Bargmann-type transforms and modified harmonic oscillators

TL;DR

This paper explores Bargmann-type transforms and their associated eigenfunctions for harmonic oscillators, analyzing both commutative and non-commutative cases, and introduces a complete orthogonal system related to phase plane ellipses.

Contribution

It introduces a new class of orthonormal systems derived from Bargmann-type transforms and provides explicit eigenvalues and eigenfunctions for certain non-commutative harmonic oscillators.

Findings

Eigenfunctions form complete orthonormal systems

Explicit eigenvalues for commutative non-commutative harmonic oscillators

Complete orthogonal system related to phase plane ellipses

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.

Equations

$$T_{h}u(z)=C_{\varphi }{h}^{-3/4}{\int }_{\mathbb{R}}{e}^{i\varphi (z,x)/h}u(x)dx,z\in \mathbb{C},$$

$$\varphi (z,x)=\frac{A}{2}{z}^2+Bzx+\frac{C}{2}{x}^2,A,B,C\in \mathbb{C}$$

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

$$\Phi (z)=\frac{|Bz{|}^2}{4\mathrm{Im}C}-\mathrm{Re}\left\{\frac{(Bz{)}^2}{4\mathrm{Im}C}+\frac{A{z}^2}{2i}\right\},$$

$$\kappa_{T}(\mathbb{R}^{2})=\left\{\left(z,\frac{2}{i}\frac{\partial\Phi}{\partial z}(z)\right)\ \bigg{|}\ z\in\mathbb{C}\right\}.$$

$$(T_{h}u,T_{h}v{)}_{L_{\Phi }^2}=(u,v{)}_{L^2}u,v\in L^2(\mathbb{R}),$$

$$(U,V{)}_{L_{\Phi }^2}={\int }_{\mathbb{C}}U(z)\overline{V(z)}{e}^{-2\Phi (z)/h}L(dz),U,V\in L_{\Phi }^2(\mathbb{C}),$$

$$(u,v{)}_{L^2}={\int }_{\mathbb{R}}u(x)\overline{v(x)}dxu,v\in L^2(\mathbb{R}).$$

$$T_h^{\ast }U(x)=C_{\varphi }{\int }_{\mathbb{C}}{e}^{-i\overline{\varphi (z,x)}/h}U(z)e^{-2\Phi (z)/h}L(dz),x\in \mathbb{R},U\in {\mathcal{H}}_{\Phi }(\mathbb{C}).$$

$$T_h\circ T_h^{\ast }U(z)=\frac{C_{\Phi }}{h}{\int }_{\mathbb{C}}{e}^{2\Psi (z,\bar{\zeta })/h}U(\zeta )e^{-2\Phi (\zeta )/h}L(d\zeta ),U\in L_{\Phi }^2(\mathbb{C}),$$

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

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

$$L^{2}_{B}(\mathbb{C})=\left\{U(z)\ \bigg{|}\ \int_{\mathbb{C}}\lvert{U(z)}\rvert^{2}e^{-\lvert{z}\rvert^{2}/2h}L(dz)<\infty\right\},$$

$$B_h\circ B_h^{\ast }U(z)=\frac{1}{2\pi h}{\int }_{\mathbb{C}}{e}^{z\bar{\zeta }/2h}U(\zeta )e^{-|\zeta {|}^2/2h}L(d\zeta ),U\in L_B^2(\mathbb{C}).$$

$$U(z)=\sum _{n=0}^{\infty }(U,{\phi }_{B,n}{)}_{{\mathcal{H}}_B}{\phi }_{B,n}(z),{\phi }_{B,n}(z)=\frac{z^n}{\sqrt{\pi (2h{)}^{n+1}n!}},n=0,1,2,\dots .$$

$${\Lambda }_{B}U(z):=2h\frac{\partial U}{\partial z}(z),{\Lambda }_B^{\ast }U(z):=zU(z).$$

$$({\Lambda }_B^{\ast }\circ {\Lambda }_B+h){\phi }_{B,n}=({\Lambda }_B\circ {\Lambda }_B^{\ast }-h){\phi }_{B,n}=(2n+1)h{\phi }_{B,n},n=0,1,2,\dots .$$

$${\varphi }_{B,n}(x):=B_h^{\ast }{\phi }_{B,n}(x),P_B:=B_h^{\ast }\circ {\Lambda }_B\circ B_h,P_B^{\ast }:=B_h^{\ast }\circ {\Lambda }_B^{\ast }\circ B_h.$$

