Bargmann-Type Transforms and Modified Harmonic Oscillators

Abstract

We study some complete orthonormal systems on the real line. These systems are determined by Bargmann-type transforms, which are Fourier integral operators with complex-valued quadratic phase functions. Each system consists of eigenfunctions for a second-order elliptic differential operator like the Hamiltonian of the harmonic oscillator. We also study the commutative case of a certain class of systems of second-order differential operators called the non-commutative harmonic oscillators. By using the diagonalization technique, we compute the eigenvalues and eigenfunctions for the commutative case of the non-commutative harmonic oscillators. Finally, we study a family of functions associated with an ellipse in the phase plane. We show that the family is a complete orthogonal system on the real line.

1 Introduction

We are concerned with the Bargmann-type transform defined by

$$\begin{aligned} T_hu(z) = C_\phi h^{-3/4} \int _{\mathbb {R}} e^{i\phi (z,x)/h} u(x) \mathrm{d}x, \quad z\in {\mathbb {C}}, \end{aligned}$$

(1)

where $\phi (z,x)$ is a complex-valued quadratic phase function of the form

$$\begin{aligned} \phi (z,x) = \frac{A}{2}z^2 + Bzx + \frac{C}{2}x^2, \quad A,B,C \in {\mathbb {C}} \end{aligned}$$

with assumptions $B\ne 0$ and ${\text {Im}}\,C>0$, and $C_\phi =2^{-1/2}\pi ^{-3/4}|{B}|({\text {Im}}\,C)^{-1/4}$. Throughout the present paper, we deal with only the one-dimensional case for the sake of simplicity. It is possible to discuss higher-dimensional case, and we omit the detail. Note that the integral transform (1) is well defined for tempered distributions on ${\mathbb {R}}$ since ${\text {Re}}(i\phi (z,x))={\mathcal {O}}(-{\text {Im}}\, C x^2/3)$ for $|{x}|\rightarrow \infty$.

These integral transforms were introduced by Sjöstrand (See, e.g., [13]). He developed microlocal analysis based on them. One can see (1) as a global Fourier integral operator associated with a linear canonical transform $\kappa _T:{\mathbb {C}}^2\ni (x,-\phi ^\prime _x(z,x))\mapsto (z,\phi ^\prime _z(z,x))\in {\mathbb {C}}^2$, that is,

$$\begin{aligned} \kappa _T: {\mathbb {C}}^2 \ni (x,\xi ) \mapsto \left( -\,\frac{Cx+\xi }{B},Bx-\frac{A(Cx+\xi )}{B}\right) \in {\mathbb {C}}^2. \end{aligned}$$

(2)

If we set $\Phi (z)=\max _{x\in {\mathbb {R}}}{\text {Re}}(i\phi (z,x))$, then we have

$$\begin{aligned} \Phi (z)= & {} \frac{|{Bz}|^2}{4\,{\text {Im}}\,C} - {\text {Re}} \left\{ \frac{(Bz)^2}{4\,{\text {Im}}\,C} + \frac{Az^2}{2i} \right\} ,\nonumber \\ \kappa _T({\mathbb {R}}^2)= & {} \left\{ \left( z,\frac{2}{i}\frac{\partial \Phi }{\partial z}(z)\right) \ \bigg \vert \ z\in {\mathbb {C}} \right\} . \end{aligned}$$

(3)

This means that the singularities of a tempered distribution u described in the phase plane ${\mathbb {R}}^2$ are translated into those of $T_hu$ in the I-Lagrangian submanifold $\kappa _T({\mathbb {R}}^2)$. The microlocal analysis of Sjöstrand is based on the equivalence of the Weyl quantization on ${\mathbb {R}}$, the Weyl quantization on $\kappa _T({\mathbb {R}})$, and the Berezin–Toeplitz quantization on ${\mathbb {C}}$. For more detail about them, see [13] or [4].

Let $L^2({\mathbb {R}})$ be the set of all square-integrable functions on ${\mathbb {R}}$, and let $L^2_\Phi ({\mathbb {C}})$ be the set of all square-integrable functions on ${\mathbb {C}}$ with respect to a weighted measure $e^{-2\Phi (z)/h}L(\mathrm{d}z)$, where L is the Lebesgue measure on ${\mathbb {C}}\simeq {\mathbb {R}}^2$. We denote by ${\mathscr {H}}_\Phi ({\mathbb {C}})$ the set of all entire functions in $L^2_\Phi ({\mathbb {C}})$. It is well known that $T_h$ gives a Hilbert space isomorphism of $L^2({\mathbb {R}})$ onto ${\mathscr {H}}_\Phi ({\mathbb {C}})$, that is,

$$\begin{aligned} (T_hu,T_hv)_{L^2_\Phi } = (u,v)_{L^2} \quad u,v \in L^2({\mathbb {R}}), \end{aligned}$$

where

$$\begin{aligned} (U,V)_{L^2_\Phi }= & {} \int _{\mathbb {C}} U(z)\overline{V(z)} e^{-2\Phi (z)/h} L(\mathrm{d}z), \quad U,V \in L^2_\Phi ({\mathbb {C}}), \\ (u,v)_{L^2}= & {} \int _{{\mathbb {R}}} u(x)\overline{v(x)} \mathrm{d}x \quad u,v \in L^2({\mathbb {R}}). \end{aligned}$$

We sometimes denote $(U,V)_{L^2_\Phi }$ for $U,V \in {\mathscr {H}}_\Phi ({\mathbb {C}})$ by $(U,V)_{{\mathscr {H}}_\Phi }$. The inverse mapping $T_h^*$ is given by

$$\begin{aligned} T_h^*U(x) = C_\phi \int _{\mathbb {C}} e^{-i \overline{\phi (z,x)}/h} U(z)e^{-2\Phi (z)/h} L(\mathrm{d}z), \quad x\in {\mathbb {R}}, \quad U \in {\mathscr {H}}_\Phi ({\mathbb {C}}). \end{aligned}$$

Note that $T_h^*$ is well defined for $U \in L^2_\Phi ({\mathbb {C}})$. $T_h{\circ }T_h^*$ becomes an orthogonal projector of $L^2_\Phi ({\mathbb {C}})$ onto ${\mathscr {H}}_\Phi ({\mathbb {C}})$. More concretely,

$$\begin{aligned} T_h{\circ }T_h^*U(z) = \frac{C_\Phi }{h} \int _{\mathbb {C}} e^{2\Psi (z,{\bar{\zeta }})/h} U(\zeta ) e^{-2\Phi (\zeta )/h} L(\mathrm{d}\zeta ), \quad U \in L^2_\Phi ({\mathbb {C}}), \end{aligned}$$

(4)

where $C_\Phi =|{B}|^2/(2\pi \,{\text {Im}}\,C)$, and $\Psi (z,\zeta )$ is a holomorphic quadratic function defined by the critical value of $-\{\phi (z,X)-\overline{\phi ({\bar{\zeta }},{\bar{X}})}\}/2i$ for $X\in {\mathbb {C}}$, that is,

$$\begin{aligned} \Psi (z,\zeta ) = \frac{|{B}|^2z\zeta }{4\,{\text {Im}}\, C} - \frac{B^2z^2+{\bar{B}}^2\zeta ^2}{8\,{\text {Im}}\, C} - \frac{Az^2-{\bar{A}}\zeta ^2}{4i}. \end{aligned}$$

(5)

In particular, $U=T_h{\circ }T_h^*U$ for $U \in {\mathscr {H}}_\Phi ({\mathbb {C}})$, and ${\mathscr {H}}_\Phi ({\mathbb {C}})$ becomes a reproducing kernel Hilbert space.

Here, we recall elementary facts related with the classical Bargmann transform $B_h$ which is the most important example of $T_h$. This was introduced by Bargmann in [3]. We can also refer [8] for this. The Bargmann transform $B_h$ on ${\mathbb {R}}$ is defined by

$$\begin{aligned} B_h(z) = 2^{-1/2}(\pi h)^{-3/4} \int _{\mathbb {R}} e^{-(z^2/4-zx+x^2/2)/h} u(x) \mathrm{d}x, \quad z\in {\mathbb {C}}. \end{aligned}$$

Note that the integral kernel of $B_h$ is the generating function of Hermite functions. We denote $L_\Phi ({\mathbb {C}})$ and ${\mathscr {H}}_\Phi ({\mathbb {C}})$ for $B_h$ by $L^2_B({\mathbb {C}})$ and ${\mathscr {H}}_B({\mathbb {C}})$, respectively, those are,

$$\begin{aligned} L^2_B({\mathbb {C}}) = \left\{ U(z) \ \bigg \vert \ \int _{\mathbb {C}} |{U(z)}|^2 e^{-|{z}|^2/2h} L(\mathrm{d}z) < \infty \right\} , \end{aligned}$$

and ${\mathscr {H}}_B({\mathbb {C}})=\{U(z) \in L^2_B({\mathbb {C}})\ \vert \ \partial U/\partial {\bar{z}}=0\}$. The Bargmann projector, which is the orthogonal projection of $L_B^2({\mathbb {C}})$ onto ${\mathscr {H}}_B({\mathbb {C}})$, is given by

$$\begin{aligned} B_h{\circ }B_h^*U(z) = \frac{1}{2\pi h} \int _{\mathbb {C}} e^{z{\bar{\zeta }}/2h} U(\zeta ) e^{-|\zeta |^2/2h} L(\mathrm{d}\zeta ), \quad U \in L^2_B({\mathbb {C}}). \end{aligned}$$

In view of the Taylor expansion of the reproducing kernel $e^{z{\bar{\zeta }}/2h}/(2\pi h)$, the formula $U=B_h{\circ }B_h^*U$ for $U \in {\mathscr {H}}_B({\mathbb {C}})$ becomes

$$\begin{aligned} U(z) = \sum _{n=0}^\infty (U,\varphi _{B,n})_{{\mathscr {H}}_B} \varphi _{B,n}(z), \quad \varphi _{B,n}(z) = \frac{z^n}{\sqrt{\pi (2h)^{n+1} n!}}, \quad n=0,1,2,\ldots . \end{aligned}$$

A family of functions $\{\varphi _{B,n}\}_{n=0}^\infty$ is a complete orthonormal system of ${\mathscr {H}}_B({\mathbb {C}})$ since ${\mathscr {H}}_B({\mathbb {C}})$ is the set of all entire functions belonging to $L^2_B({\mathbb {C}})$.

