
transition probability for any orthogonal pair and give its actual value. We
start calculating the eigenvalues of $\ A$:

$\lambda(\varepsilon)=\frac{1}{2}\Bigg{[}\frac{2\varepsilon+\varepsilon^{2}}{\varepsilon^{2}+\varepsilon^{3}}\pm\sqrt{\bigg{(}\frac{2\varepsilon+\varepsilon^{2}}{\varepsilon^{2}+\varepsilon^{3}}\bigg{)}^{2}-4\bigg{[}\frac{1}{\varepsilon^{2}+\varepsilon^{3}}-\frac{1}{(\varepsilon-2\varepsilon^{2})^{2}}\bigg{]}}\Bigg{]}$

From this it follows that

$\alpha\lambda=\Bigg{[}\varepsilon\longrightarrow\frac{1}{2}\Bigg{[}\frac{2+\varepsilon}{1+\varepsilon}\pm\sqrt{\bigg{(}\frac{2+\varepsilon}{1+\varepsilon}\bigg{)}^{2}-4\bigg{[}\frac{1}{1+\varepsilon}-\frac{1}{(1-2\varepsilon)^{2}}\bigg{]}}\Bigg{]}\Bigg{]}$

and hence $\alpha\cdot\lambda\approx\frac{1}{2}[2\pm 2]$. We denote by $\
\lambda_{1}$ the eigenvalue with the $\ +$ sign chosen and by $\ \lambda_{2}$
the one with the $\ -$ sign chosen. From this and the use of a graphic
calculator one proves that $\|\lambda_{1}\|=e$ and
$\|\lambda_{2}\|<e^{-18}<1$. Hence $\ \lambda_{1}$ is a pure infinity and $\
\lambda_{2}$ is an infinitesimal. We also have that

$\alpha(\lambda_{1}-\lambda_{2})=\sqrt{\bigg{(}\frac{2+\varepsilon}{1+\varepsilon}\bigg{)}^{2}-4\bigg{[}\frac{1}{1+\varepsilon}-\frac{1}{(1-2\varepsilon)^{2}}\bigg{]}}\approx\sqrt{2}$

proving that $\ \lambda_{1}-\lambda_{2}$ is a pure infinity. A basis of
eigenvectors of $\ A$ is $\\{(1,k),(-k,1)\\}$, where

$\
k(\varepsilon)=(\varepsilon-2\varepsilon^{2})(\lambda_{1}-(\varepsilon+\varepsilon^{2})^{-1})=$
$(\varepsilon-2\varepsilon^{2})\frac{1}{2}\Bigg{[}\frac{1}{\varepsilon}+\sqrt{\bigg{(}\frac{2\varepsilon+\varepsilon^{2}}{\varepsilon^{2}+\varepsilon^{3}}\bigg{)}^{2}-4\bigg{[}\frac{1}{\varepsilon^{2}+\varepsilon^{3}}-\frac{1}{(\varepsilon-2\varepsilon^{2})^{2}}\bigg{]}}\Bigg{]}=$
$\frac{1}{2}\Bigg{[}1+\sqrt{\bigg{(}\frac{2+\varepsilon}{1+\varepsilon}\bigg{)}^{2}-4\bigg{[}\frac{1}{1+\varepsilon}-\frac{1}{(1-2\varepsilon)^{2}}\bigg{]}}\Bigg{]}\approx
1.5:=k_{0}$

Hence $\ M=\frac{1}{\sqrt{1+k^{2}}}\left(\begin{array}[]{ll}1&-k\\\
k&1\end{array}\right)$ conjugates $\ A$ into diagonal form $\
D=\left(\begin{array}[]{ll}e^{i\lambda_{1}}&0\\\
0&e^{i\lambda_{2}}\end{array}\right)$ and $\ M\approx
M_{0}=\frac{1}{\sqrt{1+k_{0}^{2}}}\left(\begin{array}[]{ll}1&-k_{0}\\\
k_{0}&1\end{array}\right)$. From this it follows that

$\langle u\mid exp(iA)v\rangle=\langle Mu\mid
DMv\rangle=:g_{1}g_{3}e^{i\lambda_{1}}+g_{2}g_{4}e^{i\lambda_{2}}$

where $\ Mu=(g_{1},g_{2}),Mv=(g_{3},g_{4})$, $g_{i}\approx
y_{i}\in\mathbb{R},\ i=1,4$, $g_{1}g_{3}+g_{2}g_{4}=0$ and $\
\\{e^{i\lambda_{1}},e^{i\lambda_{2}}\\}$ consists of pure vibrations. We
already saw that $\ \lambda_{1}$ is a pure infinity. As in the previous
example, one gets that

