${\mathcal{D}}$-Pseudo-Bosons, Complex Hermite Polynomials, and Integral
Quantization

$\boldsymbol{\mathcal{D}}$-Pseudo-Bosons, Complex Hermite Polynomials,
and Integral Quantization

S. Twareque ALI ${}^{{\dagger}^{1}}$, Fabio BAGARELLO ${}^{{\dagger}^{2}}$ and
Jean Pierre GAZEAU ${}^{{\dagger}^{3}{\dagger}^{4}}$

S.T. Ali, F. Bagarello and J.P. Gazeau

${}^{{\dagger}^{1}}$ Department of Mathematics and Statistics, Concordia
University,
${}^{{\dagger}^{1}}$ Montréal, Québec, Canada H3G 1M8
<EMAIL_ADDRESS>
${}^{{\dagger}^{2}}$ Dipartimento di Energia, ingegneria dell’Informazione e
modelli Matematici,
${}^{{\dagger}^{2}}$ Scuola Politecnica, Università di Palermo, I-90128
Palermo, and INFN, Torino, Italy<EMAIL_ADDRESS>http://www.unipa.it/fabio.bagarello

${}^{{\dagger}^{3}}$ APC, UMR 7164, Univ Paris Diderot, Sorbonne Paris-Cité,
75205 Paris, France<EMAIL_ADDRESS>${}^{{\dagger}^{4}}$ Centro
Brasileiro de Pesquisas Físicas, Rio de Janeiro, 22290-180 Rio de Janeiro,
Brazil

Received March 28, 2015, in final form September 21, 2015; Published online
October 01, 2015

The ${\mathcal{D}}$-pseudo-boson formalism is illustrated with two examples.
The first one involves deformed complex Hermite polynomials built using
finite-dimensional irreducible representations of the group ${\rm
GL}(2,{\mathbb{C}})$ of invertible $2\times 2$ matrices with complex entries.
It reveals interesting aspects of these representations. The second example is
based on a pseudo-bosonic generalization of operator-valued functions of a
complex variable which resolves the identity. We show that such a
generalization allows one to obtain a quantum pseudo-bosonic version of the
complex plane viewed as the canonical phase space and to understand functions
of the pseudo-bosonic operators as the quantized versions of functions of a
complex variable.

pseudo-bosons; coherent states; quantization; complex Hermite polynomials;
finite group representation

81Q12; 47C05; 81S05; 81R30; 33C45

##  1 Introduction

Two new illustrations of the ${\mathcal{D}}$-pseudo-boson (${\mathcal{D}}$-pb)
formalism [13] are presented in this paper. Both display original and non-
trivial results. The first one involves a family of biorthogonal polynomials,
named deformed complex Hermite polynomials, various properties of which have
been worked out during the past years (see for instance [8, 16] and references
therein). Their construction involves a deformation of the well-known
bivariate complex Hermite polynomials [4, 25, 29, 30, 31, 32, 33] using
finite-dimensional irreducible representations of the group ${\rm
GL}(2,{\mathbb{C}})$ of invertible $2\times 2$ matrices with complex entries
and reveals interesting aspects of these representations. The second example
introduces families of vectors and operators in the underlying Hilbert space
built in the same same way as standard coherent states, as orbits in the
Hilbert space of the projective Weyl–Heisenberg group. An appealing
consequence of this construction is the resolution of the identity satisfied
by these families, possibly on a dense subspace. Hence, it becomes possible to
proceed with integral quantizations [17, 24], which then yield the correct
pseudo-bosonic commutation rules. This unorthodox path to the quantum world
can give rise to interesting developments regarding the possibility of
building self-adjoint operators from a non-real classical function on the
phase space while real functions could have non-symmetric quantum operator
counterparts.

The organization of the article is as follows. In Section 2 we present the
necessary mathematical material for understanding the ${\mathcal{D}}$-pb
formalism. In Section 3 we start with a pair of bosonic operators to build
orthonormal bases in a Hilbert space. We next make use of finite-dimensional
representations of the group ${\rm GL}(2,{\mathbb{C}})$ to deform this 2-boson
algebra into a pair of pseudo-bosons. In Section 4 we illustrate the procedure
with deformed complex Hermite polynomials. Note that a set of bi-orthogonal
Hermite polynomials were presented, with not much interest in mathematical
rigour, in [38] via the pseudo-boson operators. Useful inequalities/estimates
are then proved in Section 5. They are necessary for characterizing
${\mathcal{D}}$-pb in this ${\rm GL}(2,{\mathbb{C}})$ context. They are also
necessary to get total families in the relevant Hilbert space. In Section 6 we
introduce two “displacement operators” depending on a complex variable, and
arising as a consequence of the existence of a pair of ${\mathcal{D}}$-pb as
introduced in Section 2. By using these operators we derive two types of
“oblique” resolutions of the identity (see also [38] for previous works in
this direction). Based on these results, we proceed in Section 7 to the
integral quantization of functions (or distributions) of a complex variable,
obtaining thereby a set of original results. In particular, the quantized
version of the canonical Poisson bracket of conjugate pairs $z$ and $\bar{z}$
is precisely the pseudo-bosonic commutation rule. In Section 8 we sketch what
could be done in the future, starting from the results presented in this
paper. The three appendices give a set of technical formulae used in the main
body of the paper. Those concerning the finite-dimensional irreducible
representations of the group ${\rm GL}(2,{\mathbb{C}})$ are given in Appendix
A, while those concerning some of the asymptotic behaviours of the
corresponding matrix elements are given in Appendix B. Finally, in Appendix C
we deduce the matrix elements of the bi-displacement operators introduced in
connection with bi-coherent states and our pseudo-bosonic integral
quantization.

##  2 The mathematics of $\boldsymbol{\mathcal{D}}$-pbs

Let ${\mathcal{H}}$ be a Hilbert space with scalar product
$\langle\cdot,\cdot\rangle$ and associated norm $\|\cdot\|$. Further, let $a$
and $b$ be two operators on ${\mathcal{H}}$, with domains $D(a)$ and $D(b)$
respectively, $a^{\dagger}$ and $b^{\dagger}$ their respective adjoints; we
assume the existence of a dense set ${\mathcal{D}}$ in ${\mathcal{H}}$ such
that $a^{\sharp}{\mathcal{D}}\subseteq{\mathcal{D}}$ and
$b^{\sharp}{\mathcal{D}}\subseteq{\mathcal{D}}$, where $x^{\sharp}$ is either
$x$ or $x^{\dagger}$: ${\mathcal{D}}$ is assumed to be stable under the action
of $a$, $b$, $a^{\dagger}$ and $b^{\dagger}$. Note that we are not requiring
here that ${\mathcal{D}}$ coincide with, e.g., $D(a)$ or $D(b)$. However due
to the fact that $a^{\sharp}f$ is well defined, and belongs to ${\mathcal{D}}$
for all $f\in{\mathcal{D}}$, it is clear that ${\mathcal{D}}\subseteq
D(a^{\sharp})$. Analogously, we conclude that ${\mathcal{D}}\subseteq
D(b^{\sharp})$.

######  Definition 2.1.

The operators $(a,b)$ are ${\mathcal{D}}$-pseudo-bosonic (${\mathcal{D}}$-pb)
if, for all $f\in{\mathcal{D}}$, we have

$\displaystyle abf-baf=f.$ (2.1)

To simplify the notation at many places in the sequel, instead of (2.1) we
will simply write $[a,b]=I$, where $I$ is the identity operator on
${\mathcal{H}}$ having in mind that both sides of this equation have to act on
a certain $f\in{\mathcal{D}}$.

Our working assumptions are the following:

######  Assumption ${\mathcal{D}}$-pb 2.2.

There exists a non-zero $\varphi_{0}\in{\mathcal{D}}$ such that
$a\varphi_{0}=0$.

######  Assumption ${\mathcal{D}}$-pb 2.3.

There exists a non-zero $\Psi_{0}\in{\mathcal{D}}$ such that
$b^{\dagger}\Psi_{0}=0$.

We then define the vectors

$\displaystyle\varphi_{n}:=\frac{1}{\sqrt{n!}}b^{n}\varphi_{0},\qquad\Psi_{n}:=\frac{1}{\sqrt{n!}}{a^{\dagger}}^{n}\Psi_{0},\qquad
n\geq 0,$ (2.2)

and introduce the sets ${\mathcal{F}}_{\Psi}=\\{\Psi_{n},\,n\geq 0\\}$ and
${\mathcal{F}}_{\varphi}=\\{\varphi_{n},\,n\geq 0\\}$. Since ${\mathcal{D}}$
is stable in particular under the action of $a^{\dagger}$ and $b$, we deduce
that each $\varphi_{n}$ and each $\Psi_{n}$ belongs to ${\mathcal{D}}$ and,
therefore, to the domains of $a^{\sharp}$, $b^{\sharp}$ and $N^{\sharp}$,
where $N=ba$.

It is now simple to deduce the following lowering and raising relations:

$\displaystyle b\varphi_{n}=\sqrt{n+1}\varphi_{n+1},\qquad$ $\displaystyle
n\geq 0,$ $\displaystyle a\varphi_{0}=0,\qquad
a\varphi_{n}=\sqrt{n}\varphi_{n-1},\qquad$ $\displaystyle n\geq 1,$
$\displaystyle a^{\dagger}\Psi_{n}=\sqrt{n+1}\Psi_{n+1},\qquad$ $\displaystyle
n\geq 0,$ $\displaystyle b^{\dagger}\Psi_{0}=0,\qquad
b^{\dagger}\Psi_{n}=\sqrt{n}\Psi_{n-1},\qquad$ $\displaystyle n\geq 1,$

as well as the following eigenvalue equations: $N\varphi_{n}=n\varphi_{n}$ and
$N^{\dagger}\Psi_{n}=n\Psi_{n}$, $n\geq 0$, where
$N^{\dagger}=a^{\dagger}b^{\dagger}$. As a consequence of these equations,
choosing the normalization of $\varphi_{0}$ and $\Psi_{0}$ in such a way
$\langle\varphi_{0},\Psi_{0}\rangle=1$, we also deduce that

$\displaystyle\langle\varphi_{n},\Psi_{m}\rangle=\delta_{n,m},$

for all $n,m\geq 0$.

The conclusion is, therefore, that ${\mathcal{F}}_{\varphi}$ and
${\mathcal{F}}_{\Psi}$ are biorthonormal sets of eigenstates of $N$ and
$N^{\dagger}$, respectively. This, in principle, does not allow us to conclude
that they are also bases for ${\mathcal{H}}$, or even Riesz bases. However,
let us introduce for the time being the following assumption:

######  Assumption ${\mathcal{D}}$-pb 2.4.

${\mathcal{F}}_{\varphi}$ is a basis for ${\mathcal{H}}$.

Notice that this automatically implies that ${\mathcal{F}}_{\Psi}$ is a basis
for ${\mathcal{H}}$ as well [41]. However, examples are known in which this
natural assumption is not satisfied; see, for instance, [9, 11, 12, 14, 21,
23]. In view of this fact, a weaker version of Assumption ${\mathcal{D}}$-pb
2.4 has been introduced recently: for that the concept of
${\mathcal{G}}$-quasi bases is necessary.

######  Definition 2.5.

Let ${\mathcal{G}}$ be a suitable dense subspace of ${\mathcal{H}}$. Two
biorthogonal sets ${\mathcal{F}}_{\eta}=\\{\eta_{n}\in{\mathcal{H}}$, $n\geq
0\\}$ and ${\mathcal{F}}_{\Phi}=\\{\Phi_{n}\in{\mathcal{H}},\,n\geq 0\\}$ are
called ${\mathcal{G}}$-quasi bases if, for all $f,g\in{\mathcal{G}}$, the
following holds:

$\displaystyle\langle f,g\rangle=\sum_{n\geq 0}\langle
f,\eta_{n}\rangle\langle\Phi_{n},g\rangle=\sum_{n\geq 0}\langle
f,\Phi_{n}\rangle\langle\eta_{n},g\rangle.$ (2.3)

Is is clear that, while Assumption ${\mathcal{D}}$-pb 2.4 implies (2.3), the
reverse is false. However, if ${\mathcal{F}}_{\eta}$ and
${\mathcal{F}}_{\Phi}$ satisfy (2.3), we still have some (weak) form of the
resolution of the identity and we can still deduce interesting results,
specially in view of quantization as presented in Section 7. For the sake of
simplicity, we will often use in the sequel the popular shorthand notation

$\displaystyle\sum_{n\geq 0}|\eta_{n}\rangle\langle\Phi_{n}|=I,$ (2.4)

to be understood in the weak sense on a dense subspace. Incidentally we see
that if $f\in{\mathcal{G}}$ is orthogonal to all the $\Phi_{n}$’s (or to all
the $\eta_{n}$’s), then $f$ is necessarily zero: we say that
${\mathcal{F}}_{\Phi}$ (or ${\mathcal{F}}_{\eta}$) is total in
${\mathcal{G}}$. Note that this does not imply that these families are total
in the whole Hilbert space ${\mathcal{H}}$ since we suppose that (2.3) holds
for $f,g\in{\mathcal{G}}$, but not, in general, for $f,g\in{\mathcal{H}}$.
Therefore we cannot conclude that each vector of ${\mathcal{H}}$ orthogonal
to, say, all the $\varphi_{n}$ is necessarily zero, while we can conclude this
for each vector of ${\mathcal{G}}$.

With this in mind, we now consider the aforementioned weaker form of
Assumption ${\mathcal{D}}$-pb 2.4:

######  Assumption ${\mathcal{D}}$-pbw 2.6.

${\mathcal{F}}_{\varphi}$ and ${\mathcal{F}}_{\Psi}$ are ${\mathcal{G}}$-quasi
bases, for some dense subspace ${\mathcal{G}}$ in ${\mathcal{H}}$.

