Coherent states on the unit ball of C n and asymptotic expansion of their associated covariant symbol

Abstract

Starting from a complete family for the unit sphere $S^{n}$ in the complex n-space $C^{n}$ (whose elements are coherent states attached to the Barut–Girardello space), we obtain an asymptotic expansion for the associated Berezin transform. The proof involves the computation of the asymptotic behaviour of functions in the complete family. Furthermore, in an analogous manner (slightly weaker) we obtain the asymptotic expansion of the covariant symbol of a pseudo-differential operator on $L^{2} (S^{n})$ .

Keywords

Holomorphic space Berezin transform Bessel function asymptotic approximation semiclassical pseudo-differential operator coherent states

1. Introduction and summary

Let $B_{C^{n}}$ be the Bargmann spaces of all entire functions on $C^{n}$ square integrable with respect to the Gaussian measure $d v_{n}^{ℏ} (z) = {(π ℏ)}^{- n} e^{- | z |^{2} / ℏ} d z d \overline{z}$ , $ℏ > 0$ , with $z = (z_{1}, \dots, z_{n})$ , $| z |^{2} = | z_{1} |^{2} + \dots + | z_{n} |^{2}$ and $d z d \overline{z}$ Lebesgue measure on $C^{n}$ .

It is know that the Bargmann space $B_{C^{n}}$ enjoys the property of having a reproducing kernel $k_{w}^{ℏ} (z) = e^{z \cdot w / ℏ}$ , where $z \cdot w = z_{1} {\overline{w}}_{1} + \dots + z_{n} {\overline{w}}_{n}$ denotes the usual inner product in $C^{n}$ , and for all $f \in B_{C^{n}}$ the following equation holds: $\begin{matrix} f (z) = \int_{C^{n}} f (w) k_{w}^{ℏ} (z) d v_{n}^{ℏ} (w) = ⟨ f, k_{z}^{ℏ} ⟩ . \end{matrix}$

Recall that for $f \in L^{\infty} (C^{n})$ , the Berezin transform $B^{ℏ} (f)$ of f is the function on $C^{n}$ defined by $\begin{matrix} B^{ℏ} (f) (z) = \frac{⟨ f k_{z}^{ℏ}, k_{z}^{ℏ} ⟩}{⟨ k_{z}^{ℏ}, k_{z}^{ℏ} ⟩} = {(k_{z} (z))}^{- 1} \int_{C^{n}} f (w) {| k_{z} (w) |}^{2} d v_{n}^{ℏ} (w) . \end{matrix}$ Explicitly, $B^{ℏ} (f) (z) = {(π ℏ)}^{- n} \int f (w) e^{- | z - w |^{2} / ℏ} d w d \overline{w}$ which is just the standard formula for the solution at time $t = ℏ / 4$ of the heat equation on $C^{n} = R^{2 n}$ with initial data f. It follows that for $f \in L^{\infty} (C^{n})$ a smooth function in a neighbourhood of z, its Berezin transform has the following asymptotic expansion when ℏ goes to zero $\begin{matrix} B^{ℏ} (f) (z) = f (z) + ℏ \frac{1}{1! 4} \partial_{w \overline{w}} + ℏ^{2} \frac{1}{2! 4^{2}} \partial_{w \overline{w}}^{2} + \dots, \end{matrix}$ with $\partial_{w \overline{w}} = \sum_{j = 1}^{n} \partial^{2} / \partial w_{j} \partial {\overline{w}}_{j}$ denoting the Laplace operator.

It turns out that this kind of situation prevails in much greater generality. Namely, consider a domain $Ω \subset C^{n}$ equipped with a Kähler form ω, i.e. there must exist a strictly plurisubharmonic real-valued smooth function Φ such that $ω = \partial \overline{\partial} Φ$ . For any $ℏ > 0$ , we take the weighted Bergman spaces $H L^{2} (Ω, e^{- Φ / ℏ} d μ)$ of all holomorphic functions in $L^{2} (Ω, e^{- Φ / ℏ} d μ)$ , where $d μ (z) = \det ([g_{i \bar{j}}]) d z d \overline{z}$ with $g_{i \bar{j}} = \partial^{2} Φ / \partial z_{i} \partial {\overline{z}}_{j}$ . These spaces enjoy the property of having a reproducing kernel $K^{ℏ} (z, w)$ ; and one may define Berezin transform $B^{ℏ}$ in the same way as before.

Now, assume that Ω is a bounded symmetric domain and $e^{Φ}$ is the Bergman kernel of Ω; or that Ω is smoothly bounded and strictly pseudoconvex, and $e^{- Φ}$ is a defining function for $Ω^{2}$ ; or that $Ω = C^{n}$ and $Φ (z) = | z |^{2}$ . Then as $ℏ ↘ 0$ , there are asymptotic expansions [2–4,7] $\begin{array}{l} (1) & K^{ℏ} (z, z) = e^{Φ (z) / ℏ} ℏ^{- n} \sum_{ℓ = 0}^{\infty} ℏ^{ℓ} b_{ℓ} (z), \\ (2) & B^{ℏ} (f) = \sum_{ℓ = 0}^{\infty} ℏ^{ℓ} Q_{ℓ} f, \end{array}$ for some functions $b_{ℓ} \in C^{\infty} (Ω)$ , with $b_{0} = 1$ ; some differential operators $Q_{ℓ}$ , with $Q_{0}$ the identity operator and $Q_{1}$ the Laplace–Beltrami operator with respect to the metric $g_{i \bar{j}}$ .

On the other hand, in Ref. [2] Berezin defined the covariant symbol of a bounded linear operator A on a Hilbert space H endowed with the inner product $(\cdot, \cdot)$ as the complex-valued function $B (A)$ on a set M by $\begin{matrix} B (A) (α) = \frac{(A K (\cdot, α), K (\cdot, α))}{(K (\cdot, α), K (\cdot, α))}, α \in M, \end{matrix}$ where the family ${K (\cdot, α) \in H | α \in M}$ forms a complete system for H. The covariant symbol have an application to quantization on Kähler manifolds. This is the so-called Berezin quantization which in turn is equivalent to the asymptotic expansion of the Berezin transform (2) [14]. For introductory papers on Berezin quantization we refer to [8].

Under the above definition of covariant symbol, it not only makes sense to consider the complete family ${K (\cdot, α)}$ as the reproducing kernel with one freezing variable, as in the weighted Bergman spaces, but also any other complete family can be so considered. Hence, it is of interest to investigate if the spaces that have a complete family satisfy properties similar to (1) and (2).

Coherent states are not only mathematical tools which provide a close connection between classical and quantum formalism, but they are also a specific complete set of vectors satisfying a certain resolution of the identity condition. Formally it is possible to construct a set of coherent states in any Hilbert space and with them the associated covariant symbol.

One such Hilbert space, namely, is the Hardy space $O$ on the complex unit sphere $S^{n}$ , consisting of all functions in $L^{2} (S^{n})$ whose Poisson extension into the interior of $S^{n}$ is holomorphic, where $S^{n} = {x \in C^{n} | | x_{1} |^{2} + \dots + | x_{n} |^{2} = 1}$ and $L^{2} (S^{n})$ denotes the Hilbert space of square integrable functions with respect to the normalized surface measure $d S^{n} (x)$ on $S^{n}$ and endowed with the usual inner product $\begin{matrix} (3) & {⟨ ϕ, ψ ⟩}_{S^{n}} = \int_{S^{n}} ϕ (x) \overline{ψ (x)} d S^{n} (x), ϕ, ψ \in L^{2} (S^{n}) . \end{matrix}$ Starting from the Barut–Girardello space ([1,10]) we build a complete family $K_{p} = {K_{n, p}^{ℏ} (\cdot, z) | z \in C^{n}}$ for $O$ which are defined in terms of a suitable power series of the inner product $x \cdot z / ℏ$ which is an infinite series that is not in a closed form like an exponential function.

The aim of the present paper is to show an analogue of the asymptotic expansions (1) and (2) for the functions $K_{n, p}^{ℏ} (\cdot, z)$ , $z \in C^{n}$ , and the associated Berezin transform, respectively.

The paper is organized as follows. In Section 2 we review briefly the formalism of coherent states, describe the Barut–Girardello space as well as some of its properties. From this we construct the complete family $K_{p}$ and the associated covariant symbol.

An analogue of formula (1) for the asymptotic behaviour as $h ↘ 0$ of a function in $K_{p}$ is established in Section 3. This asymptotic expression is obtained following the work of Thomas and Wassell [17]. Moreover, to give rigorous proofs of subsequent theorems, we estimate the derivative of any order of the series defining $K_{n, p}^{ℏ} (x, z)$ .

Using the asymptotic expansion of functions in $K_{p}$ and the stationary phase method, in Section 4 we get the asymptotic behaviour of the Berezin transform.

Moreover, since the covariant symbol is not only defined for Toeplitz operators but for any bounded linear operator, it is natural to ask if there exists an analogue of the asymptotic expansion obtained in Section 4 for the covariant symbol of more general operators than Toeplitz operators. For this reason, in Section 5 we relate the principal symbol of a given semiclassical pseudo-differential operator of order zero acting on $L^{2} (S^{n})$ (see Appendix A where we give a brief description of what we mean by semiclassical pseudo-differential operators on a manifold) with its covariant symbol in the semiclassical limit $ℏ ↘ 0$ .

Throughout the paper, we will use the following basic notation. For every $z, w \in C^{k}$ , $z = (z_{1}, \dots, z_{k})$ , $w = (w_{1}, \dots, w_{k})$ , and for every multi-index $ℓ = (ℓ_{1}, \dots, ℓ_{k}) \in Z_{+}^{k}$ of length k, where $Z_{+}$ is the set of non-negative integers, let $\begin{array}{l} z \cdot w = \sum_{s = 1}^{k} z_{s} {\overline{w}}_{s}, | z | = \sqrt{z \cdot z}, | ℓ | = \sum_{s = 1}^{k} ℓ_{s}, ℓ! = \prod_{s = 1}^{k} ℓ_{s}!, z^{ℓ} = \prod_{s = 1}^{k} z_{s}^{ℓ_{s}} . \end{array}$

Given $ω \in C$ , let us denote its real and imaginary parts by $ℜ (ω)$ and $ℑ (ω)$ respectively.

Whenever convenient, we will abbreviate $\partial / \partial v_{j}$ , $\partial / \partial {\overline{v}}_{j}$ , etc., to $\partial_{v_{j}}$ , $\partial_{{\overline{v}}_{j}}$ , etc., respectively, $\partial_{v_{1}} \partial_{v_{2}} \dots \partial_{v_{k}}$ to $\partial_{v_{1} v_{2} \dots v_{k}}$ , and $\partial_{z}^{ℓ} = \partial_{z_{1}}^{ℓ_{1}} \dots \partial_{z_{k}}^{ℓ_{k}}$ with $\partial_{z_{j}}^{ℓ_{j}} = \underset{ℓ_{j}}{\underset{︸}{\partial_{z_{j}} \cdot \dots \cdot \partial_{z_{j}}}}$ .

2. The covariant symbol on $O$

In this section we introduce the covariant symbol of a bounded linear operator with domain in $O$ and some of properties it satisfies. In order to define this symbol, we build a complete family $K_{p} = {K_{n, p}^{ℏ} (\cdot, z) | z \in C^{n}}$ in $O$ whose elements are coherent states in $O$ which in turn stem from the reproducing kernel of the Barut–Girardello space.

2.1. Coherent states

Let us recall a well known general formalism about coherent states (see pp. 72-76 of Ref. [11]).

Let $(X, d μ)$ be a measure space and let $A^{2} \subset L^{2} (X, d μ)$ be a closed subspace of infinite dimension. Let ${Φ_{ℓ}}_{ℓ = 0}^{\infty}$ be an orthonormal basis of $A^{2}$ satisfying, for arbitrary $ξ \in X$ , $\sum_{ℓ = 0}^{\infty} | Φ_{ℓ} (ξ) |^{2} < \infty$ . Then, the function $\begin{matrix} T (ξ, η) : = \sum_{ℓ = 0}^{\infty} Φ_{ℓ} (ξ) \overline{Φ_{ℓ} (η)}, ξ, η \in X \end{matrix}$ defined on $X \times X$ is a reproducing kernel of the Hilbert space $A^{2}$ . Let $H$ be another (functional) Hilbert space with $dim H = dim A^{2}$ and ${ϕ_{ℓ}}_{ℓ = 0}^{\infty}$ an orthonormal basis of $H$ . The coherent states in $H$ labelled by points $ξ \in X$ are defined as $\begin{matrix} (4) & K (\cdot, ξ) : = \sum_{ℓ = 0}^{\infty} ϕ_{ℓ} (\cdot) \overline{Φ_{ℓ} (ξ)} . \end{matrix}$

By definition, the coherent state transform $U : H \to A^{2}$ defined by $\begin{matrix} U ϕ (ξ) : = {⟨ ϕ, K (\cdot, ξ) ⟩}_{H} \end{matrix}$ is an isometry. Thus, for $ϕ, ψ \in H$ , we have $\begin{matrix} {⟨ ψ, ϕ ⟩}_{H} = {⟨ U ψ, U ϕ ⟩}_{L^{2} (X, d μ)} = \int_{X} {⟨ ψ, K (\cdot, ξ) ⟩}_{H} {⟨ K (\cdot, ξ), ϕ ⟩}_{H} d μ (ξ) . \end{matrix}$ Thereby, we have a resolution of the identity of $H$ .

