Integral Representation and Asymptotic Expansion for Hypergeometric Coherent States

Abstract

An integral representation is found for hypergeometric coherent states. It contains a generalized hypergeometric function. An asymptotic expansion of hypergeometric coherent states near $z=1$ is constructed. This expansion is used to find asymptotic eigenfunctions of the Hamiltonian of the hydrogen atom in a magnetic field near the lower boundaries of spectral clusters.

1. Introduction: Hypergeometric Coherent States

Hypergeometric coherent states were introduced in [1,2]. Let us recall their definition. Let an arbitrary representation of quadratic commutation relations be given in some Hilbert space L:

$$\begin{array}{c}[{\mathbf{B}}_{\mathbf{1}},{\mathbf{B}}_{\mathbf{2}}]=i\hslash {\mathbf{B}}_{\mathbf{0}}{\mathbf{B}}_{\mathbf{3}},[{\mathbf{B}}_{\mathbf{0}},{\mathbf{B}}_{\mathbf{1}}]=2i\hslash {\mathbf{B}}_{\mathbf{2}},\\ [{\mathbf{B}}_{\mathbf{2}},{\mathbf{B}}_{\mathbf{3}}]=-\frac{i\hslash }{2}({\mathbf{B}}_{\mathbf{0}}{\mathbf{B}}_{\mathbf{1}}+{\mathbf{B}}_{\mathbf{1}}{\mathbf{B}}_{\mathbf{0}}),[{\mathbf{B}}_{\mathbf{0}},{\mathbf{B}}_{\mathbf{2}}]=-2i\hslash {\mathbf{B}}_{\mathbf{1}},\\ [{\mathbf{B}}_{\mathbf{3}},{\mathbf{B}}_{\mathbf{1}}]=-\frac{i\hslash }{2}({\mathbf{B}}_{\mathbf{0}}{\mathbf{B}}_{\mathbf{2}}+{\mathbf{B}}_{\mathbf{2}}{\mathbf{B}}_{\mathbf{0}}),[{\mathbf{B}}_{\mathbf{0}},{\mathbf{B}}_{\mathbf{3}}]=0.\end{array}$$

(1)

The generators of this representation are denoted by ${\widehat{B}}_j$, $j=0,1,2,3$. Define the operators

$${\widehat{B}}_{\pm }={\widehat{B}}_2\mp i{\widehat{B}}_1.$$

Let m, n be integer numbers, $n>\left|m\right|\ge 0$, and ${\chi }_0\in L$ be a normalized “vacuum” vector subjected to the equations

$${\widehat{B}}_{-}{\chi }_0=0,{\widehat{B}}_0{\chi }_0=\hslash \left(\right|m|+1-n){\chi }_0,2{\widehat{B}}_3{\chi }_0={\hslash }^2(n-1)\left(\right|m|+1){\chi }_0.$$

We define hypergeometric coherent states as

$${\mathfrak{H}}_z=I_m\left(\frac{2}{\hslash }{\left(z{\widehat{B}}_{+}\right)}^{1/2}\right){\chi }_0,$$

where $z\in \mathbb{C}$ and the function

$$I_m\left(r\right)=\frac{\left|m\right|!2^{\left|m\right|-1}}{\pi r^{\left|m\right|}}{\int }_0^{2\pi }{e}^{rcos\phi +im\phi }d\phi$$

differs from the standard Bessel function of an imaginary argument [3] by the additional multiplier $r^{-\left|m\right|}$ and the normalization, so that $I_m\left(r\right)\to 1$ as $r\to 0$. Then,

$${\mathfrak{H}}_z=\sum _{j=0}^{n-\left|m\right|-1}\overline{P_j\left(\overline{z}\right)}{\chi }_j.$$

(2)

Here, $\left\{P_j\right\}$, $j=0,\dots ,n-\left|m\right|-1$ is an orthonormal basis in $\mathcal{P}[m,n]$

$$P_j\left(\overline{z}\right)=\sqrt{k_j}{\overline{z}}^j,$$

(3)

where the numbers

$$k_j=\frac{{(n-j)}_j{(n-|m|-j)}_j}{j!{(1+|m\left|\right)}_j},$$

(4)

and

$${\left(\alpha \right)}_j\equiv \alpha (\alpha +1)\dots (\alpha +j-1),j=1,2,\dots ,{\left(\alpha \right)}_0\equiv 1$$

is the Pochhammer symbol. The space $\mathcal{P}[m,n]$ is the space of polynomials over $\mathbb{C}$ of a degree at most of $n-\left|m\right|-1$ endowed with the norm

$$\mid \mid \mathsf{\Phi }\mid {\mid }_{m,n}={\left(\int \mid \mathsf{\Phi }\left(\overline{z}\right){\mid }^{2}d{\mu }_{m,n}\left(z\right)\right)}^{1/2},$$

where

$$d{\mu }_{m,n}\left(z\right)={\varrho \left(\right|z|}^2)d\overline{z}dz.$$

The function $\varrho \left(r\right)$ has the form

$$\varrho \left(r\right)=\frac{n(n-|m\left|\right){\left(\right|m|+1)}_n}{2\pi {(n+1)}_{n+1}}{r}^{\left|m\right|}F(n+1,n+|m|+1;2n+2;1-r),$$

where F is a hypergeometric series [4]

$$F(a,b;c;y)=\sum _{k=0}^{\infty }\frac{{\left(a\right)}_k{\left(b\right)}_k}{{\left(c\right)}_{k}k!}{y}^k.$$

Finally, the vectors $\left\{{\chi }_j\right\}$, $j=0,\dots ,n-\left|m\right|-1$ form an orthonormal basis in some subspace $L[m,n]\subset L$ on which an irreducible representation of the Karasev–Novikova algebra ${\mathcal{F}}_{quant}$ (1) is realized.

