Nonlinear Coherent States Associated with a Measure on the Positive Real Half Line

Abstract

We construct a class of generalized nonlinear coherent states by means of a newly obtained class of 2D complex orthogonal polynomials. The associated Bargmann-type transform is discussed. A polynomials realization of the basis of the quantum states Hilbert space is also obtained. Here, the entire structure owes its existence to a certain measure on the positive real half line, of finite total mass, together with all its moments. We illustrate this method with the measure $r^\beta e^{-r}dr$, where $\beta$ is a non-negative constant, which leads to a new generalization of the true-polyanalytic Bargmann transform.

Appendices

Appendix A. Proof of Proposition 3.2.1

Assuming the finiteness of the sum

$$\begin{aligned} {\mathcal {N}}_{\beta ,m}(z{\overline{z}} )=\sum _{n=0}^{\infty } \frac{\vert P_{n,m}(z,{\overline{z}};\beta )\vert ^2}{x_{n}^{\beta ,m}!} <+\infty , \end{aligned}$$

(A.1)

means that

$$\begin{aligned} \sum _{n=0}^{m-1 } \frac{\vert P_{n,m}(z,{\overline{z}};\beta )\vert ^2}{x_{n}^{\beta ,m}!}+\sum _{n=m}^{\infty } \frac{\vert P_{n,m}(z,{\overline{z}};\beta )\vert ^2}{x_{n}^{\beta ,m}!} <+\infty \end{aligned}$$

(A.2)

this implies the finiteness of the infinite sum in (A.2) which may be written (up to a multiplicative constant) as

$$\begin{aligned}&\sum _{n=m}^{\infty } \frac{|\phi _{m}(z{\overline{z}};n-m+\beta )|^2}{\zeta _{n\wedge m}\left( \left| n-m\right| +\beta \right) }(z{\overline{z}})^{n-m}\\&\quad = \sum _{n=m}^{\infty } \left( \sum _{i,j=0}^{m}\frac{c_i(m;n-m+\beta )c_j(m;n-m+\beta )}{\zeta _{n\wedge m}\left( \left| n-m\right| +\beta \right) }(z{\overline{z}})^{2m-i-j} \right) (z{\overline{z}})^{n-m}\\&\quad = \sum _{n=0}^{\infty } \left( \sum _{i,j=0}^{m}\frac{c_i(m;n+\beta )c_j(m;n+\beta )}{\zeta _{m}( n +\beta )}(z{\overline{z}})^{2m-i-j} \right) (z{\overline{z}})^{n}\\&\quad = \sum _{i,j=0}^{m} (z{\overline{z}})^{2m-i-j} \left( \sum _{n=0}^{\infty } \frac{c_i(m;n+\beta )c_j(m;n+\beta )}{\zeta _{m}( n +\beta )} (z{\overline{z}})^{n}\right) . \end{aligned}$$

The radius of convergence of the series

$$\begin{aligned} \sum _{n=0}^{\infty } \frac{c_i(m;n+\beta )c_j(m;n+\beta )}{\zeta _{m}( n +\beta )} (z{\overline{z}})^{n} \end{aligned}$$

(A.3)

can be found by applying the ratio test and it is given by (3.13). $\square$

Appendix B. Proof of Proposition 4.1.1

We start by writing the sum

$$\begin{aligned} S = \sum \limits _{j=0}^{+\infty }\frac{\Gamma (\beta +1) \left( j\wedge m\right) !}{\Gamma \left( \beta +j\vee m+1\right) } H_{j,m}^{\left( \beta \right) }\left( z,{\overline{z}}\right) \overline{H_{j,m}^{\left( \beta \right) }\left( w,{\overline{w}}\right) } \end{aligned}$$

as

$$\begin{aligned} S= & {} \sum \limits _{j=0}^{m-1 }\frac{ j !}{ \left( \beta + 1\right) _m }H_{j,m}^{\left( \beta \right) }\left( z,{\overline{z}}\right) \overline{H_{j,m}^{\left( \beta \right) }\left( w,{\overline{w}}\right) } \\&+\sum \limits _{j=m}^{+\infty }\frac{ m !}{\left( \beta +1\right) _j }H_{j,m}^{\left( \beta \right) }\left( z,{\overline{z}}\right) \overline{H_{j,m}^{\left( \beta \right) }\left( w,{\overline{w}}\right) }\\= & {} \sum \limits _{j=0}^{m-1 }\frac{ j !({\overline{z}}w)^{m-j}}{ \left( \beta + 1\right) _m } L_{j}^{\left( \beta +m-j\right) }\left( z{\overline{z}}\right) L_{j}^{\left( \beta +m-j\right) }\left( w{\overline{w}}\right) \\&+\sum \limits _{j=m}^{+\infty }\frac{m !}{ \left( \beta +1\right) _j}H_{j,m}^{\left( \beta \right) }\left( z,{\overline{z}}\right) \overline{H_{j,m}^{\left( \beta \right) }\left( w,{\overline{w}}\right) }. \end{aligned}$$