Two important operators, in general unbounded, are the following ones

$\displaystyle
D(S_{\varphi})=\bigg{\\{}f\in{\mathcal{H}}\colon\sum_{n}\langle\varphi_{n},f\rangle\varphi_{n}\
\mbox{exists in}\ {\mathcal{H}}\bigg{\\}}\qquad\mbox{and}\qquad
S_{\varphi}f=\sum_{n}\langle\varphi_{n},f\rangle,\varphi_{n}$

for all $f\in D(S_{\varphi})$, and, similarly,

$\displaystyle
D(S_{\Psi})=\bigg{\\{}h\in{\mathcal{H}}\colon\sum_{n}\langle\Psi_{n},h\rangle\Psi_{n}\
\mbox{exists in}\ {\mathcal{H}}\bigg{\\}}\qquad\mbox{and}\qquad
S_{\Psi}h=\sum_{n}\langle\Psi_{n},h\rangle\Psi_{n},$

for all $h\in D(S_{\Psi})$. It is clear that $\Psi_{n}\in D(S_{\varphi})$ and
$\varphi_{n}\in D(S_{\Psi})$, for all $n\geq 0$. However, since
${\mathcal{F}}_{\varphi}$ and ${\mathcal{F}}_{\Psi}$ are not required to be
bases here, it is convenient to work under the additional hypothesis that
${\mathcal{D}}\subseteq D(S_{\Psi})\cap D(S_{\varphi})$, which is often true
in concrete examples [13]. In this way $S_{\Psi}$ and $S_{\varphi}$ are
automatically densely defined. Also, since $\langle S_{\Psi}f,g\rangle=\langle
f,S_{\Psi}g\rangle$ for all $f,g\in D(S_{\Psi})$, $S_{\Psi}$ is a symmetric
operator, as well as $S_{\varphi}\colon\langle S_{\varphi}f,g\rangle=\langle
f,S_{\varphi}g\rangle$ for all $f,g\in D(S_{\varphi})$. Moreover, since they
are positive operators, they are also semibounded [36]

$\displaystyle\langle S_{\varphi}f,f\rangle\geq 0,\qquad\langle
S_{\Psi}h,h\rangle\geq 0,$

for all $f\in D(S_{\varphi})$ and $h\in D(S_{\Psi})$. Hence both these
operators admit self-adjoint (Friedrichs) extensions, $\hat{S}_{\varphi}$ and
$\hat{S}_{\Psi}$ [36], which are both also positive. Now, the spectral theorem
ensures us that we can define the square roots $\hat{S}_{\Psi}^{1/2}$ and
$\hat{S}_{\varphi}^{1/2}$, which are self-adjoint and positive and, in
general, unbounded. These operators can be used to define new scalar products
and new related notions of the adjoint, as well as new mutually orthogonal
vectors. These and other aspects, which are particularly relevant in the
present context, are discussed in some detail in [13].

##  3 Biorthogonal families of vectors and polynomials

In this section we present the first illustration of the above formalism with
an explicit group theoretical construction of pseudo-bosonic operators. We
start with a pair of bosonic operators $a_{i}$, $a_{i}^{\dagger}$, $i=1,2$,
acting (irreducibly) on the Hilbert space $\mathcal{H}$. They satisfy the
commutation relations,

$\displaystyle\big{[}a_{i},a_{j}^{\dagger}\big{]}=I\delta_{ij},\qquad[a_{i},a_{j}]=\big{[}a_{i}^{\dagger},a_{j}^{\dagger}\big{]}=0,\qquad
i,j=1,2.$ (3.1)

Starting with the (normalized) ground state vector $\varphi_{0,0}$, for which
$a_{i}\varphi_{0,0}=0$, $i=1,2$, we define the vectors,

$\displaystyle\varphi_{n_{1},n_{2}}=\frac{\big{(}a_{1}^{\dagger}\big{)}^{n_{1}}\big{(}a_{2}^{\dagger}\big{)}^{n_{2}}}{\sqrt{n_{1}!n_{2}!}}\varphi_{0,0},\qquad
n_{1},n_{2}=0,1,2,\ldots,\infty.$ (3.2)

These vectors form an orthonormal basis in $\mathcal{H}$.

We now reorder the elements of this basis as in (3.3) below. For any integer
$L\geq 0$, let us define the set of $L+1$ vectors

$\displaystyle
f^{L}_{m}=\frac{\big{(}a_{1}^{\dagger}\big{)}^{m}\big{(}a_{2}^{\dagger}\big{)}^{L-m}}{\sqrt{m!(L-m)!}}\varphi_{0,0}=\varphi_{m,L-m},\qquad
m=0,1,2,\ldots,L,$ (3.3)

and denote by $\mathcal{H}^{L}$ the $(L+1)$-dimensional subspace of
$\mathcal{H}$ spanned by these vectors. Clearly

$\displaystyle\langle
f^{L}_{m},f^{M}_{n}\rangle=\delta_{LM}\delta_{mn}\qquad\text{and}\qquad\mathcal{H}=\oplus_{L=0}^{\infty}\mathcal{H}^{L}.$

Hence, the $f^{L}_{m}$ are a relabeling of the vectors $\varphi_{n_{1},n_{2}}$
which will be useful in the sequel.

###  3.1 A second basis and a Cuntz algebra

Using the vectors $f^{L}_{m}$, we now introduce a second relabeling, this time
using a single index. We set

$\displaystyle F_{n}=f^{L}_{m}=\varphi_{m,L-m},\qquad\text{where}\qquad
n=\frac{L(L+1)}{2}+m.$ (3.4)

Note that in making this relabeling, we have used the bijective map
$\beta\colon\mathbb{N}\times\mathbb{N}\to\mathbb{N}$, defined by

$\displaystyle
n=\beta(n_{1},n_{2})=\frac{(n_{1}+n_{2})(n_{1}+n_{2}+1)}{2}+n_{1}.$ (3.5)

The inverse map $(n_{1},n_{2})=\beta^{-1}(n)$ is obtained by taking

$\displaystyle
L=\sup_{\ell\in\mathbb{N}}\left\\{\ell\colon\frac{\ell(\ell+1)}{2}\leq
n\right\\}$

and then writing

$\displaystyle n_{1}=n-\frac{L(L+1)}{2}\qquad\text{and}\qquad n_{2}=L-n_{1}.$

These successive relabelings of the point set ${\mathbb{N}}^{2}$ are
graphically described in Fig. 1 below. They just illustrate the well-known
countability of ${\mathbb{N}}^{2}$, or, equivalently, of the positive rational
numbers.

Figure 1: Three successive relabelings of the point set ${\mathbb{N}}^{2}$. On
the left: point set ${\mathbb{N}}^{2}$ of pairs $(n_{1},n_{2})$ with their
corresponding non-negative label $n$. On the right $2d$-relabeling
$(n_{1},n_{2})\mapsto(m,L)$ where $m=n_{1}$, $L=n_{1}+n_{2}$ and eventually
$1d$-relabeling $n=\frac{L(L+1)}{2}+m$ where dots in the sector
$\\{(m,L),\,L\in{\mathbb{N}},\,m=0,1,\dots,L\\}$ are marked with their
corresponding non-negative label $n$.

We next define two bosonic operators $B$, $B^{\dagger}$, in the standard
manner, using the vectors $F_{n}$:

$\displaystyle BF_{n}=\sqrt{n}F_{n-1},\qquad BF_{0}=0,\qquad
B^{\dagger}F_{n}=\sqrt{n+1}F_{n+1},\qquad[B,B^{\dagger}]=I,$ (3.6)

and from (3.4) we find their actions on the vectors $f^{L}_{m}$:

$\displaystyle
Bf^{L}_{m}=\begin{cases}\sqrt{\dfrac{L(L+1)}{2}+m}f^{L}_{m-1},&\text{if}\
m>0,\vspace{1mm}\\\ \sqrt{\dfrac{L(L+1)}{2}}f^{L-1}_{L-1},&\text{if}\
m=0,\end{cases}$ $\displaystyle
B^{\dagger}f^{L}_{m}=\begin{cases}\sqrt{\dfrac{L(L+1)}{2}+m+1}f^{L}_{m+1},&\text{if}\
m<L,\vspace{1mm}\\\ \sqrt{\dfrac{(L+1)(L+2)}{2}}f^{L+1}_{0},&\text{if}\
m=L.\end{cases}$ (3.7)

This means that, writing the vectors $f^{L}_{m}$ and $F_{n}$ in ascending
order

$\displaystyle\begin{array}[]{@{}c}f^{0}_{0}\\\ \updownarrow\\\
F_{0}\end{array},\underbrace{\begin{array}[]{c}f^{1}_{0}\\\ \updownarrow\\\
F_{1}\end{array},\begin{array}[]{c}f^{1}_{1}\\\ \updownarrow\\\
F_{2}\end{array}},\underbrace{\begin{array}[]{c}f^{2}_{0}\\\ \updownarrow\\\
F_{3}\end{array},\begin{array}[]{c}f^{2}_{1}\\\ \updownarrow\\\
F_{4}\end{array},\begin{array}[]{c}f^{2}_{2}\\\ \updownarrow\\\
F_{5}\end{array}},\underbrace{\begin{array}[]{c}f^{3}_{0}\\\ \updownarrow\\\
F_{6}\end{array},\begin{array}[]{c}f^{3}_{1}\\\ \updownarrow\\\
F_{7}\end{array},\begin{array}[]{c}f^{3}_{2}\\\ \updownarrow\\\
F_{8}\end{array},\begin{array}[]{c}f^{3}_{3}\\\ \updownarrow\\\
F_{9}\end{array}},\ldots,$ (3.38)

the operators $B^{\dagger}$ and $B$ move them up and down this array,
respectively (see right part of Fig. 1).

As a direct consequence of the maps $(m,L)\mapsto n$ introduced in (3.4) and
the above correspondence (3.38), there is an interesting set of isometries
$S_{n}$, $n=0,1,2,\ldots,\infty$, of the Hilbert space $\mathcal{H}$
associated to the two sets of basis vectors $\\{F_{n}\\}$ and
$\\{\varphi_{n,m}\\}$. We define these operators as

$\displaystyle S_{n}F_{m}=\varphi_{m,n}=F_{k(m,n)},$ (3.39)

where

$\displaystyle k(m,n):=\frac{(m+n)(m+n+1)}{2}+m,\qquad
n,m=0,1,2,\ldots,\infty.$

Clearly, $\|S_{n}\|=1$, $n=0,1,2,\ldots,\infty$. These operators were
introduced in [7], where they were used to construct coherent states on
$C^{*}$-Hilbert modules. The following properties are easily proved.

######  Proposition 3.1.

  1. $(i)$

The isometries $S_{n}$ are not unitary maps. Indeed, one has

$\displaystyle S_{m}^{\dagger}S_{n}=\delta_{mn}I\qquad\text{and}\qquad
S_{n}S_{n}^{\dagger}=\mathbb{P}_{n},$

$\mathbb{P}_{n}$ being the projection operator onto the subspace
$\mathcal{H}_{n}$ of $\mathcal{H}$ spanned by the vectors $\varphi_{m,n}$,
$m=0,1,2,\ldots,\infty$.

  2. $(ii)$

The kernel of $S_{n}^{\dagger}$ is the set of all vectors of the type
$\varphi_{m,k}$, $m=0,1,2,\ldots$, and $k\neq n$.

  3. $(iii)$

$S_{m}S_{n}^{\dagger}$ is a partial isometry from $\mathcal{H}_{n}$ to
$\mathcal{H}_{m}$.

  4. $(iv)$

The positive operators $S_{n}S_{n}^{\dagger}$ resolve the identity

$\displaystyle\sum_{n=0}^{\infty}S_{n}S_{n}^{\dagger}=I,$ (3.40)

the sum converging strongly.

  5. $(v)$

There exist the following relationships between the operators $a_{1}$,
$a_{1}^{\dagger}$ in (3.1) and the operators $B$, $B^{\dagger}$ in (3.7)
through $S_{n}$, $S_{n}^{\dagger}$:

$\displaystyle S_{n}^{\dagger}a_{1}S_{n}=B,\qquad
S_{n}^{\dagger}a_{1}^{\dagger}S_{n}=B^{\dagger},$

while $S_{n}^{\dagger}a_{2}S_{n}=S_{n}^{\dagger}a_{2}^{\dagger}S_{n}=0$.

The $S_{n}$ generate a $C^{*}$-algebra $\mathcal{O}_{\infty}$, known as a
Cuntz algebra [20], which is a subject of independent interest. Note also,
that we have used here a very specific bijection (3.5) to define the vectors
$F_{n}$. Of course, there are many other possible bijections, which will also
give rise to associated Cuntz algebras. But this particular one will be useful
for our subsequent analysis.