We shall see more detail about $\{\varphi _{B,n}\}_{n=0}^\infty$. We set for $U \in {\mathscr {H}}_B({\mathbb {C}})$

$$\begin{aligned} \Lambda _BU(z) := 2h \frac{\partial U}{\partial z}(z), \quad \Lambda _B^*U(z) := zU(z). \end{aligned}$$

Actually, $\Lambda _B^*$ is the adjoint of $\Lambda _B$ on ${\mathscr {H}}_B({\mathbb {C}})$. Elementary computation gives

$$\begin{aligned} (\Lambda _B^*\circ \Lambda _B+h)\varphi _{B,n} = (\Lambda _B\circ \Lambda _B^*-h)\varphi _{B,n} = (2n+1)h\varphi _{B,n}, \quad n=0,1,2,\ldots .\nonumber \\ \end{aligned}$$

(6)

We shall pullback these facts on ${\mathbb {R}}$ by using $B_h^*$. Set

$$\begin{aligned} \phi _{B,n}(x):=B_h^*\varphi _{B,n}(x), \quad P_B:=B_h^*\circ \Lambda _B \circ B_h, \quad P_B^*:=B_h^*\circ \Lambda _B^*\circ B_h. \end{aligned}$$

$\phi _{B,n}$ is said to be the nth Hermite function, and a family $\{\phi _{B,n}\}_{n=0}^\infty$ is a complete orthonormal system of $L^2({\mathbb {R}})$ since $B_h$ is a Hilbert space isomorphism of $L^2({\mathbb {R}})$ onto ${\mathscr {H}}_B({\mathbb {C}})$. Operators

$$\begin{aligned} P_B = h\frac{\mathrm{d}}{\mathrm{d}x}+x, \quad P_B^*= -h\frac{\mathrm{d}}{\mathrm{d}x}+x \end{aligned}$$

are said to be annihilation and creation operators, respectively. Note that

$$\begin{aligned} \varphi _{B,n}= & {} ((2h)^nn!)^{-1/2}(\Lambda _B^*)^n\varphi _{B,0}, \quad n=0,1,2,\ldots , \\ \phi _{B,0}(x):= & {} B_h^*\varphi _{B,0}(x)=\frac{1}{(\pi h)^{1/4}}e^{-x^2/2h}. \end{aligned}$$

Then, we have so-called Rodrigues formula

$$\begin{aligned} \phi _{B,n}(x) = \frac{1}{\sqrt{(2h)^nn!}} (P_B^*)^n \phi _{B,0}(x) = \frac{(-1)^n}{(\pi h)^{1/4}\sqrt{(2h)^nn!}} e^{-x^2/2h} \left( h\frac{\mathrm{d}}{\mathrm{d}x}\right) ^n e^{-x^2/h}. \end{aligned}$$

Set $H_B=P_B^*\circ P_B + h = P_B \circ P_B^*- h$. Then

$$\begin{aligned} H_B= -h^2\frac{\mathrm{d}^2}{\mathrm{d}x^2}+x^2 \end{aligned}$$

which is said to be the Hamiltonian of the harmonic oscillator. Equation (6) becomes

$$\begin{aligned} H_B\phi _{B,n}=(2n+1)h\phi _{B,n}, \quad n=0,1,2,\ldots . \end{aligned}$$

Thus, the nth Hermite function $\phi _{B,n}$ is an eigenfunction of $H_B$ for the nth eigenvalue $(2n+1)h$.

The purpose of the present paper is to study the generalization of the known facts on the usual Bargmann transform $B_h$. The plan of this paper is as follows. In Sect. 2, we study the general Bargmann-type transform (1) and obtain generalized annihilation and creation operators, the Hamiltonian of the generalized harmonic oscillator and its eigenvalues, generalized Hermite functions and the Rodrigues formula. In Sect. 3, we study a $2\times 2$-system of second-order ordinary differential operators, which is said to be a non-commutative harmonic oscillators. More precisely, we study the commutative case of the non-commutative harmonic oscillators and obtain the eigenvalues and eigenfunctions by using our original elementary computation. Finally, in Sect. 4, we study the general Bargmann-type transform (1) which might be related with ellipses in the phase plane ${\mathbb {R}}^2$.

2 Modified Harmonic Oscillators and Hermite Functions

In this section, we study the general form of the Bargmann-type transform (1). We remark that the choice of the constant A in the phase function is not essential. We can choose

$$\begin{aligned} A = - \frac{iB^2}{2\,{\text {Im}}\, C}. \end{aligned}$$

Then, (3) and (5) become very simple as

$$\begin{aligned} \Phi (z) = \frac{|{Bz}|^2}{4\,{\text {Im}}\, C}, \quad \Psi (z,{\bar{\zeta }}) =\frac{Bz \cdot \overline{B\zeta }}{4\,{\text {Im}}\, C} \end{aligned}$$

respectively. Moreover, the orthogonal projector (4) of $L^2_\Phi ({\mathbb {C}})$ onto ${\mathscr {H}}_\Phi ({\mathbb {C}})$, and the I-Lagrangian submanifold (2) become

$$\begin{aligned} T_h \circ T_h^*U(z)= & {} \frac{C_\Phi }{h} \int _{{\mathbb {C}}} e^{Bz \cdot \overline{B\zeta }/2h \,{\text {Im}}\, C} U(\zeta ) e^{-|{Bz}|^2/2h \,{\text {Im}}\, C} L(\mathrm{d}\zeta ), \\ \kappa _T({\mathbb {R}}^2)= & {} \left\{ \left( z,\frac{|{B}|^2}{2i\,{\text {Im}}\, C}{\bar{z}}\right) \ \bigg \vert \ z\in {\mathbb {C}} \right\} \end{aligned}$$

respectively. Recall $U(z)=T_h \circ T_h^*U(z)$ for all $U \in {\mathscr {H}}_\Phi ({\mathbb {C}})$. If we consider the Taylor expansion of $e^{-|{Bz}|^2/2h \,{\text {Im}}\, C}$, we have for $U \in {\mathscr {H}}_\Phi ({\mathbb {C}})$,

$$\begin{aligned} U(z)= & {} \sum _{n=0}^\infty (U,\varphi _n)_{{\mathscr {H}}_\Phi } \varphi (z), \\ \varphi _n(z)= & {} \frac{|{B}|}{\sqrt{2\pi h \,{\text {Im}}\, C}} \cdot \frac{1}{\sqrt{n!}} \cdot \left( \frac{Bz}{\sqrt{2h \,{\text {Im}}\, C}}\right) ^n, \quad n=0,1,2,\ldots . \end{aligned}$$

Theorem 2.1

The family of monomials $\{\varphi _n\}_{n=0}^\infty$ is a complete orthonormal system of ${\mathscr {H}}_\Phi ({\mathbb {C}})$.

Proof

The completeness is obvious. We have only to show that $(\varphi _m,\varphi _n)_{{\mathscr {H}}_\Phi }=\delta _{mn}$, where $\delta _{mn}$ is Kronecker’s delta. Without loss of generality, we may assume that $m \geqslant n$. By using the integration by parts and the change of variable $\zeta =Bz/\sqrt{h \,{\text {Im}}\, C}$, we deduce that

$$\begin{aligned} (\varphi _m,\varphi _n)_{{\mathscr {H}}_\Phi }&= \frac{|{B}|^2}{2\pi h \,{\text {Im}}\, C} \cdot \frac{1}{\{m!n!(2h \,{\text {Im}}\, C)^{m+n}\}^{1/2}} \\&\quad \times \int _{\mathbb {C}} (Bz)^m({\overline{Bz}})^n e^{-|{Bz}|^2/2h \,{\text {Im}}\, C} L(\mathrm{d}z) \\&= \frac{|{B}|^2}{2\pi h \,{\text {Im}}\, C} \cdot \frac{1}{\{m!n!(2h \,{\text {Im}}\, C)^{m+n}\}^{1/2}} \\&\quad \times \int _{\mathbb {C}} \left\{ \left( -\frac{2h \,{\text {Im}}\, C}{{\bar{B}}} \frac{\partial }{\partial {\bar{z}}}\right) ^m e^{-|{Bz}|^2/2h \,{\text {Im}}\, C} \right\} ({\overline{Bz}})^n L(\mathrm{d}z) \\&= \frac{|{B}|^2}{2\pi h \,{\text {Im}}\, C} \cdot \frac{1}{\{m!n!(2h \,{\text {Im}}\, C)^{m+n}\}^{1/2}} \\&\quad \times \int _{\mathbb {C}} \left\{ \left( -\frac{2h \,{\text {Im}}\, C}{{\bar{B}}} \frac{\partial }{\partial {\bar{z}}}\right) ^m ({\overline{Bz}})^n \right\} e^{-|{Bz}|^2/2h \,{\text {Im}}\, C} L(\mathrm{d}z) \\&= \delta _{mn} \frac{|{B}|^2}{2\pi h \,{\text {Im}}\, C} \cdot \frac{1}{m!(2h \,{\text {Im}}\, C)^m} \\&\quad \times \int _{\mathbb {C}} \left\{ \left( -\frac{2h \,{\text {Im}}\, C}{{\bar{B}}} \frac{\partial }{\partial {\bar{z}}}\right) ^m ({\overline{Bz}})^m \right\} e^{-|{Bz}|^2/2h \,{\text {Im}}\, C} L(\mathrm{d}z) \\&= \delta _{mn} \frac{|{B}|^2}{2\pi h \,{\text {Im}}\, C} \int _{\mathbb {C}} e^{-|{Bz}|^2/2h \,{\text {Im}}\, C} L(\mathrm{d}z) \\&= \delta _{mn} \frac{1}{2\pi } \int _{\mathbb {C}} e^{-|\zeta |^2/2} L(\mathrm{d}\zeta ) = \delta _{mn}. \end{aligned}$$

This completes the proof. $\square$

Set $\phi _n(x)=T_h^*\varphi _n(x)$, $n=0,1,2,\ldots$. Since $T_h$ is a Hilbert space isomorphism of $L^2({\mathbb {R}})$ onto ${\mathscr {H}}_\Phi ({\mathbb {C}})$, we have the following.