$\ |\langle u\mid
exp(iA)v\rangle|=2|g_{1}g_{3}|\cdot\bigg{|}sin\bigg{(}\frac{\lambda_{1}-\lambda_{2}}{2}\bigg{)}\bigg{|}$

and hence the transition probability is

$\frac{4}{\pi}\cdot|g_{1}(0)g_{3}(0)|=\frac{4}{\pi}\cdot\bigg{|}\frac{1}{\sqrt{10}}\bigg{(}\frac{\sqrt{2}}{2}-k_{0}\frac{\sqrt{2}}{2}\bigg{)}\bigg{(}\frac{\sqrt{2}}{2}+k_{0}\frac{\sqrt{2}}{2}\bigg{)}\bigg{|}=\frac{2(k_{0}^{2}-1)}{(k_{0}^{2}+1)\pi}\approx
0.244853758603$

Clearly these two examples show how to solve the generic case of a $\ 2\times
2$ generalized matrix. We now compute the actual value reported in [5, 6]. In
this case,
$A=\Bigg{[}\varepsilon\longrightarrow\left(\begin{array}[]{ll}\frac{1}{\epsilon+\epsilon^{2}}&\frac{0.4}{\epsilon-2\epsilon^{2}}\\\
\frac{0.4}{\epsilon-2\epsilon^{2}}&\ \
\frac{1.3}{\epsilon^{2}}\end{array}\right)\Bigg{]}$ with eigenvalues

$\lambda(\varepsilon)=\frac{1}{2}\Bigg{[}\frac{1.3+2.3\varepsilon}{\varepsilon^{2}+\varepsilon^{3}}\pm\sqrt{\bigg{(}\frac{1.3+2.3\varepsilon}{\varepsilon^{2}+\varepsilon^{3}}\bigg{)}^{2}-4\bigg{[}\frac{1.3}{\varepsilon^{3}+\varepsilon^{4}}-\frac{0.16}{(\varepsilon-2\varepsilon^{2})^{2}}\bigg{]}}\Bigg{]}$

From this it follows that

$\alpha^{2.6}\cdot\lambda(\varepsilon)=\frac{1}{2}\Bigg{[}\varepsilon^{.06}\cdot\frac{1.3+2.3\varepsilon}{1+\varepsilon}\pm\sqrt{\varepsilon^{1.2}\bigg{(}\frac{1.3+2.3\varepsilon}{1+\varepsilon}\bigg{)}^{2}-4\varepsilon^{0.2}\bigg{[}\frac{1.3(1-2\varepsilon)2-0.16(\varepsilon+\varepsilon^{2})}{(1+\varepsilon)(1-2\varepsilon)}\bigg{]}}\Bigg{]}$

and hence $\alpha^{2.6}\cdot\lambda\approx 0$. We denote by $\ \lambda_{1}$
the eigenvalue with the $\ +$ sign chosen and by $\ \lambda_{2}$ the one with
the $\ -$ sign chosen. Then $\alpha^{2.5}\cdot\lambda_{1}\not\approx 0$. And
hence $\ \|\lambda_{1}\|\geq e^{2.5}$ and $\ \lambda_{1}$ is a pure infinity.
For $\ \lambda_{2}$, using a graphic calculator, we find that $\
\alpha^{-18.2}\cdot\lambda_{2}\approx 0$ but $\
\alpha^{-18.2}\cdot\lambda_{2}\not\approx 0$. It follows that $\alpha_{2}$ is
an infinitesimal of norm less the $\ e^{-18.2}$.

$\alpha^{2}(\lambda_{1}-\lambda_{2})=\Bigg{[}\varepsilon\longrightarrow\sqrt{\bigg{(}\frac{1.3+2.3\varepsilon}{1+\varepsilon}\bigg{)}^{2}-4\bigg{[}\frac{1.3\varepsilon}{1+\varepsilon}-\frac{0.16\varepsilon^{2}}{(1-2\varepsilon)^{2}}\bigg{]}\
}\Bigg{]}\approx 1$

proving that $\ \lambda_{1}-\lambda_{2}$ is a pure infinity. A basis of
eigenvectors of $\ A$ is $\\{(1,k),(-k,1)\\}$, where

$\
k(\varepsilon)=\frac{5}{2}(\varepsilon-2\varepsilon^{2})(\lambda_{1}-(\varepsilon+\varepsilon^{2})^{-1})=$
$\frac{5}{2}(\varepsilon-2\varepsilon^{2})\cdot\frac{1}{2}\Bigg{[}\frac{1.3}{\varepsilon^{2}}+\sqrt{\bigg{(}\frac{1.3+2.3\varepsilon}{\varepsilon^{2}+\varepsilon^{3}}\bigg{)}^{2}-4\bigg{[}\frac{1.3}{\varepsilon^{3}+\varepsilon^{4}}-\frac{0.16}{(\varepsilon-2\varepsilon)^{2}}\bigg{]}}\Bigg{]}$

$\frac{5}{4}(1-2\varepsilon)\Bigg{[}\frac{1.3}{\varepsilon}+\sqrt{\bigg{(}\frac{1.3+2.3\varepsilon}{\varepsilon+\varepsilon^{2}}\bigg{)}^{2}-4\bigg{[}\frac{1.3}{\varepsilon+\varepsilon^{2}}-\frac{0.16}{(1-2\varepsilon)^{2}}\bigg{]}}\Bigg{]}$