###  3.2 Deformed operators and bases

To proceed further, let

$\displaystyle g=\begin{pmatrix}g_{11}&g_{12}\\\ g_{21}&g_{22}\end{pmatrix},$

be an element of the GL$(2,\mathbb{C})$ group (i.e., $g$ is a complex $2\times
2$ matrix with $\det[g]\neq 0$), using which we define the new operators

$\displaystyle A^{g}_{1}=\bar{g}_{11}a_{1}+\bar{g}_{21}a_{2},\qquad
A^{g}_{2}=\bar{g}_{12}a_{1}+\bar{g}_{22}a_{2},$

and the corresponding adjoint operators $A^{g{\dagger}}_{i}$, $i=1,2$, i.e.,
in matrix notations

$\displaystyle\begin{pmatrix}A^{g}_{1}\\\
A^{g}_{2}\end{pmatrix}\equiv\mathbf{A}^{g}=g^{{\dagger}}\cdot\mathbf{a},\qquad\mathbf{a}:=\begin{pmatrix}a_{1}\\\
a_{2}\end{pmatrix},\qquad\begin{pmatrix}A^{g{\dagger}}_{1}\\\
A^{g{\dagger}}_{2}\end{pmatrix}\equiv\mathbf{A}^{g+}={}^{t}g\cdot\mathbf{a}^{+},\qquad\mathbf{a}^{+}:=\begin{pmatrix}a^{{\dagger}}_{1}\\\
a^{{\dagger}}_{2}\end{pmatrix}.$

We call these operators deformed bosonic operators; they satisfy
$[A^{g}_{1},A^{g}_{2}]=[A^{g{\dagger}}_{1},A^{g{\dagger}}_{2}]=0$, however,
the other commutators are in general different from those of the undeformed
operators $a_{i}$, $a_{j}$, $i=1,2$. Indeed, we have the general commutation
relations

$\displaystyle[A_{i}^{g},A^{g{\dagger}}_{j}]=\overline{g_{1i}}g_{1j}+\overline{g_{2i}}g_{2j},\qquad
i,j=1,2,$

so that the matrix elements of $g$ would have to satisfy
$\overline{g_{1i}}\;g_{1j}+\overline{g_{2i}}g_{2j}=\delta_{ij}$ (which is
equivalent to having $g^{\dagger}g=I_{2}$, i.e., a unitary matrix) in order to
recover the standard commutation relations (3.1). However we leave aside this
condition, which is not relevant for us.

Using the operators $A^{g{\dagger}}_{i}$, $i=1,2$, and noting that
$A^{g}_{i}\varphi_{0,0}=0$, we now construct a set of $g$-deformed basis
vectors in a manner analogous to the construction of the vectors
$\varphi_{n_{1},n_{2}}$ in (3.2). We define

$\displaystyle\varphi_{n_{1},n_{2}}^{g}=\frac{(A^{g{\dagger}}_{1})^{n_{1}}(A^{g{\dagger}}_{2})^{n_{2}}}{\sqrt{n_{1}!n_{2}!}}\varphi_{0,0},\qquad
n_{1},n_{2}=0,1,2,\ldots,\infty.$ (3.41)

Adopting the group representation notations (A.2), we rewrite (3.41) as

$\displaystyle\varphi_{n_{1},n_{2}}^{g}=e_{n_{1},n_{2}}\big{(}{}^{t}g\cdot\mathbf{a}^{+}\big{)}\varphi_{0,0},\qquad
e_{n_{1},n_{2}}(\mathbf{a}^{+}):=\frac{{a^{{\dagger}}_{1}}^{n_{1}}{a^{{\dagger}}_{2}}^{n_{2}}}{\sqrt{n_{1}!n_{2}!}}.$
(3.42)

It is obvious that, in general, these vectors are not mutually orthogonal,
since they are not eigenstates (with different eigenvalues) of some self-
adjoint operator. To continue, for each $L\geq 0$ let us define the set of
$L+1$ vectors $f^{g,L}_{m}$ in a manner analogous to (3.3)

$\displaystyle
f^{g,L}_{m}=e_{m,L-m}({}^{t}g\cdot\mathbf{a}^{+})\varphi_{0,0}=\varphi_{m,L-m}^{g},\qquad
m=0,1,2,\ldots,L,$ (3.43)

It is clear that these vectors are linear combinations of the $f^{L}_{m}$,
hence they also span the subspace $\mathcal{H}^{L}$ of $\mathcal{H}$. This is
simply due to the ${\rm GL}(2,{\mathbb{C}})$ representation operator defined
in (A.1) with $L=s$, and acting in the present context as the map [16]
$\mathcal{T}^{L}(g)\colon\mathcal{H}^{L}\to\mathcal{H}^{L}$ for which

$\displaystyle\mathcal{T}^{L}(g)f^{L}_{m}=f^{g,L}_{m},\qquad
m=0,1,2,\ldots,L,\qquad g\in\mathrm{GL}(2,\mathbb{C}).$ (3.44)

The matrix elements of the operators $\mathcal{T}^{L}(g)$ in the basis (3.42)
are given in (A.4). In the $f^{L}_{m}$ basis they read as [16]

$\displaystyle\mathcal{T}_{m^{\prime}m}^{L}(g)=\sum_{q}\binom{m}{q}\binom{L-m}{m^{\prime}-q}g_{11}^{q}g_{21}^{m-q}g_{12}^{m^{\prime}-q}g_{22}^{L-m+q-m^{\prime}},\qquad
0\leq m^{\prime},m\leq L.$ (3.45)

The range of values assumed by $q$ in the above sum is determined by the
cancellation of the binomial coefficients involved, i.e.,
$\max\\{0,m^{\prime}+m-L\\}\leq q\leq\min\\{m^{\prime},m\\}$. These matrix
elements are discussed in greater detail in Section 5.

###  3.3 Biorthogonal families of vectors and pseudobosons

Corresponding to the vectors $f^{g,L}_{m}$, let us define a dual family of
vectors $\widetilde{f}^{g,L}_{m}$ by the relation

$\displaystyle\widetilde{f}^{g,L}_{m}=\mathcal{T}^{L}(\widetilde{g})f^{L}_{m}=f^{\widetilde{g},L}_{m},\qquad\widetilde{g}:=(g^{\dagger})^{-1}.$
(3.46)

Clearly these vectors are also elements of the subspace $\mathcal{H}^{L}$.
From (3.44), (A.6) and the representation theoretical property of
$\mathcal{T}^{L}(g)$, by which
$\mathcal{T}^{L}(g^{-1})=(\mathcal{T}^{L}(g))^{-1}$, we see that

$\displaystyle\big{\langle}\widetilde{f}^{g,L}_{m},f^{g,M}_{n}\big{\rangle}=\delta_{LM}\delta_{mn}.$

This means that on each subspace $\mathcal{H}^{L}$ the vectors $f^{g,L}_{m}$
and $\widetilde{f}^{g,L}_{m}$ form two biorthogonal bases, while they are, in
general, biorthogonal sets in ${\mathcal{H}}$.

Consider now the operator
$\mathcal{T}(g)=\oplus_{L=0}^{\infty}\mathcal{T}^{L}(g)$. This operator is in
general unbounded and densely defined in $\mathcal{H}$, since
$\mathcal{T}^{L}(g)$ is bounded on each subspace $\mathcal{H}^{L}$. In
particular $\mathcal{T}(g)$ is well defined on the vectors $F_{n}$ in (3.4).
We thus define the two sets of vectors

$\displaystyle
F^{g}_{n}=\mathcal{T}(g)F_{n},\qquad\text{and}\qquad\widetilde{F}^{g}_{n}=\mathcal{T}(\widetilde{g})F_{n}=F^{\widetilde{g}}_{n},\qquad
n=0,1,\ldots,\infty,$

in duality, for which

$\displaystyle\big{\langle}\widetilde{F}^{g}_{m},F^{g}_{n}\big{\rangle}=\delta_{mn}.$

Note that the existence of the inverse operator $(\mathcal{T}(g))^{-1}$, as a
densely defined operator on $\mathcal{H}$ is guaranteed by the property
$(\mathcal{T}^{L}(g))^{-1}=\mathcal{T}^{L}(g^{-1})$ on each subspace
$\mathcal{H}^{L}$.

It is now possible to construct families of pseudobosons using the vectors
$F^{g}_{n}$ and $\widetilde{F}^{g}_{n}$. The following proposition is easily
derived from the above material.

######  Proposition 3.2.

Given the operators $B$, $B^{\dagger}$ in (3.6), for any $g\in{\rm
GL}(2,{\mathbb{C}})$ let us define the deformed operators

$\displaystyle
B(g)=\mathcal{T}(g)B(\mathcal{T}(g))^{-1},\qquad\widetilde{B}(g)=B(\widetilde{g}),$

and their adjoints $B(g)^{\dagger}$, $\widetilde{B}(g)^{\dagger}$. Then, as
operators on the full Hilbert space $\mathcal{H}$, they satisfy, at least
formally, the pseudo-bosonic commutation relations

$\displaystyle\big{[}B(g),\widetilde{B}(g)^{\dagger}\big{]}=\big{[}\widetilde{B}(g),B(g)^{\dagger}\big{]}=I.$

Their actions on the vectors $F^{g}_{n}$, $\widetilde{F}^{g}_{n}$ read as

$\displaystyle B(g)F^{g}_{n}=\sqrt{n}F^{g}_{n-1},\qquad$ $\displaystyle
B(g)^{\dagger}\widetilde{F}^{g}_{n}=\sqrt{n+1}\widetilde{F}^{g}_{n+1},$
$\displaystyle\widetilde{B}(g)\widetilde{F}^{g}_{n}=\sqrt{n}\widetilde{F}^{g}_{n-1},\qquad$
$\displaystyle\widetilde{B}(g)^{\dagger}F^{g}_{n}=\sqrt{n+1}F^{g}_{n+1}.$

Notice that, all throughout this section, $g$ is a fixed element in ${\rm
GL}(2,{\mathbb{C}})$. This is important since, if we take $g_{1},g_{2}\in{\rm
GL}(2,{\mathbb{C}})$, with $g_{1}\neq g_{2}$, then nothing can be said about
$[B(g_{1}),\widetilde{B}(g_{2})^{\dagger}]$, for instance.

To relate the equations above with the general structure discussed in Section
2, we start by observing that
$B(g)F^{g}_{0}=0=\widetilde{B}(g)\widetilde{F}^{g}_{0}$. This shows that the
two non zero vacua required in Assumptions ${\mathcal{D}}$-pb 2.2 and
${\mathcal{D}}$-pb 2.3 of Section 2 do exist and coincide111Here $B(g)$ and
$\widetilde{B}(g)$ respectively play the role of $a$ and $b$ in Section 2.. In
fact $F^{g}_{0}=\widetilde{F}^{g}_{0}=\varphi_{0,0}$. Moreover, calling
${\mathcal{D}}$ the linear span of the vectors $\varphi_{n_{1},n_{2}}$ in
(3.2), it is clear that (i) $F^{g}_{0}$,
$\widetilde{F}^{g}_{0}\in{\mathcal{D}}$, (ii) that ${\mathcal{D}}$ is dense in
${\mathcal{H}}$ and (iii) ${\mathcal{D}}$ is left invariant by $B(g)$,
$\widetilde{B}(g)$ and by their adjoints. In fact these operators map each
finite linear combination of the $\varphi_{n_{1},n_{2}}$’s into a different,
but still finite, linear combination of the same vectors. Thus we conclude
that the present setup reflects, at least in part, what was discussed in
Section 2. However, in Section 5 we will show that Assumption
${\mathcal{D}}$-pb 2.4 is not satisfied, while its weaker version Assumption
${\mathcal{D}}$-pbw 2.6 holds true.

##  4 Deformed complex Hermite polynomials

In this section we give a concrete realization of the kind of pseudo-bosons
discussed above. Let us consider the irreducible representation of the
operators $a_{i}$, $a_{i}^{\dagger}$, $i=1,2$, on the Hilbert space
$\mathcal{H}({\mathbb{C}})={\mathcal{L}}^{2}({\mathbb{C}},\mathrm{d}\nu(z,\overline{z}))$,
where

$\displaystyle\mathrm{d}\nu(z,\overline{z})=e^{-|z|^{2}}\frac{\mathrm{d}z\wedge\mathrm{d}\overline{z}}{2\pi
i}=\frac{1}{\pi}e^{-(x^{2}+y^{2})}\mathrm{d}x\mathrm{d}y,\qquad z=x+iy,$

where they are realized as follows

$\displaystyle a_{1}=\partial_{z},\quad
a_{1}^{\dagger}=z-\partial_{\overline{z}},\qquad
a_{2}=\partial_{\overline{z}},\quad
a_{2}^{\dagger}={\overline{z}}-\partial_{z}.$ (4.1)

The basis vectors $\varphi_{n_{1},n_{2}}$, given in (3.2), now turn out to be
the normalized complex Hermite polynomials in the variables $z$,
$\overline{z}$, which we shall denote by $h_{n_{1},n_{2}}({\mathfrak{z}})$,
where we adopt the vector notation for group theoretical reasons

$\displaystyle{\mathfrak{z}}:=\begin{pmatrix}z\\\ \bar{z}\end{pmatrix}.$

The normalized vacuum state $\varphi_{0,0}$, satisfying
$a_{i}\varphi_{0,0}=0$, $i=1,2,$ is simply the constant function
$h_{0,0}({\mathfrak{z}})=1$. These polynomials have been discussed extensively
in the literature (see, for example, [6, 19, 25, 34], and very recently in [4,
29, 30, 31, 32, 33]). Their expression can be directly inferred from (3.2)

$\displaystyle
h_{n_{1},n_{2}}({\mathfrak{z}})=\frac{(z-\partial_{\overline{z}})^{n_{1}}({\overline{z}}-\partial_{z})^{n_{2}}}{\sqrt{n_{1}!n_{2}!}}h_{0,0}$
$\displaystyle\hphantom{h_{n_{1},n_{2}}({\mathfrak{z}})=}{}=\frac{1}{\sqrt{n_{1}!n_{2}!}}\sum_{k=0}^{\min(n_{1},n_{2})}(-1)^{k}k!{n_{1}\choose
k}{n_{2}\choose k}z^{n_{1}-k}\overline{z}^{n_{2}-k}.$ (4.2)

Alternatively, they can also be obtained from the expression

$\displaystyle
h_{n_{1},n_{2}}({\mathfrak{z}})=e^{-\partial_{z}\partial_{\overline{z}}}\frac{z^{n_{1}}\overline{z}^{n_{2}}}{\sqrt{n_{1}!n_{2}!}}=e^{-\partial_{z}\partial_{\overline{z}}}e_{n_{1},n_{2}}(\mathbf{\mathfrak{z}}).$
(4.3)

Note that these complex Hermite polynomials are of particular interest in the
study of physical systems constituted by several layers. Indeed, such systems
can be modeled by spaces of polyanalytic functions generated by complex
Hermite polynomials. This has recently found several applications in signal
analysis [1, 3, 27] and in the statistics of higher Landau levels [28].

