Integral Representation and Asymptotic Expansion for Hypergeometric Coherent States

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.

In particular, if the representationB = (B 0 , B 1 , then the basis {χ j }, j = 0, . . . , n − |m| − 1 has the form Here, L M N (y) are Laguerre polynomials [3] and the normalization constants have the form Note that, in the implementation specified above, the algebra F quant consists of operators on R 3 commuting with S 0 and M 3 . The coherent transformation H : P [m, n] → L[m, n] is given by the formula 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 Φ(z) ∈ 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 parameterh → 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 H z at z = 1: Here, 2 F 2 (α 1 , α 2 ; β 1 , β 2 ; y) is a generalized hypergeometric function [4], and the equation |ω| = ρ, where ω ∈ C, 0 < ρ < 1, is a constant defining a circle oriented counterclockwise. For z = 1, hypergeometric coherent states are expressed in terms of Gegenbauer polynomials C λ n (z) [3]: 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 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 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 − (R 3 ) and the operators contained in the quantum averages can be represented as functions of the generators of the algebra 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 that was constructed; this function is a solution of the equation 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.

Asymptotic Expansion of Hypergeometric Coherent States Near z = 1
As is known [4], the function u = 2 F 2 (−N, 1; |m| + 1, 2|m| + 1; z) satisfies the differential equation Here, 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, where |m| 1, and the constant a satisfies the inequalities In addition, in Equation (10), we make the change r = |m| −3/2 z. As a result, we obtain the equation It follows from (21), (22) that, for |z| |m| 2 , the function u has the asymptotics Here, |m| 1 and Let us construct an asymptotic solution of Equation (26) in the form of WKB-approximation u = (y 0 (r) + y 1 (r) |m| We substitute (29) in Equation (26) and equate the summands at the equal powers of |m| to zero. We obtain −r 2 Therefore, The functions y 0 , y 1 are determined from the equations that, with (30) taken into account, are written as We have Here, c (0) , c (1) are constants. As a result, expansion (29) becomes The constants c (0) , c (1) are determined from the matching condition for expansions (27), (31) for |r| of the order of |m| 3/4 . We obtain from which, it follows that Thus, we have proved the following lemma.

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 Φ(z) and (|z| 2 ) were constructed near the point z = z = 1. Together with expansion (34), they allow one to obtain the asymptotics of the coherent transformation H(Φ) defined by relation (7), and to use it further to calculate norms and quantum averages in problems related to the hydrogen atom.

Funding:
The results were obtained in the framework of the state assignments of the Russian Ministry of Education and Science (project FSWF-2020-0022).
Informed Consent Statement: Informed consent was obtained from all subjects involved in the study.
Data Availability Statement: Not applicable.