We denote the infinite sum in the last equation by

$$\begin{aligned} S_{(\infty )}=\sum \limits _{j=m}^{+\infty }\frac{m !}{ \left( \beta +1\right) _j }H_{j,m}^{\left( \beta \right) }\left( z,{\overline{z}}\right) \overline{H_{j,m}^{\left( \beta \right) }\left( w,{\overline{w}}\right) }. \end{aligned}$$

(B.1)

By replacing $H_{j,m}^{\left( \beta \right) }\left( z,{\overline{z}}\right)$ by their expressions in (4.1) and making some manipulations, we obtain

$$\begin{aligned} S_{(\infty )}= & {} \frac{1}{m!}\sum \limits _{k=0}^{m}\sum \limits _{l=0}^{m} \left( \begin{array}{c} m \\ k \end{array} \right) \left( \begin{array}{c} m \\ l \end{array} \right) (-1)^{k+l} \nonumber \\&\times \sum \limits _{j=0}^{+\infty }(\beta +1)_{j+m}\frac{z^{j+k}{\overline{z}}^{k}{\overline{w}}^{j+l}w^{l}}{(\beta +1)_{j+k}(\beta +1)_{j+l}}. \end{aligned}$$

(B.2)

Using the identities

$$\begin{aligned} (a)_{k+m}=(a+m)_k(a)_m \quad \text {and} \quad (-m)_k=(-1)^k\frac{m!}{(m-k)!}=(-1)^kk!\left( \begin{array}{c} m \\ k \end{array} \right) , \end{aligned}$$

we arrive at the result

$$\begin{aligned} S_{(\infty )}= \frac{(\beta +1)_m}{m!}\sum \limits _{k=0}^{m}\sum \limits _{l=0}^{m}\frac{(-m)_k(-m)_l(z{\overline{z}})^k(w{\overline{w}})^{l}}{k!l!(\beta +1)_k(\beta +1)_{l}}{}_{2}F_{2}\left( \begin{array}{c} 1,m+\beta +1 \\ k+\beta +1,l+\beta +1 \end{array} \big |z{\overline{w}}\right) .\\ \end{aligned}$$

$\square$

Appendix C. Generalized Lauricella Series

The generalized Lauricella series in several variables is defined by [39, p. 36]:

$$\begin{aligned}&F_{C:D^{(1)};\cdots ;D^{(n)}}^{A:B^{(1)};\cdots ;B^{(n)}}\left[ \begin{array}{c} {[}(a):\theta ^{(1)},\ldots ,\theta ^{(n)}]:[(b^{(1)}):\phi ^{(1)}];\cdots [(b^{(n)}):\phi ^{(n)}] \\ {[}(c):\psi ^{(1)},\ldots ,\psi ^{(n)}]:[(b^{(1)}):\delta ^{(1)}];\cdots {[}(b^{(n)}):\delta ^{(n)}] \end{array} z_1,\ldots ,z_n \right] \end{aligned}$$

(C.1)

$$\begin{aligned}&\quad :=\sum \limits _{m_1,\ldots ,m_n=0}^{+\infty }\frac{\prod \nolimits _{j=1}^{A}(a_j)_{m_1\theta _j^{(1)}+\cdots +m_n\theta _j^{(n)}}\prod \nolimits _{j=1}^{B^{(1)}}(b_j^{(1)})_{m_1\phi _1^{(1)}}\cdots \prod \nolimits _{j=1}^{B^{(n)}}(b_j^{(n)})_{m_n\phi _j^{(n)}}}{\prod \nolimits _{j=1}^{C}(c_j)_{m_1\psi _j^{(1)}+\cdots +m_n\psi _j^{(n)}}\prod \nolimits _{j=1}^{D^{(1)}}(d_j^{(1)})_{m_1\delta _1^{(1)}}\cdots \prod \nolimits _{j=1}^{D^{(n)}}(d_j^{(n)})_{m_n\delta _j^{(n)}}},\frac{z_1^{m_1}}{m_1!}...\frac{z_n^{m_n}}{m_n!}\nonumber \\ \end{aligned}$$