Theorem 2.2

$\{\phi _n\}_{n=0}^\infty$ is a complete orthonormal system of $L^2({\mathbb {R}})$.

In what follows we study the family of functions $\{\phi _n\}_{n=0}^\infty$ in detail. Let $\Lambda$ be a linear operator on ${\mathscr {H}}_\Phi ({\mathbb {C}})$ defined by

$$\begin{aligned} \Lambda U(z) = \frac{2h \,{\text {Im}}\,C}{|{B}|^2} \frac{\mathrm{d}U}{\mathrm{d}z}(z), \quad U \in {\mathscr {H}}_\Phi ({\mathbb {C}}). \end{aligned}$$

Its Hilbert adjoint is

$$\begin{aligned} \Lambda ^*U(z) = zU(z), \quad U \in {\mathscr {H}}_\Phi ({\mathbb {C}}). \end{aligned}$$

We call $\Lambda$ and $\Lambda ^*$ annihilation and creation operators on ${\mathscr {H}}_\Phi ({\mathbb {C}})$, respectively. Since $\varphi _n$ is a monomial of degree n, we have for $n=0,1,2,\ldots$

$$\begin{aligned} \varphi _n(z)= & {} \frac{1}{\sqrt{n!}} \left( \frac{B}{\sqrt{2h\,{\text {Im}}\,C}}\right) ^n \bigl (\Lambda ^*\bigr )^n \varphi _0(z), \end{aligned}$$

(7)

$$\begin{aligned} \left( \Lambda ^*\circ \Lambda + \frac{h\,{\text {Im}}\,C}{|{B}|^2} \right) \varphi _n(z)= & {} \left( \Lambda \circ \Lambda ^*- \frac{h\,{\text {Im}}\,C}{|{B}|^2} \right) \varphi _n(z) = \frac{h\,{\text {Im}}\,C}{|{B}|^2} (2n+1) \varphi _n(z). \nonumber \\ \end{aligned}$$

(8)

We shall pullback these facts by using $T_h^*$. Set

$$\begin{aligned}&P := T_h^*\circ \Lambda \circ T_h, \quad P^*:= T_h^*\circ \Lambda ^*\circ T_h, \\&H := P^*\circ P + \frac{h\,{\text {Im}}\,C}{|{B}|^2} = P \circ P^*- \frac{h\,{\text {Im}}\,C}{|{B}|^2}. \end{aligned}$$

To state the concrete form of H, we introduce the Weyl pseudodifferential operators. For an appropriate function $a(x,\xi )$ of $(x,\xi )\in {\mathbb {R}}^2$, its Weyl quantization is defined by

$$\begin{aligned} {\text {Op}}_h^\text {W}(a)u(x) = \frac{1}{2\pi h} \iint _{{\mathbb {R}}^2} e^{i(x-y)\xi /h} a\left( \frac{x+y}{2},\xi \right) u(y) \mathrm{d}y\mathrm{d}\xi \end{aligned}$$

for $u \in {\mathscr {S}}({\mathbb {R}})$, where ${\mathscr {S}}({\mathbb {R}})$ denotes the Schwartz class on ${\mathbb {R}}$. Set $D_x=-i\mathrm{d}/\mathrm{d}x$ for short.

Here, we give the concrete forms of operators P, $P^*$ and H on ${\mathbb {R}}$.

Proposition 2.3

We have

$$\begin{aligned} P&= - \frac{1}{{\bar{B}}} (hD_x+{\bar{C}}x), \quad P^*= - \frac{1}{B} (hD_x+Cx), \\ H&= \frac{1}{|{B}|^2} \left\{ h^2D_x^2 + |{C}|^2x^2 + (C+{\bar{C}}) \left( xhD_x+\frac{h}{2i} \right) \right\} \\&= \frac{1}{|{B}|^2} {\text {Op}}_h^\text {W} \Bigl ( \xi ^2 + |{C}|^2x^2 + (C+{\bar{C}})x\xi \Bigr ). \end{aligned}$$

Proof

We first compute P and $P^*$. Since $\Lambda \circ T_h = T_h \circ P$, we deduce that for any $u \in {\mathscr {S}}({\mathbb {R}})$,

$$\begin{aligned} \Lambda \circ T_h u(z)&= \frac{2h\,{\text {Im}}\,C}{|{B}|^2} \frac{\mathrm{d}}{\mathrm{d}z} C_\phi h^{-3/4} \int _{\mathbb {R}} e^{i\phi (z,x)/h} u(x) \mathrm{d}x \\&= C_\phi h^{-3/4} \int _{\mathbb {R}} e^{i\phi (z,x)/h} \left\{ \frac{2h\,{\text {Im}}\,C}{|{B}|^2} \frac{i}{h} \frac{\partial \phi }{\partial z}(z,x) \right\} u(x) \mathrm{d}x \\&= C_\phi h^{-3/4} \int _{\mathbb {R}} e^{i\phi (z,x)/h} \frac{2i\,{\text {Im}}\,C}{|{B}|^2} \Bigl ( Az+Bx \Bigr ) u(x) \mathrm{d}x \\&= C_\phi h^{-3/4} \int _{\mathbb {R}} e^{i\phi (z,x)/h} \frac{2i\,{\text {Im}}\,C}{|{B}|^2} \left( -\frac{iB^2}{2 \,{\text {Im}}\,C}z+Bx \right) u(x) \mathrm{d}x \\&= C_\phi h^{-3/4} \int _{\mathbb {R}} e^{i\phi (z,x)/h} \frac{1}{{\bar{B}}} \Bigl ( Bz+2i\,{\text {Im}}\,Cx \Bigr ) u(x) \mathrm{d}x \\&= C_\phi h^{-3/4} \int _{\mathbb {R}} \Bigl ( (hD_x-Cx) e^{i\phi (z,x)/h} \Bigr ) \frac{1}{{\bar{B}}} u(x) \mathrm{d}x \\&\quad +\, C_\phi h^{-3/4} \int _{\mathbb {R}} e^{i\phi (z,x)/h} \frac{2i\,{\text {Im}}\,Cx}{{\bar{B}}} u(x) \mathrm{d}x \\&= C_\phi h^{-3/4} \int _{\mathbb {R}} e^{i\phi (z,x)/h} \frac{-1}{{\bar{B}}} (hD_x+Cx) u(x) \mathrm{d}x \\&\quad + \,C_\phi h^{-3/4} \int _{\mathbb {R}} e^{i\phi (z,x)/h} \frac{2i\,{\text {Im}}\,Cx}{{\bar{B}}} u(x) \mathrm{d}x \\&= C_\phi h^{-3/4} \int _{\mathbb {R}} e^{i\phi (z,x)/h} \frac{-1}{{\bar{B}}} (hD_x+Cx-2i\,{\text {Im}}\,Cx) u(x) \mathrm{d}x \\&= C_\phi h^{-3/4} \int _{\mathbb {R}} e^{i\phi (z,x)/h} \frac{-1}{{\bar{B}}} (hD_x+{\bar{C}}x) u(x) \mathrm{d}x, \end{aligned}$$

which shows that $P=-(hD_x+{\bar{C}}x)/{\bar{B}}$. In the same way, we can obtain $P^*=-(hD_x+Cx)/B$, which is certainly the adjoint of P on $L^2({\mathbb {R}})$. Next, we compute H. Simple computation gives

$$\begin{aligned} H&= P^*\circ P + \frac{h\,{\text {Im}}\,C}{|{B}|^2} = \frac{1}{|{B}|^2} (hD_x+Cx)(hD_x+{\bar{C}}x) + \frac{h\,{\text {Im}}\,C}{|{B}|^2} \\&= \frac{1}{|{B}|^2} \left\{ h^2D_x^2 + |{C}|^2x^2 + (C+{\bar{C}})xhD_x + \frac{h{\bar{C}}}{i} + h\,{\text {Im}}\,C \right\} \\&= \frac{1}{|{B}|^2} \left\{ h^2D_x^2 + |{C}|^2x^2 + (C+{\bar{C}})xhD_x + \frac{h\,{\text {Re}}\,C}{i} \right\} \\&= \frac{1}{|{B}|^2} \left\{ h^2D_x^2 + |{C}|^2x^2 + (C+{\bar{C}}) \left( xhD_x + \frac{h}{2i} \right) \right\} , \end{aligned}$$

which completes the proof. $\square$

By using the pullback of (7) and (8), we have for $n=0,1,2,\ldots$

$$\begin{aligned} \phi _n(x)&= \frac{1}{\sqrt{n!}} \left( \frac{B}{\sqrt{2h\,{\text {Im}}\,C}}\right) ^n \bigl (P^*\bigr )^n \phi _0(x) \\&= \frac{1}{\sqrt{n!}} \left( \frac{-1}{\sqrt{2h\,{\text {Im}}\,C}}\right) ^n \bigl (hD_x+Cx\bigr )^n \phi _0(x) \\&= \frac{1}{\sqrt{n!}} \left( \frac{-1}{\sqrt{2h\,{\text {Im}}\,C}}\right) ^n e^{-iCx^2/2h} (hD_x)^n \bigl (e^{iCx^2/2h}\phi _0(x)\bigr ), \\ H\phi _n&= \frac{h\,{\text {Im}}\,{C}}{|{B}|^2} (2n+1)\phi _n. \end{aligned}$$

If we compute the concrete form of $\phi _0$, then we obtain the Rodrigues formula for $\{\phi _n\}_{n=0}^\infty$.