In particular, if the representation $\widehat{B}=({\mathbf{B}}_{\mathbf{0}},{\mathbf{B}}_{\mathbf{1}},{\mathbf{B}}_{\mathbf{2}},{\mathbf{B}}_{\mathbf{3}})$ is implemented in the Hilbert space $L=L_{-}^2\left({\mathbb{R}}^3\right)$ [2], where $L_{-}^2\left({\mathbb{R}}^3\right)$ is a space with inner product

$${\langle \phi ,{\phi }^{\prime }\rangle }_{-}=\frac{\pi }{4}{\int }_{{\mathbb{R}}^3}\frac{\phi \left(q\right){\overline{\phi }}^{\prime }\left(q\right)}{\left|q\right|}dq$$

and $L[m,n]\subset L_{-}^2\left({\mathbb{R}}^3\right)$ is a joint eigensubspace of the operators

$$S_0=\left|q\right|\left(\frac{1}{4}+{(-i\hslash \frac{\partial }{\partial q})}^2\right),M_3=q_1(-i\hslash \frac{\partial }{\partial q_2})-q_2(-i\hslash \frac{\partial }{\partial q_1}),$$

where

$$q=(q_1,q_2,q_3),-i\hslash \frac{\partial }{\partial q}=(-i\hslash \frac{\partial }{\partial q_1},-i\hslash \frac{\partial }{\partial q_2},-i\hslash \frac{\partial }{\partial q_3}),$$

then the basis $\left\{{\chi }_j\right\}$, $j=0,\dots ,n-\left|m\right|-1$ has the form

$$\begin{array}{c}{\chi }_j\left(q\right)=c_j{(q_1+i\mathrm{sgn}\left(m\right)q_2)}^{\left|m\right|}\\ \times exp\left(-\frac{\left|q\right|}{2\hslash }\right)L_j^{\left|m\right|}\left(\frac{\left|q\right|+q_3}{2\hslash }\right)L_{n-\left|m\right|-1-j}^{\left|m\right|}\left(\frac{\left|q\right|-q_3}{2\hslash }\right).\end{array}$$

(5)

Here, $L_N^M\left(y\right)$ are Laguerre polynomials [3] and the normalization constants have the form

$$c_j={(-1)}^j{\left(2^{\left|m\right|}\pi {\hslash }^{\left|m\right|+1}\sqrt{{(n-|m|-j)}_{\left|m\right|}{(1+j)}_{\left|m\right|}}\right)}^{-1},j=0,\dots ,n-\left|m\right|-1.$$

(6)

Note that, in the implementation specified above, the algebra ${\mathcal{F}}_{quant}$ consists of operators on ${\mathbb{R}}^3$ commuting with $S_0$ and $M_3$. The coherent transformation $\mathfrak{H}:\mathcal{P}[m,n]\to L[m,n]$ is given by the formula

$$\mathfrak{H}\left(\mathsf{\Phi }\right)=\int \mathsf{\Phi }\left(\overline{z}\right){\mathfrak{H}}_{z}d{\mu }_{m,n}\left(z\right),\mathsf{\Phi }\in \mathcal{P}[m,n].$$

(7)

Hypergeometric coherent states play an important role in quantum mechanics [2,5] and quantum optics [6,7]. For example, in [2], using coherent transformation (7), the global formulas were constructed for the asymptotic eigenfunctions of the Hamiltonian of the hydrogen atom in a homogeneous magnetic field in which the polynomials $\mathsf{\Phi }\left(\overline{z}\right)\in \mathcal{P}[m,n]$ are solutions of the spectral problem for the Heun equation. Coherent transformation (7) turns out to be very convenient from the point of view of the semiclassical approximation with respect to the parameter $\hslash \to 0$. For more information on the theory of coherent transformations, see [8,9]. The papers [10,11,12,13,14,15] are devoted to generalized hypergeometric coherent states, as well as nonlinear f-coherent states.

In this paper, an integral representation is found for ${\mathfrak{H}}_z$ at $z\ne 1$:

$$\begin{array}{cc}\hfill {\mathfrak{H}}_z& =\frac{\sqrt{(n-1)!}{(1-z)}^{n-\left|m\right|-1}}{\sqrt{(n-|m|-1)!}{\left(\right|m|!)}^{3/2}{2}^{\left|m\right|}\pi {\hslash }^{\left|m\right|+1}}{(q_1+i\mathrm{sgn}\left(m\right)q_2)}^{\left|m\right|}exp\left(-\frac{\left|q\right|}{2\hslash }\right)\hfill \\ & \times \frac{1}{2\pi i}{\int }_{\left|\omega \right|=\rho }{\omega }^{-1}(1-2\left[\frac{1-z}{1+z}-\frac{q_3}{\left|q\right|}\right]\omega /\left[1-\frac{(1-z)q_3}{(1+z)\left|q\right|}\right]+{\omega }^2{)}^{-\left|m\right|-1/2}\hfill \\ & {\times }_2{F}_2\left(-n+\left|m\right|+1,1;\left|m\right|+1,2\left|m\right|+1;\frac{(z+1)\left|q\right|}{(1-z)2\hslash \omega }\left[1+\frac{(z-1)q_3}{(z+1)\left|q\right|}\right]\right)d\omega .\end{array}$$

