2. Position operator

Let ψ ¼ ψ<sup>n</sup> � �<sup>∞</sup> <sup>n</sup>¼<sup>0</sup> be a fixed Carleman sequence in <sup>L</sup><sup>2</sup> <sup>X</sup>; <sup>μ</sup> � �. It is clear from the foregoing that <sup>ψ</sup> is not a complete sequence in L<sup>2</sup> X; μ � �. We denote by L<sup>ψ</sup> the closure of the linear span of the sequence ψpð Þx n o<sup>∞</sup> p¼0 :

$$\mathfrak{L}\_{\psi} = \overline{\text{span}\{\psi\_n, n \in \mathbb{N}\}}.\tag{11}$$

We start this section by defining some formal spaces.

#### 2.1. Formal elements

Definition 1. (see [7]) We call formal element any expression of the form

$$f = \sum\_{n \in \mathbb{N}} a\_n \psi\_{n'} \tag{12}$$

where the coefficients anð Þ n∈ N are scalars.

The sequence að Þ<sup>n</sup> <sup>n</sup> is said to generate the formal element f .

Definition 2. We say that f is the zero formal element, and we note f ¼ 0 if an ¼ 0 for all n∈ N:

We say that two formal elements f <sup>¼</sup> <sup>P</sup> <sup>n</sup><sup>∈</sup> <sup>N</sup> anψ<sup>n</sup> and g <sup>¼</sup> <sup>P</sup> <sup>n</sup> <sup>∈</sup> <sup>N</sup> bnψ<sup>n</sup> are equal if an ¼ bn for all n∈ N.

If φ is a scalar function defined for each an, we set

$$\left\langle \phi \left( \sum\_{n} a\_{n} \psi\_{n} \right) \right\rangle = \sum\_{n} \phi(a\_{n}) \psi\_{n'} \tag{13}$$

or in another form,

$$
\varphi(a\_1, a\_2, \dots, a\_n, \dots) = (\varphi(a\_1), \varphi(a\_2), \dots, \varphi(a\_n), \dots). \tag{14}
$$

For example, let

$$
\varphi(\mathbf{x}) = \frac{1}{\mathbf{x}} \quad (\mathbf{x} \neq \mathbf{0}). \tag{15}
$$

We define a linear form h i :; : on F<sup>ψ</sup> by setting:

with the series converging on the right side of (21).

<sup>f</sup> <sup>¼</sup> <sup>X</sup> n

<sup>n</sup> anbn <sup>¼</sup> <sup>P</sup>

n

n

a1 <sup>n</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> n � �bn <sup>¼</sup> <sup>X</sup>

Here, we introduce the crucial tool of our work.

<sup>f</sup> <sup>∘</sup> <sup>g</sup> <sup>¼</sup> <sup>X</sup> n

<sup>¼</sup> <sup>X</sup> n

2.3. The multiplication operation

denoted " ∘ " and defined by:

anψ<sup>n</sup> !

ð Þ <sup>λ</sup>an bn <sup>¼</sup> <sup>λ</sup> <sup>X</sup>

a1 <sup>n</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> n � �ψn;

\* +

; X n

Proof. Indeed, let

in Fψ.

2.

3.

■

We have then:

1. h i <sup>f</sup> ; <sup>g</sup> <sup>¼</sup> <sup>P</sup>

4. h i <sup>f</sup> ; <sup>f</sup> <sup>¼</sup> <sup>P</sup>

h i <sup>λ</sup><sup>f</sup> ; <sup>g</sup> <sup>¼</sup> <sup>λ</sup> <sup>X</sup>

<sup>¼</sup> <sup>X</sup> n

<sup>f</sup> <sup>1</sup> <sup>þ</sup> <sup>f</sup> <sup>2</sup>; <sup>g</sup> � � <sup>¼</sup> <sup>X</sup>

X n

Proposition 4. The form (21) verifies the properties of scalar product.