2.2. The Barut–Girardello space

Let us briefly describe the Barut–Girardello space $E_{n, p}$ , $n ⩾ 1$ , $p > - n$ . The spaces $E_{1, p}$ were discovered by Barut and Girardello [1], while introducing a class of coherent states associated with noncompact groups, which is why we refer to these spaces with their name (this space were also obtained by Ali and Engliš in [9] but they called it Laguerre Fock space).

Let us consider the following measure $d m_{n, p}^{ℏ}$ on $C^{n}$ , $n ⩾ 1$ , $\begin{matrix} (5) & d m_{n, p}^{ℏ} (z) = \frac{1}{Γ (n + p)} \frac{2}{{(π ℏ^{2})}^{n}} {(\frac{| z |}{ℏ})}^{p} K_{p} (2 \frac{| z |}{ℏ}) d z d \overline{z}, p > - n, \end{matrix}$ where $z = (z_{1}, \dots, z_{n}) \in C^{n}$ , $d z d \overline{z}$ stands for the Lebesgue measure on $C^{n}$ and Γ, $K_{ν}$ denote the Gamma and MacDonald–Bessel function of order ν, respectively (see Sections 8.4 and 8.5 of Ref. [12] for definition and expressions for these special functions). We remark that the measure $d m_{n, p}^{ℏ}$ is the measure considered by Barut and Girardello, $n = 1$ (see section III-4 of Ref. [1]) and by Fujii and Funahashi, $n > 1$ (see section III of Ref. [10]), but rescaled by the factor $\frac{1}{ℏ}$ .

Follows Ref. [1], let us define the Barut–Girardello space $E_{n, p}$ , $n ⩾ 1$ , $p > - n$ , as the space of all entire functions on $C^{n}$ square integrable with respect to the measure $d m_{n, p}^{ℏ}$ and endowed with the inner product $\begin{matrix} {(f, g)}_{p} = \int_{z \in C^{n}} f (z) \overline{g (z)} d m_{n, p}^{ℏ} (z), f, g \in L^{2} (C^{n}, d m_{n, p}^{ℏ}) . \end{matrix}$

The space $E_{n, p}$ enjoys the property of having a reproducing kernel $T_{n, p}$ (the inner product of two Barut–Girardello coherent states), given in turn by the following expression: $\begin{matrix} (6) & T_{n, p} (z, w) = \sum_{ℓ = 0}^{\infty} \frac{1}{{(n + p)}_{ℓ}} \frac{1}{ℓ!} {(\frac{z \cdot w}{ℏ^{2}})}^{ℓ}, z, w \in C^{n} . \end{matrix}$ For details on the latter see section IV of Ref. [1] and Section 2.2 of Ref. [5].

By using the modified Bessel functions of the first kind (see Section 5.7 of Ref. [15]) $\begin{matrix} (7) & I_{ν} (ω) = \sum_{ℓ = 0}^{\infty} \frac{{(ω / 2)}^{ν + 2 ℓ}}{Γ (ℓ + 1) Γ (ℓ + ν + 1)}, | Arg (ω) | < π, ω \neq 0, \end{matrix}$ we have, for $z, w \in C^{n}$ satisfying $z \cdot w \neq 0$ , the following expression for the reproducing kernel of $E_{n, p}$ $\begin{matrix} (8) & T_{n, p} (z, w) = Γ (n + p) {(\frac{z \cdot w}{ℏ^{2}})}^{\frac{1}{2} (- p - n + 1)} I_{n + p - 1} (\frac{2 \sqrt{z \cdot w}}{ℏ}) . \end{matrix}$

Moreover, the space $E_{n, p}$ admits an orthonormal basis (see Section 2.2 of Ref. [5]) given by the functions $\begin{matrix} (9) & Φ_{k, p} (z) = \frac{1}{ℏ^{| k |}} \sqrt{\frac{1}{{(n + p)}_{| k |} k!}} z^{k}, z \in C^{n}, k \in Z_{+}^{n}, \end{matrix}$ where ${(a)}_{ℓ}$ stands for the Pochhammer symbol (raising factorial), ${(a)}_{ℓ} = a (a + 1) \dots (a + ℓ - 1) = \frac{Γ (a + ℓ)}{Γ (a)}$ .

2.3. Coherent states in $O$ : The functions $K_{n, p}^{ℏ}$

Let us apply the formalism in Section 2.1 in order to define a set of coherent states in $O$ . Let us consider

$(X, d μ) = (C^{n}, d m_{n, p}^{ℏ})$ , with $d m_{n, p}^{ℏ}$ the measure defined in Eq. (5),

$A^{2} = E_{n, p}$ the Barut–Girardello space defined in Section 2.2,

$H = O$ the Hardy space on $S^{n}$ ,

${Φ_{k, p} | k \in Z_{+}^{n}}$ the orthonormal basis of $E_{n, p}$ defined in Eq. (9) and ${ϕ_{k} | k \in Z_{+}^{n}}$ the orthonormal basis of $O$ given by the functions $\begin{matrix} ϕ_{k} (x) = \sqrt{\frac{{(n)}_{| k |}}{k!}} x^{k}, x \in S^{n}, k \in Z_{+}^{n} . \end{matrix}$

According to Eq. (4) we may consider, for each $p > - n$ , a set of coherent states labelled by points $z \in C^{n}$ and belonging to the Hilbert space $O$ , which are defined by $\begin{matrix} (10) & K_{n, p}^{ℏ} (x, z) : = \sum_{ℓ = 0}^{\infty} \frac{c_{ℓ, p}}{ℓ!} {(\frac{x \cdot z}{ℏ})}^{ℓ}, {(c_{ℓ, p})}^{2} = \frac{{(n)}_{ℓ}}{{(n + p)}_{ℓ}}, x \in S^{n} . \end{matrix}$

The family of coherent states $K_{p} = {K_{n, p}^{ℏ} (\cdot, z) | z \in C^{n}}$ satisfies the conditions to define a Berezin symbolic calculus on $O$ , i.e. the family $K_{p}$ satisfies the following two properties (see [5] for details about rigorous proof of the following statements):

The family $K_{p}$ forms a complete system for $O$ (see proposition 4.3 of Ref. [5]): for all $Φ, Ψ \in O$ , Parseval’s identity is valid $\begin{array}{l} {⟨ Φ, Ψ ⟩}_{S^{n}} & = \int_{C^{n}} {⟨ Φ, K_{n, p}^{ℏ} (\cdot, z) ⟩}_{S^{n}} ⟨ K_{n, p}^{ℏ} (\cdot, z), Ψ) ⟩_{S^{n}} d m_{n, p}^{ℏ} (z), \end{array}$

The operator $U_{n, p} : L^{2} (S^{n}) \to E_{n, p} \subset L^{2} (C^{n}, d m_{n, p}^{ℏ})$ defined by $\begin{matrix} (11) & U_{n, p} Ψ (z) = {⟨ Ψ, K_{n, p}^{ℏ} (\cdot, z) ⟩}_{S^{n}} \end{matrix}$ is unitary (see Theorem 3.1 of Ref. [5]).

Moreover, for any $z, w \in C^{n}$ (see Theorem 4.1 of Ref. [5]) $\begin{matrix} (12) & U_{n, p} (K_{n, p}^{ℏ} (\cdot, w)) (z) = T_{n, p} (z, w) . \end{matrix}$

2.4. The associated covariant symbol to the family $K_{p}$

Since the functions in $K_{p}$ satisfy the properties (I) and (II), the Berezin’s theory allow us to consider the following

Definition 2.1.
The covariant symbol $B_{ℏ, p} (A)$ of an operator A with domain in $O$ is defined as $\begin{matrix} (13) & B_{ℏ, p} (A) (z) = \frac{{⟨ A K_{n, p}^{ℏ} (\cdot, z), K_{n, p}^{ℏ} (\cdot, z) ⟩}_{S^{n}}}{{⟨ K_{n, p}^{ℏ} (\cdot, z), K_{n, p}^{ℏ} (\cdot, z) ⟩}_{S^{n}}}, z \in C^{n} . \end{matrix}$

Note that this definition makes sense since the denominator is positive by the relation $‖ K_{n, p}^{ℏ} (\cdot, z) ‖_{S^{n}}^{2} = T_{n, p} (z, z)$ (see Eqs (11) and (12)) and Eq. (6). Moreover, since the functions in $K_{p}$ are continuous, if $A : O \to O$ is a bounded operator, its covariant symbol can be extended uniquely to a function defined on a neighbourhood of the diagonal in $C^{n} \times C^{n}$ in such a way that it is holomorphic in the first factor and anti-holomorphic in the second. In fact, such an extension is given explicitly by $\begin{matrix} (14) & B_{ℏ, p} (A) (w, z) : = \frac{{⟨ A K_{n, p}^{ℏ} (\cdot, z), K_{n, p}^{ℏ} (\cdot, w) ⟩}_{S^{n}}}{{⟨ K_{n, p}^{ℏ} (\cdot, z), K_{n, p}^{ℏ} (\cdot, w) ⟩}_{S^{n}}} . \end{matrix}$

On the other hand, let us denote by $C^{\infty} (S^{n})$ the algebra of complex-valued $C^{\infty}$ functions on $S^{n}$ . For $Φ \in C^{\infty} (S^{n})$ , the Toeplitz operator with symbol Φ is, by definition, the operator $T_{Φ} : O \to O$ given by $\begin{array}{l} T_{Φ} (ψ) : = P (Φ ψ), ψ \in O, \end{array}$ where $P : L^{2} (S^{n}) \to O$ is the orthogonal projection.

Starting from $Φ \in C^{\infty} (S^{n})$ , we can assign to it its Toeplitz operator $T_{Φ}$ and then assign to $T_{Φ}$ the covariant symbol $B_{ℏ, p} (T_{Φ})$ . It is an element of $C^{\infty} (C^{n})$ . Altogether we obtain a map $Φ \mapsto B_{p}^{ℏ} (Φ) : = B_{ℏ, p} (T_{Φ})$ .
Definition 2.2.
The map $B_{p}^{ℏ} : C^{\infty} (S^{n}) \to C^{\infty} (C^{n})$ defined for $Φ \in C^{\infty} (S^{n})$ by $\begin{matrix} (15) & B_{p}^{ℏ} (Φ) = B_{ℏ, p} (T_{Φ}) \end{matrix}$ is called Berezin transform.

We end this section by showing some properties of the extended covariant symbol and the Berezin transform that we will use to obtain the asymptotic expansion of the covariant symbol. Proposition 2.3.
Let $U \in SU (n)$ (the group of $n \times n$ unitary matrices with unit determinant) and $T_{U} : L^{2} (S^{n}) \to L^{2} (S^{n})$ be the operator defined by $T_{U} Ψ (x) = Ψ (U^{- 1} x)$ with $Ψ \in L^{2} (S^{n})$ . Let A be a bounded linear operator with domain in $O$ . Then:
For $w, z \in C^{n}$ $\begin{matrix} B_{ℏ, p} (A) (w, z) = B_{ℏ, p} (T_{U^{- 1}} A T_{U}) (U^{- 1} w, U^{- 1} z) . \end{matrix}$ In particular, $B_{ℏ, p} (A) = T_{U} B_{ℏ, p} (T_{U^{- 1}} A T_{U})$ if $z = w$ .

The Berezin transform $B_{p}^{ℏ}$ is invariant under the orthogonal transformations $T_{U}$ , i.e. $B_{p}^{ℏ} \circ T_{U} = T_{U} \circ B_{p}^{ℏ}$ .

Proof.
Since the inner product in $C^{n}$ is $SU (n)$ -invariant, we have $K_{n, p}^{ℏ} (\cdot, z) = T_{U} K_{n, p}^{ℏ} (\cdot, U^{- 1} z)$ . From the $SU (n)$ -invariance of $d S^{n}$ and definitions of the extended covariant symbol and the Berezin transform (see Eq. (14) and Definition 2.2) we conclude the proof of Proposition 2.3. □

3. Semiclassical properties of the functions in $K_{p}$

Consider the set of functions $K_{p}$ defined in Section 2.3. The main goal of this section is to show semiclassical properties of the functions in $K_{p}$ , which in turn will allow us to obtain asymptotic expansions of the Berezin transform and the covariant symbol.