(8)

Here, ${}_2{F}_2({\alpha }_1,{\alpha }_2;{\beta }_1,{\beta }_2;y)$ is a generalized hypergeometric function [4], and the equation $\left|\omega \right|=\rho$, where $\omega \in \mathbb{C}$, $0<\rho <1$, is a constant defining a circle oriented counterclockwise. For $z=1$, hypergeometric coherent states are expressed in terms of Gegenbauer polynomials $C_n^{\lambda }\left(z\right)$ [3]:

$$\begin{array}{c}{\mathfrak{H}}_1=\frac{\sqrt{(n-|m|-1)!}{\left(2\right|m\left|\right)!\left|q\right|}^{n-\left|m\right|-1}}{\sqrt{\left|m\right|!(n-1)!}(n+|m|-1)!2^{\left|m\right|}\pi {\hslash }^n}{(q_1+i\mathrm{sgn}\left(m\right)q_2)}^{\left|m\right|}\\ \times exp\left(-\frac{\left|q\right|}{2\hslash }\right)C_{n-\left|m\right|-1}^{\left|m\right|+1/2}\left(\frac{q_3}{\left|q\right|}\right).\end{array}$$

(9)

In the papers [5,16,17], a general method was proposed for finding the asymptotics of the spectrum and asymptotic eigenfunctions near the boundaries of spectral clusters, which are formed near the eigenvalues of the unperturbed operator in the case of frequency resonance. It is based on a new integral representation. Using this method, the asymptotic behavior of the spectrum of the hydrogen atom in a magnetic field near the lower boundaries of spectral clusters was found in [5]. Coherent transformation (7) was used to construct the corresponding asymptotic eigenfunctions. At the same time, the main contribution to the norm of the asymptotic solution, as well as to the asymptotics of quantum averages, is made by a small neighborhood of the point $\overline{z}=z=1$, which plays the key role in constructing the asymptotics near the lower boundaries of spectral clusters (in the case of upper boundaries, such a point is $\overline{z}=z=-1$).

However, the methods developed in [5] use the unitarity of coherent transformation (7) and can only be applied when the norm and quantum averages are calculated in the space $L_{-}^2\left({\mathbb{R}}^3\right)$ and the operators contained in the quantum averages can be represented as functions of the generators of the algebra ${\mathcal{F}}_{quant}$. These requirements are not met in a number of problems. This is the case, for example, when studying the spectrum of the hydrogen atom in a self-consistent field near the boundaries’ spectral clusters. Therefore, the problem is to find the asymptotics of hypergeometric coherent states in a neighborhood of the point $z=1$. This asymptotics is obtained in this paper (see (34)). It is derived using a semiclassical approximation of the function

$${}_2{F}_2(-n+|m|+1,1;|m|+1,2|m|+1;z)$$

that was constructed; this function is a solution of the equation

$$\begin{array}{c}{z}^2\frac{d^{3}u}{d{z}^3}+[-z^2+3\left(\right|m|+1\left)z\right]\frac{d^{2}u}{d{z}^2}{+\left[\right(n-\left|m\right|-3)z+2|m|}^2+3\left|m\right|+1]\frac{du}{dz}\\ +(n-|m|-1)u=0.\end{array}$$

(10)

Asymptotic expansion (34), together with the expansions found in [5], make it possible to approximately calculate the norms and quantum averages of the hydrogen atom in a self-consistent field near the lower boundaries of spectral clusters. In addition, we note that (34) can be used in a number of other problems related to the hydrogen atom.

2. Integral Representation of Hypergeometric Coherent States

Let

$$N=n-\left|m\right|-1.$$

(11)

From the Formulas (2)–(6), it follows that

$$\begin{array}{c}{\mathfrak{H}}_z=\frac{\sqrt{N!\left|m\right|!(N+|m\left|\right)!}}{2^{\left|m\right|}\pi {\hslash }^{\left|m\right|+1}}{(q_1+i\mathrm{sgn}\left(m\right)q_2)}^{\left|m\right|}\\ \times exp\left(-\frac{\left|q\right|}{2\hslash }\right){\mathsf{\Phi }}_{N,\left|m\right|}\left(z,\frac{\left|q\right|+q_3}{2\hslash },\frac{\left|q\right|-q_3}{2\hslash }\right).\end{array}$$

(12)

Here,

$${\mathsf{\Phi }}_{N,\left|m\right|}(z,x,y)=\sum _{j=0}^N\frac{{(-z)}^j{L}_j^{\left|m\right|}\left(x\right)L_{N-j}^{\left|m\right|}\left(y\right)}{\left(\right|m|+j)!(N+|m|-j)!},$$

(13)

$$x=\frac{\left|q\right|+q_3}{2\hslash },y=\frac{\left|q\right|-q_3}{2\hslash }.$$

(14)