It follows that $\alpha k\approx 2.6$ and thus $\ k$ is a pure infinity.
Consequently, $\ M=\frac{1}{\sqrt{1+k^{2}}}\left(\begin{array}[]{ll}1&-k\\\
k&1\end{array}\right)$ conjugates $\ A$ into diagonal form $\
D=\left(\begin{array}[]{ll}e^{i\lambda_{1}}&0\\\
0&e^{i\lambda_{2}}\end{array}\right)$ and $\ M\approx
M_{0}=\left(\begin{array}[]{ll}0&-1\\\ 1&0\end{array}\right)$. From this it
follows that

$\langle u\mid exp(iA)v\rangle=\langle Mu\mid
DMv\rangle=:g_{1}g_{3}e^{i\lambda_{1}}+g_{2}g_{4}e^{i\lambda_{2}}$

where $\ Mu=(g_{1},g_{2}),Mv=(g_{3},g_{4})$, $g_{i}\approx
y_{i}\in\mathbb{R},\ i=1,4$, $g_{1}g_{3}+g_{2}g_{4}=0$ and $\
\\{e^{i\lambda_{1}},e^{i\lambda_{2}}\\}$ consists of pure vibrations. If
follows from Corollary 3.8 that $\langle u\mid exp(iA)v\rangle$ has a mean
value. Once gain $\ \lambda_{1}$ is a pure infinity and, consequently,

$\ |\langle u\mid
exp(iA)v\rangle|=2|g_{1}g_{3}|\cdot\bigg{|}sin\bigg{(}\frac{\lambda_{1}-\lambda_{2}}{2}\bigg{)}\bigg{|}$

concluding that the transition probability is

$\frac{4}{\pi}\cdot|g_{1}(0)g_{3}(0)|=\frac{4}{\pi}\bigg{|}\bigg{(}-\frac{\sqrt{2}}{2}\bigg{)}\bigg{(}\frac{\sqrt{2}}{2}\bigg{)}\bigg{|}=\frac{2}{\pi}\approx
0.636619772368$

This value is different from the one found in [5, 6]. From our calculations,
in the dimension $\ 2$ case, it follows that if the support of the generalized
transition probability has more dan one element then the same will be true for
the element $\ |g_{1}\cdot g_{3}|$.

Consider
$A=\Bigg{[}\varepsilon\longrightarrow\left(\begin{array}[]{ll}\frac{1}{\sqrt{\epsilon}+\epsilon^{2}}&\frac{2}{\epsilon-2\epsilon^{2}}\\\
\frac{2}{\epsilon-2\epsilon^{2}}&\ \
\frac{1}{\epsilon}\end{array}\right)\Bigg{]}$ and let us calculate the mean
value $|\langle u\mid\exp(-iA)v\rangle|$.

The eigenvalues of $\ -A$ are

$\lambda(\varepsilon)=\frac{1}{2}\Bigg{[}\frac{1+\sqrt{\varepsilon}+\varepsilon\sqrt{\varepsilon}}{\varepsilon(1+\varepsilon\sqrt{\varepsilon})}\pm\sqrt{\bigg{(}\frac{1+\sqrt{\varepsilon}+\varepsilon\sqrt{\varepsilon}}{\varepsilon(1+\varepsilon\sqrt{\varepsilon})}\bigg{)}^{2}-4\bigg{[}\frac{\sqrt{\varepsilon}(1-2\varepsilon)^{2}-\varepsilon\sqrt{\varepsilon}-1}{\varepsilon^{2}(1+\varepsilon\sqrt{\varepsilon})(1-2\varepsilon)^{2}}\bigg{]}}\
\Bigg{]}$

From we get that

$\alpha\lambda=\Bigg{[}\varepsilon\longrightarrow\frac{1}{2}\Bigg{[}\frac{1+\sqrt{\varepsilon}+\varepsilon\sqrt{\varepsilon}}{1+\varepsilon\sqrt{\varepsilon}}\pm\sqrt{\bigg{(}\frac{1+\sqrt{\varepsilon}+\varepsilon\sqrt{\varepsilon}}{1+\varepsilon\sqrt{\varepsilon}}\bigg{)}^{2}-4\bigg{[}\frac{\sqrt{\varepsilon}(1-2\varepsilon)^{2}-\varepsilon\sqrt{\varepsilon}-1}{(1+\varepsilon\sqrt{\varepsilon})(1-2\varepsilon)^{2}}\bigg{]}}\
\Bigg{]}\Bigg{]}$

and hence $\alpha\cdot\lambda\approx\frac{1}{2}[1\pm\sqrt{5}]$ We denote by $\
\lambda_{1}$ the eigenvalue with the $\ +$ sign chosen and by $\ \lambda_{2}$
the one with the $\ -$ sign chosen. From this it follows that $\ \lambda_{1}$
and $\ \lambda_{2}$ are pure infinities. Also

$\ \alpha(\lambda_{1}-\lambda_{2})\approx\sqrt{5}$

and hence $\lambda_{1}-\lambda_{2}$ is an infinity whose norm is $\ \geq
e^{-1}$. A basis of eigenvectors of $\ A$ is $\\{(1,k),(k,-1)\\}$, where