The $g$-deformed basis vectors $\varphi^{g}_{n_{1},n_{2}}$ in (3.41), which we
now denote by $h^{g}_{n_{1},n_{2}}$, are also polynomials in $z$,
$\overline{z}$, which are linear combinations of the $h_{n_{1},n_{2}}$ [8, 16,
40]. Within the ${\rm GL}(2,{\mathbb{C}})$ representation framework, they are
obtainable from a formula analogous to (3.42)

$\displaystyle
h^{g,L}_{n_{1},n_{2}}({\mathfrak{z}})=e^{-\partial_{z}\partial_{\overline{z}}}e_{n_{1},n_{2}}\big{(}{}^{t}g\cdot{\mathfrak{z}}\big{)}:=e^{-\partial_{z}\partial_{\overline{z}}}\big{(}\mathcal{T}^{L}(g)e_{n_{1},n_{2}}\big{)}({\mathfrak{z}}),\qquad
L=n_{1}+n_{2}.$ (4.4)

Similarly, with the notation introduced in (3.46), we define the dual
polynomials

$\displaystyle\widetilde{h}^{g,L}_{n_{1},n_{2}}({\mathfrak{z}})=h^{\widetilde{g},L}_{n_{1},n_{2}}({\mathfrak{z}})=e^{-\partial_{z}\partial_{\overline{z}}}e_{n_{1},n_{2}}\big{(}{}^{t}\widetilde{g}\cdot{\mathfrak{z}}\big{)}:=e^{-\partial_{z}\partial_{\overline{z}}}\big{(}\mathcal{T}^{L}(\widetilde{g}\big{)}e_{n_{1},n_{2}})({\mathfrak{z}}).$
(4.5)

We derive from (4.3) and the definition given in (A.3) of the matrix elements
of the representation operator $\mathcal{T}^{L}(g)$, the following expansions
in which the apparent double summation is actually reduced a single summation
because of the restriction $n_{1}+n_{2}=L=n^{\prime}_{1}+n^{\prime}_{2}$

$\displaystyle\begin{split}&h^{g,L}_{n_{1},n_{2}}({\mathfrak{z}})=\sum_{n^{\prime}_{1},n^{\prime}_{2}=L-n^{\prime}_{1}}\mathcal{T}^{L}_{n^{\prime}_{1},n^{\prime}_{2};n_{1},n_{2}}(g)h_{n^{\prime}_{1},n^{\prime}_{2}}({\mathfrak{z}}),\\\
&\widetilde{h}^{g,L}_{n_{1},n_{2}}({\mathfrak{z}})=\sum_{n^{\prime}_{1},n^{\prime}_{2}=L-n^{\prime}_{1}}\mathcal{T}^{L}_{n^{\prime}_{1},n^{\prime}_{2};n_{1},n_{2}}(\widetilde{g})h_{n^{\prime}_{1},n^{\prime}_{2}}({\mathfrak{z}}).\end{split}$

Similarly, writing now $h^{L}_{m}({\mathfrak{z}})$ for the relabeled vectors
$f^{L}_{m}$ in (3.3) and $h^{g,L}_{m}$ for the $f^{g,L}_{m}$ in (3.43) and
using (3.44) and (3.45) we get

$\displaystyle
h^{g,L}_{m}({\mathfrak{z}})=\sum_{m^{\prime}=0}^{L}\mathcal{T}^{L}(g)_{m^{\prime}m}h^{L}_{m^{\prime}}({\mathfrak{z}}),\qquad
h^{\widetilde{g},L}_{m}({\mathfrak{z}})=\sum_{m^{\prime}=0}^{L}\mathcal{T}^{L}(\widetilde{g})_{m^{\prime}m}\;h^{L}_{m^{\prime}}({\mathfrak{z}}).$

We refer to the polynomials $h^{g,L}_{m}({\mathfrak{z}})$ as deformed complex
Hermite polynomials. It is now a routine matter to go over to a basis $H_{n}$,
$n=0,1,2,\ldots,\infty,$ which would be the analogous relabeling of the
$h^{L}_{m}$ as the $F_{n}$ in (3.4) are the relabeled versions of the
$f^{L}_{m}$. Similarly we may define the deformed polynomials
$H^{g}_{n}({\mathfrak{z}})$ and
$\widetilde{H}^{g}_{n}({\mathfrak{z}})=H^{\widetilde{g}}_{n}({\mathfrak{z}})$.
The biorthonormality of these polynomials is expressed via the integral
relation

$\displaystyle\int_{\mathbb{C}}\overline{\widetilde{H}^{g}_{n}({\mathfrak{z}})}H^{g}_{n^{\prime}}({\mathfrak{z}})d\nu(z,\overline{z})=\delta_{nn^{\prime}},$

which then are the pseudo-bosonic complex polynomial states.

Before leaving this section let us note that for fixed $n$, the polynomials
$h_{m,n}({\mathfrak{z}})$, $m=0,1,2,\ldots,\infty$ (see (4.2)) are
polyanalytic functions of order $n$ (see, for example [2]). More precisely,
the subspace $\mathcal{H}_{n}$ of $\mathcal{H}(\mathbb{C})$, consisting of
such polyanalytic functions spanned by the complex Hermite polynomials with a
fixed degree is the $n$th polyanalytic sector (corresponding to the $n$th
Landau level, also called a true [39] or pure [28] polyanalytic space) and the
direct sum of the first $k$ such spaces is the $k$th polyanalytic space. A
detailed proof of this fact, also valid for Banach spaces, can be found in
[3]. It should also be mentioned that the decomposition of
$\mathcal{H}(\mathbb{C})$ into true-polyanalytic spaces was first introduced
by Vasilevski in [39], where the action of the operators in (4.1) was
explored.

Since by (3.39) $S_{n}H_{m}=h_{m,n}$, the isometry $S_{n}$ maps the whole
Hilbert space $\mathcal{H}(\mathbb{C})$ to its $n$th polyanalytic sector and
(3.40) is then the statement that $\mathcal{H}(\mathbb{C})$ decomposes into an
orthogonal direct sum of polyanalytic subspaces. (Note that a function
$f(z,\overline{z})$ is polyanalytic of order $n$ if
$\frac{\partial^{n+1}f}{\partial\overline{z}^{n+1}}=0$). Such functions have
also found much use recently in signal analysis.

##  5 Norm estimates for biorthogonal families of polynomials

In this section, we take advantage of the group representation properties and
of the orthonormality of the complex Hermite polynomials to estimate the
respective norms of these new (deformed) vectors. This is useful in
determining whether the sets $\\{h^{g,L}_{n_{1},n_{2}}\\}$ and
$\\{h^{\widetilde{g},L}_{n_{1},n_{2}}\\}$ constitute bases in the Hilbert
space
$\mathcal{H}({\mathbb{C}})={\mathcal{L}}^{2}({{\mathbb{C}}},\mathrm{d}\nu(z,\overline{z}))$
or not. (Some similar, though less sharp, estimates were given in [16].)

######  Proposition 5.1.

Let $g\in\mathrm{GL}(2,{\mathbb{C}})$ and
$\big{\\{}h^{g,L}_{n_{1},n_{2}}\big{\\}}$ and
$\big{\\{}h^{\widetilde{g},L}_{n_{1},n_{2}}\big{\\}}$ be the deformed complex
Hermite polynomials defined in (4.4) and (4.5) respectively.

  1. $(i)$

We have the following upper bounds for their respective norms

$\displaystyle\big{\|}h^{g,L}_{n_{1},n_{2}}\big{\|}^{2}\leq\big{(}\operatorname{tr}g^{{\dagger}}g\big{)}^{(n_{1}+n_{2})/2},\qquad\big{\|}\widetilde{h}^{g,L}_{n_{1},n_{2}}\big{\|}^{2}\leq\big{(}\operatorname{tr}\big{(}g^{{\dagger}}g\big{)}^{-1}\big{)}^{(n_{1}+n_{2})/2}.$
(5.1)

  2. $(ii)$

More precisely

$\displaystyle\frac{a^{n_{1}}d^{L-n_{1}}}{\sqrt{\pi\min(n_{1},L-n_{1})}}\leq\big{\|}h^{g,L}_{n_{1},n_{2}}\big{\|}^{2}\leq\binom{L}{n_{1}}a^{n_{1}}d^{L-n_{1}},$
(5.2) $\displaystyle\frac{1}{|\det
g|^{2L}}\frac{d^{n_{1}}a^{L-n_{1}}}{\sqrt{\pi\min(n_{1},L-n_{1})}}\leq\big{\|}\widetilde{h}^{g,L}_{n_{1},n_{2}}\big{\|}^{2}\leq\frac{1}{|\det
g|^{2L}}\binom{L}{n_{1}}d^{n_{1}}a^{L-n_{1}},$ (5.3)

where we have introduced the notations

$\displaystyle g^{{\dagger}}g\equiv\begin{pmatrix}a&b\\\
\bar{b}&d\end{pmatrix}\quad\Rightarrow\quad\big{(}g^{{\dagger}}g\big{)}^{-1}=\frac{1}{|\det
g|^{2}}\begin{pmatrix}d&-b\\\ -\bar{b}&a\end{pmatrix},\qquad|\det
g|^{2}=ad-|b|^{2},$

for the positive invertible matrix $g^{{\dagger}}g$.

###### Proof.

By using group representation properties for the operator $\mathcal{T}^{L}$,
we find the following expressions for the squared norms of the deformed
complex Hermite polynomials

$\displaystyle\big{\|}h^{g,L}_{n_{1},n_{2}}\big{\|}^{2}=\sum_{n^{\prime}_{1},n^{\prime}_{2}}\big{|}\mathcal{T}^{L}_{n^{\prime}_{1},n^{\prime}_{2};n_{1},n_{2}}(g)\big{|}^{2}=\sum_{n^{\prime}_{1},n^{\prime}_{2}}\mathcal{T}^{s}_{n^{\prime}_{1},n^{\prime}_{2};n_{1},n_{2}}(g)\overline{\mathcal{T}^{L}_{n^{\prime}_{1},n^{\prime}_{2};n_{1},n_{2}}(g)}$
$\displaystyle\hphantom{\big{\|}h^{g,L}_{n_{1},n_{2}}\big{\|}^{2}}{}=\sum_{n^{\prime}_{1},n^{\prime}_{2}}\mathcal{T}^{L}_{n_{1},n_{2};n^{\prime}_{1},n^{\prime}_{2}}\big{(}\bar{g}^{{\dagger}}\big{)}\mathcal{T}^{L}_{n^{\prime}_{1},n^{\prime}_{2};n_{1},n_{2}}(\bar{g})=\mathcal{T}^{L}_{n_{1},n_{2};n_{1},n_{2}}\big{(}\bar{g}^{{\dagger}}\bar{g}\big{)}=\mathcal{T}^{L}_{n_{1},n_{2};n_{1},n_{2}}\big{(}g^{{\dagger}}g\big{)},\\!\\!\\!$
(5.4)
$\displaystyle\big{\|}\widetilde{h}^{g,L}_{n_{1},n_{2}}\big{\|}^{2}=\mathcal{T}^{L}_{n_{1},n_{2};n_{1},n_{2}}\big{(}\widetilde{g}^{{\dagger}}\widetilde{g}\big{)}=\mathcal{T}^{L}_{n_{1},n_{2};n_{1},n_{2}}\big{(}\big{(}g^{{\dagger}}g\big{)}^{-1}\big{)}.$
(5.5)

Hence we are led to studying the asymptotic behavior of the expressions
arising from (A.4) and (A.5) respectively

$\displaystyle\mathcal{T}^{L}_{n_{1},n_{2};n_{1},n_{2}}(h)=(\det
h)^{n_{1}}h_{22}^{n_{2}-n_{1}}P_{n_{1}}^{(0,n_{2}-n_{1})}\left(1+2\frac{|h_{12}|^{2}}{\det
h}\right)$ (5.6)
$\displaystyle\hphantom{\mathcal{T}^{L}_{n_{1},n_{2};n_{1},n_{2}}(h)=}{}=(\det
h)^{n_{1}}h_{22}^{n_{2}-n_{1}}\,{}_{2}F_{1}\left(-n_{1},n_{2}+1;1;\frac{|h_{12}|^{2}}{\det
h}\right),$

where $h$ is positive and Hermitian.