3.1. Estimate of the inner product of functions in the complete family $K_{p}$

From Eqs (11), (12), (8) and the fact that the modified Bessel function $I_{ϑ}$ , $ϑ \in R$ , has the following asymptotic expression when $| ω | \to \infty$ (see formula 8.451-5 of Ref. [12]) $\begin{array}{l} (16) & I_{ϑ} (ω) & = \frac{e^{ω}}{\sqrt{2 π ω}} \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{{(2 ω)}^{k}} \frac{Γ (ϑ + k + \frac{1}{2})}{k! Γ (ϑ - k + \frac{1}{2})}, | Arg (ω) | < \frac{π}{2}, \end{array}$ we can obtain the asymptotic expansion for the inner product of two functions in $K_{p}$ :

Proposition 3.1.
Let $n ⩾ 1$ , $p > - n$ and $z, w \in C^{n}$ . Assume $z \cdot w \neq 0$ and $| Arg (z \cdot w) | < π$ , then for $ℏ ↘ 0$ $\begin{array}{rcl} (17) & {⟨ K_{n, p}^{ℏ} (\cdot, w), K_{n, p}^{ℏ} (\cdot, z) ⟩}_{S^{n}} & = & T_{n, p} (z, w) \\ = & \frac{Γ (n + p)}{2 \sqrt{π}} {(\frac{z \cdot w}{ℏ^{2}})}^{\frac{1}{2} (- p - n + \frac{1}{2})} e^{\frac{2}{ℏ} \sqrt{z \cdot w}} [1 \\ (18) & - \frac{(n + p - \frac{3}{2}) (n + p - \frac{1}{2})}{4 \sqrt{z \cdot w}} ℏ + O (ℏ^{2})] . \end{array}$ In particular, for $z \in C^{n} - {0}$ : $\begin{array}{l} (19) & {‖ K_{n, p}^{ℏ} (\cdot, z) ‖}_{S^{n}}^{2} & = \frac{Γ (n + p)}{2 \sqrt{π}} {(\frac{| z |}{ℏ})}^{- p - n + \frac{1}{2}} e^{\frac{2}{ℏ} | z |} [1 - \frac{(n + p - \frac{3}{2}) (n + p - \frac{1}{2})}{4 | z |} ℏ + O (ℏ^{2})] . \end{array}$
Remark 1.
We are mainly interested in using Proposition 3.1 for the case $z = w$ (and then $Arg (z \cdot w) = 0$ ) in this paper. The case when $| Arg (z \cdot w) | = π$ requires the use of an asymptotic expression valid in a different region than the one we are considering in Proposition 3.1. Thus if we take the branch of the square root function given by $\sqrt{z} = | z |^{1 / 2} exp (ı Arg (z) / 2)$ with $0 < Arg (z) < 2 π$ then by using formula 8.451-5 in Ref. [12] we obtain the asymptotic expression, $\begin{array}{l} {⟨ K_{n, p}^{ℏ} (\cdot, w), K_{n, p}^{ℏ} (\cdot, z) ⟩}_{S^{n}} = & \frac{Γ (n + p)}{2 \sqrt{π}} {(\frac{z \cdot w}{ℏ^{2}})}^{\frac{1}{2} (- p - n + \frac{1}{2})} \\ (20) & \times [e^{\frac{2}{ℏ} \sqrt{z \cdot w}} + e^{- \frac{2}{ℏ} \sqrt{z \cdot w}} e^{π ı (n + p - \frac{1}{2})}] [1 + O (ℏ)] . \end{array}$ Note that both asymptotic expressions in Eqs (18) and (20) coincide in the common region where they are valid.

3.2. Asymptotic of the functions in $K_{- 1}$

In order to give a rigorous proof of Theorems 4.4 and 5.1 we need to estimate the derivative of any order of the series defining $K_{n, p}^{ℏ} (x, z)$ . The definition of our functions $K_{n, p}^{ℏ} (x, z)$ in terms of an infinite series (not in a closed form like an exponential function) might look like it could be difficult to deal with them. However, from the work of Thomas and Wassell (see Appendix B of Ref. [17]) we can obtain an asymptotic expansion of $K_{n, - 1}^{ℏ} (x, z)$ , i.e. $n ⩾ 2$ and $p = - 1$ .

First, based on the definition of $K_{n, - 1}^{ℏ}$ (see Eq. (10)) let us define, for $a \in R - {0}$ , the function $g_{a} : C \to C$ by $\begin{matrix} (21) & g_{a} (z) = \sum_{ℓ = 0}^{\infty} \frac{\sqrt{a ℓ + 1}}{ℓ!} z^{ℓ} . \end{matrix}$

Note that for $z \in C^{n}$ , $K_{n, - 1}^{ℏ} (x, z) = g_{\frac{1}{n - 1}} (\frac{x \cdot z}{ℏ})$ , $x \in S^{n}$ . In this subsection we obtain asymptotics of the sth derivative of $g_{a}$ as a function of z for $ℜ (z) \to + \infty$ and $| ℑ (z) | ⩽ C ℜ (z)$ with C a positive constant, which in turn will allow us to obtain asymptotics of the sth derivative of $K_{n, - 1}^{ℏ} (\cdot, z)$ for ℏ small and x in either of the following regions on $S^{n}$ $\begin{matrix} (22) & W_{z} = {x \in S^{n} | C \frac{ℜ (x \cdot z)}{| z |} ⩾ 1} and V_{z} = S^{n} - W_{z} . \end{matrix}$

The proof of the lemma below follows the work of Thomas and Wassel (see Appendix B of Ref. [17] and Lemma 10.1 of Ref. [6] for more details).

Lemma 3.2.
Let $z \in C$ , s be any non-negative integer number and $a \in R$ with $0 < a ⩽ 2$ . The $s th$ derivative of $g_{a}$ has the following asymptotic expansion, for $ℜ (z) \to + \infty$ and $| ℑ (z) | ⩽ C ℜ (z)$ with C a positive constant $\begin{matrix} (23) & \frac{d^{s} g_{a} (z)}{d z^{s}} = \sqrt{a} z^{1 / 2} exp (z) [1 + \frac{a_{1, s}}{z} + \frac{a_{2, s}}{z^{2}} + \dots + \frac{a_{N, s}}{z^{N}} + O (z^{- (N + 1)})] \end{matrix}$ with $a_{1, s}, a_{2, s}, \dots, a_{N, s}$ some constants.
Proof.
We denote by $K^{(ℓ)} (z)$ , $ℓ = 0, 1, \dots$ , the ℓst derivative of a given function $K : C \to C$ evaluated in $z \in C$ , i.e. $K^{(ℓ)} (z) = \frac{d^{ℓ} K (z)}{d z^{ℓ}}$ .

Using the fact $\frac{1}{\sqrt{a ℓ + 1}} = \frac{1}{\sqrt{π}} \int_{- \infty}^{\infty} e^{- (a ℓ + 1) t^{2}} d t$ and the Taylor series for the exponential function we obtain $\begin{array}{l} g_{a} (z) & = \frac{1}{\sqrt{π}} \sum_{ℓ = 0}^{\infty} \frac{a ℓ + 1}{ℓ!} z^{ℓ} \int_{- \infty}^{\infty} e^{- (a ℓ + 1) t^{2}} d t \\ (24) & = \frac{2}{\sqrt{π}} \int_{0}^{\infty} [a z e^{- a t^{2}} + 1] exp (z e^{- a t^{2}}) e^{- t^{2}} d t . \end{array}$

Let us consider the change of variables $e^{- a t^{2}} = 1 - w^{2}$ with $w \in [0, 1]$ . From Eq. (24) $\begin{matrix} (25) & g_{a} (z) = \frac{2}{\sqrt{a π}} e^{z} \int_{0}^{1} [a z (1 - w^{2}) + 1] e^{- z w^{2}} m (w) d w \end{matrix}$ where $\begin{array}{l} m (w) : = \frac{w {(1 - w^{2})}^{\frac{1}{a} - 1}}{\sqrt{- ln (1 - w^{2})}} . \end{array}$

Let us write the integral in Eq. (25) as an integral over the region $0 ⩽ t < 1 / 2$ plus an integral over the region $1 / 2 ⩽ t ⩽ 1$ and let us call them $J_{a} (z)$ and $I_{a} (z)$ respectively.

Now we claim that for any $k \in Z_{+}$ , $I_{a}^{(k)} (z)$ is $O (| z |^{- \infty})$ when $ℜ (z) \to + \infty$ . In order to estimate $I_{a}^{(k)} (z)$ , first write $\begin{array}{l} m (w) = \frac{w {(1 - w^{2})}^{\frac{1}{a} - \frac{1}{2}}}{{[- (1 - w^{2}) ln (1 - w^{2})]}^{1 / 2}} . \end{array}$

Since $0 < a ⩽ 2$ , then the numerator ${(1 - w^{2})}^{\frac{1}{a} - \frac{1}{2}}$ in the last expression is a bounded function in the interval $[1 / 2, 1]$ . Notice also that the function $- 1 / ln (1 - w^{2})$ is bounded from above by $- 1 / ln (3 / 4)$ in the same interval. Since the integral $\int_{1 / 2}^{1} 1 / \sqrt{1 - w^{2}} d w$ is finite and the absolute value of the derivative (respect to z) of any order of the integrand in Eq. (25) is bounded by a constant times the function ${(1 - w^{2})}^{- 1 / 2}$ , then by the dominated convergence theorem we have that for any natural number r $\begin{array}{l} | z^{r} I_{a}^{(k)} (z) | & ⩽ M e^{- ℜ (z) / 4} | z |^{r} (| z | + 1), \end{array}$ with M a constant number independent of z. Since $| ℑ (z) | ⩽ C ℜ (z)$ , then $| z | ⩽ C_{1} ℜ (z)$ for some constant $C_{1}$ . Therefore, $| z^{r} I_{a}^{(k)} (z) | ⩽ M | ℜ (z) |^{r + 1} e^{- ℜ (z) / 4} < \infty$ if $ℜ (z) \to \infty$ . Thus we have $I_{a}^{(k)} (z) = O (| z |^{- r})$ for all $r \in N$ , i.e. $\begin{matrix} (26) & I_{a}^{(k)} (z) = O (| z |^{- \infty}), for ℜ (z) \to + \infty . \end{matrix}$

Let us now study the term $J_{a} (z)$ and its kth derivative. First note that the function $m$ has an even and $C^{\infty}$ extension to the interval $(- 1 / 2, 1 / 2)$ which implies that $m$ has an asymptotic expansion in the interval $[0, 1 / 2)$ in terms of even powers of the variable w (see Ref. [6]). Namely, there exist numbers $b_{2 j}$ , $j = 0, 1, \dots$ such that for $w \in [0, 1 / 2)$ $\begin{array}{l} m (w) - \sum_{j = 0}^{N} b_{2 j} w^{2 j} = O (w^{2 N + 2}), b_{0} = m (0) = 1 . \end{array}$

Let us write $J_{a}^{(k)} (z) = G_{a}^{(k)} (z) + H_{a}^{(k)} (z)$ with $\begin{matrix} (27) & \begin{matrix} G_{a} (z) = \int_{0}^{1 / 2} e^{- z w^{2}} m (w) d w, \\ H_{a} (z) = \int_{0}^{1 / 2} a z (1 - w^{2}) e^{- z w^{2}} m (w) d w . \end{matrix} \end{matrix}$

Let us study $G_{a}^{(k)}$ . Let N be a natural number and define $\begin{array}{l} (28) & G_{a, 1} (z) = \int_{0}^{1 / 2} e^{- z w^{2}} \sum_{j = 0}^{N} b_{2 j} w^{2 j} d w, \\ (29) & G_{a, 2} (z) = \int_{0}^{1 / 2} e^{- z w^{2}} (m (w) - \sum_{j = 0}^{N} b_{2 j} w^{2 j}) d w . \end{array}$ Notice that $G_{a}^{(k)} (z) = G_{a, 1}^{(k)} (z) + G_{a, 2}^{(k)} (z)$ . We claim that $G_{a, 2}^{(k)} (z) = O (| z |^{- N - k - \frac{3}{2}})$ . Since for any $ℓ \in Z_{+}$ the absolute value of the ℓst derivative of the integrand in Eq. (29) is bounded by the function $w^{2 N + 2 + 2 ℓ}$ , which in turn is integrable, then by the dominated convergence theorem we have $\begin{matrix} | G_{a, 2}^{(k)} (z) | ⩽ M \int_{0}^{1 / 2} e^{- ℜ (z) w^{2}} w^{2 N + 2 + 2 k} d w ⩽ \frac{M}{| z |^{(2 N + 2 k + 3) / 2}}, \end{matrix}$ with M a constant depending only on N and k, and where to obtain the last inequality we have considered the change of variables $η = \sqrt{ℜ (z)} w$ and that $| ℑ (z) | ⩽ C ℜ (z)$ .