anψn, g <sup>¼</sup> <sup>X</sup>

<sup>n</sup> anbn ¼ h i f ; g :

n

\* +

<sup>n</sup> j j an <sup>2</sup> <sup>≥</sup> 0andh i <sup>f</sup> ; <sup>f</sup> <sup>&</sup>gt; 0 if <sup>f</sup> 6¼ <sup>0</sup>:

anψ<sup>n</sup>

anψn; X n

> X n

> > n a1 <sup>n</sup>bn <sup>þ</sup><sup>X</sup> n a2

Definition 6. We call multiplication with respect to the Carleman sequence ψ<sup>n</sup>

f ;ψ<sup>n</sup> � � <sup>g</sup>;ψ<sup>n</sup>

bnψ<sup>n</sup>

Remark 5. On <sup>L</sup>ψ, the scalar product h i :; : coincides with the scalar product ð Þ :; : of L<sup>2</sup> <sup>X</sup>; <sup>μ</sup> � �.

� � <sup>¼</sup> <sup>X</sup>

Definition 7. We call position operator in L<sup>ψ</sup> any unitary self-adjoint (see [1]) operator satisfying

n

\* +

n

anψn; X n

\* +

bnψ<sup>n</sup>

bnψn, f <sup>1</sup> <sup>¼</sup> <sup>X</sup>

<sup>¼</sup> <sup>X</sup> n

bnψ<sup>n</sup>

n a1

ð Þ λan ψn;

\* +

¼ λh i f ; g :

X n

bnψ<sup>n</sup>

<sup>n</sup>bn <sup>¼</sup> <sup>f</sup> <sup>1</sup>; <sup>g</sup> � � <sup>þ</sup> <sup>f</sup> <sup>2</sup>; <sup>g</sup> � �:

anbnψn, <sup>∀</sup>ð Þ <sup>f</sup> ; <sup>g</sup> <sup>∈</sup> <sup>F</sup><sup>2</sup>

� �

<sup>n</sup>, the operation

<sup>ψ</sup>: (22)

<sup>n</sup>ψnand <sup>f</sup> <sup>2</sup> <sup>¼</sup> <sup>X</sup>

n a2 <sup>n</sup>ψ<sup>n</sup>

<sup>¼</sup> <sup>X</sup> n

anbn (21)

53

A Formal Perturbation Theory of Carleman Operators http://dx.doi.org/10.5772/intechopen.79022

If an 6¼ 0 for all n∈ N, then the formal element

$$\varphi\left(\sum\_{n} a\_{n}\psi\_{n}\right) = \sum\_{n} \frac{1}{a\_{n}}\psi\_{n} \tag{16}$$

is called inverse of the formal element <sup>f</sup> <sup>¼</sup> <sup>P</sup> <sup>n</sup> anψn.

Furthermore, we define the conjugate of a formal element f by

$$
\overline{f} = \sum\_{n} \overline{a\_n} \psi\_n. \tag{17}
$$

Denote by F<sup>ψ</sup> the set of all formal elements (12).

On Fψ, we define the following algebraic operations:

the sum

$$\begin{aligned} + &: \quad \mathcal{F}\_{\psi} \times \mathcal{F}\_{\psi} \\ & \quad \left(\sum\_{n} a\_{n} \psi\_{n}\right) + \left(\sum\_{n} b\_{n} \psi\_{n}\right) \\ \end{aligned} \qquad \rightarrow \quad \begin{aligned} \mathcal{F}\_{\psi} \\ &\quad \sum\_{n} (a\_{n} + b\_{n}) \psi\_{n} \end{aligned} \tag{18}$$

and the product

$$\begin{aligned} \cdot &: \quad \mathbb{C} \times \mathcal{F}\_{\psi} \\ & \quad \cdot \left( \sum\_{n} a\_{n} \psi\_{n} \right) \quad = \sum\_{n} (\lambda.a\_{n}) \psi\_{n} . \end{aligned} \tag{19}$$

Hence, we obtain a complex vector space structure for Fψ.