$\
k(\varepsilon)=\frac{(\varepsilon-2\varepsilon^{2})}{2}\bigg{(}\lambda_{1}-\frac{1}{\sqrt{\epsilon}+\epsilon^{2}}\bigg{)}=$
$\frac{1-2\varepsilon}{2}\Bigg{[}1+\sqrt{\bigg{(}\frac{1+\sqrt{\varepsilon}+\varepsilon\sqrt{\varepsilon}}{1+\varepsilon\sqrt{\varepsilon}}\bigg{)}^{2}-4\bigg{[}\frac{\sqrt{\varepsilon}(1-2\varepsilon)^{2}-\varepsilon\sqrt{\varepsilon}-1}{(1+\varepsilon\sqrt{\varepsilon})(1-2\varepsilon)^{2}}\bigg{]}}\
\Bigg{]}\approx\frac{1+\sqrt{5}}{2}=:k_{0}$

Hence $\ M=\frac{1}{\sqrt{1+k^{2}}}\left(\begin{array}[]{ll}1&-k\\\
k&1\end{array}\right)$ conjugates $\ A$ into diagonal form $\
D=\left(\begin{array}[]{ll}e^{i\lambda_{1}}&0\\\
0&e^{i\lambda_{2}}\end{array}\right)$ and $\ M\approx
M_{0}=\frac{1}{\sqrt{1+k^{2}_{0}}}\left(\begin{array}[]{ll}1&-k_{0}\\\
k_{0}&1\end{array}\right)$. From this it follows that

$\langle u\mid exp(iA)v\rangle=\langle Mu\mid
DMv\rangle=:g_{1}g_{3}e^{i\lambda_{1}}+g_{2}g_{4}e^{i\lambda_{2}}$

where $\ Mu=(g_{1},g_{2}),Mv=(g_{3},g_{4})$, $g_{i}\approx
y_{i}\in\mathbb{R},\ i=1,4$, $g_{1}g_{3}+g_{2}g_{4}=0$ and $\
\\{e^{i\lambda_{1}},e^{i\lambda_{2}}\\}$ consists of pure vibrations. If
follows from Corollary 3.8 that $\langle u\mid exp(iA)v\rangle$ has a mean
value. Note that $\ \lambda_{1}-\lambda_{2}$ is a pure infinity. As in the
previous example, one gets that

$\ |\langle u\mid
exp(iA)v\rangle|=2|g_{1}g_{3}|\cdot\bigg{|}sin\bigg{(}\frac{\lambda_{1}-\lambda_{2}}{2}\bigg{)}\bigg{|}$

As before, we conclude that the transition probability is

$\frac{4}{\pi}\cdot|g_{1}(0)g_{3}(0)|=\frac{4}{\pi}\cdot\bigg{|}\frac{1}{1+k_{0}^{2}}\bigg{(}\frac{\sqrt{2}}{2}+k_{0}\frac{\sqrt{2}}{2}\bigg{)}\bigg{(}\frac{\sqrt{2}}{2}-k_{0}\frac{\sqrt{2}}{2}\bigg{)}\bigg{|}=\frac{2(k_{0}^{2}-1)}{\pi(k_{0}^{2}+1)}\approx
0.284705017367$

The reported value in [5, 6] is $\ 0.1539$.

We now consider the general finite dimensional case. Proceeding in the same
way for an $\ n\times n$ symmetric matrix $\ A$, with $\
Spec(A)=\\{\lambda_{1},\cdots,\lambda_{n}\\}$, one gets that

$|\langle u\mid
exp(iA)v\rangle|=\sqrt{\|a\|^{2}+2\sum\limits_{i<j}a_{i}a_{j}cos(\lambda_{i}-\lambda_{j})}$
$|\langle u\mid
exp(iA)v\rangle|=2\sqrt{-\sum\limits_{i<j}a_{i}a_{j}sin^{2}\bigg{(}\frac{\lambda_{i}-\lambda_{j}}{2}\bigg{)}}$
$|\langle u\mid exp(iA)v\rangle|\leq
2\bigg{(}\sum\limits_{i<j}\sqrt{|a_{i}|\cdot|a_{j}|\
}\cdot\bigg{|}sin\bigg{(}\frac{\lambda_{i}-\lambda_{j}}{2}\bigg{)}\bigg{|}\bigg{)}$
$a=(a_{1},\cdots,a_{n})\in({\overline{\mathbb{C}}_{as}})^{n}\ \mbox{and}\
\sum\limits_{i}a_{i}=0,\|a\|^{2}<1$ (9)

If the set $\ \\{\lambda_{i}-\lambda_{j}\ :\ \lambda_{i}\neq\lambda_{j}\in
Spec(A)\\}$ consists of pure infinities and infinitesimals and $\
supp(a)=\\{a_{0}\\}\in{\mathbb{R}^{n}}$ then $\ |\langle u\mid
exp(iA)v\rangle|^{2}$ has a mean value (similar to the case $\ n=2$). One can
use Phasor addition to reduz this sum and be able to see if $\ |\langle u\mid
exp(iA)v\rangle|$ has a mean value. Probably $\ \\{\lambda_{i}-\lambda_{j}\ :\
\lambda_{i},\lambda_{j}\in Spec(A)\\}$ consisting of pure infinities and
infinitesimals generically imply all other conditions.