On the other hand, let us write $G_{a, 1}^{(k)} (z)$ as follows $\begin{array}{l} (30) & G_{a, 1}^{(k)} & = \sum_{j = 0}^{N} b_{2 j} \int_{0}^{\infty} e^{- z w^{2}} w^{2 j} {(- w^{2})}^{k} d w - \sum_{j = 0}^{N} b_{2 j} \int_{\frac{1}{2}}^{\infty} e^{- z w^{2}} w^{2 j} {(- w^{2})}^{k} d w . \end{array}$

The second term of $G_{a, 1}^{(k)} (z)$ is $O (| z |^{- \infty})$ because for any natural number r we have $\begin{array}{l} | z^{r} \sum_{j = 0}^{N} b_{2 j} \int_{\frac{1}{2}}^{\infty} e^{- z w^{2}} w^{2 (k + j)} d w | \\ (31) & ⩽ M {| ℜ (z) |}^{r} \sum_{j = 0}^{N} \frac{| b_{2 j} | e^{- ℜ (z) / 4}}{| ℜ (z) |^{k + j + 1 / 2}} \int_{0}^{\infty} e^{- t^{2}} {(t + \sqrt{ℜ (z)} / 2)}^{2 (k + j)} d t \\ (32) & ⩽ C_{1} {| ℜ (z) |}^{r} e^{- ℜ (z) / 4} \end{array}$ with M, $C_{1}$ constants independent of z and $C_{1}$ only depends on N, r and k. To obtain the inequality (31) we have considered the changes of variables $v = w - 1 / 2$ and $t = \sqrt{ℜ (z)} v$ . The inequality (32) is a consequence of the fact that the integrals $\int_{0}^{\infty} e^{- t^{2}} {(t + \sqrt{ℜ (z)} / 2)}^{2 (k + j)} d t$ are polynomials in the variable $\sqrt{ℜ (z)}$ .

On the other hand, the integral $\int_{0}^{\infty} e^{- z w^{2}} w^{2 (k + j)} d w$ is equal to the integral along the straight line ${t / \sqrt{z} | t ⩾ 0}$ in the complex plane (where we are considering the principal branch of the square root function and then $\sqrt{z}$ is well defined in the set $| ℑ (z) | ⩽ C ℜ (z)$ ) with $z \neq 0$ . Namely, $\int_{0}^{\infty} e^{- z w^{2}} w^{2 (k + j)} d w = \frac{1}{z^{k + j} \sqrt{z}} \int_{0}^{\infty} e^{- t^{2}} t^{2 (k + j)} d t$ . Thus from Eq. (30) $\begin{array}{l} (33) & G_{a, 1}^{(k)} & = \sum_{j = 0}^{N} {(- 1)}^{k} b_{2 j} \frac{d_{j + k}}{z^{k + j} \sqrt{z}} + O (| z |^{- \infty}) \end{array}$ with $d_{r} = \int_{0}^{\infty} e^{- t^{2}} t^{2 r} d t$ . In particular $d_{0} = \frac{\sqrt{π}}{2}$ .

We conclude that $\begin{array}{l} (34) & G_{a}^{(k)} (z) = {(- 1)}^{k} \frac{\sqrt{z}}{z^{k + 1}} [\sum_{j = 0}^{N} \frac{d_{j + k}}{z^{j}} b_{2 j} + O (| z |^{- N - 1})] . \end{array}$

Let us now study $H_{a}^{(k)} (z)$ , defined in Eq. (27). From the equality $H_{a} (z) = a z [G_{a} (z) + G_{a}^{(1)} (z)]$ and Eq. (34) $\begin{array}{rcl} H_{a}^{(k)} (z) & = & a [z G^{(k)} (z) + z G^{(k + 1)} (z) + k G^{(k - 1)} (z) + k G^{(k)} (z)] \\ = & a {(- 1)}^{k} \frac{\sqrt{z}}{z^{k}} [b_{0} (d_{k} - k d_{k - 1}) \\ (35) & + \sum_{j = 1}^{N} \frac{1}{z^{j}} (b_{2 j} - b_{2 (j - 1)}) (d_{j + k} - k d_{j + k - 1}) + O (| z |^{- N - 1})] \end{array}$ where we are taking the convention that $k d_{k - 1} = 0$ when $k = 0$ .

Thus, from Eqs (25), (26), (34) and (35) we obtain $\begin{array}{l} g_{a}^{(s)} (z) = & 2 e^{z} {(\frac{z}{a π})}^{\frac{1}{2}} \sum_{k = 0}^{s} (\binom{s}{k}) \frac{{(- 1)}^{k}}{z^{k}} [a b_{0} (d_{k} - k d_{k - 1}) \\ + \sum_{j = 1}^{N} \frac{1}{z^{j}} (a (b_{2 j} - b_{2 (j - 1)}) (d_{j + k} - k d_{j + k - 1}) \\ (36) & + b_{2 (j - 1)} d_{k + j - 1}) + O (| z |^{- N - 1})] . \end{array}$ The asymptotic expansion of $g_{a}^{(s)}$ given in Eq. (23) follows from Eq. (36). Moreover, constants $a_{k, s}$ , $k = 1, 2, \dots$ , are given by: $\begin{array}{l} a_{k, s} = \frac{2}{\sqrt{π}} [(\binom{s}{k}) {(- 1)}^{k} (d_{k} - k d_{k - 1}) + \sum_{r = 0}^{k - 1} (\binom{s}{r}) {(- 1)}^{r} (\frac{1}{a} b_{2 (k - r - 1)} d_{k - 1} \\ + (b_{2 (k - r)} - b_{2 (k - r - 1)}) (d_{k} - r d_{k - 1}))], k = 1, \dots, s \\ a_{k, s} = \frac{2}{\sqrt{π}} \sum_{r = 0}^{s} (\binom{s}{r}) {(- 1)}^{s} [\frac{1}{a} b_{2 (k - r - 1)} d_{k - 1} \\ + (b_{2 (k - r)} - b_{2 (k - r - 1)}) (d_{k} - r d_{k - 1})], k = s + 1, \dots \end{array}$ □

Using Lemma 3.2 with $z = x \cdot z / ℏ$ we obtain the following asymptotic expansion of the sth derivative of $g_{a}$ evaluated at $z = x \cdot z / ℏ$ for $ℏ ↘ 0$ : Proposition 3.3.
Let $z \in C^{n} - {0}$ , C be a constant greater than one, $W_{z}$ , $V_{z}$ be the regions defined in Eq. ( 22 ) and s a non-negative integer number. Then for $ℏ ↘ 0$ we have $\begin{array}{l} (37) & g_{a}^{(s)} (x \cdot z / ℏ) = \sqrt{a} {[\frac{x \cdot z}{ℏ}]}^{\frac{1}{2}} e^{\frac{x \cdot z}{ℏ}} [1 + \frac{a_{1, s}}{x \cdot z} ℏ + O (ℏ^{2})], for x \in W_{z}, \\ (38) & | g_{a}^{(s)} (x \cdot z / ℏ) | ⩽ C_{1} e^{| z | / ℏ} e^{μ | z | / ℏ} (\frac{| z |}{ℏ} + 1), for x \in V_{z}, \end{array}$ with $C_{1}$ a constant and $μ = \frac{1}{C} - 1 < 0$ . In particular, for $x \in W_{z}$ $\begin{array}{l} K_{n, - 1}^{ℏ} (x, z) \\ (39) & = {[\frac{x \cdot z}{ℏ (n - 1)}]}^{\frac{1}{2}} exp (\frac{x \cdot z}{ℏ}) [1 + \frac{a_{1}}{x \cdot z} ℏ + O (ℏ^{2})] \end{array}$ with $a_{1} = a_{1, 0} = \frac{1}{2} (n - \frac{5}{4})$ .
Proof.
First we note that for $x \in W_{z}$ , $| ℑ (x \cdot z) | ⩽ | z | ⩽ C ℜ (x \cdot z)$ , therefore we can use the asymptotic expansion of the function $g_{a}^{(s)}$ indicated in Lemma 3.2. Thus, from Eq. (23) $\begin{array}{l} g_{a}^{(s)} (x \cdot z / ℏ) \\ (40) & = \sqrt{a} {[\frac{x \cdot z}{ℏ}]}^{\frac{1}{2}} exp (\frac{x \cdot z}{ℏ}) [1 + \frac{a_{1, s}}{x \cdot z} ℏ + E_{s} (x, ℏ)], \end{array}$ where the error term $E_{s} (x, ℏ)$ is $O (ℏ^{2})$ uniformly with respect to $x \in W_{z}$ because in such a region we have $1 / | x \cdot z | ⩽ C / | z |$ .

To show Eq. (38), let us consider the integral expression (25) to find that the sth derivative of $g_{a}$ evaluated at $z = x \cdot z / ℏ$ can be written as $\begin{array}{l} g_{a}^{(s)} (\frac{x \cdot z}{ℏ}) \\ = \int_{0}^{1} [P_{s, 1} (w) \frac{x \cdot z}{ℏ} + P_{s, 2} (w)] e^{\frac{x \cdot z}{ℏ} (1 - w^{2})} m (w) d w, \end{array}$ with $P_{s, 1}$ and $P_{s, 2}$ polynomial functions on the variable w. Thus for $x \in V_{z}$ $\begin{array}{rcl} | g_{a}^{(s)} (\frac{x \cdot z}{ℏ}) | & ⩽ & e^{| z | / ℏ} \int_{0}^{1} | P_{s, 1} (w) \frac{x \cdot z}{ℏ} + P_{s, 2} (w) | | m (w) | \\ \times exp (\frac{| z |}{ℏ} [\frac{ℜ (x \cdot z)}{| z |} (1 - w^{2}) - 1]) d w \\ ⩽ & C_{1} e^{| z | / ℏ} e^{μ | z | / ℏ} (\frac{| z |}{ℏ} + 1) \end{array}$ with $C_{1}$ a constant and $μ = \frac{1}{C} - 1 < 0$ . □

4. Asymptotic expansion of the Berezin transform

In this section we show an analogue of the asymptotic expansion (2) for the associated Berezin transform to the family $K_{p}$ (see Eq. (15)).

It is possible to obtain all the asymptotic expansion of the Berezin transform $B_{p}^{ℏ} (ϕ)$ , $ϕ \in C^{\infty} (S^{n})$ , $p > - n$ , when ϕ is a polynomial. For the general case, when ϕ is a smooth function, we consider only the case $n ⩾ 2$ and $p = 0$ or $p = - 1$ .

Theorem 4.1.
Let $k \in Z_{+}^{n}$ be a multi-index and $φ_{k} (x) = x^{k}$ , with $x \in S^{n}$ . Then for $w, z \in C^{n} - {0}$ $\begin{array}{l} B_{ℏ, p} (T_{φ_{k}}) (w, z) = & \frac{{(\frac{w \cdot z}{ℏ^{2}})}^{\frac{1}{2} (n + p - 1)}}{I_{n + p - 1} (2 \sqrt{w \cdot z} / ℏ)} {(\frac{w}{ℏ})}^{k} \sum_{ℓ = 0}^{\infty} {(\frac{w \cdot z}{ℏ^{2}})}^{ℓ} \frac{c_{ℓ, p}}{c_{| k | + ℓ, p}} \\ \times \frac{1}{ℓ! Γ (| k | + ℓ + n + p)} \end{array}$ where $c_{ℓ, p}$ are defined in Eq. ( 10 ).
Proof.
From Eq. (14) and properties of the orthogonal projection $\begin{array}{l} (41) & B_{ℏ, p} (T_{φ_{k}}) (w, z) & = \frac{{⟨ x^{k} K_{n, p}^{ℏ} (\cdot, z), K_{n, p}^{ℏ} (\cdot, w) ⟩}_{S^{n}}}{{⟨ K_{n, p}^{ℏ} (\cdot, z), K_{n, p}^{ℏ} (\cdot, w) ⟩}_{S^{n}}} . \end{array}$

From the explicit expression of the function $K_{n, p}^{ℏ} (\cdot, z)$ given in Eq. (10), the dominated convergence theorem and Lemma A1 in Ref. [5], which states that $\begin{matrix} \int_{S^{n}} x^{k} {(x \cdot z)}^{ℓ} {(w \cdot x)}^{s} d S^{n} (x) = δ_{s, | k | + ℓ} \frac{(n - 1)! (ℓ + | k |)!}{(| k | + ℓ + n - 1)!} w^{k} {(w \cdot z)}^{ℓ} \end{matrix}$ with $δ_{i, j}$ denoting the Kronecker symbol, we have $\begin{matrix} (42) & {⟨ x^{k} K_{n, p}^{ℏ} (\cdot, z), K_{n, p}^{ℏ} (\cdot, w) ⟩}_{S^{n}} = {(\frac{w}{ℏ})}^{k} \sum_{ℓ = 0}^{\infty} \frac{c_{ℓ, p} c_{| k | + ℓ, p} (n - 1)!}{ℓ! (| k | + ℓ + n - 1)!} {(\frac{w \cdot z}{ℏ^{2}})}^{ℓ} . \end{matrix}$

From Eqs (41), (42), (11), (12) and (8) we conclude the proof of Theorem 4.1. □
Corollary 4.2.
Let $z \in C^{n} - {0}$ , $k \in Z_{+}^{n}$ be a multi-index and $φ_{k}$ as above. Then for $ℏ ↘ 0$ $\begin{matrix} (43) & B_{0}^{ℏ} (φ_{k}) (z) = {(\frac{z}{| z |})}^{k} [1 - \frac{| k | (| k | + 2 n - 2)}{4 | z |} ℏ + O (ℏ^{2})] . \end{matrix}$
Proof.
From Eq. (15), Theorem 4.1 and the equality $c_{s, 0} = 1$ (which is valid for any non-negative integer number s, see Eq. (10)), we have $\begin{array}{l} B_{0}^{ℏ} (φ_{k}) (z) & = \frac{1}{I_{n - 1} (2 | z | / ℏ)} {(\frac{z}{| z |})}^{k} \sum_{ℓ = 0}^{\infty} \frac{{(| z | / ℏ)}^{2 ℓ + | k | + n - 1}}{Γ (ℓ + 1) Γ (ℓ + | k | + n)} = {(\frac{z}{| z |})}^{k} \frac{I_{n + | k | - 1} (2 | z | / ℏ)}{I_{n - 1} (2 | z | / ℏ)} \end{array}$ where we have used the expression of the modified Bessel function $I_{ν}$ , $ν \in R$ , as a power series (see Eq. (7)) to obtain the last equality.