#### 2.2. Bounded formal elements

Definition 3. A formal element f <sup>¼</sup> <sup>P</sup> n ∈ N anψ<sup>n</sup> is bounded if its sequence að Þ<sup>n</sup> <sup>n</sup> is bounded.

We denote by B<sup>ψ</sup> the set of all bounded formal elements.

It is clear that B<sup>ψ</sup> is a subspace of Fψ.

We claim that:


$$
\mathcal{L}\_{\psi} \subset \mathcal{B}\_{\psi} \subset \mathcal{F}\_{\psi}. \tag{20}
$$

A Formal Perturbation Theory of Carleman Operators http://dx.doi.org/10.5772/intechopen.79022 53

We define a linear form h i :; : on F<sup>ψ</sup> by setting:

$$\left\langle \sum\_{n} a\_{n} \psi\_{n}, \sum\_{n} b\_{n} \psi\_{n} \right\rangle = \sum\_{n} a\_{n} \overline{b\_{n}} \tag{21}$$

with the series converging on the right side of (21).

Proposition 4. The form (21) verifies the properties of scalar product.

Proof. Indeed, let

$$f = \sum\_{n} a\_{n} \psi\_{n'} \mathcal{g} = \sum\_{n} b\_{n} \psi\_{n'} f\_{1} = \sum\_{n} a\_{n}^{1} \psi\_{n} \text{and} \, f\_{2} = \sum\_{n} a\_{n}^{2} \psi\_{n'}$$

in Fψ.

φð Þ¼ x

φ X n

Furthermore, we define the conjugate of a formal element f by

If an 6¼ 0 for all n∈ N, then the formal element

52 Perturbation Methods with Applications in Science and Engineering

is called inverse of the formal element <sup>f</sup> <sup>¼</sup> <sup>P</sup>

Denote by F<sup>ψ</sup> the set of all formal elements (12).

the sum

and the product

We claim that:

2.2. Bounded formal elements

Definition 3. A formal element f <sup>¼</sup> <sup>P</sup>

It is clear that B<sup>ψ</sup> is a subspace of Fψ.

2. Furthermore we have the strict inclusions:

1. L<sup>ψ</sup> is a subspace of Bψ:

On Fψ, we define the following algebraic operations:

X n

Hence, we obtain a complex vector space structure for Fψ.

We denote by B<sup>ψ</sup> the set of all bounded formal elements.

anψ<sup>n</sup> ! 1 x

anψ<sup>n</sup> !

> <sup>f</sup> <sup>¼</sup> <sup>X</sup> n

þ : F<sup>ψ</sup> � F<sup>ψ</sup> ! F<sup>ψ</sup>

<sup>þ</sup> <sup>X</sup> n

� : C � F<sup>ψ</sup> ! F<sup>ψ</sup>

anψ<sup>n</sup> !

<sup>λ</sup> � <sup>X</sup> n

n ∈ N

bnψ<sup>n</sup> !

> <sup>¼</sup> <sup>X</sup> n

<sup>¼</sup> <sup>X</sup> n

anψ<sup>n</sup> is bounded if its sequence að Þ<sup>n</sup> <sup>n</sup> is bounded.

L<sup>ψ</sup> ⊂B<sup>ψ</sup> ⊂ Fψ: (20)

<sup>¼</sup> <sup>X</sup> n

<sup>n</sup> anψn.