Note that the alternative forms of this expression, obtained by exploiting the
symmetry with respect to the interchange $1\rightarrow 2$, may be easer to
manipulate

$\displaystyle\mathcal{T}^{L}_{n_{1},n_{2};n_{1},n_{2}}(h)=(\det
h)^{n_{2}}h_{11}^{n_{1}-n_{2}}\,{}_{2}F_{1}\left(n_{1}+1,-n_{2};1;-\frac{|h_{12}|^{2}}{\det
h}\right)$
$\displaystyle\hphantom{\mathcal{T}^{L}_{n_{1},n_{2};n_{1},n_{2}}(h)}{}=h_{11}^{n_{1}}h_{22}^{n_{2}}\,{}_{2}F_{1}\left(-n_{1},-n_{2};1;\frac{|h_{12}|^{2}}{h_{11}h_{22}}\right)$
(5.7)
$\displaystyle\hphantom{\mathcal{T}^{L}_{n_{1},n_{2};n_{1},n_{2}}(h)}{}=(\det
h)^{n_{1}+n_{2}+1}h_{11}^{-n_{2}-1}h_{22}^{-n_{1}-1}\,{}_{2}F_{1}\left(n_{1}+1,n_{2}+1;1;\frac{|h_{12}|^{2}}{h_{11}h_{22}}\right).$

The most symmetrical and simplest form is clearly (5.7), which, once expanded,
reads as

$\displaystyle\mathcal{T}^{L}_{n_{1},n_{2};n_{1},n_{2}}(h)=h_{11}^{n_{1}}h_{22}^{n_{2}}\sum_{m=0}^{n_{1}\curlyvee
n_{2}}\binom{n_{1}}{m}\binom{n_{2}}{m}\left(-\frac{|h_{12}|^{2}}{h_{11}h_{22}}\right)^{m}.$
(5.8)

From this expression, from the fact that for any positive hermitian matrix
$|h_{12}|^{2}\leq h_{11}h_{22}$ (strict inequality if the matrix is
nonsingular), and from the well-known summation formula (e.g., see [26])

$\displaystyle\sum_{m}\binom{n_{1}}{m}\binom{n_{2}}{p-m}=\binom{n_{1}+n_{2}}{p},$

we easily derive the following upper bound (keeping in mind $L=n_{1}+n_{2}$),

$\displaystyle\mathcal{T}^{L}_{n_{1},n_{2};n_{1},n_{2}}(h)\leq\binom{L}{n_{1}}h_{11}^{n_{1}}h_{22}^{n_{2}}\leq(\operatorname{tr}h)^{L}.$
(5.9)

From this follow the upper bounds for the norms of the vectors in question
given in (5.1).

Next, we prove in Appendix B the following estimates of the diagonal matrix
elements $\mathcal{T}^{L}_{n_{1},n_{2};n_{1},n_{2}}(h)$:

$\displaystyle\frac{h_{11}^{n_{1}}h_{22}^{n_{2}}}{\sqrt{\pi\min(n_{1},n_{2})}}\leq\mathcal{T}^{L}_{n_{1},n_{2};n_{1},n_{2}}(h)\leq\frac{L!}{n_{1}!n_{2}!}h_{11}^{n_{1}}h_{22}^{n_{2}}\leq(\operatorname{tr}h)^{L},$
(5.10)

the lower bound being asymptotic at large $n_{1}$, $n_{2}$, whereas the upper
bound is valid for any $n_{1}$, $n_{2}$. The application of these estimates to
the norms of the polynomials $\\{h^{g,L}_{n_{1},n_{2}}\\}$ and
$\\{h^{\widetilde{g},L}_{n_{1},n_{2}}\\}$ (with $L=n_{1}+n_{2}$) given by
(5.4) and (5.5) yields the inequalities (5.2) and (5.3). ∎

As we have seen previously, the operators and the vectors introduced so far
satisfy Assumptions ${\mathcal{D}}$-pb 2.2 and ${\mathcal{D}}$-pb 2.3. On the
other hand, the estimates above suggest that, using the explicit
representation of our vectors in terms of our deformed complex Hermite
polynomials, Assumption ${\mathcal{D}}$-pb 2.4 might be not satisfied. In
fact, in order for ${\mathcal{F}}_{g}:=\\{h^{g,L}_{n_{1},n_{2}}\\}$ and
${\mathcal{F}}_{\tilde{g}}:=\\{\widetilde{h}^{g,L}_{n_{1},n_{2}}\\}$ (or
equivalently $\\{\widetilde{H}^{g}_{n}({\mathfrak{z}})\\}$ and
$\\{\widetilde{H}^{g}_{n}({\mathfrak{z}})\\}$) to be bases for
${\mathcal{H}}$, the product of their norms should be bounded in $n_{1}$ and
$n_{2}$, see, for instance, [22, Lemma 3.3.3]. Now we derive from (5.2) and
(5.3)

$\displaystyle\frac{1}{\pi\min(n_{1},n_{2})}\left(\frac{ad}{|\det
g|^{2}}\right)^{L}\leq\big{\|}h^{g,L}_{n_{1},n_{2}}\big{\|}^{2}\big{\|}\widetilde{h}^{g,L}_{n_{1},n_{2}}\big{\|}^{2}\leq\binom{L}{n_{1}}^{2}\left(\frac{ad}{|\det
g|^{2}}\right)^{L}.$

So unless $g$ is diagonal, we see from $ad>|\det g|^{2}$ that the product is
not bounded in $L$.

However, it is possible to check that ${\mathcal{F}}_{g}$ and
${\mathcal{F}}_{\tilde{g}}$ are ${\mathcal{G}}$-quasi bases, with
${\mathcal{G}}:=D(T(g)^{\dagger})\cap D(T^{-1}(g))$, which is dense in
${\mathcal{H}}$ since it contains the linear span of the original polynomials
$h_{n_{1},n_{2}}$. This is a consequence of the fact that the vectors in
${\mathcal{F}}_{g}$ and ${\mathcal{F}}_{\tilde{g}}$ are the image, via
$(T(g)^{\dagger}$ and $(T^{-1}(g)$, of the $h_{n_{1},n_{2}}$’s. So, this is
enough to conclude that we are fully within the general pseudo-bosonic
framework.

##  6 Bi-displacement operators and bi-coherent states

We now consider how a pair $(a^{\sharp},b^{\sharp})$ of pseudo-bosonic
operators, behaving as in Section 2, can be used to construct a generalized
version of the canonical coherent states. Our analysis extends that originally
contained in [10, 38]. First of all we introduce, at least formally, the two
$z$-dependent operators

$\displaystyle\mathfrak{D}(z)=\exp\\{zb-\overline{z}a\\},\qquad\widetilde{\mathfrak{D}}(z)=\exp\big{\\{}za^{\dagger}-\overline{z}b^{\dagger}\big{\\}}.$

They will be named bi-displacement operators, by analogy with the
Weyl–Heisenberg case.

######  Assumption ${\mathcal{D}}$-pbw 6.1.

With the notations of Section 2, for all $z\in{\mathbb{C}}$, $\mathfrak{D}(z)$
and $\widetilde{\mathfrak{D}}(z)$ are defined in the dense subspace
${\mathcal{D}}$ of ${\mathcal{H}}$ such that
$a^{\sharp}{\mathcal{D}}\subseteq{\mathcal{D}}$ and
$b^{\sharp}{\mathcal{D}}\subseteq{\mathcal{D}}$.

The Weyl formula, $e^{A}\,e^{B}=e^{\left(\frac{1}{2}[A,B]\right)}\,e^{(A+B)}$,
(arising from the Baker–Campbell–Hausdorff relation) which is valid for any
pair of operators that commute with their commutator, $[A,[A,B]]=0=[B,[A,B]]$,
yields the following alternative, factorized forms of these operators

$\displaystyle{\mathfrak{D}}(z)=e^{-|z|^{2}/2}e^{zb}e^{-\bar{z}a}=e^{|z|^{2}/2}e^{-\bar{z}a}e^{zb},$
(6.1)
$\displaystyle\widetilde{\mathfrak{D}}(z)=e^{-|z|^{2}/2}e^{za^{{\dagger}}}e^{-\bar{z}b^{{\dagger}}}=e^{|z|^{2}/2}e^{-\bar{z}b^{{\dagger}}}e^{za^{{\dagger}}}.$
(6.2)

The operator-valued maps $z\mapsto{\mathfrak{D}}(z)$ and
$z\mapsto\widetilde{\mathfrak{D}}(z)$ are (possibly local) projective
representations of the abelian group of translations of the complex plane.
Indeed, let us apply the Weyl formula to the product
${\mathfrak{D}}(z_{1}){\mathfrak{D}}(z_{2})$. One gets the composition rules

$\displaystyle{\mathfrak{D}}(z_{1}){\mathfrak{D}}(z_{2})=e^{-iz_{1}\wedge
z_{2}}{\mathfrak{D}}(z_{1}+z_{2}),$ (6.3)
$\displaystyle\widetilde{\mathfrak{D}}(z_{1})\widetilde{\mathfrak{D}}(z_{2})=e^{-iz_{1}\wedge
z_{2}}\widetilde{\mathfrak{D}}(z_{1}+z_{2}),$ (6.4)

where $z_{1}\wedge z_{2}:=x_{1}y_{2}-x_{2}y_{1}$ for
$x_{i}=\operatorname{Re}z_{i}$, $y_{i}=\operatorname{Im}z_{i}$. In particular,
since ${\mathfrak{D}}(0)=I=\widetilde{\mathfrak{D}}(0)$,

$\displaystyle({\mathfrak{D}}(z))^{-1}={\mathfrak{D}}(-z),\qquad\big{(}\widetilde{\mathfrak{D}}(z)\big{)}^{-1}=\widetilde{\mathfrak{D}}(-z).$

These relations also give

$\displaystyle{\mathfrak{D}}(-z)={\mathfrak{D}}(z)^{-1}=\widetilde{\mathfrak{D}}^{\dagger}(z).$
(6.5)

Let $\varpi(z)$ be a function on the complex plane obeying the (normalization)
condition

$\displaystyle\varpi(0)=1,$ (6.6)

and being assumed to define the two bounded operators ${\sf M}$ and
$\widetilde{\sf M}$ on $\mathcal{H}$ through the operator-valued integrals

$\displaystyle{\sf
M}=\int_{\mathbb{C}}\varpi(z)\mathfrak{D}(z)\frac{\mathrm{d}^{2}z}{\pi},$
$\displaystyle\widetilde{\sf
M}=\int_{\mathbb{C}}\varpi(z)\widetilde{\mathfrak{D}}(z)\frac{\mathrm{d}^{2}z}{\pi}=\int_{\mathbb{C}}\varpi(-z)\mathfrak{D}^{{\dagger}}(z)\frac{\mathrm{d}^{2}z}{\pi}.$

Note that if we explicitly express the dependence of ${\mathsf{M}}$ on the
weight function, ${\mathsf{M}}\equiv{\mathsf{M}}^{\varpi}$, then
$\widetilde{\mathsf{M}}\equiv\left({\mathsf{M}}^{{\sf
P}{\varpi}}\right)^{{\dagger}}$, where ${\sf P}$ is the parity operator, ${\sf
P}f(z)=f(-z)$. Hence, we have the interesting relation

$\displaystyle\varpi(z)=\overline{\varpi(-z)}\quad\forall\,z\quad\Rightarrow\quad{\sf
M}^{{\dagger}}=\widetilde{\sf M}.$

We now give the following proposition, where the fact that ${\mathfrak{D}}(z)$
and $\widetilde{\mathfrak{D}}(z)$ are defined for each $z\in{\mathbb{C}}$ is
crucial:

######  Proposition 6.2.

If ${\mathfrak{D}}(z)$, $\widetilde{\mathfrak{D}}(z)$, and $\varpi(z)$, are
such that

$\displaystyle{\mathfrak{D}}(z)\left[\int_{\mathbb{C}}{\mathfrak{D}}(z^{\prime})\varpi(z^{\prime})\frac{\mathrm{d}^{2}z^{\prime}}{\pi}\right]{\mathfrak{D}}(-z)=\int_{\mathbb{C}}{\mathfrak{D}}(z){\mathfrak{D}}(z^{\prime}){\mathfrak{D}}(-z)\varpi(z^{\prime})\frac{\mathrm{d}^{2}z^{\prime}}{\pi},$
(6.7)
$\displaystyle\widetilde{\mathfrak{D}}(z)\left[\int_{\mathbb{C}}\widetilde{\mathfrak{D}}(z^{\prime})\varpi(z^{\prime})\frac{\mathrm{d}^{2}z^{\prime}}{\pi}\right]\widetilde{\mathfrak{D}}(-z)=\int_{\mathbb{C}}\widetilde{\mathfrak{D}}(z)\widetilde{\mathfrak{D}}(z^{\prime})\widetilde{\mathfrak{D}}(-z)\varpi(z^{\prime})\frac{\mathrm{d}^{2}z^{\prime}}{\pi}$

hold, for all $z$, in a weak sense on the dense subspace ${\mathcal{D}}$ of
${\mathcal{H}}$, then the families

$\displaystyle{\sf M}(z):={\mathfrak{D}}(z){\sf
M}{\mathfrak{D}}(-z)={\mathfrak{D}}(z){\sf
M}\widetilde{\mathfrak{D}}^{{\dagger}}(z),$ (6.8) $\displaystyle\widetilde{\sf
M}(z):=\widetilde{\mathfrak{D}}(z)\widetilde{\sf
M}\widetilde{\mathfrak{D}}(-z)=\widetilde{\mathfrak{D}}(z){\sf
M}{\mathfrak{D}}^{{\dagger}}(z)$

of bi-displaced operators under the respective actions of ${\mathfrak{D}}(z)$
and $\widetilde{\mathfrak{D}}(z)$ resolve the identity in the sense given in
(2.4)

$\displaystyle\int_{\mathbb{C}}{\sf M}(z)\frac{\mathrm{d}^{2}z}{\pi}=I,$ (6.9)
$\displaystyle\int_{\mathbb{C}}\widetilde{\sf
M}(z)\frac{\mathrm{d}^{2}z}{\pi}=I.$ (6.10)