Using the asymptotic expression of the modified Bessel function $I_{ν} (ω)$ when $| ω | \to \infty$ (see Eq. (16)) and the identity $Γ (b + 1) = b Γ (b)$ we obtain the asymptotic expansion given in Eq. (43). □
Theorem 4.3.
Let $z \in C^{n} - {0}$ and ϕ be a smooth function on $S^{n}$ . Then for $ℏ ↘ 0$ $\begin{array}{l} (44) & B_{0}^{ℏ} (ϕ) (z) & = ϕ (z / | z |) + \frac{ℏ}{4 | z |} [4 Δ_{x \overline{x}} - R^{2} - (2 n - 2) R] ϕ (z / | z |) + O (ℏ^{2}), \end{array}$ where $Δ_{x \overline{x}} = \sum_{j = 1}^{n} \partial_{x_{j} {\overline{x}}_{j}}$ and $R : = \sum_{j = 1}^{n} (x_{j} \partial_{x_{j}} + {\overline{x}}_{j} \partial_{{\overline{x}}_{j}})$ denote the Laplace operator and the radial derivative on $R^{2 n} ≅ C^{n}$ , respectively.
Proof.
Let us first note that it is enough to show that (44) holds for all points z of the form $z = r {\hat{u}}_{1}$ , with $r > 0$ , and ${\hat{u}}_{1} = (1, 0, \dots, 0) \in R^{n}$ .

Indeed, let $U \in SU (n)$ and $T_{U}$ be the unitary transformation defined in Proposition 2.3. Since the Laplace operator $Δ_{x \overline{x}}$ as well as the radial derivative $R$ are clearly invariant under unitary transformation $T_{U}$ , the right hand side of Eq. (44) is likewise invariant under $T_{U}$ .

By (2) of Proposition 2.3, the validity of Eq. (44) for ϕ at z is therefore equivalent to its validity for $T_{U^{- 1}} ϕ = ϕ \circ U$ at $U^{- 1} z$ .

Since any given z can be mapped by a suitable $U \in SU (n)$ into a point of the form $r {\hat{u}}_{1}$ , with $r = | z |$ , it is indeed enough to prove (44) only for points z of the latter form.

For the rest of the proof of Theorem 4.3, we thus assume that $z = r {\hat{u}}_{1}$ with $r > 0$ .

Since $K_{n, 0}^{ℏ} (x, z) = e^{x \cdot z / ℏ}$ (see Eq. (10), with $p = 0$ ), from Definition 2.2 and the norm estimate of the function $K_{n, 0}^{ℏ} (\cdot, z)$ (see Eq. (19)) we obtain $\begin{array}{l} B_{0}^{ℏ} (ϕ) (z) = & B_{ℏ, 0} T_{ϕ} (z) \\ = & \frac{2 \sqrt{π}}{Γ (n)} {(\frac{r}{ℏ})}^{n - \frac{1}{2}} \int_{S^{n}} exp (\frac{2 r}{ℏ} [ℜ (x_{1}) - 1]) ϕ (x) (1 \\ (45) & + \frac{ℏ}{16 r} (2 n - 3) (2 n - 1) + O (ℏ^{2})) d S_{n} (x) . \end{array}$

Writing the coordinates $x_{j}$ of a point $x \in S^{n}$ in the form $x_{j} = y_{j} + ı y_{n + j}$ , with $y_{j}$ and $y_{n + j}$ real numbers. Note that the vector $y = (y_{1}, \dots, y_{2 n}) \in S^{2 n - 1} = {a \in R^{2 n} | | a | = 1}$ .

In order to estimate (45), let us define $\begin{array}{l} (46) & A Ψ & = {(\frac{r}{π ℏ})}^{n - \frac{1}{2}} \int_{S^{2 n - 1}} Ψ (ω) exp (\frac{ı}{ℏ} f (ω)) d Ω_{2 n - 1} (ω), \end{array}$ where $d Ω_{2 n - 1}$ is the normalized surface measure on $S^{2 n - 1}$ and $f (ω) = - 2 ı r [ω_{1} - 1]$ .

Let us introduce spherical coordinates for the variables $(ω_{1}, \dots, ω_{2 n}) \in S^{2 n - 1}$ : $\begin{array}{l} ω_{1} = sin (θ_{2 n - 1}) \dots sin (θ_{2}) cos (θ_{1}), \\ ω_{2} = sin (θ_{2 n - 1}) \dots sin (θ_{2}) sin (θ_{1}), \\ ⋮ \\ ω_{2 n - 1} = sin (θ_{2 n - 1}) cos (θ_{2 n - 2}), \\ ω_{2 n} = cos (θ_{2 n - 1}) \end{array}$ with $θ_{1} \in (- π, π)$ , $θ_{2}, θ_{3}, \dots, θ_{2 n - 1} \in (0, π)$ .

The function f appearing in the argument of the exponential function in Eq. (46) has only one critical point (as a function of the angles) which contributes to the asymptotic, namely $θ_{0} = (0, π / 2, \dots, π / 2)$ . In addition, since $\begin{matrix} \frac{\partial^{2} f (ω (θ))}{\partial θ_{ℓ} \partial θ_{j}} |_{θ = θ_{0}} = 2 ı r δ_{j, ℓ}, \end{matrix}$ then the Hessian matrix of f evaluated at the critical point $θ_{0}$ is equal to $f^{″} (ω (θ_{0})) = 2 ı r I_{2 n - 1}$ , where $I_{s}$ denotes the identity matrix of size s. Moreover, $\det (f^{″} (ω (θ_{0}))) = {(2 ı r)}^{2 n - 1}$ . From the stationary phase method (see Ref. [13]) we obtain that $\begin{matrix} (47) & A Ψ = \frac{1}{N} [\sum_{ℓ < k} ℏ^{ℓ} M_{ℓ} Ψ |_{cp} + O (ℏ^{k})], \end{matrix}$ where N is the surface area of $S^{2 n - 1}$ and $\begin{array}{l} M_{ℓ} Ψ |_{cp} = & \sum_{s = ℓ}^{3 ℓ} \frac{ı^{- ℓ} 2^{- s}}{s! (s - ℓ)!} {[(- {(f^{″} (θ_{0}))}^{- 1}) \hat{D} \cdot \hat{D}]}^{s} [Ψ (ω (θ)) \\ (48) & \times {(sin θ_{2 n - 1})}^{2 n - 2} \dots sin θ_{2} {(p_{cp})}^{s - ℓ}] |_{cp} \end{array}$ with $g |_{cp}$ denoting the evaluation at the critical point $θ_{0}$ of a given function $g$ , $\begin{array}{l} (49) & p_{cp} = p_{cp} (θ) & = - 2 ı r (sin θ_{2 n - 1} \dots sin θ_{2} cos θ_{1} - 1) - ı r (θ_{1}^{2} + \sum_{j = 2}^{2 n - 1} {(θ_{j} - \frac{π}{2})}^{2}), \end{array}$ and $\hat{D}$ the column vector of size $2 n - 1$ whose j entry is $\partial_{θ_{j}}$ (i.e. ${(\hat{D})}_{j} = \partial_{θ_{j}}$ ).

Since $(- {(f^{″} (ω (θ_{0})))}^{- 1}) \hat{D} \cdot \hat{D} = \frac{ı}{2 r} \sum_{j = 1}^{2 n - 1} \partial_{θ_{j} θ_{j}}$ we obtain from Eq. (48) $\begin{array}{l} (50) & M_{0} Ψ |_{cp} = Ψ ({\hat{e}}_{1}), \\ (51) & M_{1} Ψ |_{cp} = \frac{1}{4 r} [\sum_{j = 1}^{2 n - 1} \partial_{θ_{j}} \partial_{θ_{j}} Ψ (ω (θ)) - \frac{1}{4} (2 n - 1) (2 n - 3) Ψ (ω (θ))] |_{cp} \\ (52) & = \frac{1}{4 r} [Δ_{y y} - \partial_{y_{1} y_{1}} - (2 n - 1) \partial_{y_{1}} - \frac{1}{4} (2 n - 1) (2 n - 3)] Ψ ({\hat{e}}_{1}), \end{array}$ with ${\hat{e}}_{1} = (1, 0, \dots, 0) \in R^{2 n}$ . Equation (51) is obtained using that $\partial_{θ}^{k} p_{cp} (θ_{0}) = 0$ for any multi index $k \in Z_{+}^{2 n - 1}$ that satisfies $| k | ⩽ 3$ or $| k |$ is odd or $θ_{j}$ is odd for some $j = 1, \dots, 2 n - 1$ . Equation (52) is obtained using the chain rule.

From Eqs (47), (50) and (52) $\begin{array}{l} A Ψ = & \frac{1}{N} [Ψ ({\hat{e}}_{1}) + \frac{ℏ}{4 r} (Δ_{y y} - \partial_{y_{1} y_{1}} - (2 n - 1) \partial_{y_{1}} \\ (53) & - \frac{1}{4} (2 n - 1) (2 n - 3)) Ψ ({\hat{e}}_{1})] + O (ℏ^{2}) . \end{array}$

Thus if we take $Ψ (ℜ (x), ℑ (x)) = ϕ (x)$ , $x \in S^{n}$ , in Eq. (46), then from Eqs (45), (53), the chain rule and the fact that $N = 2 π^{n} / Γ (n)$ we have $\begin{array}{l} B ϕ (z) : = & \frac{2 \sqrt{π}}{Γ (n)} {(\frac{r}{ℏ})}^{n - \frac{1}{2}} \int_{S^{n}} exp (\frac{2 r}{ℏ} [ℜ (x_{1}) - 1]) ϕ (x) \\ = & ϕ (z / | z |) + \frac{ℏ}{4 | z |} [4 Δ_{x \overline{x}} - R^{2} - (2 n - 2) R \\ (54) & - \frac{1}{4} (2 n - 1) (2 n - 3)] ϕ (z / | z |) + O (ℏ^{2}) \end{array}$

From Eqs (45) and (54) we conclude the proof of this theorem. □
Remark 2.
For $n = 1$ , the asymptotic expansion found in Theorem 4.3 (with ϕ a smooth function on S) coincides, up to an error of the order $O (ℏ^{2})$ , with that found by Ali and Engliš in theorem 8 of Ref. [9] (considering $f (z) = ϕ (z / | z |)$ , $z \in C$ ). In the latter, Ali and Engliš considered the Berezin transform on the space $E_{1, 0}$ (see Section 2.2) taking the complete family ${T_{1, 0} (\cdot, z) | z \in C}$ as the reproducing kernel of $E_{1, 0}$ with one freezing variable.

In Section 3.2 we obtained an asymptotic expression of the functions $K_{n, - 1}^{ℏ} (\cdot, z)$ , $z \in C^{n}$ , this result will allow us to obtain an asymptotic expansion for the Berezin transform $B_{- 1}^{ℏ}$ similar to that obtained in Theorem 4.3. Theorem 4.4.
Let $n ⩾ 2$ , $z \in C^{n} - {0}$ and ϕ be a smooth function on $S^{n}$ . Then for $ℏ ↘ 0$ $\begin{array}{l} (55) & B_{- 1}^{ℏ} (ϕ) (z) = ϕ (z / | z |) + \frac{ℏ}{4 | z |} [4 Δ_{x \overline{x}} - R^{2} - (2 n - 2) R] ϕ (z / | z |) + O (ℏ^{2}), \end{array}$ where $Δ_{x \overline{x}} = \sum_{j = 1}^{n} \partial_{x_{j} {\overline{x}}_{j}}$ and $R : = \sum_{j = 1}^{n} (x_{j} \partial_{x_{j}} + {\overline{x}}_{j} \partial_{{\overline{x}}_{j}})$ denote the Laplace operator and the radial derivative on $R^{2 n} ≅ C^{n}$ , respectively.
Proof.
By the same argument that was used in the proof of Theorem 4.3, we only need to show this theorem for points z in the form $(r, 0, \dots, 0)$ , with $r > 0$ . For the rest of the proof of Theorem 4.4, we thus assume that $z = (r, 0, \dots, 0)$ with $r > 0$ .