Further, we use the relation [18]

$$\sum _{k+\ell =n}\frac{{(-1)}^k{L}_k^{\alpha }\left(x\right)L_{\ell }^{\beta }\left(y\right)}{{(\alpha +1)}_k{(\beta +1)}_k}=\frac{{(-1)}^n{(x+y)}^n}{{(\alpha +1)}_n{(\beta +1)}_n}{P}_n^{(\alpha ,\beta )}\left(\frac{y-x}{y+x}\right),$$

where $P_n^{(\alpha ,\beta )}\left(z\right)$ are Jacobi polynomials [3], from which, using (13), we obtain

$${\mathsf{\Phi }}_{N,\left|m\right|}(1,x,y)=\frac{{(-1)}^N{(x+y)}^N}{\left(\right(\left|m\right|+{N)!)}^2}{P}_N^{\left(\right|m|,|m\left|\right)}\left(\frac{y-x}{y+x}\right).$$

(15)

For $N=0,1,2,\dots$, we define the function

$$\begin{array}{c}{F}_{N,\left|m\right|}(z,x,y)=\frac{{(-1)}^N{(zx+y)}^N}{\left(\right(\left|m\right|+{N)!)}^2}{P}_N^{\left(\right|m|,|m\left|\right)}\left(\frac{y-zx}{y+zx}\right)\\ =\frac{{(-1)}^N{(zx+y)}^N\left(2\right|m\left|\right)!}{\left(\right|m|+N)!\left|m\right|!\left(2\right|m|+N)!}{C}_N^{\left|m\right|+1/2}\left(\frac{y-zx}{y+zx}\right).\end{array}$$

(16)

Then, taking into account (13), (15), and the relation [3]

$$L_j^{\left|m\right|}\left(zx\right)=\sum _{\ell =0}^j{c}_{\left|m\right|+j}^{\ell }{z}^{j-\ell }{(1-z)}^{\ell }{L}_{j-\ell }^{\left|m\right|}\left(x\right),$$

where $c_k^{\ell }$ are binomial coefficients, we have

$$\begin{array}{c}{F}_{N,\left|m\right|}(z,x,y)=\sum _{j=0}^N\frac{{(-1)}^j}{(N+|m|-j)!}\sum _{\ell =0}^j\frac{z^{j-\ell }{(1-z)}^{\ell }{L}_{j-\ell }^{\left|m\right|}\left(x\right)L_{N-j}^{\left|m\right|}\left(y\right)}{\ell !\left(\right|m|+j-\ell )!}\\ =\sum _{\ell =0}^N\frac{{(1-z)}^{\ell }}{\ell !}\sum _{j=\ell }^N\frac{{(-1)}^j{z}^{j-\ell }{L}_{j-\ell }^{\left|m\right|}\left(x\right)L_{N-j}^{\left|m\right|}\left(y\right)}{(N+|m|-j)!\left(\right|m|+j-\ell )!}=\sum _{\ell =0}^N\frac{{(z-1)}^{\ell }}{\ell !}{\mathsf{\Phi }}_{N-\ell ,\left|m\right|}(z,x,y).\end{array}$$

(17)

From (17), we express the function ${\mathsf{\Phi }}_{N,\left|m\right|}(z,x,y)$ in terms of Gegenbauer polynomials.

Lemma 1.

The following relation holds:

$$\begin{array}{c}{\mathsf{\Phi }}_{N,\left|m\right|}(z,x,y)=\frac{{(-1)}^N\left(2\right|m\left|\right)!{(zx+y)}^N}{\left|m\right|!}\sum _{\ell =0}^N\frac{1}{\ell !(N+|m|-\ell )!(N+2|m|-\ell )!}\\ \times {\left(\frac{z-1}{zx+y}\right)}^{\ell }{C}_{N-\ell }^{\left|m\right|+1/2}\left(\frac{y-zx}{y+zx}\right).\end{array}$$

(18)

Here, $N=0,1,2,\dots$.

Proof. 

We will use the method of mathematical induction. For $N=0$, from (17), we obtain

$$F_{0,\left|m\right|}(z,x,y)={\mathsf{\Phi }}_{0,\left|m\right|}(z,x,y).$$

Suppose that, for all $0\le k<N$, we have

$${\mathsf{\Phi }}_{k,\left|m\right|}(z,x,y)=\sum _{s=0}^k\frac{{(1-z)}^s}{s!}{F}_{k-s,\left|m\right|}(z,x,y).$$

Then, by (17),

$${\mathsf{\Phi }}_{N,\left|m\right|}(z,x,y)=F_{N,\left|m\right|}(z,x,y)-\sum _{\ell =1}^N\frac{{(z-1)}^{\ell }}{\ell !}\sum _{s=0}^{N-\ell }\frac{{(1-z)}^s}{s!}{F}_{N-\ell -s,\left|m\right|}(z,x,y)$$

$$=F_{N,\left|m\right|}(z,x,y)+\sum _{k=1}^N\sum _{\ell =1}^k\frac{{(-1)}^{\ell +1}{(1-z)}^k}{\ell !(\ell -k)!}{F}_{N-k,\left|m\right|}(z,x,y)$$

$$=\sum _{k=0}^N\frac{{(1-z)}^k}{k!}{F}_{N-k,\left|m\right|}(z,x,y).$$

Finally, taking (16) into account, we obtain (18). The lemma is proved.

Note that Formulas (11), (12), (14), and (18) imply relation (9).