Theorem 2.4

We have for $n=0,1,2,\ldots$

$$\begin{aligned} \phi _0(x)&= \left( \frac{{\text {Im}}\,C}{\pi h}\right) ^{1/4} e^{-i{\bar{C}}x^2/2h}, \\ \phi _n(x)&= \left( \frac{{\text {Im}}\,C}{\pi h}\right) ^{1/4} \frac{1}{\sqrt{n!}} \left( \frac{-1}{\sqrt{2h\,{\text {Im}}\,C}}\right) ^n e^{-i\,{\text {Re}}\,C x^2/2h} e^{{\text {Im}}\,C x^2/2h} (hD_x)^n e^{-{\text {Im}}\,C x^2/h}. \end{aligned}$$

Proof

Recall the definition of $\phi _0$. We have

$$\begin{aligned} \phi _0(x)&= T_h^*\varphi _0(x) = C_\phi h^{-3/4} \int _{\mathbb {C}} e^{-i\overline{\phi (z,x)}/h} \varphi _0(z) e^{-2\Phi (z)/h} L(\mathrm{d}z) \\&= 2^{-1}(\pi h)^{-5/4}({\text {Im}}\,C)^{-3/4}|{B}|^2 \int _{\mathbb {C}} e^{F_1(z,x)/h} L(\mathrm{d}z), \end{aligned}$$

where

$$\begin{aligned} F_1(z,x) = -i \left\{ \frac{i({\overline{Bz}})^2}{4\,{\text {Im}}\,C} + {\overline{Bz}} + \frac{{\bar{C}}x^2}{2} \right\} - \frac{|{Bz}|^2}{2\,{\text {Im}}\,C}. \end{aligned}$$

Change the variable $\zeta =\xi +i\eta :={\overline{Bz}}$, $(\xi ,\eta )\in {\mathbb {R}}^2$. We deduce

$$\begin{aligned} \phi _0(x)&= 2^{-1}(\pi h)^{-5/4}({\text {Im}}\,C)^{-3/4} \int _{\mathbb {C}} e^{F_2(\zeta ,x)/h} L(\mathrm{d}\zeta ), \\ F_2(\zeta ,x)&= \frac{\zeta ^2}{4\,{\text {Im}}\,C} - i\zeta x - \frac{i{\bar{C}}x^2}{2} - \frac{|{\zeta }|^2}{2\,{\text {Im}}\,C} \\&= - \frac{\bigl \{\xi -i(\eta -2\,{\text {Im}}\,Cx)\bigr \}^2}{4\,{\text {Im}}\,C} - \frac{(\eta -{\text {Im}}\,Cx)^2}{{\text {Im}}\,C} - \frac{i{\bar{C}}x^2}{2}. \end{aligned}$$

Then, we can obtain

$$\begin{aligned} \phi _0(x)&= 2^{-1}(\pi h)^{-5/4}({\text {Im}}\,C)^{-3/4} e^{-i{\bar{C}}x^2/2h} \int _{\mathbb {R}} e^{-\xi ^2/4h\,{\text {Im}}\,C} \mathrm{d}\xi \int _{\mathbb {R}} e^{-\eta ^2/h\,{\text {Im}}\,C} \mathrm{d}\eta \\&= \left( \frac{{\text {Im}}\,C}{\pi h} \right) ^{1/4} e^{-i{\bar{C}}x^2/2h}. \end{aligned}$$

This completes the proof. $\square$

3 The Commutative Case of Non-commutative Harmonic Oscillators

Consider a $2\times 2$ system of second-order differential operators of the form

$$\begin{aligned} Q_{(\alpha ,\beta )} = \frac{1}{2} A_{(\alpha ,\beta )} \,{\text {Op}}_h^\text {W} (\xi ^2+x^2) + J \,{\text {Op}}_h^\text {W} (ix\xi ), \end{aligned}$$

where $\alpha$ and $\beta$ are positive constants satisfying $\alpha \beta >1$, and

$$\begin{aligned} A_{(\alpha ,\beta )} = \begin{bmatrix} \alpha&\quad 0 \\ 0&\quad \beta \end{bmatrix}, \quad J = \begin{bmatrix} 0&\quad -\,1 \\ 1&\quad 0 \end{bmatrix}. \end{aligned}$$

A matrix $A_{(\alpha ,\beta )}(\xi ^2+x^2)/2+J(ix\xi )$, which is the symbol of the operator $Q_{(\alpha ,\beta )}$, is a Hermitian matrix, and all its eigenvalues are real valued. Note that all its eigenvalues are positive for $(x,\xi )\ne (0,0)$ if and only if $\alpha \beta >1$. In other words, $Q_{(\alpha ,\beta )}$ is a system of semiclassical elliptic differential operators if and only if $\alpha \beta >1$. The system of differential operators $Q_{(\alpha ,\beta )}$ was mathematically introduced in [10] by Parmeggiani and Wakayama. They call $Q_{(\alpha ,\beta )}$ a Hamiltonian of non-commutative harmonic oscillator. The word “non-commutative” comes from the non-commutativity $A_{(\alpha ,\beta )}J \ne JA_{(\alpha ,\beta )}$ for $\alpha \ne \beta$. It is not known that the system of differential equations for $Q_{(\alpha ,\beta )}$ describes a physical phenomenon.

Parmeggiani and Wakayama intensively studied spectral properties of $Q_{(\alpha ,\beta )}$ in [10,11,12]. See also a monograph [9]. They proved that if $\alpha \beta >1$, then $Q_{(\alpha ,\beta )}$ is a self-adjoint and positive operator, and its spectra consists of positive eigenvalues whose multiplicities are at most three. In case of $\alpha =\beta$, they obtained more detail.

The purpose of the present section is to give alternative proof of the results of Parmeggiani and Wakayama for the commutative case $\alpha =\beta$. More precisely, we study $Q_{(\alpha ,\alpha )}$ by using the results in the previous section.

In what follows we assume that $\alpha =\beta$. Then, $\alpha >1$ since $\alpha >0$ and $\alpha ^2=\alpha \beta >1$. Let I be the $2\times 2$ identity matrix. Set $Q_\alpha =Q_{(\alpha ,\alpha )}$ for short. Let U be a $2\times 2$ unitary matrix defined by

$$\begin{aligned} U = \frac{1}{\sqrt{2}} \begin{bmatrix} 1&\quad -\,i \\ 1&\quad i \end{bmatrix} \end{aligned}$$

which diagonalize J as

$$\begin{aligned} UJU^*= \begin{bmatrix} -\,i&\quad 0\\ 0&\quad i \end{bmatrix}. \end{aligned}$$

Then, we have

$$\begin{aligned} U Q_\alpha U^*= & {} \frac{\alpha }{2} I\, {\text {Op}}_h^\text {W} (\xi ^2+x^2) + iUJU^*\,{\text {Op}}_h^\text {W}(x\xi ) = \begin{bmatrix} H_{\alpha ,+}&\quad 0 \\ 0&\quad H_{\alpha ,-} \end{bmatrix}, \\ H_{\alpha ,\pm }= & {} {\text {Op}}_h^\text {W} (\xi ^2+x^2) \pm {\text {Op}}_h^\text {W}(x\xi ) = {\text {Op}}_h^\text {W} \left( \left|\sqrt{\frac{\alpha }{2}} (\nu _{\alpha ,\pm }\xi +x) \right|^2 \right) , \\ \nu _{\alpha ,\pm }= & {} \frac{\pm 1+i\sqrt{\alpha ^2-1}}{\alpha }. \end{aligned}$$

Note that $|\nu _{\alpha ,\pm }|=1$ and ${\text {Im}}\,\nu _{\alpha ,\pm }>0$.

Here, we make use of the results in the previous section by setting

$$\begin{aligned} B = \sqrt{\frac{2}{\alpha }} \nu _{\alpha ,\pm }, \quad C=\nu _{\alpha ,\pm }, \quad A = \frac{B^2}{2i\,{\text {Im}}\,C} = \frac{\nu _{\alpha ,\pm }^2}{i\sqrt{\alpha ^2-1}}. \end{aligned}$$

Note that the requirement ${\text {Im}}\,C>0$ is satisfied. Set

$$\begin{aligned} \phi _{\alpha ,\pm ,n}(x)&= e^{\mp ix^2/2\alpha h} h_{\alpha ,n}(x), \\ h_{\alpha ,n}(x)&= \left( \frac{\sqrt{\alpha ^2-1}}{\pi h \alpha }\right) ^{1/4} \frac{1}{\sqrt{n!}} \left( -\sqrt{\frac{\alpha }{2\sqrt{\alpha ^2-1}h}}\right) ^n \\&\quad \times e^{\sqrt{\alpha ^2-1} x^2/2\alpha h} (hD_x)^n e^{-\sqrt{\alpha ^2-1} x^2/\alpha h} \end{aligned}$$

for $n=0,1,2,\ldots$. Then, we deduce that $\{\phi _{\alpha ,\pm ,n}\}_{n=0}^\infty$ is a complete orthonormal system of $L^2({\mathbb {R}})$, and

$$\begin{aligned} H_{\alpha ,\pm } \phi _{\alpha ,\pm ,n} = \frac{\sqrt{\alpha ^2-1}}{2}h(2n+1)\phi _{\alpha ,\pm ,n}, \quad n=0,1,2,\ldots . \end{aligned}$$

In order to get the eigenfunctions of $Q_\alpha$, we set

$$\begin{aligned} \Phi _{\alpha ,+,n}(x) = U^*\begin{bmatrix} \phi _{\alpha ,+,n} \\ 0 \end{bmatrix}, \quad \Phi _{\alpha ,-,n}(x) = U^*\begin{bmatrix} 0 \\ \phi _{\alpha ,-,n} \end{bmatrix}, \quad n=0,1,2,\ldots , \end{aligned}$$

those are,

$$\begin{aligned} \Phi _{\alpha ,\pm ,n}(x) = h_{\alpha ,n}(x) \cdot \frac{e^{\mp ix^2/2\alpha h}}{\sqrt{2}} \begin{bmatrix} 1 \\ \pm i \end{bmatrix}, \quad n=0,1,2,\ldots . \end{aligned}$$

We have proved the results of this section as follows.

Theorem 3.1

A system of ${\mathbb {C}}^2$-valued functions $\bigl \{\Phi _{\alpha ,\mu ,n} \ \big \vert \ \mu =\pm , n=0,1,2,\ldots .\bigr \}$ is a complete orthonormal system of $L^2({\mathbb {R}}) \oplus L^2({\mathbb {R}})$, and satisfies

$$\begin{aligned} Q_\alpha \Phi _{\alpha ,\pm ,n} = \frac{\sqrt{\alpha ^2-1}}{2} h(2n+1) \Phi _{\alpha ,\pm ,n}, \quad n=0,1,2,\ldots . \end{aligned}$$

This is not a new result. This was first proved by Parmeggiani and Wakayama in [10]. We believe that our method of proof is easier than that of [10].