(C.2)

where the coefficients

$$\begin{aligned} \left\{ \begin{array}{c} \theta _j^{(k)},\; j=1,\ldots ,A,\quad \phi _j^{(k)},\; j=1,\ldots ,B^{(k)},\\ \psi _j^{(k)},\; j=1,\ldots ,C, \quad \delta _j^{(k)},\; j=1,\ldots ,D^{(k)}, \end{array} k=1,\ldots ,n \right. \end{aligned}$$

(C.3)

are real and positive, and (a) abbreviates the array of A parameters $a_1,\ldots ,a_A$; $(b^{(k)})$ abbreviates the array of $(B^{(k)})$ parameters $b_j^{(k)}, j=1,\ldots ,B^{(k)}, k=1,\ldots ,n$. Similar interpretation holds for the remaining parameters. For precise conditions under which the generalized Lauricella function converges, see [38, pp. 153–157].

Appendix D. Computation of the Integral Kernel $B_{\beta ,m}(x,z)$

The $\mu _\beta$-NLCS in (4.7) evaluating in $x\in {\mathbb {R}}$ are given by

$$\begin{aligned} \vartheta _{z}^{\beta ,m}(x) = \left( {\mathcal {N}}_{\beta ,m} (z{\overline{z}})\;\right) ^{-\frac{1}{2}}\;B_{\beta ,m}(x,z), \quad \ z\in {\mathbb {C}} \end{aligned}$$

(D.1)

where

$$\begin{aligned} B_{\beta ,m}(x,z)= \sum _{j=0}^{+\infty } \sqrt{\frac{\Gamma (\beta +1) \left( j\wedge m\right) !}{\Gamma \left( \beta +j\vee m+1\right) }} \frac{2^{j/2}}{\sqrt{(\beta +1)_j}}\overline{H_{j,m}^{\left( \beta \right) }\left( z,{\overline{z}}\right) }H_j(x,\beta ). \end{aligned}$$

(D.2)

We now express $H_{j,m}^{\left( \beta \right) }\left( z,{\overline{z}}\right)$ in terms of Laguerre polynomials as in (4.2) and we split the sum $\sum _{j=0}^{+\infty }$ in two parts $S_{m-1}=\sum _{j=0}^{m-1}$ and $S_{(\infty )}=\sum _{j=m}^{+\infty }$ as

$$\begin{aligned} B_{\beta ,m}(x,z)= & {} \sum _{j=0}^{m-1} (-1)^{j}\sqrt{\frac{j !}{\left( \beta +1\right) _m }} \frac{2^{j/2}z^{m-j}}{\sqrt{(\beta +1)_j}}L^{(m-j+\beta )}_{j}(z{\overline{z}} )H_j(x,\beta )\\&+\,(-1)^{m}\sqrt{m!}\sum _{j=m}^{+\infty } \frac{2^{j/2}\bar{z}^{j-m}}{(\beta +1)_j}L^{(j-m+\beta )}_{m}(z{\overline{z}} )H_j(x,\beta )=S_{m-1}+S_{(\infty )}. \end{aligned}$$

We write the infinite sum

$$\begin{aligned} S_{(\infty )}= & {} (-1)^m\sqrt{m!}\sum _{j=0}^{+\infty } \frac{2^{j/2}\bar{z}^{j-m}}{(\beta +1)_j}L^{(j-m+\beta )}_{m}(z{\overline{z}} )H_j(x,\beta )\\&-\, (-1)^m\sqrt{m!}\sum _{j=0}^{m-1} \frac{2^{j/2}\bar{z}^{j-m}}{(\beta +1)_j}L^{(j-m+\beta )}_{m}(z{\overline{z}} )H_j(x,\beta )=S_{(\infty )}^*-S_{m-1}^*. \end{aligned}$$

Next, we write the Laguerre polynomial ([25, p. 242]) as

$$\begin{aligned} L^{(j-m+\beta )}_{m}(z{\overline{z}} )=\frac{1}{m!}\sum _{k=0}^{m}\left( \begin{array}{c} m \\ k \end{array} \right) \frac{(\beta +1-k)_{k}(\beta +1)_{j}}{(\beta +1-k)_{j}}\left( -z\bar{z}\right) ^{m-k} \end{aligned}$$

(D.3)

to present the infinite sum as

$$\begin{aligned} S_{(\infty )}^* =\frac{z^m}{\sqrt{m!}}\sum _{k=0}^{m}\left( \begin{array}{c} m \\ k \end{array} \right) \frac{(-1)^k(\beta +1-k)_{k}}{\left( z\bar{z}\right) ^{k}}\sum _{j=0}^{+\infty } \frac{ (\bar{z}/\sqrt{2})^{j}}{(\beta +1-k)_{j}}H_j(x,\beta ) \end{aligned}$$

(D.4)

To obtain a closed form for $S_{(\infty )}^*$ we make use of the following lemma.