Let

$$\xi =\frac{z-1}{zx+y},\eta =\frac{y-zx}{y+zx}.$$

(19)

Since the following integral representation holds for Gegenbauer polynomials [3]:

$$C_n^{\lambda }\left(\eta \right)=\frac{1}{2\pi i}{\int }_{\left|\omega \right|=\rho }\frac{d\omega }{{\omega }^{n+1}{(1-2\eta \omega +{\omega }^2)}^{\lambda }},\omega \in \mathbb{C},$$

where $\left|\omega \right|=\rho$, $0<\rho <1$ is a circle oriented counterclockwise, it follows that the function ${\mathsf{\Phi }}_{N,\left|m\right|}(z,x,y)$ can be represented as

$${\mathsf{\Phi }}_{N,\left|m\right|}(z,x,y)=\frac{{(-1)}^N\left(2\right|m\left|\right)!{(zx+y)}^N}{\left|m\right|!2\pi i}{\int }_{\left|\omega \right|=\rho }\frac{f_{N,\left|m\right|}\left(\xi \omega \right)d\omega }{{\omega }^{N+1}{(1-2\eta \omega +{\omega }^2)}^{\left|m\right|+1/2}}.$$

(20)

Here, the polynomial

$$f_{N,\left|m\right|}\left(p\right)=\sum _{\ell =0}^N\frac{p^{\ell }}{\ell !(N+|m|-\ell )!(N+2|m|-\ell )!}$$

(21)

is expressed in terms of a generalized hypergeometric series

$${}_2{F}_2({\alpha }_1,{\alpha }_2;{\beta }_1,{\beta }_2;y)=\sum _{n=0}^{\infty }\frac{{\left({\alpha }_1\right)}_n{\left({\alpha }_2\right)}_n{y}^n}{{\left({\beta }_1\right)}_n{\left({\beta }_2\right)}_{n}n!}.$$

For $p\ne 0$, the following relation holds:

$$f_{N,\left|m\right|}\left(p\right)=\frac{p^N}{N!\left(2\right|m\left|\right)!\left|m\right|!}{}_2{F}_2(-N,1;|m|+1,2|m|+1;-\frac{1}{p}).$$

(22)

Therefore, by (14), (19), and (20) for $z\ne 1$, we have the integral representation

$$\begin{array}{c}{\mathsf{\Phi }}_{N,\left|m\right|}\left(z,\frac{\left|q\right|+q_3}{2\hslash },\frac{\left|q\right|-q_3}{2\hslash }\right)=\frac{{(1-z)}^N}{N!{\left(\right|m|!)}^{2}2\pi i}\\ \times {\int }_{\left|\omega \right|=\rho }{\omega }^{-1}(1-2\left[\frac{1-z}{1+z}-\frac{q_3}{\left|q\right|}\right]\omega /\left[1-\frac{(1-z)q_3}{(1+z)\left|q\right|}\right]+{\omega }^2{)}^{-\left|m\right|-1/2}\\ {\times }_2{F}_2\left(-N,1;\left|m\right|+1,2\left|m\right|+1;\frac{(z+1)\left|q\right|}{(1-z)2\hslash \omega }\left[1+\frac{(z-1)q_3}{(z+1)\left|q\right|}\right]\right)d\omega .\end{array}$$

(23)

The following assertion follows from (11), (12), and (2). □

Theorem 1.

Hypergeometric coherent states at $z\ne 1$ can be represented as(1).

3. Asymptotic Expansion of Hypergeometric Coherent States near $\mathbf{z}=\mathbf{1}$

As is known [4], the function $u={}_2{F}_2(-N,1;|m|+1,2|m|+1;z)$ satisfies the differential equation

$$\left\{\delta \right(\delta +\left|m\right|\left)\right(\delta +2\left|m\right|)-z(\delta -N\left)\right(\delta +1\left)\right\}u=0.$$

Here,

$$\delta =z\frac{d}{dz}.$$

Taking (11) into account, one can transform this equation into the form (10).

Following [5], we consider the numbers m and n, which correspond to the lower boundaries of spectral clusters. In this case,

$$n=\sqrt{a}\left|m\right|,$$

(24)

where $\left|m\right|\gg 1$, and the constant a satisfies the inequalities

$$1<a<5.$$

(25)

In addition, in Equation (10), we make the change $r={\left|m\right|}^{-3/2}z$. As a result, we obtain the equation

$$\begin{array}{c}\frac{r^2}{{\left|m\right|}^{3/2}}\frac{d^{3}u}{d{r}^3}+\left[-r^2+\frac{3r}{\sqrt{\left|m\right|}}+\frac{3r}{{\left|m\right|}^{3/2}}\right]\frac{d^{2}u}{d{r}^2}\\ +\left[\left(\sqrt{a-1}\right)\left|m\right|r+2\sqrt{\left|m\right|}-3r+\frac{3}{\sqrt{\left|m\right|}}+\frac{1}{{\left|m\right|}^{3/2}}\right]\frac{du}{dr}+[\left(\sqrt{a-1}\right)\left|m\right|-1]u=0.\end{array}$$

(26)

It follows from (21), (22) that, for $\left|z\right|\gg {\left|m\right|}^2$, the function u has the asymptotics

$$\begin{array}{c}u=\frac{{N!\left|m\right|!\left(2\right|m\left|\right)!(-z)}^N}{(N+|m\left|\right)!(N+2|m\left|\right)!}\{1-\frac{(N+|m\left|\right)(N+2|m\left|\right)}{z}\\ +\frac{{(N+|m|-1)}_2{(N+2|m|-1)}_2}{2z^2}-\frac{{(N+|m|-2)}_3{(N+2|m|-2)}_3}{6z^3}+O\left(\frac{{\left|m\right|}^8}{z^4}\right)\}.\end{array}$$