The occurrence of phasor addition and the relations $\
a=(a_{1},\cdots,a_{n})\in({\overline{\mathbb{C}}_{as}})^{n}$, $\
\sum\limits_{i}a_{i}=0$, $\ \|a\|^{2}<1$ in Equation 6 might be of interest.
We shall call it the perfect destructive interference condition.

We shall now prove that the perfect destructive interference condition also
holds for the transition probability of normal matrices. In fact, let $\ U\in
M_{n}(\overline{\mathbb{K}})$ be a normal matrix, i.e., $\ UU^{*}=U^{*}U$. We
may write $\ U=A+iB$ where $\ A=\frac{1}{2}(U+U^{*})$ and $\
B=\frac{1}{2i}(U-U^{*})$ are selfadjoint commuting matrices. Let $\ M$ be a
unitary matrix such that $MAM^{*}=D_{1}$ is a diagonal matrix, $\ \lambda\in
Spec(A)$ and $\ W=V_{\lambda}$. Since $\ W$ is generated by orthogonal free
vectors, it is in fact generated by some of the columns of $\ M$, it is a free
$\ \overline{\mathbb{K}}-$module. Since $\ AB=BA$, it follows that $\ B$
leaves $\ W$ invariant and $\ B_{|_{W}}$ is a symmetric operator. $\ W$ being
a free $\ \overline{\mathbb{K}}-$module and $\ B_{|_{W}}$ being symmetric,
implies that we may decompose $\ W$ as a direct sum of eigen modules of $\ B$.
Hence, modifying some columns of $\ M$ if necessary, we may suppose that $\
MBM^{*}=D_{2}$ is also a diagonal matrix and thus $\
MUM^{*}=D=diag(\lambda_{1},\cdots,\lambda_{n})$ is diagonal with $\
MM^{*}=Id$. Let $\ u,v$ be orthogonal vectors of $\ {\mathbb{K}}^{n}$, then $\
\langle exp(iU)u\mid v\rangle=\langle M^{*}exp(iD)Mu\mid v\rangle=\langle
exp(iD)Mu\mid
Mv\rangle=\sum\limits_{k}exp(i\lambda_{k})g_{k}\overline{f_{k}}$, with $\
\sum\limits_{k}g_{k}f_{k}=\langle Mu\mid Mv\rangle=\langle u\mid v\rangle=0$,
proving the claim. It follows that the results of Section 5 also holds for
normal matrices. In [5, 6] the following expression is considered:

$\bigg{|}\frac{1}{\sqrt{6}}\bigg{[}\exp\bigg{(}\frac{i}{\varepsilon}\bigg{)}-2\exp\bigg{(}\frac{i}{\varepsilon^{2}}\bigg{)}+(1+\varepsilon)\exp\bigg{(}\frac{i}{\sqrt{\varepsilon}}\bigg{)}\bigg{]}\bigg{|}$

Since this expression does not satisfy the perfect destructive interference
condition, it does not represent a transition probability coming from a normal
matrix.

###### Theorem 6.1

Let $\ f=\sqrt{\sum\limits_{i}g_{i}\cdot(f_{i}\circ\mu_{i})^{2}\ }$, where
$g=(g_{1},\cdots,g_{n})\in({\overline{\mathbb{C}}_{as}})^{n}$ or $\
|g_{i}|\leq C^{2}\ \forall\ i$ for some real constant $\ C>0$, the $\
\mu_{i}$’s are pure infinities or infinitesimals and the $\ f_{i}$’s are a.p.
functions. Then $\ supp(\nu(f))\neq\emptyset$.

Proof. We have that $\
f=|f|=\sqrt{\bigg{|}\sum\limits_{i}g_{i}\cdot(f_{i}\circ\mu_{i})^{2}\bigg{|}\
}\leq C\bigg{(}\sum\limits_{i}|f_{i}\circ\mu_{i}|\bigg{)}$. The result follows
by Theorem 3.6 and Theorem 3.9.

It follows from the previous theorem that
$\bigg{|}\frac{1}{\sqrt{6}}\bigg{[}\exp\bigg{(}\frac{i}{\varepsilon}\bigg{)}-2\exp\bigg{(}\frac{i}{\varepsilon^{2}}\bigg{)}+(1+\varepsilon)\exp\bigg{(}\frac{i}{\sqrt{\varepsilon}}\bigg{)}\bigg{]}\bigg{|}$
has generalized transition probabilities and the same holds for every normal
matrix $\ A$ whose eigenvalues are either pure infinities or infinitesimals.
The last assertion follows from Equation 6, Theorem 3.9 and Theorem 6.1. With
this, we gave an affirmative answer for the existence of generalized
transition probabilities in the finite dimensional case. The transition
probability defined by Colombeau-Gsponer exists if and only the support of the
generalized transition probabilities has a unique element.