1 an

, xð Þ 6¼ 0 : (15)

anψn: (17)

ð Þ an þ bn ψ<sup>n</sup>

ð Þ <sup>λ</sup>:an <sup>ψ</sup>n: (19)

(18)

ψ<sup>n</sup> (16)

We have then:

$$\begin{split} \mathbf{1.} \quad \langle f, g \rangle &= \sum\_{n} a\_{n} \overline{b\_{n}} = \sum\_{n} a\_{n} \overline{b\_{n}} = \overline{\langle f, g \rangle}. \\ \langle \lambda f, g \rangle &= \left\langle \lambda \left( \sum\_{n} a\_{n} \psi\_{n} \right), \sum\_{n} a\_{n} \psi\_{n} \right\rangle = \left\langle \sum\_{n} (\lambda a\_{n}) \psi\_{n}, \sum\_{n} b\_{n} \psi\_{n} \right\rangle \\ \mathbf{2.} \quad \mathbf{2.} \quad \langle \lambda a\_{n} \rangle \overline{b\_{n}} &= \lambda \left\langle \sum\_{n} a\_{n} \psi\_{n}, \sum\_{n} b\_{n} \psi\_{n} \right\rangle = \lambda \langle f, g \rangle. \\ \left\langle f\_{1} + f\_{2}, g \right\rangle &= \left\langle \sum\_{n} \left( a\_{n}^{1} + a\_{n}^{2} \right) \psi\_{n}, \sum\_{n} b\_{n} \psi\_{n} \right\rangle \\ \mathbf{3.} \quad \quad \quad = \sum\_{n} \left( a\_{n}^{1} + a\_{n}^{2} \right) \overline{b\_{n}} = \sum\_{n} a\_{n}^{1} \overline{b\_{n}} + \sum\_{n} a\_{n}^{2} \overline{b\_{n}} = \left\langle f\_{1}, g \right\rangle + \left\langle f\_{2}, g \right\rangle. \\ \mathbf{4.} \quad \quad \left\langle f, f \right\rangle = \sum\_{n} |a\_{n}|^{2} \ge 0 \text{and} \langle f, f \rangle > 0 \text{ if } f \neq 0. \end{split}$$

■

Remark 5. On <sup>L</sup>ψ, the scalar product h i :; : coincides with the scalar product ð Þ :; : of L<sup>2</sup> <sup>X</sup>; <sup>μ</sup> � �.

#### 2.3. The multiplication operation

Here, we introduce the crucial tool of our work.

Definition 6. We call multiplication with respect to the Carleman sequence ψ<sup>n</sup> � � <sup>n</sup>, the operation denoted " ∘ " and defined by:

$$f \bullet \mathbf{g} = \sum\_{n} \langle f, \psi\_{n} \rangle \langle \mathbf{g}, \psi\_{n} \rangle = \sum\_{n} a\_{n} b\_{n} \psi\_{n'} \quad \forall (f, \mathbf{g}) \in \mathcal{F}\_{\psi}^{2}. \tag{22}$$

Definition 7. We call position operator in L<sup>ψ</sup> any unitary self-adjoint (see [1]) operator satisfying

$$
\mathcal{U}\mathcal{U}(f\circ\mathcal{g}) = (\mathcal{U}f\circ\mathcal{U}\mathcal{g})\_{\prime} \quad \text{for all } f,\mathcal{g}\in\mathcal{L}\_{\psi}.\tag{23}
$$

It's clear that j is injective.

Hence,

■

and

If f <sup>¼</sup> <sup>P</sup>

Let <sup>α</sup> <sup>¼</sup> <sup>P</sup>

It is clear that A

with domain

Now let <sup>m</sup> <sup>∈</sup> <sup>N</sup>: Since <sup>U</sup><sup>2</sup> <sup>¼</sup> I, then

Finally j is well defined as involution.