$$P_B=h\frac{d}{dx}+x,P_B^{\ast }=-h\frac{d}{dx}+x$$

$${\phi }_{B,n}=((2h{)}^{n}n!{)}^{-1/2}({\Lambda }_B^{\ast }{)}^n{\phi }_{B,0},n=0,1,2,\dots ,$$

$${\varphi }_{B,0}(x):=B_h^{\ast }{\phi }_{B,0}(x)=\frac{1}{(\pi h{)}^{1/4}}{e}^{-x^2/2h}.$$

$${\varphi }_{B,n}(x)=\frac{1}{\sqrt{(2h{)}^{n}n!}}(P_B^{\ast }{)}^n{\varphi }_{B,0}(x)=\frac{(-1{)}^n}{(\pi h{)}^{1/4}\sqrt{(2h{)}^{n}n!}}{e}^{-x^2/2h}{\left(h\frac{d}{dx}\right)}^n{e}^{-x^2/h}.$$

$$H_B=-h^2\frac{d^2}{d{x}^2}+x^2$$

$$H_B{\varphi }_{B,n}=(2n+1)h{\varphi }_{B,n},n=0,1,2,\dots .$$

$$A=-\frac{i{B}^2}{2\mathrm{Im}C}.$$

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

$$T_h\circ T_h^{\ast }U(z)=\frac{C_{\Phi }}{h}{\int }_{\mathbb{C}}{e}^{Bz\cdot \overline{B\zeta }/2h\mathrm{Im}C}U(\zeta )e^{-|Bz{|}^2/2h\mathrm{Im}C}L(d\zeta ),$$

$$\kappa_{T}(\mathbb{R}^{2})=\left\{\left(z,\frac{\lvert{B}\rvert^{2}}{2i\operatorname{Im}C}\bar{z}\right)\ \bigg{|}\ z\in\mathbb{C}\right\}$$

$$U(z)=\sum _{n=0}^{\infty }(U,{\phi }_n{)}_{{\mathcal{H}}_{\Phi }}\phi (z),$$

$${\phi }_n(z)=\frac{|B|}{\sqrt{2\pi h\mathrm{Im}C}}\cdot \frac{1}{\sqrt{n!}}\cdot {\left(\frac{Bz}{\sqrt{2h\mathrm{Im}C}}\right)}^n,n=0,1,2,\dots .$$

Full text

Supported by the JSPS Grant-in-Aid for Scientific Research #19K03569.

1. Introduction

We are concerned with the Bargmann-type transform defined by

[TABLE]

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

[TABLE]

with assumptions $B\neq 0$ and $\operatorname{Im}C>0$, and $C_{\phi}=2^{-1/2}\pi^{-3/4}\lvert{B}\rvert(\operatorname{Im}C)^{-1/4}$. Throughout of 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 \operatorname{Re}\bigl{(}i\phi(z,x)\bigr{)}=\mathcal{O}(-\operatorname{Im}Cx^{2}/3) for $\lvert{x}\rvert\rightarrow\infty$.

These integral transforms were introduced by Sjöstrand (See, e.g., [14]). 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\bigl{(}x,-\phi^{\prime}_{x}(z,x)\bigr{)}\mapsto\bigl{(}z,\phi^{\prime}_{z}(z,x)\bigr{)}\in\mathbb{C}^{2}, that is,

[TABLE]

If we set \Phi(z)=\displaystyle\max_{x\in\mathbb{R}}\operatorname{Re}\bigl{(}i\phi(z,x)\bigr{)}, then we have

[TABLE]

[TABLE]

This means that the singularities of a tempered distribution $u$ described in the phase plane $\mathbb{R}^{2}$ are translated into those of $T_{h}u$ 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 [14] 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 $e^{-2\Phi(z)/h}L(dz)$, 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,

[TABLE]

where

[TABLE]

[TABLE]

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 $T_{h}^{\ast}$ is given by

[TABLE]

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

[TABLE]

where $C_{\Phi}=\lvert{B}\rvert^{2}/(2\pi\operatorname{Im}C)$, 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,

[TABLE]

In particular, $U=T_{h}{\circ}T_{h}^{\ast}U$ 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 [9] for this. The Bargmann transform $B_{h}$ on $\mathbb{R}$ is defined by