Acknowledgements: This paper is part of the second author Ph.D. thesis done
under the guidance of the first author. He is grateful to IME-USP and the
Centro de Ciências de Balsas-UFMA for their hospitality and the possibility to
carry out his Ph.D. as programmed. The first author is grateful to J. Aragona
and J.F. Colombeau for previous collaborations resulting in [5] and [6] and to
J.F. Colombeau for introducing him into the subject contained in [30, 42].

## References

  * [1] Antosik, P., Mikusinski, J., Sikorski, R., Theory of Distributions: The sequential approach, Elsevier, 1957.
  * [2] Aragona, J., Biagioni. H.A., An intrinsic definition of the Colombeau algebra of generalized functions, Analysis Mathematica 17,1991, p. 75-132.
  * [3] Aragona, J., Catuogno, P., Colombeau, J.F., Juriaans, S.O., Olivera, Ch., Multiplication of distributions in Mathematical Physics, V.K. Dobrev eds Lie Theory and its Applications in Physics. Springer Proceedings in Math. and Statistics 191, 2017.
  * [4] Aragona, J., Fernandez, R., Juriaans, S.O., A discontinuous Colombeau differential calculus, Monatsh. Math. 144, 2005,pp. 13-29.
  * [5] Aragona, J., Colombeau, J.F., Juriaans, S.O., Multiplication of distributions and nonperturbative transition probabilities in QFT, unpublshed (2017).
  * [6] Aragona, J., Colombeau, J.F., Catuogno, P., Juriaans, S.O., Olivera, C., Multiplication of Distributions and Nonperturbative Calculations of Transition Probabilities, Quantum Theory And Symmetries, 393–401, 2017, Springer.
  * [7] Aragona, J., Fernandez, R., Juriaans, S. O., Oberguggenberger, M., Differential calculus and integration of generalized functions over membranes, Monatshefte f’́ur Mathematik (2012).
  * [8] Aragona, J., Fernandez, R., Juriaans, S.O.,  The sharp topology on the full Colombeau algebra of generalized functions (2005), Integral Transforms and Special Functions vol. 17, Nos. 2-3, February-March 2006, 165-170.
  * [9] Aragona, J., Fernandez, R., Juriaans, S.O., Natural topologies on Colombeau algebras, Topol. Methods Nonlinear Anal. 34(1), 161-180 (2009)
  * [10] Aragona, J., Garcia, A. R. G., Juriaans, S. O., Algebraic theory of Colombeau’s generalized numbers, Journal of Algebra 384 (2013), 194-211.
  * [11] Aragona, J., Garcia, A. R. G., Juriaans, S. O., Generalized solutions of nonlinear parabolic equation with generalized functions as initial data, Nonlinear Analysis 71(2009), 5187-5207.
  * [12] Aragona, J., Juriaans, S.O.,  Some Structural Properties of the Topological Ring of Colombeau Generalized Numbers Comm. Alg. 29(5), (2001), 2201-2230.
  * [13] Aragona, J., Juriaans, S. O., Oliveira, O. R. B, Scarpalézos, D., _Algebraic and geometric theory of the topological ring of Colombeau generalized functions_ , Proceedings of the Edinburgh Mathematical Society (Series 2) 51 (2008), 545-564.
  * [14] Aragona, J., Soares, M.,  An Existence Theorem for an Analytic first order PDE in the Framework of Colombeau’s theory Monatsh. Math. 134, (2001), 9-17.
  * [15] Besicovitch, A.S., Almost Periodic Functions, Cambridge University Press, 1954.
  * [16] Bhatia, R., Perturbation bounds for matrix eigenvalues, SIAM, 2007.
  * [17] Bhatia, R., The Hoffman-Wielandt inequality in infinite dimensions, Proceedings of the Indian Academy of Sciences-Mathematical Sciences, vol.104(3), 483–494, 1994.
  * [18] Biagioni, H.A., A Nonlinear Theory of Generalized Functions Lecture Notes in Math. 1421. Springer, 1990.
  * [19] Bohr, H., Almost Periodic Functions, Chelsea Pub Company, New York, 1947.
  * [20] Christyakov, V.V., The Colombeau generalized nonlinear analysis and the Schwartz linear distribution theory, J. Math. Sci. 93, 1999, p. 3-40.
  * [21] Colombeau, J.F., A multiplication of distributions, J. Math. Anal. Appl. 94, 1, 1983, p. 96-115.
  * [22] Colombeau, J.F., A general multiplication of distributions, Comptes rendus Acad. Sci. Paris 296,1983, pp. 357-360.
  * [23] Colombeau, J.F., New Generalized Functions and Multiplication of Distributions, Noth-Holland-Elsevier, 1984.
  * [24] Colombeau, J.F., Elementary Introduction to New Generalized Functions, Noth-Holland-Elsevier, 1985.
  * [25] Colombeau, J.F., Multiplication of Distributions, Lecture Notes in Math. 1532, SpringerVerlag, Berlin-Heidelberg-New York, 1992.