4 Orthogonal Systems Associated with Ellipses in the Phase Plane

Throughout of the present section, we assume that $h=1$ for the sake of simplicity. We begin with recalling the relationship between the standard Bargmann transform $B_1$ and circles in the phase plane. Here, we introduce a Berezin–Toeplitz quantization on ${\mathbb {C}}$. Let b(z) be an appropriate function on ${\mathbb {C}}$. Set

$$\begin{aligned} a(x,\xi ) = \frac{1}{\pi } \iint _{{\mathbb {R}}^2} e^{-(x-y)^2-(\xi -\eta )^2} b(y-i\eta ) \mathrm{d}y\mathrm{d}\eta . \end{aligned}$$

It is known that

$$\begin{aligned} \left( {\text {Op}}^\text {W}_1(a)u,v\right) _{L^2} = (bB_1u,B_1v)_{L^2_B} = \left( {\tilde{T}}_bB_1u,B_1v\right) _{{\mathscr {H}}_B} \end{aligned}$$

for $u,v \in {\mathscr {S}}({\mathbb {R}})$. See, e.g., [13] and [4]. The operator ${\tilde{T}}_b$ is said to be the Berezin–Toeplitz quantization of b, which acts on ${\mathscr {H}}_B({\mathbb {C}})$. If b is a characteristic function on ${\mathbb {C}}$, then ${\text {Op}}^\text {W}_1(a)$ is said to be a Daubechies’ localization operator introduced in [7]. Moreover, Daubechies proved that if b(z) is radially symmetric, that is, b is of the form $b(x-i\xi )=c(x^2+\xi ^2)$ with some function c(s) for $s\geqslant 0$, then all the usual Hermite functions $\phi _{B,n}$ ($n=0,1,2,\ldots$) are the eigenfunctions of ${\tilde{T}}_b$:

$$\begin{aligned} {\tilde{T}}_b\phi _{B,n}=\lambda _n\phi _{B,n}, \quad \lambda _n = \frac{1}{n!} \int _0^\infty c(2s)s^ne^{-s} \mathrm{d}s, \quad n=0,1,2,\ldots . \end{aligned}$$

Recently, Daubechies’ results have been developed. Here, we quote two interesting results of inverse problems studied in [1] and [15]. On one hand, in [15] Yoshino proved that radially symmetric symbols of the Berezin–Toeplitz quantization on ${\mathbb {C}}$ can be reconstructed by all the eigenvalues $\{\lambda _n\}_{n=0}^\infty$. More precisely, by using the framework of hyperfunctions, he obtained the reconstruction formula for radially symmetric symbols. On the other hand, in [1] Abreu and Dörfler studied the inverse problem for Daubechies’ localization operators. Let $\Omega$ be a bounded subset of ${\mathbb {C}}$, and let b(z) be the characteristic function of $\Omega$. They proved that if there exists a nonnegative integer m such that the mth Hermite function $\phi _{B,m}$ is an eigenfunction of ${\text {Op}}^\text {W}_1(a)$, then $\Omega$ must be a disk centered at the origin. In this case, it follows automatically that all the Hermite functions $\phi _{B,n}$ are eigenfunctions of ${\text {Op}}^\text {W}_1(a)$ associated with eigenvalues

$$\begin{aligned} \lambda _n = e^{-R} \sum _{k=n+1}^\infty \frac{R^k}{k!}, \quad n=0,1,2,\ldots ., \end{aligned}$$

respectively, where R is the radius of $\Omega$. In particular $R=-\log (1-\lambda _0)$. That is the review of the relationship between the usual Bargmann transform and circles (or disks) in ${\mathbb {C}}$.

The purpose of the present section is to consider the possibility of the extension of the above to ellipses (or elliptic disks) in ${\mathbb {C}}$. Unfortunately, however, we could not obtain the extension of the above. In what follows we introduce a family of functions which might be concerned with ellipses in ${\mathbb {C}}$. Here, it is worth to mention the interesting work [14] of van Eijndhoven and Meyers. They introduced for $0<s<1$ function spaces $\chi _s({\mathbb {C}})$, which is the set of all entire functions $\varphi (z)$ on ${\mathbb {C}}$ satisfying

$$\begin{aligned} \int _{\mathbb {C}} |\varphi (z)|^2 \exp \left( -\frac{1-s^2}{2s}|{z}|^2+\frac{1+s^2}{4s}(z^2+{\bar{z}}^2)\right) L(\mathrm{d}z) < \infty . \end{aligned}$$

As the author pointed out in [5], $\chi _s({\mathbb {C}})$ is determined by the ellipse on ${\mathbb {C}}$ of the form

$$\begin{aligned} \{x-i\xi \in {\mathbb {C}}\ \vert \ (x,\xi )\in {\mathbb {R}}^2, sx^2+\xi ^2=\rho ^2\}, \quad \rho >0, \end{aligned}$$

and $\chi _s({\mathbb {C}})$ is a special case of ${\mathscr {H}}_\Phi ({\mathbb {C}})$ with

$$\begin{aligned} A = \frac{i}{s}, \quad B = \pm i\sqrt{1-s^2}, \quad C = t+is \quad (t\in {\mathbb {R}}). \end{aligned}$$

Recently, Ali, Górska, Horzela and Szafraniec in [2] studied some kinds of generating functions of Hermite polynomials in the abstract setting and introduced some ortonormalized holomorphic Hermite functions in some function spaces including $\chi _s({\mathbb {C}})$. Here, we introduce a holomorphic Hermite functions $\{h_n\}_{n=0}^\infty$ on ${\mathbb {C}}$ and normalizing constants b(n) defined by

$$\begin{aligned} h_n(z) = e^{z^2/2} \left( -\frac{\mathrm{d}}{\mathrm{d}z}\right) ^n e^{-z^2}, \quad b(n) = \frac{\pi \sqrt{s}}{1-s} \left( 2\frac{1+s}{1-s}\right) ^n n!. \end{aligned}$$

One of the interesting results of [2] is that $\{h_n/\sqrt{b(n)}\}_{n=0}^\infty$ is a complete orthonormal system of $\chi _s({\mathbb {C}})$. See [6] for more information on general holomorphic Hermite functions and their basic properties.

Let $\alpha >0$ and let $\beta \in {\mathbb {R}}$. Suppose that $(\alpha ,\beta )\ne (1,0)$. For $\rho >0$,

$$\begin{aligned} E_\rho := \{ x-i\xi \in {\mathbb {C}} \ \vert \ (x,\xi )\in {\mathbb {R}}^2, |\alpha {x}-i(\beta {x}+\xi )|\leqslant \rho \} \end{aligned}$$

is an elliptic disk in ${\mathbb {C}}$. Note that $E_\rho$ is a usual disk if and only if $(\alpha ,\beta )=(1,0)$. Note that

$$\begin{aligned} \bigl \{ \partial {E_\rho } \ \big \vert \ \alpha>0, \beta \in {\mathbb {R}}, \rho >0 \bigr \} \end{aligned}$$

is the set of all ellipses centered at the origin, where $\partial {E_\rho }=\{x-i\xi \in {\mathbb {C}}\ \vert \ |\alpha {x}-i(\beta {x}+\xi )|= \rho \}$. Indeed, consider a function

$$\begin{aligned}&F(x,y;a,b,\theta ) := a(x\cos \theta -\xi \sin \theta )^2 + b(x\sin \theta +\xi \cos \theta )^2, \quad \\&\quad a>0, b>0, \theta \in [0,2\pi ]. \end{aligned}$$

Elementary computation gives

$$\begin{aligned} \frac{F(x,y;a,b,\theta )}{a\sin ^2\theta +b\cos ^2\theta } = \frac{ab}{(a\sin ^2\theta +b\cos ^2\theta )^2} x^2 + \left\{ \frac{(b-a)\sin \theta \cos \theta }{a\sin ^2\theta +b\cos ^2\theta }x + \xi \right\} ^2. \end{aligned}$$

Here, we introduce a function $\psi _0(z)$ which seems to be related with an elliptic disk $E_\rho$. Set $z=x-i\xi$ and $\zeta =\alpha {x}-i(\beta {x}+\xi )$ for $(x,\xi )\in {\mathbb {R}}^2$, $\alpha >0$ and $\beta \in {\mathbb {R}}$. Then,

$$\begin{aligned} \zeta = \frac{\alpha +1-i\beta }{2} z + \frac{\alpha -1-i\beta }{2} {\bar{z}}, \quad z = \frac{\alpha +1+i\beta }{2\alpha } \zeta - \frac{\alpha -1-i\beta }{2\alpha } {\bar{\zeta }}. \end{aligned}$$

We define the function $\psi _0(z)$ by

$$\begin{aligned} \psi _0(z) = \exp \left( -\frac{a}{4}z^2\right) , \quad a = \frac{\alpha ^2+\beta ^2-1+2i\beta }{\alpha ^2+\beta ^2+1}. \end{aligned}$$

Let $||\cdot ||_{{\mathscr {H}}_B}$ be the norm of ${\mathscr {H}}_B({\mathbb {C}})$ determined by the inner product $(\cdot ,\cdot )_{{\mathscr {H}}_B}$. The properties of $\psi _0(z)$ are the following.

Lemma 4.1

We have

• (i) $\psi _0\in {\mathscr {H}}_B({\mathbb {C}})$.

• (ii) $|\psi _0(z)|^2e^{-|{z}|^2/2}=e^{-|\zeta |^2/(\alpha ^2+\beta ^2+1)}$.

• (iii) $||\psi _0||_{{\mathscr {H}}_B}^2=(\alpha ^2+\beta ^2+1)\pi /\alpha$.