Lemma D.1

A generating function for the associated Hermite polynomials is given by

$$\begin{aligned} \sum _{n=0}^\infty \frac{t^n}{(c)_n}H_n(x,\beta )=F_{1:0;0;0}^{1:0;0;1}\left( \begin{array}{c} [1:1,2,1]:-;-;[\beta :1] \\ \left[ c:1,2,2\right] :-;-;- \end{array} 2xt, -t^2, -2t^2\right) \nonumber \\ \end{aligned}$$

(D.5)

in terms of the generalized Lauricella function. In particular, for $c=1$ and $\beta =0$, we recover

$$\begin{aligned} \sum _{n=0}^\infty \frac{t^n}{n!}H_n(x)=e^{2xt-t^2}, \end{aligned}$$

(D.6)

the generating function of Hermite polynomials.

We apply (D.5) for parameters $t=\frac{{\overline{z}}}{\sqrt{2}}$ and $c=\beta -k+1$ to get

$$\begin{aligned}&\sum _{n=0}^\infty \frac{\left( \frac{{\overline{z}}}{\sqrt{2}}\right) ^n}{(\beta -k+1)_n}H_n(x,\beta )\nonumber \\&\quad =F_{1:0;0;0}^{1:0;0;1}\left( \begin{array}{c} {[}1:1,2,1]:-;-;[\beta :1] \\ \left[ \beta -k+1:1,2,2\right] :-;-;- \end{array} \sqrt{2}x{\overline{z}}, -\frac{{\overline{z}}^2}{2}, -{\overline{z}}^2\right) . \end{aligned}$$

(D.7)

Summarizing the above calculations by writing $B_{\beta ,m}(z,x)=S_{m-1}-S_{m-1}^*+S_{\infty }^{*}$, we obtain the announced expression (4.16). This ends the proof of Eq. (4.16). $\square$

Proof of Lemma D.1

We first recall the associated Hermite polynomials

$$\begin{aligned} H_n(x,\beta )=\sum \limits _{k=0}^{\lfloor n/2\rfloor }\frac{(-1)^k n!}{k! (n-2k)!}\left( \sum \limits _{j=0}^{k}\frac{(-k)_j (\beta )_j}{(-n)_j}\frac{2^j}{j!}\right) (2x)^{n-2k} \end{aligned}$$

(D.8)

which also are connected to the polynomials $He_{n}^{\beta }(x)$ in [44, p. 203] through the relation $H_n(x,\beta )=2^{\frac{1}{2}n}He_{n}^{\beta }(\sqrt{2}x)$. Similar manipulations as in [21, pp. 548–549], yield

$$\begin{aligned} \sum _{n=0}^\infty \frac{t^n}{(c)_n}H_n(x,\beta )= & {} \sum _{n=0}^\infty \sum \limits _{k=0}^{\lfloor n/2\rfloor }\left( \sum \limits _{j=0}^{k}\frac{(-k)_j (\beta )_j}{(-n)_j}\frac{2^j}{j!}\right) \frac{(-1)^k n!}{k! (n-2k)!}(2x)^{n-2k}\frac{t^n}{(c)_n}\nonumber \\= & {} \sum _{n,k=0}^\infty \sum \limits _{j=0}^{k}\frac{(-k)_j (\beta )_j}{(-n-2k)_j}\frac{2^j}{j!}\frac{(-1)^k (n+2k)!}{k! n!}(2x)^{n}\frac{t^{n+2k}}{(c)_{n+2k}}\nonumber \\= & {} \sum _{n,k,j=0}^\infty \frac{(n+2k+2j)!(-k-j)_j(1)_k(\beta )_j}{(k+j)!(-n-2k-2j)_j(c)_{n+2k+2j}}\frac{(2xt)^n}{n!}\nonumber \\&\times \frac{(-t^2)^k}{k!}\frac{(-2t^2)^j}{j!}. \end{aligned}$$