  * [26] Colombeau, J.F., Multiplication of distributions, Bull. Amer. Math. Soc., 23, 2, 1990, p. 251-268.
  * [27] Colombeau, J.F., Nonlinear generalized functions., Sao Paulo J. Math. Sci. 7, 2013, 2,pp; 201-239.
  * [28] Colombeau, J.F., Mathematical problems on generalized functions and the canonical Hamiltonian formalism, arXiv.org 0708.3425.
  * [29] Colombeau, J.F.,  New Generalized Functions and Multiplication of Distributions,  North Holland, Amsterdam 1984.
  * [30] Colombeau, J.F., Mathematical problems on generalized functions and the canonical Hamiltonian formalism, arXiv, preprint arXiv:0708..3425, 2007.
  * [31] Colombeau, J.F., Gsponer, A., The Heisenberg-Pauli canonical Hamiltonian formalism of quantum field theory in the rigorous mathematical setting of nonlinear generalized functions (part 1), arXiv.org 0807.0289. 2008.
  * [32] Egorov, Y., A theory of generalized functions, Russian Math. Surveys 45, 5, 1990, p. 1-49.
  * [33] Garcia, A.R.G., Juriaans, S.O., Rodrigues, W., Silva, J.C., Off-Diagonal Condition, Monatsh. Math., to appear (2022).
  * [34] Garcia, A.R.G., Juriaans, S. O., Oliveira, J., Rodrigues, W. M., A Non-linear Theory of Generalized Functions, São Paulo Journal of Mathematical Sciences, Springer, Published online (2021).
  * [35] Garcia, A.R.G., Juriaans, S. O., Oliveira, J., Rodrigues, W. M., Sets of Uniqueness for holomorphic Nets , ResearchGate, DOI: 10.13140/RG.2.2.36425.98405, 2020.
  * [36] Garetto, C., Topological Structures in Colombeau Algebras: Topological $\tilde{\mathbb{C}}$-modules and Duality Theory, Acta Applicandae Mathematica, vol.88(1), 81-123, 2005.
  * [37] Garetto, C., Vernaeve, H., Hilbert ${\tilde{\mathbb{C}}}$-modules: structural properties and applications to variational problems, Transactions of the American Mathematical Society, vol.363(4), 2047-2090, 2011.
  * [38] Grosser, M., Farkas, E., Kunzinger, M., Steinbauer, R., On the Foundations of Nonlinear Generalized Functions I and II, Memoirs of the AMS 153, 2001.
  * [39] Grosser, M., Farkas, E., Kunzinger, M., Steinbauer, R., Geometric Theory of Generalized Functions with Applications to General Relativity, Kluwer 2001.
  * [40] Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.,  Geometric Theory of Generalized Functions with Applications to General Relativity,  Kluwer Acad. Publ. vol 537, 2001.
  * [41] Grosser M, Kunzinger M., Steinbauer R., Vickers J.A.,  A Global Theory of Algebras of Generalized Functions,  Adv. Math., 166 (1), (2002), 50-72.
  * [42] Gsponer, A., A concise introduction to Colombeau generalized functions and their application in classical electrodynamics, European J. Phys. 30, 2009, 1, pp. 109-126.
  * [43] Hoskins, R.F., Sousa Pinto, J., Distributions, Ultradistributions and other Generalized Functions, Ellis-Horwood, New York-London, 1994.
  * [44] Hoffman, A.J, Wielandt, H.W, The variation of the spectrum of a normal matrix, Duke Math. J. (20), 1, 37-39, 1953.
  * [45] Hormann, G., Konjik, S., Kunzinger, M., Symplectic modules over Colombeau-generalized numbers, Comm. Algebra, 42(8), 3558–3577, 2014.
  * [46] Juriaans, S.O., Oliveira, J., Fixed Point Theorems for Hypersequences and the Foundation of Generalized Differential Geometry I: The Simplified Algebra, arXiv preprint arXiv:2205.00114, 2022, DOI: 10.13140/RG.2.2.32732.05761.
  * [47] Juriaans, S.O., A Fixed Point Theorem for Internal Sets, Research Gate, DOI: 10.13140/RG.2.2.16367.30884, 2022.
  * [48] Kastler, D., Introduction a l’Electrodynamique Quantique., Dunod, Paris, 1961.
  * [49] Kunzinger, M.,  Lie Transformation Groups in Colombeau algebras,  Doctoral Thesis, University of Viena, 1996.
  * [50] Kunzinger, M., Oberguggenberger M.,  Characterization of Colombeau Generalized Functions by their Point Value,  Math. Nachr. 203, (1999), 147-157.
  * [51] Kunzinger, M., Oberguggenberger, M., Characterization of Colombeau generalized functions by their point values, Math. Nachr. 203, 147-157 (1999)
  * [52] Kunzinger, M., Steinbauer, R., A rigorous solution concept for geodesics and geodesic deviation equations in impulsive gravitational waves, J. Math. Phys. 40 (1999), 1479-1489.