<sup>n</sup> an ψ<sup>n</sup> ∈ Fψ, then

3. Carleman operator in F<sup>ψ</sup>

3.1. Case of defect indices 1ð Þ ; 1

∘

U Uψ<sup>m</sup>

Notation In the sequel,jnð Þ will be noted by nv. We write

Uf <sup>¼</sup> <sup>U</sup> <sup>X</sup>

Remark 9. The position operator U can be extended over F<sup>ψ</sup> as follows:

A ∘

D A<sup>∘</sup> α n

Uf ¼ f v <sup>¼</sup> <sup>X</sup> n

<sup>p</sup> αpψ<sup>p</sup> ∈ Fψ; we introduce the operator A<sup>∘</sup> <sup>α</sup> defined in L<sup>ψ</sup> by

n

X n

α<sup>n</sup> f ;ψ<sup>n</sup> � � � � �

( )

� 2 < ∞

α;ψ<sup>n</sup> � � <sup>f</sup> ;ψ<sup>n</sup>

<sup>α</sup><sup>f</sup> <sup>¼</sup> <sup>α</sup> <sup>∘</sup><sup>f</sup> <sup>¼</sup> <sup>X</sup>

<sup>α</sup> is a Carleman operator induced by the kernel

� � <sup>¼</sup> <sup>f</sup> <sup>∈</sup>L<sup>ψ</sup> :

an ψ<sup>n</sup> !

� � <sup>¼</sup> <sup>U</sup>ψj mð Þ <sup>¼</sup> <sup>ψ</sup>jjm ð Þ ð Þ <sup>¼</sup> <sup>ψ</sup>m:

jjm ð Þ¼ ð Þ m:

<sup>U</sup>ψ<sup>n</sup> <sup>¼</sup> <sup>ψ</sup>j nð Þ <sup>¼</sup> <sup>ψ</sup><sup>n</sup>

<sup>¼</sup> <sup>X</sup> n

an ψ<sup>n</sup>

an ψ<sup>n</sup> <sup>v</sup> ¼ f v:

<sup>v</sup> (28)

A Formal Perturbation Theory of Carleman Operators http://dx.doi.org/10.5772/intechopen.79022

<sup>v</sup> : (30)

� �ψn: (31)

: (33)

k xð Þ¼ ; <sup>y</sup> <sup>X</sup>αnψnð Þ<sup>x</sup> <sup>ψ</sup>nð Þ<sup>y</sup> , (32)

(29)

55

The term "position operator" comes from the fact (as it will be shown in the following theorem) that for the elements of the sequence ψ ¼ ψ<sup>n</sup> � � <sup>n</sup>, the operator U acts as operator of change of position of these elements.

#### 2.4. Main results

Theorem 8. A linear operator defined on L<sup>ψ</sup> is a position operator if and only if there exist an involution j (i.e., j<sup>2</sup> <sup>¼</sup> Id) of the set <sup>N</sup> such that for all n<sup>∈</sup> <sup>N</sup>

$$
\mathcal{U}\psi\_n = \psi\_{j(n)}.\tag{24}
$$

#### Proof.


$$
\delta \mathcal{U} \psi\_n = \sum\_k \alpha\_{n,k} \,\,\psi\_k \,\,\, with \,\sum\_k |\alpha\_{n,k}|^2 = 1 \,\,\tag{25}
$$

since Uψ<sup>n</sup> ∈Lψ:.

On the other hand, we have

$$\sum\_{k} \alpha\_{n,k} \; \psi\_k = \sum\_{k} \alpha\_{n,k}^2 \; \psi\_k \tag{26}$$

as

$$\mathcal{U}\psi\_n = \mathcal{U}(\psi\_n \circ \psi\_n) = \mathcal{U}\psi\_n \circ \mathcal{U}\psi\_n.$$

The equalities (26) lead to the resolution of the system:

$$\begin{cases} \sum \alpha\_{n,k}^2 = 1, \\ \alpha\_{n,k}^2 = \alpha\_{n,k}, \quad k \in \mathbb{N}. \end{cases} \tag{27}$$

We get then

$$(\forall n \in \mathbb{N}) \ (\exists! k\_n \in \mathbb{N}) \ : \begin{cases} \alpha\_{n,k\_n} = 1, \\ \alpha\_{n,k} = 0 \end{cases} \forall k \neq k\_n.$$

Let us now consider the following application:

$$\begin{array}{ccccc} j &:& \mathbb{N} & \to & \mathbb{N} \\ & n & \mapsto & j(n) = k\_n. \end{array}$$

It's clear that j is injective.