###### Proof.

With the assumption (6.7), we have after applying (6.3) twice

$\displaystyle\int_{\mathbb{C}}{\sf
M}(z)\frac{\mathrm{d}^{2}z}{\pi}=\int_{\mathbb{C}}\frac{\mathrm{d}^{2}z}{\pi}\int_{\mathbb{C}}{\mathfrak{D}}(z){\mathfrak{D}}(z^{\prime}){\mathfrak{D}}(-z)\varpi(z^{\prime})\frac{\mathrm{d}^{2}z^{\prime}}{\pi}$
$\displaystyle\hphantom{\int_{\mathbb{C}}{\sf
M}(z)\frac{\mathrm{d}^{2}z}{\pi}}{}=\int_{\mathbb{C}}\frac{\mathrm{d}^{2}z}{\pi}\int_{\mathbb{C}}e^{-2iz\wedge
z^{\prime}}{\mathfrak{D}}(z^{\prime})\varpi(z^{\prime})\frac{\mathrm{d}^{2}z^{\prime}}{\pi}.$

Then (6.9) is a direct consequence of the formula (symplectic Fourier
transform of the function $1$ in the plane)

$\displaystyle\int_{\mathbb{C}}e^{-2iz\wedge
z^{\prime}}\frac{\mathrm{d}^{2}z}{\pi}=\int_{\mathbb{C}}e^{z\bar{z}^{\prime}-\bar{z}z^{\prime}}\frac{\mathrm{d}^{2}z}{\pi}=\pi\delta^{2}(z^{\prime}),$
(6.11)

and of the condition (6.6) with ${\mathfrak{D}}(0)=I$. The same demonstration
applies trivially to (6.10). ∎

Let us expand the operators ${\mathfrak{D}}$ and $\widetilde{\mathfrak{D}}$ in
terms of the biorthonormal bases or sets222Their nature depends on which one
of the Assumptions ${\mathcal{D}}$-pb 2.4 or ${\mathcal{D}}$-pbw 2.6 is
satisfied. (2.2),

$\displaystyle{\mathfrak{D}}(z)=\sum_{m,n}{\mathfrak{D}}_{mn}(z)|\varphi_{m}\rangle\langle\Psi_{n}|,$
(6.12)
$\displaystyle\widetilde{\mathfrak{D}}(z)=\sum_{m,n}\widetilde{\mathfrak{D}}_{mn}(z)|\Psi_{m}\rangle\langle\varphi_{n}|,$

where the matrix elements in (6.12), which involve associated Laguerre
polynomials $L^{(\alpha)}_{n}(t)$, are calculated in Appendix C

$\displaystyle{\mathfrak{D}}_{mn}(z):=\langle\Psi_{m}|{\mathfrak{D}}(z)|\varphi_{n}\rangle=\sqrt{\dfrac{n!}{m!}}e^{-|z|^{2}/2}z^{m-n}L_{n}^{(m-n)}(|z|^{2}),\qquad\mbox{for}\
m\geq n,$ (6.13)

with $L_{n}^{(m-n)}(t)=\frac{m!}{n!}(-t)^{n-m}L_{m}^{(n-m)}(t)$ for $n\geq m$.
From (6.5) we have

$\displaystyle\widetilde{\mathfrak{D}}_{mn}(z)=\overline{{\mathfrak{D}}}_{nm}(-z).$
(6.14)

Note that the polynomial parts of these matrix elements are, up to a factor,
complex Hermite polynomials.

As an interesting example, which is inspired from [18] (see also [17]), we
choose

$\displaystyle\varpi_{s}(z)=e^{s|z|^{2}/2},\qquad\operatorname{Re}s<1.$

Since this function is isotropic in the complex plane, the resulting operator
${\sf M}\equiv{\sf M}_{s}$ is diagonal. when expanded in terms of the
$\varphi_{n}$’s and $\Psi_{n}$’s in (2.2). From the expression (6.13) of the
matrix elements of ${\mathfrak{D}}(z)$, and the integral [35]

$\displaystyle\int_{0}^{\infty}e^{-\nu
x}x^{\lambda}L_{n}^{\alpha}(x)\mathrm{d}x=\frac{\Gamma(\lambda+1)\Gamma(\alpha+n+1)}{n!\Gamma(\alpha+1)}\nu^{-\lambda-1}{}_{2}F_{1}(-n,\lambda+1;\alpha+1;\nu^{-1}),$

we get the diagonal elements of ${\sf M}_{s}$

$\displaystyle\langle\varphi_{n}|{\sf
M}_{s}|\Psi_{n}\rangle=\frac{2}{1-s}\left(\frac{s+1}{s-1}\right)^{n},$

and so

$\displaystyle{\sf
M}_{s}=\int_{\mathbb{C}}\varpi_{s}(z){\mathfrak{D}}(z)\frac{{\mathrm{d}}^{2}z}{\pi}=\frac{2}{1-s}\exp\left[\left(\log\dfrac{s+1}{s-1}\right)ba\right].$

Then $s=-1$ corresponds to the basic operator

$\displaystyle{\sf
M}_{-1}=\lim_{s\to-1_{-}}\dfrac{2}{1-s}\exp\left[\left(\ln\dfrac{s+1}{s-1}\right)ba\right]=|\varphi_{0}\rangle\langle\Psi_{0}|,$

_Bi-coherent_ states show up when precisely this operator is bi-displaced
along (6.8)

$\displaystyle{\sf M}_{1}(z):={\mathfrak{D}}(z){\sf
M}_{1}{\mathfrak{D}}(-z)={\mathfrak{D}}(z)|\varphi_{0}\rangle\langle\Psi_{0}|\widetilde{\mathfrak{D}}^{{\dagger}}(z)\equiv|\varphi(z)\rangle\langle\Psi(z)|,$

i.e., they are defined as

$\displaystyle\varphi(z)={\mathfrak{D}}(z)\varphi_{0},\qquad\Psi(z)=\widetilde{\mathfrak{D}}(z)\Psi_{0}.$
(6.15)

It is necessary to check that these vectors are well defined in
${\mathcal{H}}$ for some $z\in{\mathbb{C}}$. Using the factorizations (6.1)
and (6.2) together with the properties of $\varphi_{0}$ and $\Psi_{0}$, we get

$\displaystyle\varphi(z)=e^{-|z|^{2}/2}\sum_{n=0}^{\infty}\frac{z^{n}}{\sqrt{n!}}\varphi_{n},\qquad\Psi(z)=e^{-|z|^{2}/2}\sum_{n=0}^{\infty}\frac{z^{n}}{\sqrt{n!}}\Psi_{n}.$
(6.16)

Since ${\mathfrak{D}}(z)$ and $\widetilde{\mathfrak{D}}(z)$ are not unitary
operators, or alternatively since $\varphi_{n}$ and $\Psi_{n}$ are not
normalized in general, we should concretely check that the series in (6.16)
both converge, at least for some reasonably large set of $z$’s. In fact, so
far we have assumed that the states exist for all $z\in{\mathbb{C}}$. This is
clear whenever ${\mathcal{F}}_{\varphi}$ and ${\mathcal{F}}_{\Psi}$ are o.n.
bases (in this case, in fact, convergence is for all $z\in{\mathbb{C}}$), or
when $\varphi_{0}\in D({\mathfrak{D}}(z))$ and $\Psi_{0}\in
D(\widetilde{\mathfrak{D}}(z))$, but it is not evident in general. However, it
is possible to prove the following.

######  Proposition 6.3.

Suppose that $r_{\varphi},r_{\Psi}>0$ and
$0\leq\alpha_{\varphi},\alpha_{\Psi}<\frac{1}{2}$ exist such that

$\displaystyle\|\varphi_{n}\|\leq
r_{\varphi}^{n}(n!)^{\alpha_{\varphi}},\qquad\|\Psi_{n}\|\leq
r_{\Psi}^{n}(n!)^{\alpha_{\Psi}},$ (6.17)

then $\varphi(z)$ is well defined for all $z$.

###### Proof.

The proof relies upon the following estimate

$\displaystyle\|\varphi(z)\|^{2}=e^{-|z|^{2}}\sum_{n,k=0}^{\infty}\frac{z^{n}\overline{z}^{k}}{\sqrt{n!k!}}\langle\varphi_{k},\varphi_{n}\rangle\leq
e^{-|z|^{2}}\left(\sum_{n,k=0}^{\infty}\frac{(r_{\varphi}|z|)^{n}}{(n!)^{\frac{1}{2}-\alpha_{\varphi}}}\right)^{2},$

which converges for all values of $z\in{\mathbb{C}}$ ∎

Analogously, we can prove that $\Psi(z)$ is well defined for all $z$.
Moreover,
$\langle\varphi(z),\Psi(z)\rangle=\langle\varphi_{0},\Psi_{0}\rangle=1$.

Notice that the inequalities in (6.17) are surely satisfied for Riesz bases,
since in this case the norms of both $\varphi_{n}$ and $\Psi_{n}$ are
uniformly bounded in $n$. However, our assumption here does not prevent us
from considering families ${\mathcal{F}}_{\varphi}$ and ${\mathcal{F}}_{\Psi}$
of vectors with divergent norms, as often happen in explicit models [13]. In
other words, $\varphi(z)$ and $\Psi(z)$ could also be defined if
${\mathcal{F}}_{\varphi}$ and ${\mathcal{F}}_{\Psi}$ are not bases, which
happens if both $\|\varphi_{n}\|$ and $\|\Psi_{n}\|$ diverge with $n$ [13], at
least if condition (6.17) holds true.

It is interesting to notice that conditions (6.17) are indeed satisfied in
several models recently considered in the literature. For instance, in [11],
the vector $\varphi_{n}$ satisfies an inequality like
$\|\varphi_{n}\|\leq(1+|\alpha-\beta|)^{n/2}$, where $\alpha$ and $\beta$ are
two in general complex parameters of the model. A similar estimate, with a
harmless overall constant, can also be found for $\|\Psi_{n}\|$. This means
that (6.17) also are satisfied in the model originally proposed in [9], which
is a special case of that in [11], and for the model discussed in [14], which
is a two-dimensional, non commutative, version of the same model.

Also the vectors introduced in the Swanson model [9] satisfy similar
inequalities. Indeed [13], in this model we have found that

$\displaystyle\|\varphi_{n}\|^{2}=|N_{1}|^{2}\sqrt{\frac{\pi}{\cos(2\theta)}}P_{n}\left(\frac{1}{\cos(2\theta)}\right),\qquad\|\Psi_{n}\|=\left|\frac{N_{2}}{N_{1}}\right|\|\varphi_{n}\|,$

where $N_{1}$ and $N_{2}$ are normalization constants, $\theta$ is a parameter
in $\big{]}{-}\frac{\pi}{4},\frac{\pi}{4}\big{[}$, and $P_{n}$ is a Legendre
polynomial. Using [37] we deduce that, for instance,

$\displaystyle\|\varphi_{n}\|\leq A_{\theta}\alpha_{\theta}^{n},$

where $A_{\theta}$ is a constant and

$\displaystyle\alpha_{\theta}=\sqrt{\frac{1}{\cos(2\theta)}+\left(\frac{1}{\cos^{2}(2\theta)}-1\right)^{1/2}}.$

######  Remark 6.4.

The possibility remains open that $\varphi(z)$ and $\Psi(z)$ only exist for
$z\in{\mathcal{E}}$, with ${\mathcal{E}}$ a proper (sufficiently large) subset
of ${\mathbb{C}}$. When this is so, of course, the proof of Proposition 6.2
fails to work since the integral in (6.11) will only be extended to
${\mathcal{E}}$ and not to all ${\mathbb{C}}$. Therefore, our bi-coherent
states need not to resolve the identity anymore. The analysis of this
situation is postponed to another paper.

