﻿ α-Fock空间Fα<sup>2</sup> 上的测不准原理

# α-Fock空间Fα2 上的测不准原理Uncertainty Principle for the α-Fock Space Fα2

Abstract: In this paper, we mainly introduce a positive parameter α and results about uncertainty principle of two self-adjoint operators for the Fock Space F2 are generalized to the α-Fock Space 2 in the complex plane. In particular, we also do a perfect proof for the case of a，b which are complex parameters.

1. 引言

$ℂ$ 为一维复平面， $ℝ$ 为一维实平面，对任意的正实参数 $\alpha$ ，我们定义：

$\text{d}{\lambda }_{\alpha }\left(z\right)=\frac{\alpha }{\text{π}}{\text{e}}^{-\alpha {|z|}^{2}}\text{d}A\left(z\right)$

$ℂ$ 上的Gaussian测度，其中 $\text{d}A=\text{d}x\text{d}y$$ℂ$ 上的Lebesgue面积测度。

${F}_{\alpha }^{2}={L}^{2}\left(ℂ,\text{d}{\lambda }_{\alpha }\right)\cap H\left(ℂ\right)$

$〈f,g〉={\int }_{ℂ}f\left(z\right)\stackrel{¯}{g\left(z\right)}\text{ }\text{ }\text{d}{\lambda }_{\alpha }\left(z\right)$ (1.1)

${‖f‖}_{2,\alpha }={\left({\int }_{ℂ}{|f\left(z\right)|}^{2}\text{d}{\lambda }_{\alpha }\left(z\right)\right)}^{\frac{1}{2}}$ (1.2)

2. 相关引理及主要结果

$‖\left(A-a\right)f‖‖\left(B-b\right)f‖\ge \frac{1}{2}|〈\left[A,B\right]f,f〉|$ (2.1)

$‖{f}^{\prime }+zf-af‖‖{f}^{\prime }-zf+ibf‖\ge {‖f‖}^{2}$

$f\left(z\right)={C}_{1}\mathrm{exp}\left(\frac{c-1}{2\left(c+1\right)}{z}^{2}+\frac{a-ibc}{c+1}z\right)$

${e}_{n}\left(z\right)=\sqrt{\frac{{\alpha }^{n}}{n!}}{z}^{n},\text{\hspace{0.17em}}n=0,1,2,\cdots$

$f=\underset{n=0}{\overset{\infty }{\sum }}{a}_{n}{e}_{n},\text{\hspace{0.17em}}\text{\hspace{0.17em}}g=\underset{n=0}{\overset{\infty }{\sum }}{b}_{n}{e}_{n}$

$Tf\left(z\right)=\frac{1}{\alpha }\frac{\text{d}}{\text{d}z}\underset{n=0}{\overset{\infty }{\sum }}{a}_{n}\sqrt{\frac{{\alpha }^{n}}{n!}}{z}^{n}=\frac{1}{\alpha }\underset{n=1}{\overset{\infty }{\sum }}{a}_{n}\sqrt{\frac{{\alpha }^{n}}{n!}}n{z}^{n-1}=\underset{n=0}{\overset{\infty }{\sum }}\frac{1}{\sqrt{\alpha }}\sqrt{n+1}{a}_{n+1}{e}_{n}\left(z\right)$

$zg\left(z\right)=\underset{n=0}{\overset{\infty }{\sum }}{b}_{n}\sqrt{\frac{{\alpha }^{n}}{n!}}{z}^{n+1}=\underset{n=1}{\overset{\infty }{\sum }}{b}_{n-1}\sqrt{\frac{{\alpha }^{n-1}}{\left(n-1\right)!}}{z}^{n}=\underset{n=1}{\overset{\infty }{\sum }}\sqrt{\frac{n}{\alpha }}{b}_{n-1}{e}_{n}\left(z\right)$

$〈Tf,g〉=\underset{n=0}{\overset{\infty }{\sum }}\frac{1}{\sqrt{\alpha }}\sqrt{n+1}{a}_{n+1}{\stackrel{¯}{b}}_{n}=\underset{n=1}{\overset{\infty }{\sum }}{a}_{n}\sqrt{\frac{n}{\alpha }}\stackrel{¯}{{b}_{n-1}}=〈f,zg〉=〈f,{T}^{*}g〉$