[TABLE]

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,

[TABLE]

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

[TABLE]

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

[TABLE]

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 $U\in\mathscr{H}_{B}(\mathbb{C})$

[TABLE]

Actually, $\Lambda_{B}^{\ast}$ is the adjoint of $\Lambda_{B}$ on $\mathscr{H}_{B}(\mathbb{C})$. Elementary computation gives

[TABLE]

We shall pull back these facts on $\mathbb{R}$ by using $B_{h}^{\ast}$. Set

[TABLE]

$\phi_{B,n}$ is said to be the $n$-th 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

[TABLE]

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

[TABLE]

[TABLE]

Then we have so-call the Rodrigues formula

[TABLE]

Set $H_{B}=P_{B}^{\ast}\circ P_{B}+h=P_{B}\circ P_{B}^{\ast}-h$. Then

[TABLE]

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

[TABLE]

Thus the $n$-th Hermite function $\phi_{B,n}$ is an eigenfunction of $H_{B}$ for the $n$-th 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 Section 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 Section 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 Section 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

[TABLE]

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

[TABLE]

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

[TABLE]

[TABLE]

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

[TABLE]

[TABLE]

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 $\zeta=Bz/\sqrt{h\operatorname{Im}C}$, we deduce that

[TABLE]

This completes the proof. ∎

Set $\phi_{n}(x)=T_{h}^{\ast}\varphi_{n}(x)$, $n=0,1,2,\dotsc$. 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

[TABLE]

Its Hilbert adjoint is

[TABLE]

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

[TABLE]

[TABLE]

We shall pull back these facts by using $T_{h}^{\ast}$. Set

[TABLE]

[TABLE]

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

[TABLE]

for $u\in\mathscr{S}(\mathbb{R})$, where $\mathscr{S}(\mathbb{R})$ denotes the Schwartz class on $\mathbb{R}$. Set $D_{x}=-id/dx$ for short.

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

Proposition 2.3**.**

We have

[TABLE]

[TABLE]

Proof.

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

[TABLE]

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

[TABLE]

which completes the proof. ∎

By using the pull-back of (7) and (8), we have for $n=0,1,2,\dotsc$

[TABLE]

[TABLE]

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 $n=0,1,2,\dotsc$

[TABLE]

Proof.

Recall the definition of $\phi_{0}$. We have

[TABLE]

where

[TABLE]

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

[TABLE]

[TABLE]

Then we can obtain

[TABLE]

This completes the proof. ∎

3. The commutative case of non-commutative harmonic oscillators

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

[TABLE]

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

[TABLE]

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 $(x,\xi)\neq(0,0)$ 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 [11] 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 $A_{(\alpha,\beta)}J\neq JA_{(\alpha,\beta)}$ for $\alpha\neq\beta$. 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 [11], [12] and [13]. See also a monograph [10]. 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

[TABLE]

which diagonalize $J$ as

[TABLE]

Then, we have

[TABLE]

[TABLE]

[TABLE]

Note that $\lvert\nu_{\alpha,\pm}\rvert=1$ and $\operatorname{Im}\nu_{\alpha,\pm}>0$.

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

those are,

[TABLE]

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{|}\ \mu=\pm,n=0,1,2,\dotsc\bigr{\}} is a complete orthonormal system of $L^{2}(\mathbb{R})\oplus L^{2}(\mathbb{R})$, and satisfies

[TABLE]

This is not a new result. This was first proved by Parmeggiani and Wakayama in [11]. We believe that our method of proof is easier than that of [11].

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

[TABLE]

It is known that

[TABLE]

for $u,v\in\mathscr{S}(\mathbb{R})$. See, e.g., [14] 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 $\operatorname{Op}^{\text{W}}_{1}(a)$ 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 $\phi_{B,n}$ ($n=0,1,2,\dotsc$) are the eigenfunctions of $\tilde{T}_{b}$:

[TABLE]

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 $m$-th Hermite function $\phi_{B,m}$ is an eigenfunction of $\operatorname{Op}^{\text{W}}_{1}(a)$, 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 $\operatorname{Op}^{\text{W}}_{1}(a)$ associated with eigenvalues

[TABLE]

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 [8] 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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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 $(\alpha,\beta)\neq(1,0)$. For $\rho>0$,

[TABLE]

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

[TABLE]

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

[TABLE]

Elementary computation gives

[TABLE]

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

[TABLE]

We define the function $\psi_{0}(z)$ by

[TABLE]

Let $\lVert\cdot\rVert_{\mathscr{H}_{B}}$ 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)

$\lvert\psi_{0}(z)\rvert^{2}e^{-\lvert{z}\rvert^{2}/2}=e^{-\lvert\zeta\rvert^{2}/(\alpha^{2}+\beta^{2}+1)}$.