Another reason why these vectors are called bi-coherent is because they are
eigenstates of some lowering operators. Indeed we can check that

$\displaystyle a\varphi(z)=z\varphi(z),\qquad b^{\dagger}\Psi(z)=z\Psi(z),$

for all $z\in{\mathbb{C}}$. Their overlap is given by the kernel

$\displaystyle\mathcal{K}(z,z^{\prime})=\langle\varphi(z)|\Psi(z^{\prime})\rangle=e^{-|z|^{2}/2}e^{-|z^{\prime}|^{2}/2}e^{\bar{z}z^{\prime}}=e^{-|z-z^{\prime}|^{2}/2}e^{iz\wedge
z^{\prime}},$

which is the same kernel as that for the canonical coherent states. As a
particular case of (6.9), they resolve the identity

$\displaystyle\int_{\mathbb{C}}|\varphi(z)\rangle\langle\Psi(z)|\frac{\mathrm{d}^{2}z}{\pi}=I=\int_{\mathbb{C}}|\Psi(z)\rangle\langle\varphi(z)|\frac{\mathrm{d}^{2}z}{\pi},$
(6.18)

and this entails the reproducing property of the kernel

$\displaystyle\int_{\mathbb{C}}\mathcal{K}(z,z^{\prime})\mathcal{K}(z^{\prime},z^{\prime\prime})\frac{\mathrm{d}^{2}z^{\prime}}{\pi}=\mathcal{K}(z,z^{\prime\prime}).$

Finally note the projective covariance property of bi-coherent states, as a
direct consequence of (6.15) and (6.3) and (6.4)

$\displaystyle{\mathfrak{D}}(z)\varphi(z^{\prime})=e^{-iz\wedge
z^{\prime}}\varphi(z+z^{\prime}),\qquad\widetilde{\mathfrak{D}}(z)\Psi(z^{\prime})=e^{-iz\wedge
z^{\prime}}\Psi(z+z^{\prime}).$ (6.19)

##  7 Integral quantization with bicoherent families and more

We now adapt the integral quantization scheme described in [5, 17] and [15] to
the present pseudo-bosonic formalism. This is made possible when the
resolutions of the identity (6.9) and (6.10) are valid on some dense subspace
of the Hilbert space in question. Given a weight function $\varpi(z)$ with
$\varpi(0)=1$ and the resulting families of bi-displaced operators
${\mathsf{M}}(z)$ and $\widetilde{\mathsf{M}}(z)$, the quantizations of a
function $f(z)$ on the complex plane is defined by the linear maps

$\displaystyle f\mapsto A_{f}=\int_{\mathbb{C}}f(z){\sf
M}(z)\frac{\mathrm{d}^{2}z}{\pi}=\int_{\mathbb{C}}\mathcal{F}(-z){\mathfrak{D}}(z)\varpi(z)\frac{\mathrm{d}^{2}z}{\pi},$
(7.1) $\displaystyle
f\mapsto\widetilde{A}_{f}=\int_{\mathbb{C}}f(z)\widetilde{\mathsf{M}}(z)\frac{\mathrm{d}^{2}z}{\pi}=\int_{\mathbb{C}}\mathcal{F}(-z)\widetilde{\mathfrak{D}}(z)\varpi(z)\frac{\mathrm{d}^{2}z}{\pi},$

where $\mathcal{F}$ is the symplectic Fourier transform of $f$,

$\displaystyle\mathcal{F}[f](z)\equiv\mathcal{F}(z)=\int_{\mathbb{C}}f(\xi)e^{z\bar{\xi}-\bar{z}\xi}\frac{\mathrm{d}^{2}\xi}{\pi}=\int_{\mathbb{C}}f(\xi)e^{2i\xi\wedge
z}\frac{\mathrm{d}^{2}\xi}{\pi}.$

Covariance with respect to translations reads
$A_{f(z-z_{0})}={\mathfrak{D}}(z_{0})A_{f(z)}{\mathfrak{D}}(z_{0})^{\dagger}$.
In the case of a real even weight function we have the relation
$\widetilde{A}_{f}=A_{\bar{f}}^{{\dagger}}$, and then, if the function $f$ is
real, the adjoint of $A_{f}$ is $\widetilde{A}_{f}$. A more delicate question
is to find pairs $(f,\varpi)$ for which $A_{f}$ is symmetric.

In the sequel we focus on the quantizations using ${\mathsf{M}}(z)$ only,
since there are well-defined relations between $A_{f}$ and
$\widetilde{A}_{f}$.

We now show that the generic pseudo-boson commutation rule (2.1) is always the
outcome of the above quantization, whatever the chosen complex function
$\varpi(z)$, provided integrability and differentiability at the origin is
ensured. For this let us calculate $A_{z}$ and $A_{\bar{z}}$. Taking into
account that the symplectic Fourier transform of the function $z$,
$\mathcal{F}[z](z)$, is equal to $-\partial_{\bar{z}}\pi\delta^{2}(z)$, where
$\pi\delta^{2}(z)=\int_{\mathbb{C}}e^{z\bar{\xi}-\bar{z}\xi}\frac{d^{2}\xi}{\pi}$,
one has from (7.1)
$A_{z}=-\left[\varpi(z)\partial_{\bar{z}}{\mathfrak{D}}(z)+{\mathfrak{D}}(z)\partial_{\bar{z}}\varpi(z)\right]_{z=0}$.
Then, using
$\partial_{\bar{z}}{\mathfrak{D}}(z)=-\left(a-\frac{z}{2}\right){\mathfrak{D}}(z)$
we obtain finally

$\displaystyle
A_{z}=a\varpi(0)-\left.\partial_{\bar{z}}\varpi\right|_{z=0}=a-\left.\partial_{\bar{z}}\varpi\right|_{z=0}.$

Similarly, we obtain for $A_{\bar{z}}$ the following expression

$\displaystyle A_{\bar{z}}=b+\left.\partial_{z}\varpi\right|_{z=0},$

after using the relation
$\partial_{z}{\mathfrak{D}}(z)=\left(b-\frac{\bar{z}}{2}\right){\mathfrak{D}}(z)$.

Defining the Poisson bracket for functions $f(z)$ (actually $f(z,\bar{z}))$ as

$\displaystyle\\{f,g\\}=\frac{\partial f}{\partial z}\frac{\partial
g}{\partial\bar{z}}-\frac{\partial g}{\partial z}\frac{\partial
f}{\partial\bar{z}},$

we thus check that the map (7.1) is “pseudo-canonical” in the sense that

$\displaystyle\\{z,\bar{z}\\}=1\mapsto[a,b]=I.$

##  8 Conclusion

In this paper we have discussed two illustrations of the
${\mathcal{D}}$-pseudo-bosonic formalism, the biorthogonal complex Hermite
polynomials, and a second using families of vectors and operators in the
underlying Hilbert space, built in way similar to that of the standard
coherent states, i.e., as orbits of projective representations of the
Weyl–Heisenberg group. We have also considered the resolutions of the identity
satisfied by these families and the related integral quantizations naturally
arising from them. In particular, these quantizations yield exactly the
genuine pseudo-bosonic commutation rules.

These results can be of some interest in connection with PT or pseudo-
hermitian quantum mechanics, where the role of self-adjoint operators is
usually not so relevant. In [13] several connections have been already
established between ${\mathcal{D}}$-pseudo-bosons and this extended quantum
mechanics, and our results on complex Hermite polynomials and on dual integral
quantizations suggest that more can be established. This is, in fact, part of
our future work.

##  Appendix A Irreducible finite-dimensional representations of
$\boldsymbol{{\rm GL}(2,{\mathbb{C}})}$

The linear action of a $2\times 2$ complex matrix
$\gamma=\begin{pmatrix}\gamma_{11}&\gamma_{12}\\\
\gamma_{21}&\gamma_{22}\end{pmatrix}$ on vector space ${\mathbb{C}}^{2}$ is
defined in the usual way as

$\displaystyle\gamma\cdot\mathbf{x}=\gamma\binom{x_{1}}{x_{2}}=\binom{\gamma_{11}x_{1}+\gamma_{12}x_{2}}{\gamma_{21}x_{1}+\gamma_{22}x_{2}}.$

We now consider the linear representation $\mathcal{T}^{s}$ of ${\rm
GL}(2,{\mathbb{C}})$ carried on by the complex vector space $\mathcal{V}^{s}$
of two-variable homogeneous polynomials $p(\mathbf{x})$ of fixed degree $s$ in
the following way

$\displaystyle\left(\mathcal{T}^{s}(\gamma)p\right)(\mathbf{x})=p\big{(}{}^{t}\gamma\cdot\mathbf{x}\big{)},$
(A.1)

where ${}^{t}\gamma$ is the transpose of $\gamma$. In the monomial basis of
fixed degree $s$

$\displaystyle
e_{n_{1},n_{2}}(\mathbf{x}):=\frac{x_{1}^{n_{1}}x_{2}^{n_{2}}}{\sqrt{n_{1}!n_{2}!}},\qquad
s=n_{1}+n_{2}$ (A.2)

the matrix elements
$\mathcal{T}^{s}_{n^{\prime}_{1},n^{\prime}_{2};n_{1},n_{2}}(\gamma)$ of
$\mathcal{T}^{s}(\gamma)$, defined by

$\displaystyle
e_{n_{1},n_{2}}({}^{t}\gamma\cdot\mathbf{x})=\sum_{n^{\prime}_{1},n^{\prime}_{2}=s-n^{\prime}_{1}}\mathcal{T}^{s}_{n^{\prime}_{1},n^{\prime}_{2};n_{1},n_{2}}(\gamma)e_{n^{\prime}_{1},n^{\prime}_{2}}(\mathbf{x}),$
(A.3)

are given by

$\displaystyle\mathcal{T}^{s}_{n^{\prime}_{1},n^{\prime}_{2};n_{1},n_{2}}(\gamma)=\sqrt{\frac{n^{\prime}_{1}!n^{\prime}_{2}!}{n_{1}!n_{2}!}}\gamma_{21}^{n_{1}}\gamma_{12}^{n^{\prime}_{1}}\gamma_{22}^{n_{2}-n^{\prime}_{1}}\sum_{j=0}^{n^{\prime}_{1}}\binom{n_{1}}{j}\binom{n_{2}}{n^{\prime}_{1}-j}\rho^{j}$
$\displaystyle\hphantom{\mathcal{T}^{s}_{n^{\prime}_{1},n^{\prime}_{2};n_{1},n_{2}}(\gamma)}{}=\sqrt{\frac{n^{\prime}_{1}!n^{\prime}_{2}!}{n_{1}!n_{2}!}}(\det\gamma)^{n^{\prime}_{1}}\gamma_{21}^{n_{1}-n^{\prime}_{1}}\gamma_{22}^{n_{2}-n^{\prime}_{1}}P_{n^{\prime}_{1}}^{(n_{1}-n^{\prime}_{1},n_{2}-n^{\prime}_{1})}\left(\frac{\rho+1}{\rho-1}\right),$
(A.4)
$\displaystyle\rho:=\frac{\gamma_{11}\gamma_{22}}{\gamma_{12}\gamma_{21}}.$

We impose the constraints $n_{1}+n_{2}=s=n^{\prime}_{1}+n^{\prime}_{2}$, which
have to be satisfied in all these expressions. However we keep the two
summation indices for notational convenience. The polynomials
$P_{n}^{(\alpha,\beta)}$ are the Jacobi polynomials given [35] by

$\displaystyle
P_{n}^{(\alpha,\beta)}(x)=2^{-n}(x-1)^{n}\sum_{j=0}^{n}\binom{n+\alpha}{j}\binom{n+\beta}{n-j}\left(\frac{x+1}{x-1}\right)^{j}$
$\displaystyle\hphantom{P_{n}^{(\alpha,\beta)}(x)}{}=\binom{n+\alpha}{n}\,{}_{2}F_{1}\left(-n,\alpha+\beta+n+1;\alpha+1;\frac{1-x}{2}\right).$

Note the alternative and simpler form of (A.4) in terms of the above
hypergeometric function, due to the relation
$\frac{\rho+1}{\rho-1}=1+2\dfrac{\gamma_{12}\gamma_{21}}{\det\gamma}$:

$\displaystyle\mathcal{T}^{s}_{n^{\prime}_{1},n^{\prime}_{2};n_{1},n_{2}}(\gamma)=\sqrt{\frac{n^{\prime}_{1}!n^{\prime}_{2}!}{n_{1}!n_{2}!}}(\det\gamma)^{n^{\prime}_{1}}\gamma_{21}^{n_{1}-n^{\prime}_{1}}\gamma_{22}^{n_{2}-n^{\prime}_{1}}\binom{n_{1}}{n^{\prime}_{1}}$
$\displaystyle\hphantom{\mathcal{T}^{s}_{n^{\prime}_{1},n^{\prime}_{2};n_{1},n_{2}}(\gamma)=}{}\times{}_{2}F_{1}\left(-n^{\prime}_{1},n^{\prime}_{2}+1;n_{1}-n^{\prime}_{1}+1;\frac{\gamma_{12}\gamma_{21}}{\det\gamma}\right).$

In particular the diagonal elements read as

$\displaystyle\mathcal{T}^{s}_{n_{1},n_{2};n_{1},n_{2}}(\gamma)=(\det\gamma)^{n_{1}}\gamma_{22}^{n_{2}-n_{1}}\,{}_{2}F_{1}\left(-n_{1},n_{2}+1;1;\frac{\gamma_{12}\gamma_{21}}{\det\gamma}\right).$
(A.5)

Finally, note the property

$\displaystyle\mathcal{T}^{s}\big{(}g^{\dagger}\big{)}=\big{(}\mathcal{T}^{s}(g)\big{)}^{*}.$
(A.6)

##  Appendix B Asymptotic behavior of matrix elements

In this appendix we give the asymptotic behavior of the diagonal matrix
elements $\mathcal{T}^{s}_{n_{1},n_{2};n_{1},n_{2}}(h)$, for a positive matrix
$h$, for large $n_{1}$, $n_{2}$, for two types of directions in the positive
two-dimensional square lattice
$\Lambda_{++}=\\{(n_{1},n_{2})\,|\,n_{1},n_{2}\in{\mathbb{N}}\\}$.