Proof

We first show (i). We have only to show the integrability of $|\psi _0(z)|^2e^{-|{z}|^2/2}$ since $\psi _0(z)$ is an entire function. Note that

$$\begin{aligned} |{a}|^2 = \frac{(\alpha ^2+\beta ^2-1)^2+4\beta ^2}{(\alpha ^2+\beta ^2+1)^2} = 1 - \left( \frac{2\alpha }{\alpha ^2+\beta ^2+1} \right) ^2. \end{aligned}$$

We have

$$\begin{aligned} 0< \frac{2\alpha }{\alpha ^2+\beta ^2+1} <1 \end{aligned}$$

since

$$\begin{aligned} \alpha ^2+\beta ^2+1-2\alpha = (\alpha -1)^2+\beta ^2 >0 \end{aligned}$$

for $(\alpha ,\beta )\ne (1,0)$. Thus, $0<|{a}|<1$. We deduce that

$$\begin{aligned} |\psi _0(z)|^2 e^{-|{z}|^2/2}&\leqslant \exp \left( -\frac{a}{4}z^2-\frac{{\bar{a}}}{4}{\bar{z}}^2 -\frac{1}{2}|{z}|^2 \right) \leqslant \exp \left( -\frac{1-|{a}|}{2}|{z}|^2 \right) \\&= \exp \left\{ -\frac{1}{2} \left( 1 - \sqrt{1-\left( \frac{2\alpha }{\alpha ^2+\beta ^2+1}\right) ^2} \right) |{z}|^2 \right\} . \end{aligned}$$

This implies that $|\psi _0(z)|^2e^{-|{z}|^2/2}$ is integrable on ${\mathbb {C}}$ with respect to the Lebesgue measure $L(\mathrm{d}z)$ and $\psi _0\in {\mathscr {H}}_B({\mathbb {C}})$.

We show (ii) and (iii). Elementary computation shows that

$$\begin{aligned} \frac{|\zeta |^2}{\alpha ^2+\beta ^2+1} = \frac{|{z}|^2}{2} + \frac{a}{4}z^2 + \frac{{\bar{a}}}{4}{\bar{z}}^2, \end{aligned}$$

which implies (ii). Moreover, it is easy to see that $\mathrm{d}z \wedge \mathrm{d}{\bar{z}}=\alpha ^{-1}\mathrm{d}\zeta \wedge \mathrm{d}{\bar{\zeta }}$ and

$$\begin{aligned} ||\psi _0||_{{\mathscr {H}}_B}^2= & {} \int _{\mathbb {C}} |\psi _0(z)|^2 e^{-|{z}|^2/2} L(\mathrm{d}z) = \frac{1}{\alpha } \int _{\mathbb {C}} e^{-|\zeta |^2/(\alpha ^2+\beta ^2+1)} L(\mathrm{d}\zeta ) \\= & {} \frac{\alpha ^2+\beta ^2+1}{\alpha } \pi . \end{aligned}$$

This completes the proof. $\square$

The identity $|\psi _0(z)|^2e^{-|{z}|^2/2}=e^{-|\zeta |^2/(\alpha ^2+\beta ^2+1)}$ makes us to expect that $\psi _0$ might be related with an elliptic disk $E_\rho$ and generate a family of eigenfunctions for the Daubechies’ localization operators supported in $E_\rho$. Unfortunately, however, this expectation fails to hold. The purpose of the present section is to generate a family of functions by $\psi _0$ and show its properties similar to the previous sections. To state our results in the present section, we here introduce some notation. Set

$$\begin{aligned} \lambda= & {} \frac{2\alpha ^2}{(\alpha ^2+\beta ^2+1)(\alpha ^2+\beta ^2-1-2i\beta )}, \quad \Lambda _{\alpha ,\beta } = \frac{1}{a} \frac{\partial }{\partial z} + \frac{z}{2}, \quad \\ \Lambda _{\alpha ,\beta }^*= & {} \frac{\partial }{\partial z} + \frac{a+2\lambda }{2}z. \end{aligned}$$

It is easy to see that $a+2\lambda =1/{\bar{a}}$, $\Lambda _{\alpha ,\beta }\psi _0=0$, and $\Lambda _{\alpha ,\beta }^*$ is the Hilbert adjoint of $\Lambda _{\alpha ,\beta }$ on ${\mathscr {H}}_B({\mathbb {C}})$. We make use of $\psi _0$, $\Lambda _{\alpha ,\beta }$ and $\Lambda _{\alpha ,\beta }^*$ as a generating element of a family of functions, and annihilation and creation operators, respectively. Set $\psi _n=(\Lambda _{\alpha ,\beta }^*)^n\psi _0$ for $n=0,1,2,\ldots .$, and set

$$\begin{aligned} C_{\alpha ,\beta } = \frac{1-{\bar{a}}}{2{\bar{a}}} = \frac{1+i\beta }{\alpha ^2+\beta ^2-1-2i\beta } \end{aligned}$$

for short. Properties of $\Lambda _{\alpha ,\beta }$, $\Lambda _{\alpha ,\beta }^*$ and $\{\psi _n\}_{n=0}^\infty$ are the following.

Theorem 4.2

• (i) $\{\psi _n\}_{n=0}^\infty$ satisfies a formula of the form

  $$\begin{aligned} \psi _n(z) = \left\{ e^{-\lambda z^2/2} \left( \frac{\partial }{\partial z}\right) ^n e^{\lambda z^2/2} \right\} \psi _0(z), \quad n=0,1,2,\ldots . \end{aligned}$$

• (ii) For $n=1,2,3,\ldots$,

  $$\begin{aligned} \Lambda _{\alpha ,\beta } (\Lambda _{\alpha ,\beta }^*)^n = (\Lambda _{\alpha ,\beta }^*)^n \Lambda _{\alpha ,\beta } + n \frac{\lambda }{a} (\Lambda _{\alpha ,\beta }^*)^{n-1}. \end{aligned}$$

• (iii) $\{\psi _n\}_{n=0}^\infty$ is a complete orthogonal system of ${\mathscr {H}}_B({\mathbb {C}})$.

• (iv) For $n=0,1,2,\ldots .$,

  $$\begin{aligned} \left( \frac{\Lambda _{\alpha ,\beta }^*\Lambda _{\alpha ,\beta }}{|{C_{\alpha ,\beta }}|^2} + \frac{\alpha ^2}{1+\beta ^2} \right) \psi _n = \frac{\alpha ^2}{1+\beta ^2} (2n+1) \psi _n. \end{aligned}$$

Proof

First we show (i). Note that for any $c\in {\mathbb {C}}$ and for any holomorphic function f(z), we deduce that

$$\begin{aligned} \left( \frac{\partial }{\partial z} + cz \right) f(z)&= \left( \frac{\partial }{\partial z} + cz \right) \bigl \{ e^{-cz^2/2} \bigl (e^{cz^2/2}f(z)\bigr ) \bigr \} \\&= -cze^{-cz^2/2} \bigl (e^{cz^2/2}f(z)\bigr ) + e^{-cz^2/2} \frac{\partial }{\partial z} \bigl (e^{cz^2/2}f(z)\bigr ) \\&\quad +\, cze^{-cz^2/2} \bigl (e^{cz^2/2}f(z)\bigr ) \\&= e^{-cz^2/2} \frac{\partial }{\partial z} \bigl (e^{cz^2/2}f(z)\bigr ). \end{aligned}$$

Using this repeatedly, we have

$$\begin{aligned} \psi _n(z)&= (\Lambda _{\alpha ,\beta }^*)^n \psi _0(z) = \left\{ \frac{\partial }{\partial z} + \left( \frac{a}{2}+\lambda \right) \right\} ^n e^{-az^2/4} \\&= e^{-az^2/4-\lambda z^2/2} \left( \frac{\partial }{\partial z}\right) ^n \bigl (e^{az^2/4+\lambda z^2/2} \cdot e^{-az^2/4}\bigr ) = e^{-az^2/4-\lambda z^2/2} \left( \frac{\partial }{\partial z}\right) ^n e^{\lambda z^2/2} \\&= \left\{ e^{-\lambda z^2/2} \left( \frac{\partial }{\partial z}\right) ^n e^{\lambda z^2/2} \right\} e^{-az^2/4} = \left\{ e^{-\lambda z^2/2} \left( \frac{\partial }{\partial z}\right) ^n e^{\lambda z^2/2} \right\} \psi _0(z), \end{aligned}$$

which is desired.

Next we show (ii). We employ induction on n. For $n=1$, we deduce that

$$\begin{aligned} \Lambda _{\alpha ,\beta }\Lambda _{\alpha ,\beta }^*- \Lambda _{\alpha ,\beta }^*\Lambda _{\alpha ,\beta }&= \frac{1}{a} \left( \frac{\partial }{\partial z} + \frac{a}{2}z \right) \left\{ \frac{\partial }{\partial z} + \left( \frac{a}{2}+\lambda \right) z \right\} \\&\quad -\, \left\{ \frac{\partial }{\partial z} + \left( \frac{a}{2}+\lambda \right) z \right\} \frac{1}{a} \left( \frac{\partial }{\partial z} + \frac{a}{2}z \right) \\&= \frac{1}{a} \left( \frac{\partial }{\partial z} + \frac{a}{2}z \right) (\lambda z) - (\lambda z) \frac{1}{a} \left( \frac{\partial }{\partial z} + \frac{a}{2}z \right) (\lambda z) \\&= \frac{\lambda }{a} = 1 \cdot \frac{\lambda }{a} \cdot (\Lambda _{\alpha ,\beta }^*)^0. \end{aligned}$$

Here, we suppose that (ii) holds for some $n=1,2,3,\ldots .$. We show the case of $n+1$. By using the cases of n and 1, we deduce that

$$\begin{aligned} \Lambda _{\alpha ,\beta }(\Lambda _{\alpha ,\beta }^*)^{n+1}&= \Lambda _{\alpha ,\beta }(\Lambda _{\alpha ,\beta }^*)^n\Lambda _{\alpha ,\beta }^*\\&= \left\{ (\Lambda _{\alpha ,\beta }^*)^n \Lambda _{\alpha ,\beta } + n \frac{\lambda }{a} (\Lambda _{\alpha ,\beta }^*)^{n-1} \right\} \Lambda _{\alpha ,\beta }^*\\&= (\Lambda _{\alpha ,\beta }^*)^n (\Lambda _{\alpha ,\beta }\Lambda _{\alpha ,\beta }^*) + n \frac{\lambda }{a} (\Lambda _{\alpha ,\beta }^*)^n \\&= (\Lambda _{\alpha ,\beta }^*)^n \left\{ \Lambda _{\alpha ,\beta }^*\Lambda _{\alpha ,\beta } + \frac{\lambda }{a} \right\} + n \frac{\lambda }{a} (\Lambda _{\alpha ,\beta }^*)^n \\&= (\Lambda _{\alpha ,\beta }^*)^{n+1} \Lambda _{\alpha ,\beta } + (n+1) \frac{\lambda }{a} (\Lambda _{\alpha ,\beta }^*)^n, \end{aligned}$$

which is desired.