  * [53] Farkas, E., Grosser, M, Kunzinger, M., Steinbauer, R., On the foundations of nonlinear generalized functions I, II, Mem. Am. Soc. 153(729) (2001).
  * [54] M, Kunzinger, M., Steinbauer, R., Generalized Pseudo-Riemannian Geometry, T.A.M.S., Vol. 354, Number 10, 4179-4199.
  * [55] Mayerhofer, E. , On Lorentz geometry in algebras of generalized functions, Proc. R. Soc. Edinb. Sect. A, Math., 138(4):843-871, 2008.
  * [56] Nedeljkov, M., Pilipovic, S., Scarpalezos. D., The Linear Theory of Colombeau Generalized Functions, Pitman Research Notes in Math. 1998.
  * [57] Oberguggenberger, M.  Multiplication of Distributions and Applications to Partial Differential Equations,  Pitman, 1992.
  * [58] Oberguggenberger, M., Pilipovic S., Scarpalezos D.,  Local Properties of Colombeau Generalized Functions,  Math. Nachr. 256, (2003), 88-99.
  * [59] Oberguggenberger, M., Pilipovic S., Valmorin V.,  Global Representatives of Colombeau holomorphic Generalized Functions,  preprint, 2005.
  * [60] Oberguggenberger, M., Vernaeve, H., Internal sets and internal functions in Colombeau theory, J. Math. Anal. Appl. 341, 649-659 (2008).
  * [61] Oberguggenberger, M., Multiplication of Distributions and Partial Differential Equations, Pitman Research Notes in Math. 259, Longman, Harlow, 1992.
  * [62] Penrose, R., The Road of Reality. A Complete Guide to the Laws of the Universe Jonathan Cape, 2004.
  * [63] Robinson, A., Non-standard Analysis, North-Holland, Amsterdam, (1966).
  * [64] Rosinger, E. E., Distributions and nonlinear partial differential equations, Lect. Notes Math. 684, Springer, Berlin (1978).
  * [65] Rosinger, E. E., Nonlinear partial differential equations. Sequential and weak solutions, North Holland, Amesterdam (1980).
  * [66] Rosinger, E. E., Generalized solutions of nonlinear partial differential equations, North Holland, Amesterdan (1987).
  * [67] Rosinger, E. E., Which are the maximal ideals, Prespacetime Journal, vol. 2, Issue 2, pg. 144-156, 2011.
  * [68] Rosinger, E. E., _Non-linear partial differential equations. An algebraic view of generalized solutions_ , North Holland, Amestedam (1990).
  * [69] Scarpalezos, D., Topologies dans les espaces de nouvelles fonctions generalisées de Colombeau. ${{\overline{\mathbb{C}}}}$ topologiques, Université Paris 7, 1993
  * [70] Scarpalezos, D., Colombeau’s generalized functions: topological structures micro local properties. A simplified point of view, CNRS-URA212, Université Paris 7, 1993.
  * [71] Scarpalezos, D., _Colombeau’s generalized functions: Topological structures; Microlocal properties. A simplified point of view. Part I_ , Bull. Cl. Sci. Math. Nat. Sci. Math 121 (25) (2000), 89-114.
  * [72] Schmüdgen, K., Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics 265, Springer Verlag, 2012.
  * [73] Schwartz, L., Sur l ’impossibilité de la multiplication des distributions, Comptes Rendus Acad. Sci. Paris 239, 1954, p. 847-848.
  * [74] Schwartz, L., Théorie des Distributions, Hermann, Paris, 1966.
  * [75] Steinbauer, S., Vickers, J.A., The use of Generalized Functions and Distributions in General Relativity, Class. Quant. Grav. 23, pp. R91-113, 2006.
  * [76] Vickers, J.A., Distributional geometry in general relativity, J. Geom. Phys. 62,3,2012, p. 692-705.
  * [77] Todorov, T.D., Pointwise Value and Fundamental Theorem in the Algebra of Asymptotic Functions, arXiv (2006).
  * [78] Todorov, T.D., Vernaev, H., Full algebra of generalized functions and non-standard asymptotic analysis, Log Anal, (2008) 205-234.
  * [79] Weinberg, S. The Quantum Theory of Fields Vol. 1, Cambridge, 1995.
  * [80] Vernaeve, H., Ideals in the ring of Colombeau generalized numbers, Communications in Algebra, 38, p 2199 - 2228, 2010.
  * [81] Vernaeve, H., B͡anach $\widetilde{\mathbb{C}}$-algebras https://doi.org/10.48550/arxiv.0811.1742, arXiv, 2008.
  * [82] Vernaeve, H., Optimal embeddings of Distributions into Algebras, Proc. Edinburgh Math. Soc. (2003) 46, 373-378.
  * [83] Vernaeve, H., Isomphisms of Algebras of Colombeau Generalized Functions, Monatsh. Math. 162, 225-237 (2011).
  * [84] Vernaeve, H., Banach $\widetilde{\mathbb{C}}$-algebras, arXiv, 0811.1742v1[mathFA] 11 Nov (2008).
  * [85] Wolfram, S., A New Kind of Science, Wolfram Media, Inc., 2002.