From Definition 2.2 $\begin{array}{rcl} (56) & B_{- 1}^{ℏ} (ϕ) (z) = \int_{S^{n}} ϕ (x) \frac{| K_{n, - 1}^{ℏ} (x, z) |^{2}}{‖ K_{n, - 1}^{ℏ} (\cdot, z) ‖_{S^{n}}^{2}} d S_{n} (x) . \end{array}$

Let C be the constant mentioned in Lemma 3.2 taken greater than one, and $W_{z}$ , $V_{z}$ be the regions defined in Eq. (22).

The integral in Eq. (56) can be written as an integral over the region $W_{z}$ plus an integral over $V_{z}$ denoted by the letters $J_{W} (z)$ and $J_{V} (z)$ respectively.

The integral over $W_{z}$ is the one that gives us the main asymptotic term and the integral over $V_{z}$ is actually $O (ℏ^{\infty})$ .

The assertion $J_{V} (z) = O (ℏ^{\infty})$ follows from the following inequality $\begin{matrix} \frac{| K_{n, - 1}^{ℏ} (x, z) |^{2}}{‖ K_{n, - 1}^{ℏ} (\cdot, z) ‖_{S^{n}}^{2}} ⩽ C_{1} {(\frac{| z |}{ℏ})}^{n - \frac{3}{2}} e^{2 μ \frac{| z |}{ℏ}} {[\frac{| z |}{ℏ} + 1]}^{2} (1 + O (ℏ)), x \in V_{z} \end{matrix}$ which in turn is a consequence from Eq. (38) and the norm estimate for the function $K_{n, - 1}^{ℏ} (\cdot, z)$ (see Eq. (19)).

Let us now study the term $J_{W} (z)$ . From Eq. (39) $\begin{array}{l} J_{W} (z) = & \frac{e^{2 r / ℏ}}{‖ K_{n, - 1}^{ℏ} (\cdot, z) ‖_{S^{n}}^{2}} \frac{r}{ℏ (n - 1)} \int_{W_{z}} ϕ (x) exp (\frac{2 r}{ℏ} [ℜ (x_{1}) - 1]) \\ (57) & \times [| x_{1} | + ℏ \frac{(n - \frac{5}{4}) ℜ (x_{1})}{r | x_{1} |} + E (z, ℏ)] d S_{n} (x), \end{array}$ where the error term $E (z, ℏ)$ is $O (ℏ^{2})$ uniformly with respect to $x \in W_{z}$ , because in such a region we have $1 / | x \cdot z | ⩽ C / | z |$ .

Furthermore, for each $x \in V_{z}$ , $ℜ (x_{1}) - 1 = \frac{ℜ (x \cdot z)}{| z |} - 1 < \frac{1}{C} - 1 < 0$ . Thus we can take the integral defining $J_{W} (z)$ over the whole sphere with an error $O (ℏ^{\infty})$ .

Then, from Eq. (57) $\begin{matrix} J_{W} (z) = \frac{e^{2 r / ℏ} Γ (n - 1)}{‖ K_{n, - 1}^{ℏ} (\cdot, z) ‖_{S^{n}}^{2}} \frac{1}{2 \sqrt{π}} {(\frac{r}{ℏ})}^{\frac{3}{2} - n} [B ϕ_{1} (z) + ℏ \frac{n - \frac{5}{4}}{r} B ϕ_{2} (z) + O (ℏ^{2})], \end{matrix}$ with the operator $B$ defined in Eq. (54), $ϕ_{1} (x) = ϕ (x) | x_{1} |$ and $ϕ_{2} (x) = ϕ (x) \frac{ℜ (x_{1})}{| x_{1} |}$ .

Then from Eq. (54) we have $\begin{array}{l} J_{W} (z) = & \frac{e^{2 r / ℏ} Γ (n - 1)}{‖ K_{n, - 1}^{ℏ} (\cdot, z) ‖_{S^{n}}^{2}} \frac{1}{2 \sqrt{π}} {(\frac{r}{π ℏ})}^{\frac{3}{2} - n} [ϕ (z / | z |) + \frac{ℏ}{4 | z |} (Δ_{x \overline{x}} - R^{2} \\ (58) & - (2 n - 2) R - \frac{1}{4} (2 n - 5) (2 n - 3)) ϕ (z / | z |) + O (ℏ^{2})], \end{array}$ where we have used the equalities $Δ_{x \overline{x}} ϕ_{1} (z / | z |) = (Δ_{x \overline{x}} + \frac{1}{2} R + \frac{1}{4}) ϕ (z / z)$ , $Δ_{x \overline{x}} ϕ_{2} (z / | z |) = (Δ_{x \overline{x}} - \frac{1}{4}) ϕ (z / z)$ , $R ϕ_{1} = (| x_{1} | R + | x_{1} |) ϕ$ and $R ϕ_{2} = \frac{ℜ (x_{1})}{| x_{1} |} R ϕ$ .

Finally, from Eq. (58) and the norm estimate for the function $K_{n, - 1}^{ℏ} (\cdot, z)$ (see Eq. (19)) we conclude the proof of this theorem. □

5. Asymptotic expansion of the covariant symbol of a pseudo-differential operator

The covariant symbol (see Eq. (13)) is not only defined for Toeplitz operators, but for any bounded linear operator. Hence it is natural to ask if there exists an analogue of the asymptotic expansion obtained in Theorems 4.3 and 4.4 for a more general operator than a Toeplitz operator. For this reason, in this section we prove that, for a pseudo-differential operator $A_{ℏ}$ acting on $L^{2} (S^{n})$ with principal symbol $℘ (A_{ℏ})$ (see Appendix A where we give a brief description of what we mean by semiclassical pseudo-differential operators on a manifold), the covariant symbol of $A_{ℏ}$ goes to the composition of $℘ (A_{ℏ})$ with the map $z \mapsto (z / | z |, - ı z)$ when goes ℏ to zero. Namely,

Theorem 5.1.

Let $p = 0, - 1$ and $A_{ℏ}$ be a semiclassical pseudo-differential operator of order zero with domain in $L^{2} (S^{n})$ , $n ⩾ 2$ . Then for $z \in C^{n} - {0}$ and $ℏ ↘ 0$ we have $\begin{array}{rcl} (59) & \frac{{⟨ A_{ℏ} K_{n, p}^{ℏ} (\cdot, z), K_{n, p}^{ℏ} (\cdot, z) ⟩}_{S^{n}}}{{⟨ K_{n, p}^{ℏ} (\cdot, z), K_{n, p}^{ℏ} (\cdot, z) ⟩}_{S^{n}}} = ℘ (A_{ℏ}) (z / | z |, - ı z) + O (ℏ), \end{array}$ where $℘ (A_{ℏ})$ is the principal symbol of the semiclassical pseudo-differential operator $A_{ℏ}$ .

Proof.

Let $U \in SU (n)$ and $T_{U}$ be the unitary transformation defined in Proposition 2.3. Let us first note that by Propositions 2.3 and A.6, the validity of Eq. (59) for $A_{ℏ}$ at z is equivalent to its validity for $T_{U^{- 1}} A_{ℏ} T_{U}$ at $U^{- 1} z$ . Since any given z can be mapped by a suitable $U \in SU (n)$ into a point of the form $z = λ {\hat{u}}_{1}$ with $λ = | z |$ and ${\hat{u}}_{1} = (1, 0, \dots, 0)$ a canonical unit vector in $R^{n}$ , it is indeed enough to prove (59) only for points z of the latter form.

For the rest of the proof of Theorem 5.1, we thus assume that $z = λ {\hat{u}}_{1}$ with $λ > 0$ .

Let us identify $S^{n}$ with $S^{2 n - 1}$ via the map Υ, which sends $x = (x_{1}, \dots, x_{n}) \in S^{n}$ to $(ℜ (x_{1}), ℑ (x_{1}), \dots, ℜ (x_{n}), ℑ (x_{n})) \in S^{2 n - 1}$ .

Let us consider the following particular charts $(U_{c}, κ_{c} \circ Υ)$ of $S^{n}$ , with $U_{c} = Υ^{- 1} \circ κ_{c}^{- 1} A$ , $c = 1, 2, \dots, 2 n - 1$ , where $\begin{matrix} A = {(θ, v_{3}, \dots, v_{2 n}) \in R^{2 n - 1} | - π < θ < π, \sum_{q = 3}^{2 n} v_{q}^{2} < 1 - \frac{1}{8 (n - 1)}} \end{matrix}$ and $κ_{c}^{- 1} : A \to Υ (U_{c})$ are defined by $\begin{matrix} κ_{1}^{- 1} (θ, v) = (r cos (θ), r sin (θ), v), \\ κ_{a}^{- 1} (θ, v) = R_{a} κ_{1}^{- 1} (θ, v), \end{matrix} r = {(1 - \sum_{q = 3}^{2 n} v_{q}^{2})}^{\frac{1}{2}},$ with $v = (v_{3}, \dots, v_{2 n})$ , $a = 2, \dots, 2 n - 1$ and $R_{a} \in SO (2 n)$ denoting the matrix which rotates the $(2 n - 1)$ -sphere on the $(x_{1}, x_{2})$ plane by π and then followed by another rotation on the $(x_{2}, x_{a})$ plane by $π / 2$ . More specifically, $R_{a}$ is given by the matrix $\begin{matrix} (60) & R_{a} = (- {\hat{e}}_{1}, {\hat{e}}_{a + 1}, {\hat{e}}_{3}, \dots, {\hat{e}}_{a}, {\hat{e}}_{2}, {\hat{e}}_{a + 2}, \dots, {\hat{e}}_{2 n}) \end{matrix}$ where ${{\hat{e}}_{1}, \dots, {\hat{e}}_{2 n}}$ is the canonical basis of $R^{2 n}$ regarding ${\hat{e}}_{a}$ , $a = 1, \dots, 2 n$ , as column vectors. Note that ${\hat{u}}_{1} = Υ^{- 1} ({\hat{e}}_{1}) \notin U_{a}$ for $a = 2, \dots, 2 n - 1$ , and that $S^{n} = ⋃_{c = 1}^{2 n - 1} U_{c}$ (see Appendix B for details).

Let ${t_{c}}$ be a partition of the unity associated to the open cover ${U_{c}}$ . Since ${\hat{u}}_{1}$ only belongs to $U_{1}$ then $t_{1} ({\hat{u}}_{1}) = 1$ . Consider a set of functions ${ϱ_{c}}$ with $ϱ_{c} \in C_{0}^{\infty} (U_{c})$ such that $t_{c} ϱ_{c} = t_{c}$ , for $c = 1, \dots, 2 n - 1$ . Then $\begin{array}{l} (61) & A_{ℏ} & = \sum_{c = 1}^{2 n - 1} [t_{c} A_{ℏ} ϱ_{c} + t_{c} A_{ℏ} (1 - ϱ_{c})] . \end{array}$

Moreover, since $supp (t_{c}) \cap supp (1 - ϱ_{c}) = \emptyset$ and condition $(B)$ in the definition of a semiclassical pseudo-differential operator (see Eq. (73)) we have $\begin{matrix} (62) & \frac{{⟨ t_{c} A_{ℏ} (1 - ϱ_{c}) K_{n, p}^{ℏ} (\cdot, z), K_{n, p}^{ℏ} (\cdot, z) ⟩}_{S^{n}}}{‖ K_{n, p}^{ℏ} (\cdot, z) ‖_{S^{n}}^{2}} = O (ℏ^{\infty}), c = 1, \dots, 2 n - 1 . \end{matrix}$

The basic idea of the proof is to show that the main contribution to the left hand side of Eq. (59) comes from the term $\frac{{⟨ t_{1} A_{ℏ} ϱ_{1} K_{n, p}^{ℏ} (\cdot, z), K_{n, p}^{ℏ} (\cdot, z) ⟩}_{S^{n}}}{‖ K_{n, p}^{ℏ} (\cdot, z) ‖_{S^{n}}^{2}}$ . The stationary phase method will be used to obtain such a main contribution.