(27)

Here, $\left|m\right|\gg 1$ and

$$N=(\sqrt{a}-1)\left|m\right|-1.$$

(28)

Let us construct an asymptotic solution of Equation (26) in the form of WKB-approximation

$$u=(y_0\left(r\right)+\frac{y_1\left(r\right)}{\sqrt{\left|m\right|}}+O\left(\frac{1}{\left|m\right|r}\right)+O\left(\frac{1}{\left|m\right|r^5}\right))e^{\left|m\right|S_0\left(r\right)+\sqrt{\left|m\right|}{S}_1\left(r\right)},\left|m\right|\to \infty .$$

(29)

We substitute (29) in Equation (26) and equate the summands at the equal powers of $\left|m\right|$ to zero. We obtain

$$-r^2{\left(S_0^{\prime }\right)}^2+r(\sqrt{a}-1)S_0^{\prime }=0,$$

$$-2r^2{S}_0^{\prime }{S}_1^{\prime }+r(\sqrt{a}-1)S_1^{\prime }+r^2{\left(S_0^{\prime }\right)}^3+3r{\left(S_0^{\prime }\right)}^2+2S_0^{\prime }=0.$$

Therefore,

$$S_0=(\sqrt{a}-1)lnr,S_1=-\frac{\sqrt{a}(\sqrt{a}+1)}{r}.$$

(30)

The functions $y_0$, $y_1$ are determined from the equations that, with (30) taken into account, are written as

$$-r(\sqrt{a}-1)y_0^{\prime }+\left[\frac{\sqrt{a}(a-1)(2\sqrt{a}+1)}{r^2}-\sqrt{a}+1\right]y_0=0,$$

$$-r(\sqrt{a}-1)y_1^{\prime }+\left[\frac{\sqrt{a}(a-1)(2\sqrt{a}+1)}{r^2}-\sqrt{a}+1\right]y_1+(a-2\sqrt{a}-1)y_0^{\prime }$$

$$+\left[\frac{3a^{3/2}{(\sqrt{a}+1)}^2}{r^3}-\frac{\sqrt{a}(\sqrt{a}+1)}{r}\right]y_0=0.$$

We have

$$y_0=\frac{c^{\left(0\right)}}{r}exp\left\{-\frac{\sqrt{a}(\sqrt{a}+1)(2\sqrt{a}+1)}{2r^2}\right\},$$

$$y_1=\frac{c^{\left(0\right)}}{r}\left[-\frac{\sqrt{a}(\sqrt{a}+1)(5a+5\sqrt{a}+1)}{3r^3}+\frac{2\sqrt{a}+1}{r}+c^{\left(1\right)}\right]$$

$$\times exp\left\{-\frac{\sqrt{a}(\sqrt{a}+1)(2\sqrt{a}+1)}{2r^2}\right\}.$$

Here, $c^{\left(0\right)}$, $c^{\left(1\right)}$ are constants. As a result, expansion (29) becomes

$$\begin{array}{c}u=c^{\left(0\right)}{r}^{(\sqrt{a}-1)\left|m\right|-1}\{1+\frac{2\sqrt{a}+1}{\sqrt{\left|m\right|}r}-\frac{\sqrt{a}(\sqrt{a}+1)(5a+5\sqrt{a}+1)}{3\sqrt{\left|m\right|}{r}^3}+\frac{c^{\left(1\right)}}{\sqrt{\left|m\right|}}\\ +O\left(\frac{1}{\left|m\right|}\right)+O\left(\frac{1}{\left|m\right|r^4}\right)\}exp\left\{-\frac{\sqrt{a}(\sqrt{a}+1)\sqrt{\left|m\right|}}{r}-\frac{\sqrt{a}(\sqrt{a}+1)(2\sqrt{a}+1)}{2r^2}\right\}.\end{array}$$

(31)

The constants $c^{\left(0\right)},c^{\left(1\right)}$ are determined from the matching condition for expansions (27), (31) for $\left|r\right|$ of the order of ${\left|m\right|}^{3/4}$. We obtain

$$c^{\left(0\right)}{r}^{(\sqrt{a}-1)\left|m\right|-1}\{1+\frac{c^{\left(1\right)}}{\sqrt{\left|m\right|}}-\frac{\sqrt{a}(\sqrt{a}+1)\sqrt{\left|m\right|}}{r}+\frac{a{(\sqrt{a}+1)}^2\left|m\right|}{2r^2}$$

$$-\frac{a^{3/2}{(\sqrt{a}+1)}^3{\left|m\right|}^{3/2}}{6r^3}+O\left(\frac{1}{\left|m\right|}\right)\}=\frac{N!\left|m\right|!\left(2\right|m\left|\right)!(-{\left|m\right|}^{3/2}{)}^N{r}^{(\sqrt{a}-1)\left|m\right|-1}}{(N+|m\left|\right)!(N+2|m\left|\right)!}$$

$$\times \left\{1-\frac{\sqrt{a}(\sqrt{a}+1)\sqrt{\left|m\right|}}{r}+\frac{a{(\sqrt{a}+1)}^2\left|m\right|}{2r^2}-\frac{a^{3/2}{(\sqrt{a}+1)}^3{\left|m\right|}^{3/2}}{6r^3}+O\left(\frac{1}{\left|m\right|}\right)\right\},$$

from which, it follows that

$$c^{\left(0\right)}=\frac{N!\left|m\right|!\left(2\right|m\left|\right)!(-{\left|m\right|}^{3/2}{)}^N}{(N+|m\left|\right)!(N+2|m\left|\right)!},c^{\left(1\right)}=0.$$

Thus, we have proved the following lemma.