(D.9)

These calculations can be done by applying successively the identities

$$\begin{aligned} \sum _{n=0}^\infty \sum \limits _{k=0}^{\lfloor n/2\rfloor }A(k,n)=\sum _{n=0}^\infty \sum \limits _{k=0}^{\infty }A(k,n+2k), \end{aligned}$$

(D.10)

and

$$\begin{aligned} \sum _{k=0}^\infty \sum \limits _{j=0}^{k}A(j,k)=\sum _{k=0}^\infty \sum \limits _{j=0}^{\infty }A(j,k+j), \end{aligned}$$

(D.11)

see [40, pp. 100–101]. Now, by observing that

$$\begin{aligned} \frac{(n+2k+2j)!(-k-j)_j(1)_k}{(k+j)!(-n-2k-2j)_j}=(1)_{n+2k+j} \end{aligned}$$

(D.12)

Equation (D.9) takes the form

$$\begin{aligned} \sum _{n=0}^\infty \frac{t^n}{(c)_n}H_n(x,\beta )=\sum _{n,k,j=0}^\infty \frac{(1)_{n+2k+j}(\beta )_j}{(c)_{n+2k+2j}}\frac{(2xt)^n}{n!}\frac{(-t^2)^k}{k!}\frac{(-2t^2)^j}{j!}. \end{aligned}$$

(D.13)

By recognizing the generalized Lauricella function ([39, p. 36])

$$\begin{aligned} F_{1:0;0;0}^{1:0;0;1}\left( \begin{array}{c} [1:1,2,1]:-;-;[\beta :1] \\ \left[ c:1,2,2\right] :-;-;- \end{array} 2xt, -t^2, -2t^2\right) \end{aligned}$$

(D.14)

in the R.H.S of (D.13), we complete the proof of Eq. (D.5). $\square$

Appendix E. Proof of Proposition 4.2.1

We set $d\eta _{\beta }(z)=h(z\bar{z})d\nu (z),$ where again h is a density function to be found and $d\nu (z)$ is the Lebesgue measure on ${\mathbb {C}}$. In terms of polar coordinates $z=\rho e^{i\theta } ,\ \rho >0$ and $\theta \in [0,2\pi )$, this measure reads $d\eta _{\beta }(z)=\pi ^{-1}h(\rho ^{2})\rho d\rho d\theta$. Using the expression (4.27) of coherent states, the operator ${\mathcal {O}}_{\beta }=\int _{{\mathbb {C}}}T_{z}^{\beta } \ {\mathcal {N}}_{\beta }(z\bar{z}) d\eta _{\beta }(z)$ reads successively,

$$\begin{aligned} {\mathcal {O}}_{\beta }= & {} \sum \limits _{n,j=0}^{+\infty } \left( \int _0^{+\infty } \frac{\rho ^{n+j}h(\rho ^2)\rho d\rho }{\sqrt{x_n^\beta !}\sqrt{x_j^\beta !}}\left( \int _0^{\pi } e^{i(n-j)\theta }\frac{d\theta }{2\pi } \right) \right) T_{j,n} \end{aligned}$$

(E.1)

$$\begin{aligned}= & {} \sum \limits _{n=0}^{+\infty }\left( \frac{2}{(\beta +1)_n} \int _0^{+\infty }\rho ^{2n}h(\rho ^2) \rho d\rho \right) T_{n} \end{aligned}$$

(E.2)

$$\begin{aligned}= & {} \sum \limits _{n=0}^{+\infty }\left( \frac{1}{(\beta +1)_n} \int _0^{+\infty }r^{n}h(r)dr \right) T_{n}, \end{aligned}$$

(E.3)

where $T_{j,n}$ is defined in (2.13). Now, the function h should satisfy

$$\begin{aligned} \int _0^{+\infty }r^{n}h(r)dr=\frac{\Gamma (n+\beta +1)}{\Gamma (\beta +1)}. \end{aligned}$$

(E.4)

From the Euler gamma function

$$\begin{aligned} \int _0^{+\infty }t^{s}e^{-t}dt=\Gamma (s+1), \quad Re(s)>-1, \end{aligned}$$

(E.5)

we see that the choice of $h(r)=(\Gamma (\beta +1))^{-1} r^{\beta }e^{-r}$ provides the solution of (E.4) that makes ${\mathcal {O}}_{\beta }=\mathbf{1 }_{{\mathcal {H}}}$. $\square$