• (iii)

$\lVert\psi_{0}\rVert_{\mathscr{H}_{B}}^{2}=(\alpha^{2}+\beta^{2}+1)\pi/\alpha$.

Proof.

We first show (i). We have only to show the integrability of $\lvert\psi_{0}(z)\rvert^{2}e^{-\lvert{z}\rvert^{2}/2}$ since $\psi_{0}(z)$ is an entire function. Note that

[TABLE]

We have

[TABLE]

since

[TABLE]

for $(\alpha,\beta)\neq(1,0)$. Thus $0<\lvert{a}\rvert<1$. We deduce that

[TABLE]

This implies that $\lvert\psi_{0}(z)\rvert^{2}e^{-\lvert{z}\rvert^{2}/2}$ is integrable on $\mathbb{C}$ with respect to the Lebesgue measure $L(dz)$ and $\psi_{0}\in\mathscr{H}_{B}(\mathbb{C})$.

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

[TABLE]

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

[TABLE]

This completes the proof. ∎

The identity $\lvert\psi_{0}(z)\rvert^{2}e^{-\lvert{z}\rvert^{2}/2}=e^{-\lvert\zeta\rvert^{2}/(\alpha^{2}+\beta^{2}+1)}$ 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

[TABLE]

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

[TABLE]

for short. Properties of $\Lambda_{\alpha,\beta}$, $\Lambda_{\alpha,\beta}^{\ast}$ and $\{\psi_{n}\}_{n=0}^{\infty}$ are the following.

Theorem 4.2**.**

• (i)

$\{\psi_{n}\}_{n=0}^{\infty}$* satisfies a formula of the form*

[TABLE]

• (ii)

For $n=1,2,3,\dotsc$,

[TABLE]

• (iii)

$\{\psi_{n}\}_{n=0}^{\infty}$* is a complete orthogonal system of $\mathscr{H}_{B}(\mathbb{C})$.*

• (iv)

For $n=0,1,2,\dotsc$,

[TABLE]

Proof.

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

[TABLE]

Using this repeatedly, we have

[TABLE]

which is desired.

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

[TABLE]

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

[TABLE]

which is desired.

For (iii), we here show only the orthogonality

[TABLE]

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

[TABLE]

If $m>n$, then

[TABLE]

If $m=n$, then

[TABLE]

Finally we show (iv). By using (ii) and $\Lambda_{\alpha,\beta}\psi_{0}=0$ again, we deduce that

[TABLE]

This completes the proof. ∎

Here we introduce a family of functions $\{\Psi_{n}\}_{n=0}^{\infty}$ on $\mathbb{R}$ by setting $\Psi_{n}=B_{1}^{\ast}\psi_{n}$ for $n=0,1,2,\dotsc$. In order to study basic properties on $\{\Psi_{n}\}_{n=0}^{\infty}$, we introduce notation. Set

[TABLE]

for short. Then we have

[TABLE]

[TABLE]

Set

[TABLE]

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 $n=0,1,2,\dotsc$

[TABLE]

In particular,

[TABLE]

• (ii)

$\{\Psi_{n}\}_{n=0}^{\infty}$* is a family of eigenfunctions of $H_{\alpha,\beta}$, that is,*

[TABLE]

• (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 Section 2 with

[TABLE]

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

[TABLE]

Proof.

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

[TABLE]

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

[TABLE]

[TABLE]

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

[TABLE]

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

[TABLE]

and

[TABLE]

Then we have

[TABLE]

and

[TABLE]

This completes the proof. ∎

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 $\Psi_{0}(x)$

[TABLE]

where

[TABLE]

Elementary computation gives

[TABLE]

where

[TABLE]

By using this and Lemma 4.4, we deduce that

[TABLE]

which is desired.

Finally we check the Rodrigues formula for $\Psi_{n}$. By using the definition of $\Psi_{n}$, we deduce that

[TABLE]

This completes the proof. ∎