Lemma 2.

Let

$$\zeta =\frac{(z+1)\left|q\right|}{(z-1)2\hslash \omega }\left(1+\frac{(z-1)q_3}{(z+1)\left|q\right|}\right).$$

Then, for $\left|\zeta \right|\gg \left|m\right|$, $\left|m\right|\gg 1$, the asymptotic expansion

$$\begin{array}{c}{}_2{F}_2\left(-N,1;\left|m\right|+1,2\left|m\right|+1;\frac{(z+1)\left|q\right|}{(1-z)2\hslash \omega }\left(1+\frac{(z-1)q_3}{(z+1)\left|q\right|}\right)\right)\\ =\frac{N!\left|m\right|!\left(2\right|m\left|\right)!}{(N+|m\left|\right)!(N+2|m\left|\right)!}{\zeta }^{N}exp\left\{\frac{\sqrt{a}(\sqrt{a}+1){\left|m\right|}^2}{\zeta }-\frac{\sqrt{a}(\sqrt{a}+1)(2\sqrt{a}+1){\left|m\right|}^3}{2{\zeta }^2}\right\}\\ \times \left\{1-\frac{(2\sqrt{a}+1)\left|m\right|}{\zeta }+\frac{\sqrt{a}(\sqrt{a}+1)(5a+5\sqrt{a}+1){\left|m\right|}^2}{3{\zeta }^3}+O\left(\frac{1}{\left|m\right|}\right)+O\left(\frac{{\left|m\right|}^5}{{\zeta }^4}\right)\right\}\end{array}$$

(32)

holds. Here, N has the form (28).

We put $\hslash =1/n$ as in [2,5]. In addition, we define

$$\begin{array}{c}Y=\left(\frac{1-z}{1+z}-\frac{q_3}{\left|q\right|}\right)/\left(1-\frac{(1-z)q_3}{(1+z)\left|q\right|}\right),\\ X=1+2\frac{q_3}{\left|q\right|}\omega +{\omega }^2.\end{array}$$

(33)

Further, we substitute (2) in integral representation (2) and use the Taylor formula to expand the obtained functions in power series in $z-1$. Since, for $|z-1|\ll {\left|m\right|}^{-1/3}$, $\left|m\right|\gg 1$, we have the relations

$${\left(\frac{z+1}{2}\right)}^N=exp\left\{\frac{(\sqrt{a}-1)(z-1)\left|m\right|}{2}-\frac{(\sqrt{a}-1){(z-1)}^2\left|m\right|}{8}\right\}\{1-\frac{z-1}{2}$$

$$+\frac{(\sqrt{a}-1){(z-1)}^3\left|m\right|}{24}{+O((z-1)}^4\left|m\right|)+O\left({(z-1)}^2\right)\},$$

$${\left(1+\frac{(z-1)q_3}{(z+1)\left|q\right|}\right)}^N=exp\left\{(\sqrt{a}-1)\left[\frac{(z-1)q_3}{{2\left|q\right|}^2}-{(z-1)}^2\left(\frac{q_3}{4\left|q\right|}+\frac{q_3^2}{{8\left|q\right|}^2}\right)\right]\left|m\right|\right\}$$

$$\times \{1-\frac{(z-1)q_3}{2\left|q\right|}+(\sqrt{a}-1){(z-1)}^3\left[\frac{q_3}{8\left|q\right|}+\frac{q_3^2}{{8\left|q\right|}^2}+\frac{q_3^3}{{24\left|q\right|}^3}\right]\left|m\right|$$

$${+O((z-1)}^4\left|m\right|)+O\left({(z-1)}^2\right)\},$$

$$Y=-\{\frac{q_3}{\left|q\right|}+\frac{z-1}{z+1}\left[1-\frac{q_3^2}{{\left|q\right|}^2}\right]+{\left(\frac{z-1}{z+1}\right)}^2\left[-\frac{q_3}{\left|q\right|}+\frac{q_3^3}{{\left|q\right|}^3}\right]$$

$$+{\left(\frac{z-1}{z+1}\right)}^3\left[\frac{q_3^2}{{\left|q\right|}^2}-\frac{q_3^4}{{\left|q\right|}^4}\right]+O\left({(z-1)}^4\right)\},$$

$${\{1-2Y\omega +{\omega }^2\}}^{-\left|m\right|-1/2}=X^{-\left|m\right|-1/2}exp\{-(z-1)\left(1-\frac{q_3^2}{{\left|q\right|}^2}\right)\frac{\omega \left|m\right|}{X}$$

$$+\frac{{(z-1)}^2}{2}\left(1-\frac{q_3^2}{{\left|q\right|}^2}\right)\left(1+\frac{q_3}{\left|q\right|}\right)\frac{\omega }{X^2}[1+\left(1+\frac{q_3}{\left|q\right|}\right)\omega +{\omega }^2]\left|m\right|\left\}\right\{1$$

$$-\frac{z-1}{2}\left(1-\frac{q_3^2}{{\left|q\right|}^2}\right)\frac{\omega }{X}-\frac{{(z-1)}^3}{12}\left(1-\frac{q_3^2}{{\left|q\right|}^2}\right){\left(1+\frac{q_3}{\left|q\right|}\right)}^2\frac{\omega }{X^3}[3+\left(6+\frac{6q_3}{\left|q\right|}\right)\omega$$

$$+\left(2+\frac{20q_3}{\left|q\right|}-\frac{4q_3^2}{{\left|q\right|}^2}\right){\omega }^2+\left(6+\frac{6q_3}{\left|q\right|}\right){\omega }^3+3{\omega }^4\left]\right|m|$$

$${+O((z-1)}^4\left|m\right|)+O\left({(z-1)}^2\right)\},$$

we obtain the following assertion from (12).