$\left[T,{T}^{*}\right]f=\left(T{T}^{*}-{T}^{*}T\right)f=\frac{{\left(zf\right)}^{\prime }}{\alpha }-\frac{z{f}^{\prime }}{\alpha }=\frac{f}{\alpha }$

$A=T+{T}^{*},\text{\hspace{0.17em}}\text{\hspace{0.17em}}B=i\left(T-{T}^{*}\right)$

$Af\left(z\right)=\frac{1}{\alpha }{f}^{\prime }\left(z\right)+zf\left(z\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}Bf\left(z\right)=i\left(\frac{1}{\alpha }{f}^{\prime }\left(z\right)-zf\left(z\right)\right)$ (2.2)

$\begin{array}{c}AB-BA=i\left[\left(T+{T}^{*}\right)\left(T-{T}^{*}\right)-\left(T-{T}^{*}\right)\left(T+{T}^{*}\right)\right]\\ =2i\left[{T}^{*}T-T{T}^{*}\right]=-\frac{2}{\alpha }iI\end{array}$

${‖\frac{1}{\alpha }{f}^{\prime }+zf-af‖}_{2,\alpha }{‖\frac{1}{\alpha }{f}^{\prime }-zf+ibf‖}_{2,\alpha }\ge \frac{1}{\alpha }{‖f‖}_{2,\alpha }^{2}$ (2.3)

$f\left(z\right)={C}^{\prime }\mathrm{exp}\left(\frac{\alpha \left(c-1\right)}{2\left(c+1\right)}{z}^{2}+\frac{\alpha \left(a-ibc\right)}{c+1}z\right)$

${‖\left(A-a\right)f‖}_{2,\alpha }{‖\left(B-b\right)f‖}_{2,\alpha }\ge \frac{1}{2}|〈\left[A,B\right]f,f〉|$

${‖i\left(\frac{1}{\alpha }{f}^{\prime }-zf\right)-bf‖}_{2,\alpha }={‖\frac{1}{\alpha }{f}^{\prime }-zf+ibf‖}_{2,\alpha }$

${‖\frac{1}{\alpha }{f}^{\prime }+zf-af‖}_{2,\alpha }{‖\frac{1}{\alpha }{f}^{\prime }-zf+ibf‖}_{2,\alpha }\ge \frac{1}{\alpha }{‖f‖}_{2,\alpha }^{2}$

$\frac{1}{\alpha }{f}^{\prime }+zf-af=ic\left[i\left(\frac{1}{\alpha }{f}^{\prime }-zf\right)-bf\right]$ (2.4)

$i\left(\frac{1}{\alpha }{f}^{\prime }-zf\right)-bf=ic\left(\frac{1}{\alpha }{f}^{\prime }+zf-af\right)$ (2.5)

(2.6)

1) 若，则(2.6)只有解这一零解。

2) 若，(2.6)式可变形为，则由解常微分方程的初等方法可得(2.6)的一般解为

(2.7)

(2.8)

，其中，由(2.8)式知其等价于

(2.9)

(2.10)

(2.11)

1) 对函数(2.9)计算如下：

(2.12)

(2.13)

(2.14)

(2.15)

(2.16)

2) 对函数(2.9)直接计算如下：

(2.17)

(2.18)

(2.19)

3) 对函数(2.9)直接计算如下：

(2.20)

(2.21)

(2.22)

4) 对函数(2.9)直接计算如下：

(2.23)

(2.24)

5) 对函数(2.9)直接计算如下：

(2.25)

(2.26)

6) 对函数(2.9)直接计算如下：

(2.27)

(2.28)

7) 最后给出最小值讨论。

(2.29)

(2.30)

(2.31)

(2.32)

1) 若不具有(2.31)的形式，则由定理3可知对任意

2) 若具有(2.31)的形式，为区别开来我们记为，则由定理3可知

(2.33)

(2.34)

(2.35)

(2.36)

3. 复参变量下的测不准原理及其推广

(3.1)

NOTES

*通讯作者。

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