For (iii), we here show only the orthogonality

$$\begin{aligned}&(\psi _m,\psi _n)_{{\mathscr {H}}_B} = \left( \frac{\lambda }{a}\right) ^n n! ||\psi _0||_{{\mathscr {H}}_B}^2 \delta _{mn} \nonumber \\&\quad = \frac{(\alpha ^2+\beta ^2+1)\pi }{\alpha } \left( \frac{2\alpha ^2}{(\alpha ^2+\beta ^2-1)^2+4\beta ^2} \right) ^n n! \delta _{mn}, \quad m,n=0,1,2,\ldots . \end{aligned}$$

(9)

The completeness will be automatically proved later. Recall that $\Lambda _{\alpha ,\beta }\psi _0=0$. Suppose that $m \geqslant n$. By using (ii) repeatedly, we deduce that

$$\begin{aligned} (\psi _m,\psi _n)_{{\mathscr {H}}_B}&= \bigl ( (\Lambda _{\alpha ,\beta }^*)^m\psi _0, (\Lambda _{\alpha ,\beta }^*)^n\psi _0 \bigr )_{{\mathscr {H}}_B} \\&= \bigl ( (\Lambda _{\alpha ,\beta }^*)^{m-1}\psi _0, \Lambda _{\alpha ,\beta }(\Lambda _{\alpha ,\beta }^*)^n\psi _0 \bigr )_{{\mathscr {H}}_B} \\&= \left( (\Lambda _{\alpha ,\beta }^*)^{m-1}\psi _0, (\Lambda _{\alpha ,\beta }^*)^n\Lambda _{\alpha ,\beta }\psi _0 + n\frac{\lambda }{a}(\Lambda _{\alpha ,\beta }^*)^{n-1}\psi _0 \right) _{{\mathscr {H}}_B} \\&= n\frac{\lambda }{a} \bigl ( (\Lambda _{\alpha ,\beta }^*)^{m-1}\psi _0, (\Lambda _{\alpha ,\beta }^*)^{n-1}\psi _0 \bigr )_{{\mathscr {H}}_B} \\&= \cdots b \\&= n!\left( \frac{\lambda }{a}\right) ^n \bigl ( (\Lambda _{\alpha ,\beta }^*)^{m-n}\psi _0, \psi _0 \bigr )_{{\mathscr {H}}_B}. \end{aligned}$$

If $m>n$, then

$$\begin{aligned} (\psi _m,\psi _n)_{{\mathscr {H}}_B} = n!\left( \frac{\lambda }{a}\right) ^n \bigl ( (\Lambda _{\alpha ,\beta }^*)^{m-n-1}\psi _0, \Lambda _{\alpha ,\beta }\psi _0 \bigr )_{{\mathscr {H}}_B} = 0. \end{aligned}$$

If $m=n$, then

$$\begin{aligned} (\psi _n,\psi _n)_{{\mathscr {H}}_B} = n!\left( \frac{\lambda }{a}\right) ^n (\psi _0,\psi _0)_{{\mathscr {H}}_B} = n!\left( \frac{\lambda }{a}\right) ^n ||\psi _0||_{{\mathscr {H}}_B}^2. \end{aligned}$$

Finally, we show (iv). By using (ii) and $\Lambda _{\alpha ,\beta }\psi _0=0$ again, we deduce that

$$\begin{aligned} \left( \frac{\Lambda _{\alpha ,\beta }^*\Lambda _{\alpha ,\beta }}{|{C_{\alpha ,\beta }}|^2} + \frac{\alpha ^2}{1+\beta ^2} \right) \psi _n&= \frac{\Lambda _{\alpha ,\beta }^*\Lambda _{\alpha ,\beta }(\Lambda _{\alpha ,\beta }^*)^n}{|{C_{\alpha ,\beta }}|^2} \psi _0 + \frac{\alpha ^2}{1+\beta ^2} \psi _n \\&= \frac{1}{|{C_{\alpha ,\beta }}|^2} \left\{ (\Lambda _{\alpha ,\beta }^*)^{n+1}\Lambda _{\alpha ,\beta } + n\frac{\lambda }{a} (\Lambda _{\alpha ,\beta }^*)^n \right\} \psi _0 \\&\quad +\, \frac{\alpha ^2}{1+\beta ^2} \psi _n \\&= \frac{n\lambda }{a|{C_{\alpha ,\beta }}|^2} (\Lambda _{\alpha ,\beta }^*)^n \psi _0 + \frac{\alpha ^2}{1+\beta ^2} \psi _n \\&= \left\{ \frac{n\lambda }{a|{C_{\alpha ,\beta }}|^2} + \frac{\alpha ^2}{1+\beta ^2} \right\} \psi _n = \frac{\alpha ^2}{1+\beta ^2} (2n+1) \psi _n. \end{aligned}$$

This completes the proof. $\square$

Here, we introduce a family of functions $\{\Psi _n\}_{n=0}^\infty$ on ${\mathbb {R}}$ by setting $\Psi _n=B_1^*\psi _n$ for $n=0,1,2,\ldots$. In order to study basic properties on $\{\Psi _n\}_{n=0}^\infty$, we introduce notation. Set

$$\begin{aligned} P_{\alpha ,\beta }:= & {} \frac{1}{\overline{C_{\alpha ,\beta }}} B_1^*\circ \Lambda _{\alpha ,\beta } \circ B_1, \quad P_{\alpha ,\beta }^*:= \frac{1}{C_{\alpha ,\beta }} B_1^*\circ \Lambda _{\alpha ,\beta }^*\circ B_1, \\ H_{\alpha ,\beta }:= & {} P_{\alpha ,\beta }^*\circ P_{\alpha ,\beta } + \frac{\alpha ^2}{1+\beta ^2} \end{aligned}$$

for short. Then, we have

$$\begin{aligned} P_{\alpha ,\beta }&= \frac{\mathrm{d}}{\mathrm{d}x} + \frac{\alpha ^2+i(\alpha ^2+\beta ^2+1)\beta }{1+\beta ^2}x, \quad P_{\alpha ,\beta }^*= - \frac{\mathrm{d}}{\mathrm{d}x} + \frac{\alpha ^2-i(\alpha ^2+\beta ^2+1)\beta }{1+\beta ^2}x, \\ H_{\alpha ,\beta }&= - \frac{\mathrm{d}^2}{\mathrm{d}x^2} + \frac{\alpha ^4+(\alpha ^2+\beta ^2+1)\beta ^2}{(1+\beta ^2)^2}x^2 - 2i \frac{(\alpha ^2+\beta ^2+1)\beta }{1+\beta ^2} x\frac{\mathrm{d}}{\mathrm{d}x} \\&\qquad - i \frac{(\alpha ^2+\beta ^2+1)\beta }{1+\beta ^2} \\&= {\text {Op}}^\text {W} \left( \xi ^2 + \frac{\alpha ^4+(\alpha ^2+\beta ^2+1)\beta ^2}{(1+\beta ^2)^2}x^2 + 2 \frac{(\alpha ^2+\beta ^2+1)\beta }{1+\beta ^2} x\xi \right) . \end{aligned}$$

Set

$$\begin{aligned} A_{\alpha ,\beta } = \pi ^{1/4} \left\{ \frac{\alpha ^2+\beta ^2+1}{1-i\beta } \right\} ^{1/2}, \end{aligned}$$

where we take its argument in $(-\pi /4,\pi /4)$. Our results in the present section are the following.

Theorem 4.3

• (i) We have for $n=0,1,2,\ldots .$

  $$\begin{aligned} \Psi _n(x) =&A_{\alpha ,\beta } (-C_{\alpha ,\beta })^n \exp \left( -i \frac{(\alpha ^2+\beta ^2+1)\beta }{2(1+\beta ^2)} x^2 \right) \\&\times \exp \left( \frac{\alpha ^2}{2(1+\beta ^2)} x^2 \right) \left( \frac{\mathrm{d}}{\mathrm{d}x}\right) ^n \exp \left( - \frac{\alpha ^2}{1+\beta ^2} x^2 \right) . \end{aligned}$$

  In particular,

  $$\begin{aligned} \Psi _0(x) = A_{\alpha ,\beta } \exp \left( - \frac{\alpha ^2+i(\alpha ^2+\beta ^2+1)\beta }{2(1+\beta ^2)} x^2 \right) . \end{aligned}$$

• (ii) $\{\Psi _n\}_{n=0}^\infty$ is a family of eigenfunctions of $H_{\alpha ,\beta }$, that is,

  $$\begin{aligned} H_{\alpha ,\beta }\Psi _n = \frac{\alpha ^2}{1+\beta ^2} (2n+1) \Psi _n, \quad n=0,1,2,\ldots .. \end{aligned}$$

• (iii) $\{\Psi _n\}_{n=0}^\infty$ is a complete orthogonal system of $L^2({\mathbb {R}})$.