Theorem 2.

For $|z-1|\ll {\left|m\right|}^{-1/3}$, $\left|m\right|\gg 1$, the following expansion holds for hypergeometric coherent states:

$$\begin{array}{c}{\mathfrak{H}}_z=\frac{\sqrt{\left((\sqrt{a}-1)\right|m|-1)!}\left(2\right|m\left|\right)!(-{\left|q\right|)}^{(\sqrt{a}-1)\left|m\right|-1}}{\sqrt{\left|m\right|!\left(\sqrt{a}\right|m|-1)!}((\sqrt{a}+1)\left|m\right|-1)!2^{\left|m\right|}\pi {\hslash }^{\sqrt{a}\left|m\right|}}\\ \times {(q_1+i\mathrm{sgn}\left(m\right)q_2)}^{\left|m\right|}exp\{-\frac{\left|q\right|}{2\hslash }+\frac{(\sqrt{a}-1)(z-1)}{2}\left(1+\frac{q_3}{\left|q\right|}\right)\left|m\right|\\ -\frac{(\sqrt{a}-1){(z-1)}^2}{8}{\left(1+\frac{q_3}{\left|q\right|}\right)}^2\left|m\right|\left\}\right\{1-\frac{z-1}{2}\left(1+\frac{q_3}{\left|q\right|}\right)\\ +\frac{(\sqrt{a}-1){(z-1)}^3}{24}{\left(1+\frac{q_3}{\left|q\right|}\right)}^3{\left|m\right|+O((z-1)}^4\left|m\right|)+O\left({(z-1)}^2\right)\}\frac{1}{2\pi i}\\ \times {\int }_{\left|\omega \right|=\rho }{\omega }^{-(\sqrt{a}-1)\left|m\right|}{X}^{-\left|m\right|-1/2}exp\{(z-1)\omega \left[-\left(1-\frac{q_3^2}{{\left|q\right|}^2}\right)\frac{1}{X}+\frac{\sqrt{a}+1}{\left|q\right|}\right]\left|m\right|\\ +{(z-1)}^2\omega [\frac{1}{2X^2}\left(1-\frac{q_3^2}{{\left|q\right|}^2}\right)\left(1+\frac{q_3}{\left|q\right|}\right)(1+\left(1+\frac{q_3}{\left|q\right|}\right)\omega +{\omega }^2)\\ -\frac{\sqrt{a}+1}{2\left|q\right|}\left(1+\frac{q_3}{\left|q\right|}+\frac{(2\sqrt{a}+1)\omega }{\sqrt{a}\left|q\right|}\right)\left]\left|m\right|\right\}\{1-(z-1)\omega [\frac{1}{2X}\left(1-\frac{q_3^2}{{\left|q\right|}^2}\right)\\ +\frac{2\sqrt{a}+1}{\sqrt{a}\left|q\right|}]+{(z-1)}^3\omega [\frac{\sqrt{a}+1}{\left|q\right|}[\frac{(5a+5\sqrt{a}+1){\omega }^2}{{3a\left|q\right|}^2}-\frac{2\sqrt{a}+1}{2\sqrt{a}}\left(1+\frac{q_3}{\left|q\right|}\right)\frac{\omega }{\left|q\right|}\\ +\frac{1}{4}\left(1+\frac{q_3}{\left|q\right|}+\frac{q_3^2}{{\left|q\right|}^2}\right)]-\frac{1}{12X^3}\left(1-\frac{q_3^2}{{\left|q\right|}^2}\right){\left(1+\frac{q_3}{\left|q\right|}\right)}^2[3+\left(6+\frac{6q_3}{\left|q\right|}\right)\omega \\ +\left(2+\frac{20q_3}{\left|q\right|}-\frac{4q_3^2}{{\left|q\right|}^2}\right){\omega }^2+\left(6+\frac{6q_3}{\left|q\right|}\right){\omega }^3+3{\omega }^4\left]\right]\left|m\right|\\ {+O((z-1)}^4\left|m\right|)+O\left({(z-1)}^2\right)\}d\omega .\end{array}$$

(34)

Here, X is defined by Formula (33), and the number a satisfies conditions (24), (25).

4. Conclusions

In this paper, an integral representation and an asymptotic expansion are found for hypergeometric coherent states in a neighborhood of the point $z=1$. The methods applied in this case are of general nature. They can be used not only to study hypergeometric coherent states but also to study other coherent states from [9].

In [5], in the case of lower boundaries of spectral clusters, asymptotic expansions of $\mathsf{\Phi }\left(\overline{z}\right)$ and $\varrho \left(\right|z{|}^2)$ were constructed near the point $z=\overline{z}=1$. Together with expansion (34), they allow one to obtain the asymptotics of the coherent transformation $\mathfrak{H}\left(\mathsf{\Phi }\right)$ defined by relation (7), and to use it further to calculate norms and quantum averages in problems related to the hydrogen atom.