### Behavior at large $\boldsymbol{n_{1}}$, $\boldsymbol{n_{2}}$, with fixed
$\boldsymbol{d=n_{2}-n_{1}}$

To study this behavior, we use the expression (5.6), with $d=n_{2}-n_{1}$, of
the diagonal elements in terms of the Jacobi polynomials:

$\displaystyle\mathcal{T}^{s}_{n_{1},n_{2};n_{1},n_{2}}(h)=(\det
h)^{n_{1}}h_{22}^{d}P_{n_{1}}^{(0,d)}\left(X\right),\qquad
X=1+2\frac{|h_{12}|^{2}}{\det h}.$

We suppose $d\geq 0$ with no loss of generality. From [35] we know that, at
large $n$,

$\displaystyle P_{n}^{(\alpha,\beta)}(x)\sim\frac{1}{\sqrt{2\pi
n}}(x-1)^{-\frac{\alpha}{2}}(x+1)^{-\frac{\beta}{2}}\\!\big{[}\sqrt{x+1}+\sqrt{x-1}\big{]}^{\alpha+\beta}(x-1)^{-\frac{1}{4}}\\!\big{[}x+\sqrt{x^{2}-1}\big{]}^{n+\frac{1}{2}}$

holds for $x>1$ or $x<1$. Applied to the present case this leads to the
asymptotic behavior of $\mathcal{T}^{s}_{n_{1},n_{2};n_{1},n_{2}}(h)$ for
fixed $d=n_{2}-n_{1}>0$

$\displaystyle\mathcal{T}^{s}_{n_{1},n_{2};n_{1},n_{2}}(h)\sim\frac{\left(2|h_{12}|^{2}\det
h\right)^{-1/4}}{\sqrt{2\pi
n_{1}}}\left(\frac{h_{22}}{h_{11}}\right)^{d/2}\left[\sqrt{h_{11}h_{22}}+|h_{12}|\right]^{n_{1}+n_{2}+1}$
(B.1)
$\displaystyle\hphantom{\mathcal{T}^{s}_{n_{1},n_{2};n_{1},n_{2}}(h)}{}=\frac{h_{11}^{n_{1}}h_{22}^{n_{2}}}{\sqrt{2\pi
n_{1}}}[2r(1-r)]^{-1/4}\left[1+\sqrt{r}\right]^{n_{1}+n_{2}+1},\qquad
0<r=\frac{|h_{12}|^{2}}{h_{11}h_{22}}<1.$ (B.2)

### Complete estimate for large $\boldsymbol{n_{1}}$, $\boldsymbol{n_{2}}$

Since the a priori fixed $d=n_{2}-n_{1}$ can be arbitrarily large, (B.1) is
valid for arbitrarily large $n_{1}$ and $n_{2}\geq n_{1}$. In the case
$n_{1}>n_{2}$ it is enough to permute $1\leftrightarrow 2$ in the right-hand
side of (B.2). This formula provides a lower bound to
$\mathcal{T}^{s}_{n_{1},n_{2};n_{1},n_{2}}(h)$ since for any $r\in[0,1]$,
$0\leq r(1-r)\leq 1/4$, with maximum reached for $r=1/2$, and $1\leq
1+\sqrt{r}\leq 2$.

Hence, using also (5.9), we get the estimates

$\displaystyle\frac{h_{11}^{n_{1}}h_{22}^{n_{2}}}{\sqrt{\pi\min(n_{1},n_{2})}}\leq\mathcal{T}^{s}_{n_{1},n_{2};n_{1},n_{2}}(h)\leq\frac{(n_{1}+n_{2})!}{n_{1}!n_{2}!}h_{11}^{n_{1}}h_{22}^{n_{2}}\leq(\mathrm{tr}\,h)^{n_{1}+n_{2}},$

the lower bound being asymptotic at large $n_{1}$, $n_{2}$, whereas the upper
bound is valid for any $n_{1}$, $n_{2}$.

### Another exploration: behavior at large $\boldsymbol{n_{1}}$,
$\boldsymbol{n_{2}}$, with fixed $\boldsymbol{\nu=n_{2}/n_{1}}$

Without loss of generality we suppose $\nu>1$. In order to analyze this case,
it is better to start from the simplest expression (5.8)

$\displaystyle\mathcal{T}^{s}_{n_{1},n_{2};n_{1},n_{2}}(h)=h_{11}^{n_{1}}h_{22}^{n_{2}}\sum_{m=0}^{n_{1}}\binom{n_{1}}{m}\binom{\nu
n_{1}}{m}r^{m},\qquad r=\frac{|h_{12}|^{2}}{h_{11}h_{22}}.$ (B.3)

We know already that for $r=1$ this simplifies to

$\displaystyle\mathcal{T}^{s}_{n_{1},n_{2};n_{1},n_{2}}(h)=h_{11}^{n_{1}}h_{22}^{n_{2}}\binom{n_{1}+n_{2}}{n_{1}}=h_{11}^{n_{1}}\,h_{22}^{n_{2}}\,\frac{(n_{1}+n_{2})!}{n_{1}!n_{2}!},$
(B.4)

From the Stirling formula,

$\displaystyle n!\sim\sqrt{2\pi}e^{-n}n^{n+1/2}\qquad\mbox{for large $n$},$

we see that, at large $n_{1}$, $n_{2}$, (B.4) behaves as

$\displaystyle\mathcal{T}^{s}_{n_{1},n_{2};n_{1},n_{2}}(h)\sim\sqrt{\frac{n_{1}+n_{2}}{2\pi
n_{1}n_{2}}}\frac{(n_{1}+n_{2})^{n_{1}+n_{2}}}{n_{1}^{n_{1}}n_{2}^{n_{2}}}h_{11}^{n_{1}}h_{22}^{n_{2}},$

and if $n_{2}=\nu n_{1}$ with large $\nu$, the above expression becomes

$\displaystyle\mathcal{T}^{s}_{n_{1},n_{2};n_{1},n_{2}}(h)\sim\sqrt{\frac{n_{1}+n_{2}}{2\pi
n_{1}n_{2}}}\frac{(n_{1}+n_{2})^{n_{1}+n_{2}}}{n_{1}^{n_{1}}n_{2}^{n_{2}}}h_{11}^{n_{1}}h_{22}^{n_{2}}\sim\frac{1}{\sqrt{2\pi
n_{1}\nu}}(1+\nu)^{n_{1}+1/2}(eh_{11})^{n_{1}}h_{22}^{\nu n_{1}}.$

We now consider the general case $r<1$. From the Stirling formula we derive
the asymptotic behavior of binomial coefficient at large $n$,

$\displaystyle\binom{n}{m=n\xi}\sim\frac{1}{\sqrt{2\pi
n\xi(1-\xi)}}e^{-n(\xi\log\xi+(1-\xi)\log(1-\xi))},$

where we have introduced the “continuous” variable $\xi=m/n$, $0<\xi<1$. In
the present case, we write $\xi=m/n_{1}$ and we replace the sum
$\sum\limits_{m=0}^{n_{1}}$ in (B.3) by the integral
$\int_{0}^{1}n_{1}\mathrm{d}\xi$ (or
$\int_{\epsilon}^{1-\epsilon^{\prime}}n_{1}\mathrm{d}\xi$ if some
regularization is needed). We obtain

$\displaystyle\mathcal{T}^{s}_{n_{1},n_{2};n_{1},n_{2}}(h)\sim
h_{11}^{n_{1}}h_{22}^{n_{2}}\frac{1}{2\pi}\int_{\epsilon}^{1-\epsilon^{\prime}}\left[\xi^{2}(1-\xi)\left(1-\frac{\xi}{\nu}\right)\right]^{-1/2}e^{n_{1}A(\xi)}\mathrm{d}\xi,$

with

$\displaystyle A(\xi)=-\left[2\xi\log\xi-\xi\log\nu
r+(1-\xi)\log(1-\xi)+\nu\left(1-\frac{\xi}{\nu}\right)\log\left(1-\frac{\xi}{\nu}\right)\right].$

Next we apply the Laplace method for evaluating the above integral for large
$n_{1}$ and $\nu$, ignoring the divergence at the origin. Laplace’s
approximation formula (with suitable conditions on the functions involved)
reads

$\displaystyle\int_{a}^{b}h(x)e^{nA(x)}\mathrm{d}x\approx\sqrt{\frac{2\pi}{n|A^{\prime\prime}(x_{0})|}}h(x_{0})e^{nA(x_{0})}\qquad\mbox{as}\
n\to\infty,$

where $A^{\prime}(x_{0})=0$ for $x_{0}\in[a,b]$, $A^{\prime\prime}(x_{0})<0$
and $h$ is positive. Here, we have

$\displaystyle
A^{\prime}(\xi)=\log(1-\xi)\left(1-\frac{\xi}{\nu}\right)-\log\xi^{2}+\log\nu
r,$ $\displaystyle
A^{\prime\prime}(\xi)=-\frac{1}{1-\xi}-\frac{2}{\xi}-\frac{1}{\nu-\xi}.$

We notice that $A^{\prime}(\xi)<0$ in the integration interval. The equation
$A^{\prime}(\xi)=0$ is equivalent to

$\displaystyle\xi^{2}+\frac{r(1+\nu)}{1-r}\xi-\frac{r\nu}{1-r}=0.$

The positive root is

$\displaystyle\xi_{+}=\frac{\sqrt{r}}{2(1-r)}\left[\sqrt{r(\nu-1)^{2}+4\nu}-\sqrt{r}(1+\nu)\right].$

We easily check that $\xi_{+}=0$ at $r=0$, that $\xi_{+}\to\nu/(1+\nu)$ as
$r\to 1_{-}$, and that, at fixed $\nu\geq 1$, $\xi_{+}>0$ in the range
$0<r\leq 1$. Also, for $\nu=1$, $\xi_{+}=\sqrt{r}/(1+\sqrt{r})$. Therefore,
$\xi_{+}\in(0,1)$ for all $r\in(0,1)$ and $\nu\in[1,\infty)$. Now, we have at
$\xi=\xi_{+}$

$\displaystyle
A(\xi_{+})=-\log(1-\xi_{+})\left(1-\frac{\xi_{+}}{\nu}\right)^{\nu},$
$\displaystyle
A^{\prime\prime}(\xi_{+})=-\frac{2-\left(1+\frac{1}{\nu}\right)\xi_{+}}{\xi_{+}(1-\xi_{+})\left(1-\frac{\xi_{+}}{\nu}\right)}.$

Applying the Laplace formula yields the final result

$\displaystyle\mathcal{T}^{s}_{n_{1},n_{2};n_{1},n_{2}}(h)\sim\frac{h_{11}^{n_{1}}h_{22}^{n_{2}}}{\sqrt{2\pi
n_{1}}}\left[\xi_{+}\left(2-\left(1+\frac{1}{\nu}\right)\xi_{+}\right)\right]^{-1/2}(1-\xi_{+})^{-n_{1}}\left(1-\frac{\xi_{+}}{\nu}\right)^{-n_{2}}.$

##  Appendix C Matrix elements of $\boldsymbol{{\mathfrak{D}}(z)}$ and
$\boldsymbol{\widetilde{\mathfrak{D}}(z)}$

The calculation of the matrix elements
${\mathfrak{D}}_{mn}(z):=\langle\Psi_{m}|{\mathfrak{D}}(z)|\varphi_{n}\rangle$
of ${\mathfrak{D}}(z)$ (and consequently for $\widetilde{\mathfrak{D}}(z)$
because of the relation (6.14)) can be carried out by using the resolution of
the identity (6.18), satisfied by the bi-coherent states, their projective
covariance property (6.19) and the reproducing properties of their overlap
function

$\displaystyle\langle\Psi_{m}|{\mathfrak{D}}(z)|\varphi_{n}\rangle=\iint_{{\mathbb{C}}^{2}}\frac{d^{2}z^{\prime}}{\pi}\frac{d^{2}z^{\prime\prime}}{\pi}\langle\Psi_{m}|\varphi(z^{\prime})\rangle\langle\Psi(z^{\prime})|{\mathfrak{D}}(z)|\varphi(z^{\prime\prime})\rangle\langle\Psi(z^{\prime\prime})|\varphi_{n}\rangle$
$\displaystyle\hphantom{\langle\Psi_{m}|{\mathfrak{D}}(z)|\varphi_{n}\rangle}{}=\frac{1}{\sqrt{m!n!}}\iint_{{\mathbb{C}}^{2}}\frac{d^{2}z^{\prime}}{\pi}\frac{d^{2}z^{\prime\prime}}{\pi}e^{-\frac{|z^{\prime}|^{2}}{2}}e^{-\frac{|z^{\prime\prime}|^{2}}{2}}{z^{\prime}}^{m}{\bar{z^{\prime\prime}}}^{n}e^{i\operatorname{Im}(z\bar{z^{\prime\prime}})}\langle\Psi(z^{\prime})|\varphi(z^{\prime\prime}+z)\rangle$
$\displaystyle\hphantom{\langle\Psi_{m}|{\mathfrak{D}}(z)|\varphi_{n}\rangle}{}=\frac{e^{-\frac{|z|^{2}}{2}}}{\sqrt{m!n!}}\int_{{\mathbb{C}}}\frac{d^{2}z^{\prime\prime}}{\pi}e^{-|z^{\prime\prime}|^{2}}e^{-\bar{zz^{\prime}}}{\bar{z^{\prime\prime}}}^{n}(z^{\prime\prime}+z)^{m}.$

After binomial and exponential expansions and integration, one ends up with
the following expression

$\displaystyle\langle\Psi_{m}|{\mathfrak{D}}(z)|\varphi_{n}\rangle=\begin{cases}\displaystyle\sqrt{\frac{n!}{m!}}e^{-\frac{|z|^{2}}{2}}\alpha^{m-n}L_{n}^{(m-n)}\left(|z|^{2}\right)&\mbox{for}\
\ n\leq m,\vspace{1mm}\\\
\displaystyle\sqrt{\frac{m!}{n!}}e^{-\frac{|z|^{2}}{2}}(-\bar{z})^{n-m}L_{m}^{(n-m)}\left(|z|^{2}\right)&\mbox{for}\
\ n>m,\end{cases}$

where

$\displaystyle
L_{n}^{(\mu)}(x)=\sum_{k=0}^{n}(-1)^{k}\frac{\Gamma(n+\mu+1)}{\Gamma(\mu+k+1)(n-k)!}\frac{x^{k}}{k!},$

is a generalized Laguerre polynomial.

### Acknowledgements

The authors are indebted to referees for their relevant and constructive
comments and suggestions. They acknowledge financial support from the
Università di Palermo. S.T.A. acknowledges a grant from the Natural Sciences
and Engineering Research Council (NSERC) of Canada, F.B. acknowledges support
from GNFM, J.P.G. thanks the CBPF and the CNPq for financial support and CBPF
for hospitality.

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