From the definition of a semiclassical pseudo-differential operator given in Appendix A, there exist $m \in R$ , $k \in Z_{+}$ larger than $m + n$ and symbols $a_{κ_{c}} \in S_{4 n - 2} ({⟨ p_{θ}, p_{v} ⟩}^{m})$ such that $\begin{array}{l} \frac{{⟨ t_{c} A_{ℏ} ϱ_{c} K_{n, p}^{ℏ} (\cdot, z), K_{n, p}^{ℏ} (\cdot, z) ⟩}_{S^{n}}}{‖ K_{n, p}^{ℏ} (\cdot, z) ‖_{S^{n}}^{2}} \\ = \frac{{(2 π ℏ)}^{- 2 n + 1}}{‖ K_{n, p}^{ℏ} (\cdot, z) ‖_{S^{n}}^{2}} \int_{(θ, v) \in A} \int_{(\tilde{θ}, \tilde{v}) \in A} \int_{(p_{θ}, p_{v}) \in R^{2 n - 1}} \frac{(\overline{K_{n, p}^{ℏ} (\cdot, z)} t_{c}) ({(κ_{c} \circ Υ)}^{- 1} (θ, v))}{| S^{n} |} \\ \times exp (\frac{ı}{ℏ} [(θ, v) - (\tilde{θ}, \tilde{v})] \cdot (p_{θ}, p_{v})) \\ \times {(\frac{1 + ℏ M}{1 + p_{θ}^{2} + p_{v}^{2}})}^{k} ({(a_{κ_{c}})}_{t} (θ, v, \tilde{θ}, \tilde{v}, p_{θ}, p_{v}; ℏ) (ϱ_{c} K_{n, p}^{ℏ} (\cdot, z)) ({(κ_{c} \circ Υ)}^{- 1} (\tilde{θ}, \tilde{v}))) \\ (63) & \times J (θ, v) d p_{θ} d p_{v} d \tilde{θ} d \tilde{v} d θ d v, \end{array}$ where we have used the coordinates $(θ, v)$ in $A$ and where $| S^{n} |$ denotes the surface area of the complex n-sphere, the Jacobian $J (θ, v) = 1$ and the operator M is defined by $\begin{matrix} (64) & M = \frac{1}{ı} (p_{θ} \frac{\partial}{\partial \tilde{θ}} + \sum_{j = 3}^{2 n} p_{v_{j}} \frac{\partial}{\partial {\tilde{v}}_{j}}) . \end{matrix}$

Notice that on the right hand side of Eq. (63) we have $\begin{matrix} (65) & {(1 + ℏ M)}^{k} ({(a_{κ_{c}})}_{t} ϱ_{c} K_{n, p}^{ℏ} (\cdot, z)) = \sum_{s = 0}^{k} \sum_{q = 0}^{s} (\binom{k}{s}) (\binom{s}{q}) ℏ^{s} M^{s - q} [{(a_{κ_{c}})}_{t} ϱ_{c}] M^{q} K_{n, p}^{ℏ} (\cdot, z) . \end{matrix}$ For $q ⩾ 1$ , the action of the operator $M^{q}$ on the function $K_{n, p}^{ℏ} (\cdot, z)$ can be written as (see formula 0.430-2 of Ref. [12])) $\begin{array}{l} (66) & M^{q} K_{n, p}^{ℏ} (\cdot, z) (κ_{c}^{- 1} (\tilde{θ}, \tilde{v})) = & M^{q} g (\frac{\overline{z \cdot {\tilde{x}}_{c}}}{ℏ}) = \sum_{d = 1}^{q} F_{d, q}^{c} (\tilde{θ}, \tilde{v}) \frac{1}{ℏ^{d}} g^{(d)} (\frac{\overline{z \cdot {\tilde{x}}_{c}}}{ℏ}) \end{array}$ with $g = g_{\frac{1}{n - 1}}$ the function defined in (21) if $p = - 1$ or g the exponential function if $p = 0$ , ${\tilde{x}}_{c} = {\tilde{x}}_{c} (\tilde{θ}, \tilde{v}) = {(κ_{c} \circ Υ)}^{- 1} (\tilde{θ}, \tilde{v})$ and $\begin{matrix} (67) & F_{d, q}^{c} (\tilde{θ}, \tilde{v}) = \sum \frac{q!}{p_{1}! \dots p_{ℓ}!} {(\frac{M^{1} \overline{z \cdot {\tilde{x}}_{c}}}{1!})}^{p_{1}} {(\frac{M^{2} \overline{z \cdot {\tilde{x}}_{c}}}{2!})}^{p_{2}} \dots {(\frac{M^{ℓ} \overline{z \cdot {\tilde{x}}_{c}}}{ℓ!})}^{p_{ℓ}}, \end{matrix}$ where the summation stands for non-negative numbers $p_{1}, \dots, p_{ℓ}$ such that $p_{1} + 2 p_{2} + \dots + ℓ p_{ℓ} = q$ and $d = p_{1} + p_{2} + \dots + p_{ℓ}$ .

Let us first suppose that $p = - 1$ . Let $W_{z}$ , $V_{z}$ be the regions defined in Eq. (22). Notice that if ${\tilde{x}}_{c} \in W_{z}$ , $c = 1, \dots, 2 n - 1$ , then we can use the asymptotic expression of the derivative of any order of g evaluated at ${\tilde{x}}_{c} \cdot z / ℏ$ (see Eq. (37)). For this reason, let us regard the integration with respect to the variable $(\tilde{θ}, \tilde{v})$ in Eq. (63) over the regions $κ_{c} \circ Υ (W_{z} \cap U_{c})$ and $κ_{c} \circ Υ (V_{z} \cap U_{c})$ separately and let us call them $I_{W, c} (z)$ and $I_{V, c} (z)$ , respectively. Thus we have $\begin{matrix} \frac{{⟨ t_{c} A_{ℏ} ϱ_{c} K_{n, - 1}^{ℏ} (\cdot, z), K_{n, - 1}^{ℏ} (\cdot, z) ⟩}_{S^{n}}}{‖ K_{n, - 1}^{ℏ} (\cdot, z) ‖_{S^{n}}^{2}} = I_{W, c} (z) + I_{V, c} (z) . \end{matrix}$

Now we claim that $I_{V, c} (z) = O (ℏ^{\infty})$ . Let $(\tilde{θ}, \tilde{v}) \in κ_{c} \circ Υ (V_{z} \cap U_{c})$ and ${\tilde{x}}_{c} = {(κ_{c} \circ Υ)}^{- 1} (\tilde{θ}, \tilde{v})$ .

From definition of the operator M (see Eq. (64)) we have $| M^{b} {\tilde{x}}_{c} \cdot z | ⩽ {(1 + p_{θ}^{2} + p_{v}^{2})}^{\frac{b}{2}} | N^{b} {\tilde{x}}_{c} \cdot z |$ , where $N = \frac{\partial}{\partial \tilde{θ}} + \sum_{j = 3}^{2 n} \frac{\partial}{\partial {\tilde{v}}_{j}}$ . Note that the action of the operator N on the function ${\tilde{x}}_{c} \cdot z$ involve inverse powers of the variable $\tilde{r} = {(1 - | \tilde{v} |^{2})}^{1 / 2}$ which are bounded in the support of $ϱ_{c}$ and therefore do not create any singularities; in fact $| N^{b} {\tilde{x}}_{c} \cdot z | ⩽ C_{1} {\tilde{r}}^{- b} ⩽ C_{1} {(2 \sqrt{2 n - 2})}^{b}$ for some constant $C_{1}$ . Then, from Eq. (67) $\begin{array}{l} (68) & | F_{d, q}^{c} (\tilde{θ}, \tilde{v}) | & ⩽ C_{1} {⟨ (p_{θ}, p_{v}) ⟩}^{d}, \forall {(κ_{c} \circ Υ)}^{- 1} (\tilde{θ}, \tilde{v}) \in supp (ϱ_{c}) . \end{array}$ On the other hand, since $a_{κ_{c}} \in S_{4 n - 2} ({⟨ (p_{θ}, p_{v}) ⟩}^{m})$ $\begin{matrix} (69) & | M^{b} [{(a_{κ_{c}})}_{t} ϱ_{c}] | ⩽ {⟨ (p_{θ}, p_{v}) ⟩}^{b} | N^{b} [{(a_{κ_{c}})}_{t} ϱ_{c}] | ⩽ C_{1} {⟨ (p_{θ}, p_{v}) ⟩}^{b + m} . \end{matrix}$

Then, from Eqs (63), (65), (66), (68), (69), Proposition 3.3 (specifically Eq. (38)) and the estimate of the norm of $K_{n, - 1}^{ℏ} (\cdot, z)$ (see Eq. (19)) we have $\begin{array}{l} | I_{V, c} (z) | ⩽ & \frac{C_{1} e^{μ | z | / ℏ}}{ℏ^{\frac{10 n - 3}{4}} ‖ K_{n, - 1}^{ℏ} (\cdot, z) ‖} \\ \times \int_{(θ, v) \in A} \int_{(p_{θ}, p_{v}) \in R^{2 n - 1}} \frac{| (\overline{K_{n, - 1}^{ℏ} (\cdot, z)} t_{c}) ({(κ_{c} \circ Υ)}^{- 1} (θ, v)) |}{{(1 + p_{θ}^{2} + p_{v}^{2})}^{\frac{k - m}{2}}} d p_{θ} d p_{v} d θ d v \end{array}$ with $μ < 0$ . Using the Cauchy–Schwarz inequality and that $n < k - m$ we conclude that $I_{V, c} (z) = O (ℏ^{\infty})$ .

Let us now study the terms $I_{W, c} (z)$ . From Eqs (63), (65), (66), (67), the estimate of the norm of $K_{n, - 1}^{ℏ} (\cdot, z)$ (see Eq. (19)) and Proposition 3.3 (specifically Eq. (37) with $a = \frac{1}{n - 1}$ ) we have $\begin{array}{l} I_{W, c} (z) = & \frac{ℏ | z |^{n - \frac{3}{2}} (n - 1) e^{\frac{- 2 | z |}{ℏ}}}{{(π ℏ)}^{n - \frac{1}{2}} {(2 π ℏ)}^{2 n - 1}} \\ \times \int_{(θ, v) \in κ_{c} \circ Υ (W_{z} \cap U_{c})} \int_{(\tilde{θ}, \tilde{v}) \in κ_{c} \circ Υ (W_{z} \cap U_{c})} \int_{(p_{θ}, p_{v}) \in R^{2 n - 1}} {(\frac{z \cdot x_{c}}{ℏ (n - 1)})}^{\frac{1}{2}} \\ \times t_{c} (x_{c}) exp (\frac{ı}{ℏ} [(θ, v) - (\tilde{θ}, \tilde{v})] \cdot (p_{θ}, p_{v})) exp (\frac{z \cdot x_{c}}{ℏ}) \\ \times \sum_{s = 0}^{k} \sum_{q = 0}^{s} \sum_{d = 1}^{q} (\binom{k}{s}) (\binom{s}{q}) \frac{ℏ^{s - d}}{{(1 + p_{θ}^{2} + p_{v}^{2})}^{k}} M^{s - q} [{(a_{κ_{c}})}_{t} ϱ_{c} ({\tilde{x}}_{c})] F_{d, q}^{c} (\tilde{θ}, \tilde{v}) \\ \times {(\frac{\overline{z \cdot {\tilde{x}}_{c}}}{ℏ (n - 1)})}^{\frac{1}{2}} exp (\frac{\overline{z \cdot {\tilde{x}}_{c}}}{ℏ}) [1 + O (ℏ)] d p_{θ} d p_{v} d \tilde{θ} d \tilde{v} d θ d v \end{array}$ with ${\tilde{x}}_{c} = {\tilde{x}}_{c} (\tilde{θ}, \tilde{v}) = {(κ_{c} \circ Υ)}^{- 1} (\tilde{θ}, \tilde{v})$ , $x_{c} = x_{c} (θ, v) = {(κ_{c} \circ Υ)}^{- 1} (θ, v)$ .

Then, as $z = λ {\hat{u}}_{1}$ we have $\begin{array}{l} I_{W, c} (z) = & \frac{λ^{n - \frac{3}{2}}}{{(2 π ℏ)}^{2 n - 1} {(π ℏ)}^{n - \frac{1}{2}}} \\ \times \int_{(θ, v) \in κ_{c} \circ Υ (W_{z} \cap U_{c})} \int_{(\tilde{θ}, \tilde{v}) \in κ_{c} \circ Υ (W_{z} \cap U_{c})} \int_{(p_{θ}, p_{v}) \in R^{2 n - 1}} F_{d, q}^{c} (\tilde{θ}, \tilde{v}) \\ \times \frac{ℏ^{s - d} t_{c} (x_{c})}{{(1 + p_{θ}^{2} + p_{v}^{2})}^{k}} exp (\frac{ı}{ℏ} f_{c} (θ, v, \tilde{θ}, \tilde{v}, p_{θ}, p_{v})) {(λ f^{c} (θ, v))}^{\frac{1}{2}} \\ \times {(λ f^{c} (- \tilde{θ}, - \tilde{v}))}^{\frac{1}{2}} \sum_{s = 0}^{k} \sum_{q = 0}^{s} \sum_{d = 1}^{q} (\binom{k}{s}) (\binom{s}{q}) M^{s - q} [{(a_{κ_{c}})}_{t} (θ, v, \tilde{θ}, \tilde{v}, p_{θ}, p_{v}) \\ (70) & \times ϱ_{c} ({\tilde{x}}_{c})] [1 + O (ℏ)] d p_{θ} d p_{v} d \tilde{θ} d \tilde{v} d θ d v, \end{array}$ where $\begin{matrix} f_{c} (θ, v, \tilde{θ}, \tilde{v}, p_{θ}, p_{v}) = 2 ı λ + p_{θ} (θ - \tilde{θ}) + p_{v} (v - \tilde{v}) - ı λ [f^{c} (θ, v) + f^{c} (- \tilde{θ}, - \tilde{v})] \end{matrix}$ with $f^{1} (θ, v) = r e^{ı θ}$ and $f^{c} (θ, v) = - r cos θ + ı v_{c + 1}$ for $c = 2, \dots, 2 n - 1$ .

From Eqs (70), (68), (69) and using the arguments given above to study $I_{V, c} (z)$ , it can be show that $I_{W, c} (z) = O (ℏ^{\infty})$ for $c = 2, \dots, n$ .