Now let <sup>m</sup> <sup>∈</sup> <sup>N</sup>: Since <sup>U</sup><sup>2</sup> <sup>¼</sup> I, then

$$\mathcal{U}(\mathcal{U}\psi\_m) = \mathcal{U}\psi\_{j(m)} = \psi\_{j(j(m))} = \psi\_m.$$

Hence,

U fð Þ¼ ∘ g ð Þ Uf ∘ Ug , for all f,g∈Lψ: (23)

<sup>n</sup>, the operator U acts as operator of

<sup>2</sup> <sup>¼</sup> <sup>1</sup> (25)

(27)

n, <sup>k</sup> ψ<sup>k</sup> (26)

<sup>U</sup>ψ<sup>n</sup> <sup>¼</sup> <sup>ψ</sup>j nð Þ: (24)

� �

The term "position operator" comes from the fact (as it will be shown in the following

Theorem 8. A linear operator defined on L<sup>ψ</sup> is a position operator if and only if there exist an

<sup>α</sup>n, <sup>k</sup> <sup>ψ</sup><sup>k</sup> with<sup>X</sup>

<sup>α</sup>n, <sup>k</sup> <sup>ψ</sup><sup>k</sup> <sup>¼</sup> <sup>X</sup>

k α2

� � <sup>¼</sup> <sup>U</sup>ψ<sup>n</sup> <sup>∘</sup> <sup>U</sup>ψn:

n, <sup>k</sup> ¼ αn, k, k∈ N:

�

n ↦ j nð Þ¼ kn:

αn, <sup>k</sup> ¼ 0 ∀k 6¼ kn:

k

j j αn, <sup>k</sup>

theorem) that for the elements of the sequence ψ ¼ ψ<sup>n</sup>

54 Perturbation Methods with Applications in Science and Engineering

involution j (i.e., j<sup>2</sup> <sup>¼</sup> Id) of the set <sup>N</sup> such that for all n<sup>∈</sup> <sup>N</sup>

The equalities (26) lead to the resolution of the system:

Let us now consider the following application:

1. It is easy to see that if (24) holds, then U is a position operator.

<sup>U</sup>ψ<sup>n</sup> <sup>¼</sup> <sup>X</sup> k

> X k

Uψ<sup>n</sup> ¼ U ψ<sup>n</sup> ∘ψ<sup>n</sup>

X n α2 n, <sup>k</sup> ¼ 1,

8 < :

α2

ð Þ <sup>∀</sup>n<sup>∈</sup> <sup>N</sup> ð Þ <sup>∃</sup>!kn <sup>∈</sup> <sup>N</sup> : <sup>α</sup>n, kn <sup>¼</sup> <sup>1</sup>,

j : N ! N,

2. Let U be a position operator. According to 1, we can write

change of position of these elements.

2.4. Main results

since Uψ<sup>n</sup> ∈Lψ:.

On the other hand, we have

Proof.

as

We get then

$$j(j(m)) = m.$$

Finally j is well defined as involution.

■

$$\text{Notation In the sequel, } j(n) \text{ will be noted by } n^v. \text{ We write}$$

$$
\mathcal{U}\psi\_n = \psi\_{j(n)} = \psi\_n \tag{28}
$$

and

$$\text{Ldf} = \text{U}\left(\sum\_{n} a\_n \; \psi\_n\right) = \sum\_{n} a\_n \; \psi\_{\frac{n}{n}} = \stackrel{v.}{f} \tag{29}$$

Remark 9. The position operator U can be extended over F<sup>ψ</sup> as follows: If f <sup>¼</sup> <sup>P</sup> <sup>n</sup> an ψ<sup>n</sup> ∈ Fψ, then

$$
\sharp \mathcal{U}f = \stackrel{v}{f} = \sum\_{n} a\_{n} \quad \psi\_{n} \tag{30}
$$