Recall that Theorem 4.2 was proved except for the completeness of $\{\psi _n\}_{n=0}^\infty$. Theorem 4.2 without the completeness implies (ii) of Theorem 4.3 and the orthogonality of $\{\Psi _n\}_{n=0}^\infty$ in $L^2({\mathbb {R}})$. If (i) of Theorem 4.3 holds, then the completeness of $\{\Psi _n\}_{n=0}^\infty$ in $L^2({\mathbb {R}})$ follows immediately. Indeed combining (i) of Theorem 4.3 and the results in Sect. 2 with

$$\begin{aligned} A=\frac{i(1+\beta ^2)}{2\alpha ^2}, \quad B=-i, \quad C=\frac{(\alpha ^2+\beta ^2+1)\beta +i\alpha ^2}{1+\beta ^2}, \end{aligned}$$

we can check the completeness of $\{\Psi _n\}_{n=0}^\infty$ in $L^2({\mathbb {R}})$. This implies the completeness of $\{\psi _n\}_{n=0}^\infty$ in ${\mathscr {H}}_B({\mathbb {C}})$ stated in Theorem 4.2 since $B_1$ is a Hilbert space isomorphism of $L^2({\mathbb {R}})$ onto ${\mathscr {H}}_B({\mathbb {C}})$. For this reason, we have only to show (i) of Theorem 4.3. For this purpose, we need the following.

Lemma 4.4

Let $\rho >0$ and let $2\theta \in (-\pi /2,\pi /2)$. Then we have

$$\begin{aligned} \int _{{\mathbb {R}}} \exp \Bigl (-\rho ^2e^{2i\theta }t^2\Bigr ) \mathrm{d}t = \frac{\sqrt{\pi }}{\rho e^{i\theta }}. \end{aligned}$$

Proof

The integrand is an even function of $t\in {\mathbb {R}}$. By using change of variable $t \mapsto \rho t$, we have

$$\begin{aligned} \int _{{\mathbb {R}}} \exp \Bigl (-\rho ^2e^{2i\theta }t^2\Bigr ) \mathrm{d}t = \frac{2}{\rho } \int _0^\infty \exp \Bigl (-e^{2i\theta }t^2\Bigr ) \mathrm{d}t. \end{aligned}$$

Let $R>0$. Consider a contour $\gamma _R$ which consists of $\gamma _R=\gamma _R(1)\cup \gamma _R(2)\cup \gamma _R(3)$, where

$$\begin{aligned} \gamma _R(1)= & {} \{t \ \vert \ t\in [0,R]\}, \quad \gamma _R(3) = \{e^{i\theta }(R-t) \ \vert \ t\in [0,R]\}, \\ \gamma _R(2)= & {} \{Re^{it} \ \vert \ t\in [0,\theta ]\} \quad (\theta \geqslant 0), \quad \gamma _R(2) = \{Re^{i(\theta -t)} \ \vert \ t\in [\theta ,0]\} \quad (\theta <0). \end{aligned}$$

Applying Cauchy’s theorem to the holomorphic function $e^{-z^2}$ on $\gamma _R$, we have

$$\begin{aligned} e^{i\theta } \int _0^R \exp \Bigl (-e^{2i\theta }t^2\Bigr ) \mathrm{d}t = \int _0^R e^{-t^2} \mathrm{d}t + iR \int _0^\theta \exp \Bigl (-R^2e^{2it}+it\Bigr ) \mathrm{d}t. \end{aligned}$$

Here, we note that $0<\cos (2\theta )\leqslant 1$ since $2\theta \in (-\pi /2,\pi /2)$. Then, we deduce that

$$\begin{aligned} \Bigl \vert \exp \Bigl (-R^2e^{2it}+it\Bigr ) \Bigr \vert&= \exp \Bigl (-R^2\cos (2t)\Bigr ) \leqslant \exp \Bigl (-R^2\cos (2\theta )\Bigr ) \\&= \frac{1}{\sum _{k=0}^\infty \frac{\bigl (R^2\cos (2\theta )\bigr )^k}{k!}} \leqslant \frac{1}{R^2\cos (2\theta )}, \end{aligned}$$

and

$$\begin{aligned} \left|iR \int _0^\theta \exp \Bigl (-R^2e^{2it}+it\Bigr ) \mathrm{d}t \right|&\leqslant R \left|\int _0^\theta \Bigl |\exp \Bigl (-R^2e^{2it}+it\Bigr ) \Bigr |\mathrm{d}t \right|\\&\leqslant R \left|\int _0^\theta \frac{1}{R^2\cos (2\theta )} \mathrm{d}t \right|= \frac{|\theta |}{R\cos (2\theta )}. \end{aligned}$$

Then, we have

$$\begin{aligned} \int _0^R \exp \Bigl (-e^{2i\theta }t^2\Bigr ) \mathrm{d}t = \frac{1}{e^{i\theta }} \int _0^R e^{-t^2} \mathrm{d}t + {\mathcal {O}}\left( \frac{1}{R}\right) \quad (R \rightarrow \infty ), \end{aligned}$$

and

$$\begin{aligned} \int _0^\infty \exp \Bigl (-e^{2i\theta }t^2\Bigr ) \mathrm{d}t = \frac{1}{e^{i\theta }} \int _0^\infty e^{-t^2} \mathrm{d}t = \frac{\sqrt{\pi }}{2e^{i\theta }}. \end{aligned}$$

This completes the proof. $\square$

Finally, we complete the proof of Theorem 4.3.

Proof of Theorem 4.3

It suffices to show the part (i). We first compute the concrete form of $\Psi _0(x)$. Recall the definition of $\Psi _0(x)$

$$\begin{aligned} \Psi _0(x) = B_1^*\psi _0(x) = 2^{-1/2}\pi ^{-3/4} \iint _{{\mathbb {R}}^2} e^{G(y,\eta ,x)} \mathrm{d}y\mathrm{d}\eta , \end{aligned}$$

where

$$\begin{aligned} G(y,\eta ,x) = - \frac{(y+i\eta )^2}{4} + (y+i\eta )x - \frac{x^2}{2} - \frac{a(y-i\eta )^2}{4} - \frac{y^2+\eta ^2}{2}. \end{aligned}$$

Elementary computation gives

$$\begin{aligned} G(y,\eta ,x) =&- \frac{\alpha ^2+i(\alpha ^2+\beta ^2+1)\beta }{2(1+\beta ^2)} x^2 \\&- \frac{2\alpha ^2+2\beta ^2+1+i\beta }{2(\alpha ^2+\beta ^2+1)} (y-z_1)^2 - \frac{1-i\beta }{2\alpha ^2+2\beta ^2+1+i\beta } (\eta -z_2)^2, \end{aligned}$$

where

$$\begin{aligned} z_1 = \frac{2x-i(1-a)\eta }{3+a}, \quad z_2 = i\frac{\alpha ^2+\beta ^2+i\beta }{1-i\beta }x. \end{aligned}$$

By using this and Lemma 4.4, we deduce that

$$\begin{aligned} \Psi _0(x)&= 2^{-1/2}\pi ^{-3/4} \exp \left( - \frac{\alpha ^2+i(\alpha ^2+\beta ^2+1)\beta }{2(1+\beta ^2)} x^2 \right) \\&\quad \times \int _{{\mathbb {R}}} \exp \left( - \frac{1-i\beta }{2\alpha ^2+2\beta ^2+1+i\beta } (\eta -z_2)^2 \right) \mathrm{d}\eta \\&\quad \times \int _{{\mathbb {R}}} \exp \left( - \frac{2\alpha ^2+2\beta ^2+1+i\beta }{2(\alpha ^2+\beta ^2+1)} (y-z_1)^2 \right) \mathrm{d}y \\&= 2^{-1/2}\pi ^{-3/4} \exp \left( - \frac{\alpha ^2+i(\alpha ^2+\beta ^2+1)\beta }{2(1+\beta ^2)} x^2 \right) \\&\quad \times \int _{{\mathbb {R}}} \exp \left( - \frac{1-i\beta }{2\alpha ^2+2\beta ^2+1+i\beta } \eta ^2 \right) \mathrm{d}\eta \\&\quad \times \int _{{\mathbb {R}}} \exp \left( - \frac{2\alpha ^2+2\beta ^2+1+i\beta }{2(\alpha ^2+\beta ^2+1)} y^2 \right) \mathrm{d}y \\&= 2^{-1/2}\pi ^{-3/4} \times \left\{ \frac{1-i\beta }{2\alpha ^2+2\beta ^2+1+i\beta } \cdot \frac{2\alpha ^2+2\beta ^2+1+i\beta }{2(\alpha ^2+\beta ^2+1)} \right\} ^{-1/2} \pi \\&\quad \times \exp \left( - \frac{\alpha ^2+i(\alpha ^2+\beta ^2+1)\beta }{2(1+\beta ^2)} x^2 \right) \\&= A_{\alpha ,\beta } \exp \left( - \frac{\alpha ^2+i(\alpha ^2+\beta ^2+1)\beta }{2(1+\beta ^2)} x^2 \right) , \end{aligned}$$

which is desired.

Finally, we check the Rodrigues formula for $\Psi _n$. By using the definition of $\Psi _n$, we deduce that

$$\begin{aligned} \Psi _n(x)&= B^*(\Lambda _{\alpha ,\beta }^*)^n\psi _0(x) = (C_{\alpha ,\beta }P_{\alpha ,\beta }^*)^nB^*\psi _0(x) = (C_{\alpha ,\beta })^n(P_{\alpha ,\beta }^*)^n\Psi _0(x) \\&= (-C_{\alpha ,\beta })^n \left( \frac{\mathrm{d}}{\mathrm{d}x} - \frac{\alpha ^2-i(\alpha ^2+\beta ^2+1)\beta }{1+\beta ^2}x \right) ^n \Psi _0(x) \\&= (-C_{\alpha ,\beta })^n \exp \left( \frac{\alpha ^2-i(\alpha ^2+\beta ^2+1)\beta }{2(1+\beta ^2)}x \right) \\&\qquad \times \left( \frac{\mathrm{d}}{\mathrm{d}x}\right) ^n \left\{ \exp \left( - \frac{\alpha ^2-i(\alpha ^2+\beta ^2+1)\beta }{2(1+\beta ^2)}x \right) \Psi _n(x) \right\} \\&= A_{\alpha ,\beta }(-C_{\alpha ,\beta })^n \exp \left( \frac{\alpha ^2-i(\alpha ^2+\beta ^2+1)\beta }{2(1+\beta ^2)}x \right) \left( \frac{\mathrm{d}}{\mathrm{d}x}\right) ^n \exp \left( - \frac{\alpha ^2}{1+\beta ^2}x \right) . \end{aligned}$$

This completes the proof. $\square$