Let us now study the term $I_{W, 1}$ . Since $a_{κ_{1}} \in S_{4 n - 2} ({⟨ (p_{θ}, p_{v}) ⟩}^{m})$ is a classical symbol, there exist a sequence $(a_{κ}^{(j)}) \subset S_{4 n - 2} ({⟨ (p_{θ}, p_{v}) ⟩}^{m})$ (independent of ℏ) such that $a_{κ_{1}}$ is asymptotically equivalent to the formal sum $\sum_{ℓ = 0}^{\infty} ℏ^{ℓ} a_{κ}^{(ℓ)}$ . Then, if $N > 3 n / 2$ and $Ψ_{N} = \sum_{ℓ = 0}^{N} ℏ^{ℓ} a_{κ}^{(ℓ)}$ , then for any $b \in Z_{+}^{2 n - 1}$ , there exist $ℏ_{N, b} > 0$ and $C_{N, b}$ such that $\begin{matrix} (71) & | \partial^{b} (a_{κ_{1}} - Ψ_{N}) | ⩽ C_{N, b} ℏ^{N} {⟨ (p_{θ}, p_{v}) ⟩}^{m} \end{matrix}$ uniformly in $R^{4 n - 2} \times (0, ℏ_{N, b})$ .

Let us denote by $I_{W, 1}^{1} (z)$ and $I_{W, 1}^{2} (z)$ the right side of Eq. (70) with $a_{κ_{1}} - Ψ_{N}$ and $Ψ_{N}$ instead of $a_{κ_{1}}$ respectively. Note that $I_{W, 1} (z) = I_{W, 1}^{1} (z) + I_{W, 1}^{2} (z)$ . Furthermore, from Eqs (70), (68), (69) and (71) it can be shown that ${lim}_{ℏ \to 0} I_{W, 1}^{1} (z) = 0$

Finally, let us use the stationary phase method to study the term $I_{W, 1}^{2} (z)$ .

The function $f_{1}$ has a non-negative imaginary part and one critical point $ϑ_{0}$ that contributes to the main asymptotic given by $\begin{matrix} ϑ_{0} : = (θ_{0}, v_{0}, {\tilde{θ}}_{0}, {\tilde{v}}_{0}, p_{θ_{0}}, p_{v_{0}}) = (0, 0, 0, 0, - λ, 0) . \end{matrix}$

Since $det (f_{1}^{″} (ϑ_{0}) / 2 π ı ℏ) = λ^{2 n - 1} / 2^{4 n - 2} {(π ℏ)}^{6 n - 3}$ (with $f_{1}^{″}$ the Hessian matrix of $f_{1}$ ), $f_{1} (ϑ_{0}) = 0$ , $t_{1} ({(κ_{1} \circ Υ)}^{- 1} (θ_{0}, v_{0})) = t_{1} ({\hat{u}}_{1}) = 1$ and $ϱ_{1} ({\hat{u}}_{1}) = 1$ , then we obtain from the stationary phase method (see Ref. [13]) $\begin{array}{l} I_{W, 1}^{2} (z) & = a_{κ_{1}}^{(0)} (0, 0, - λ, 0) + O (ℏ) \\ = ℘ (A_{ℏ}) ({\hat{u}}_{1}, - λ ı {\hat{u}}_{1}) + O (ℏ) \end{array}$ where we have used that $F_{s, s}^{1} ({\tilde{θ}}_{0}, {\tilde{v}}_{0}) = {(λ^{2})}^{s}$ and the definition of the principal symbol. Thus, we have proved Eq. (59) if $p = - 1$ .

The case $p = 0$ is proved in the same way. □

Footnotes

Acknowledgements

It is a pleasure to thank the referees of the present paper for their profound and careful reading, and for their valuable suggestions that substantially improved the paper.

Pseudo-differential operators

In this appendix, we give a brief description of what we mean by semiclassical pseudo-differential operators on a manifold. See [16] and [18] for details on definitions of semiclassical pseudo-differential operators on $R^{n}$ and manifolds, respectively.

Now we will give the definition of semiclassical pseudo-differential operator on $R^{n}$ . In order

Now we will give a brief description of the concept of semiclassical pseudo-differential operator on M.

Since $S^{n}$ can be identified with $S^{2 n - 1}$ via the map Υ, which sends $x = (x_{1}, \dots, x_{n}) \in S^{n}$ to $(ℜ (x_{1}), ℑ (x_{1}), \dots, ℜ (x_{n}), ℑ (x_{n})) \in S^{2 n - 1}$ , we can consider $S^{n}$ as a smooth $2 n - 1$ dimensional manifold.

Let $A_{ℏ}$ be a semiclassical pseudo-differential operator on $S^{n}$ . Based on relating corresponding properties of functions and $A_{ℏ}$ on a given chart and its rotated chart under the action of $SU (n)$ on the complex n-sphere $S^{n}$ , we can show the relationship between the principal symbols of $A_{ℏ}$ and $T_{U^{- 1}} A_{ℏ} T_{U}$ (with $U \in SU (n)$ and $T_{U}$ defined as in Proposition 2.3) as the following proposition establish it. This proposition will be useful to prove Theorem 5.1. Proposition A.6.

Let $U \in SU (n)$ and $A_{ℏ}$ be a semiclassical pseudo-differential operator on $S^{n}$ . Then $T_{U^{- 1}} A_{ℏ} T_{U}$ is a semiclassical pseudo-differential operator on $S^{n}$ with principal symbol $\begin{matrix} ℘ (T_{U^{- 1}} A_{ℏ} T_{U}) (x, η) = ℘ (A_{ℏ}) (U x, {(U^{- 1})}^{*} η), \forall (x, η) \in T^{*} (S^{n}) . \end{matrix}$

Proof.

Let $(x, η) \in T^{*} (S^{n})$ and $(X_{τ}, τ)$ be any chart such that $x \in X_{τ}$ . Let us denote by $(X_{\tilde{τ}}, \tilde{τ})$ the chart given by $X_{\tilde{τ}} = U X_{τ}$ and $\tilde{τ} = τ \circ U^{- 1}$ .

Let $v \in C^{\infty} (S^{n})$ , and $ϕ, ψ \in C_{0}^{\infty} (X_{τ})$ . Let us denote by $\tilde{v} = T_{U} v$ , $\tilde{ψ} = T_{U} ψ$ and $\tilde{ϕ} = T_{U} ϕ$ . Note that $\tilde{ψ}, \tilde{ϕ} \in C_{0}^{\infty} (X_{\tilde{τ}})$ . From Eq. (72) $\begin{array}{l} (ψ T_{U^{- 1}} A_{ℏ} T_{U} (ϕ v)) (x) & = (\tilde{ψ} A_{ℏ} \tilde{ϕ} \tilde{v}) (U x) \\ = [\tilde{ψ} {(\tilde{τ})}^{*} {Op}_{ℏ}^{t} (a_{\tilde{τ}}) {({\tilde{τ}}^{- 1})}^{*} (\tilde{ϕ} \tilde{v})] (U x) \\ = [ψ τ^{*} {Op}_{ℏ}^{t} (a_{\tilde{τ}}) {(τ^{- 1})}^{*} (ϕ v)] (x) \end{array}$ which establishes that condition $(A)$ holds for the operator $T_{U^{- 1}} A_{ℏ} T_{U}$ .

Condition $(B)$ follows from Eq. (73) and the $SU (n)$ -invariance of $d S^{n}$ .

Let us now consider another chart $(X_{τ^{'}}, τ^{'})$ such that $X_{τ} \cap X_{τ^{'}} = \emptyset$ . Condition $(C)$ follows from the equalities $τ (X_{τ}) = \tilde{τ} (X_{\tilde{τ}})$ , $τ (X_{τ^{'}}) = \tilde{τ^{'}} (X_{\tilde{τ^{'}}})$ and $\tilde{τ^{'}} \circ {(\tilde{τ})}^{- 1} = τ^{'} \circ τ^{- 1}$ .

Thus we conclude that $T_{U^{- 1}} A_{ℏ} T_{U}$ is a pseudo-differential operator with principal symbol (see Eq. (74)). $\begin{array}{l} ℘ (T_{U^{- 1}} A_{ℏ} T_{U}) (x, η) & = a_{\tilde{τ}}^{(0)} (τ x, {(τ^{- 1})}^{*} η) \\ = a_{\tilde{τ}}^{(0)} (\tilde{τ} \circ U x, {({\tilde{τ}}^{- 1})}^{*} {(U^{- 1})}^{*} η) \\ = ℘ (A_{ℏ}) (U x, {(U^{- 1})}^{*} η) . \end{array}$ □

The complex n-sphere S n as the union of the charts U c

In this appendix we prove that $S^{n} = ⋃_{c = 1}^{2 n - 1} U_{c}$ and that ${\hat{u}}_{1} \notin U_{a}$ for $a = 2, \dots, 2 n - 1$ , where ${\hat{u}}_{1} = (1, 0, \dots, 0)$ is a canonical vector in $R^{n}$ .

To prove that $S^{n} = ⋃_{c = 1}^{2 n - 1} U_{c}$ , let us first note that $\begin{matrix} S^{n} = Υ^{- 1} ({(r cos θ, r sin θ, v) | - π < θ ⩽ π, | v | ⩽ 1, v \in R^{2 n - 2}, r^{2} = 1 - | v |^{2}}) \end{matrix}$ where Υ is the map that identify $S^{n}$ with $S^{2 n - 1}$ (see its definition at the beginning of the proof of Theorem 5.1).

Consider $x \in S^{n}$ and write it as $x = Υ^{- 1} ((r cos θ, r sin θ, v))$ . Let us assume that $x \notin U_{1}$ . Then we have the following two cases:

$1 - \frac{1}{8 (n - 1)} ⩽ | v |^{2} ⩽ 1$ .

$θ = π$ and $| v |^{2} < 1 - \frac{1}{8 (n - 1)}$ .

Case (i) Let

v_{j} = max {| v_{3} |, \dots, | v_{2 n} |}

, then

| v_{j} | > \frac{1}{2 \sqrt{n - 1}}

, because otherwise

| v |^{2} ⩽ \frac{1}{2} < 1 - \frac{1}{8 (n - 1)}

Let us define $\tilde{v} = ({\tilde{v}}_{3}, \dots, {\tilde{v}}_{2 n}) \in R^{2 n - 2}$ such that ${\tilde{v}}_{i} = v_{i}$ , $i \neq j$ and ${\tilde{v}}_{j} = r sin (θ)$ . Since $\begin{matrix} {(- \frac{r}{\tilde{r}} cos (θ))}^{2} + {(\frac{v_{j}}{\tilde{r}})}^{2} = 1, where {\tilde{r}}^{2} = 1 - | \tilde{v} |^{2} = r^{2} {cos}^{2} (θ) + v_{j}^{2} > 0, \end{matrix}$ then there exist $- π ⩽ \tilde{θ} < π$ such that $cos (\tilde{θ}) = - \frac{r}{\tilde{r}} cos (θ)$ and $sin (\tilde{θ}) = \frac{v_{j}}{\tilde{r}}$ . Actually $\tilde{θ} \neq - π$ because otherwise $v_{j}$ would have to be equal to zero.

Since $| \tilde{v} |^{2} = 1 - r^{2} {cos}^{2} (θ) - v_{j}^{2} < 1 - \frac{1}{8 (n - 1)}$ then $(\tilde{θ}, \tilde{v}) \in A$ . Moreover, using the explicit expression of the matrix $R_{j - 1}$ (see Eq. (60)), one can check that $x = {(κ_{j - 1} \circ Υ)}^{- 1} (\tilde{θ}, \tilde{v}) \in U_{j - 1}$ .

Case (ii) Let us define $v_{j}$ and the vector $\tilde{v}$ as above. Note that ${\tilde{r}}^{2} = 1 - | \tilde{v} |^{2} = r^{2} + v_{j}^{2} ⩾ r^{2} = 1 - | v |^{2} > 0$ .

Consider $- π < \tilde{θ} < π$ such that $cos (\tilde{θ}) = \frac{r}{\tilde{r}}$ and $sin (\tilde{θ}) = \frac{v_{j}}{\tilde{r}}$ . Moreover, $| \tilde{v} |^{2} = 1 - r^{2} - v_{j}^{2} ⩽ | v |^{2} < 1 - \frac{1}{8 (n - 1)}$ . Therefore, $(\tilde{θ}, \tilde{v}) \in A$ and one can check that $x \in U_{j - 1}$ .

Let us now prove that ${\hat{u}}_{1}$ only belongs to $U_{1}$ . Let us assume that ${\hat{u}}_{1} \in U_{a}$ for some $a = 2, \dots, 2 n - 1$ , then there exist $(θ, v) \in A$ such that $\begin{matrix} {\hat{u}}_{1} = {(κ_{a} \circ Υ)}^{- 1} (θ, v) = Υ^{- 1} ((- r cos θ, v_{a + 1}, v_{3}, \dots, v_{a}, r sin θ, v_{a + 2}, \dots, v_{2 n})), \end{matrix}$ which is impossible since $- π < θ < π$ .

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