Wigner function and weyl transform, trace class

ABSTRACT

The connection between cross-ambiguity and cross-Wigner functions is studied. The correspondence between the class of symbols and the Weyl operators is established; we obtain the necessary and sufficient conditions on the symbol under which the Weyl operator belongs to the trace class. The theory of Kaster-Loupias-Miracle is developed and the exact conditions on the Weyl operator to be positively defined in the Hilbert space is established.

KEYWORDS:

• Wigner function
• ambiguity function
• moyal identity
• trace operator
• weyl operator

1. Introduction and definitions

This article is dedicated to Weyl correspondence and trace class operators, which is a generalization of finite rank operators and subclass of Hilbert-Schmidt operators. We consider a new approach to trace class and the cross-ambiguity, and cross-Wigner functions and consider their application quantum theory. There is extensive literature on the subject some of the newest works can be found below, in (Nokkala et al., 2021) the authors consider the classical Gaussian systems, they introduce a new paradigm for reservoir computing harnessing quantum systems; to simulate the multipartite entanglement of a complex system, the non-Gaussian generalizations are considered (Kamenshchik et al., 2021; Qin et al., 2021); the application of the pseudo-differential operators can be found in (Fan & Ying, 2020; Li, Kovachki, Azizzadenesheli, Liu, Bhattacharya, Stuart, Anandkumar et al., 2020a, 2020b; Y. Li et al., 2020). The Wigner function formalism is a relatively new approach to the problem of quantum cosmology, which helps establish a correlation between canonical momenta in quantum cosmology and their independent variables (Kamenshchik et al., 2021), (Nokkala et al., 2021; Srinivas & Wolf, 1975). The results of this work can be applied to problems of quantum information science and quantum gravity. See also reference list [11-29].

Let $\eta$ be a real non-zero parameter. The standard Fourier transform $F_{\sigma ,\eta }a\left(x,p\right)$ is given by

(1) $F_{\sigma ,\eta }a\left(x,p\right)={\left(\frac{1}{2\pi \eta }\right)}^n{⟨exp\left(-\frac{i}{\eta }\sigma \left(\left(x,p\right),\left(\tilde{x},\tilde{p}\right)\right)\right)a\left(\tilde{x},\tilde{p}\right)⟩}_{\left(\tilde{x},\tilde{p}\right)}$(1)

where $\sigma$ is a symplectic form.

The $\eta$-cross Wigner transform can be defined by

(2) $W_{\eta }\left(\psi ,\phi \right)\left(x,p\right)={\left(\frac{1}{2\pi \eta }\right)}^n{⟨exp\left(-\frac{i}{\eta }p\cdot z\right)\psi \left(x+\frac{1}{2}z\right)\bar{\phi }\left(x-\frac{1}{2}z\right)⟩}_z$(2)

that form is correctly defined for all $\psi \in L^p$ and all $\phi \in L^q$, where $p=q=2$. The equalities

(3) ${⟨W_{\mathrm{\hslash }}\left(\psi ,\phi \right)\left(x,p\right)⟩}_p=\psi \left(x\right)\bar{\phi }\left(x\right)$(3)

and

(4) ${⟨W_{\mathrm{\hslash }}\left(\psi ,\phi \right)\left(x,p\right)⟩}_x=F_{1,\mathrm{\hslash }}\left(\psi \left(p\right)\right)F_{1,\mathrm{\hslash }}\left(\bar{\phi }\left(p\right)\right)$(4)

hold at least for any $\psi \in L^2\left(R^n\right)$ and any $\phi \in L^2\left(R^n\right)$. So, we have

(5) ${⟨W_{\mathrm{\hslash }}\left(\psi ,\phi \right)\left(x,p\right)⟩}_{\left(x,p\right)}={⟨\psi |\phi ⟩}_{\left(x,p\right)}$(5)

Assume $\theta$ is a real parameter and $\mu =\theta \eta$ then the Wigner function $W_{\mu }\left(\psi ,\psi \right)\left(x,p\right)$ can be presented in the following form $W_{\mu }\left(\psi ,\psi \right)\left(x,p\right)={\left|\theta \right|}^{-n}{W}_{\eta }\left(\psi ,\psi \right)\left(x,{\left|\theta \right|}^{-1}p\right)$, in particular, if $\theta =-1$ then $W_{\eta }\left(\psi ,\psi \right)\left(x,p\right)={\left(-1\right)}^{-n}{W}_{-\eta }\left(\bar{\psi },\bar{\psi }\right)\left(x,p\right)$.

The Weyl-Heisenberg translation $T_{\eta }\left(x,p\right)$ is an operator defined by

(6) $T_{\eta }\left(x_0,p_0\right)\psi \left(x\right)=exp\left(\frac{i}{\eta }{p}_0\left(x-\frac{1}{2}{x}_0\right)\right)\psi \left(x-x_0\right)$(6)

The cross-ambiguity function $Amb\left(\psi ,\phi \right)$ is a two-dimensional function of pair $\left(\psi ,\phi \right)$ of vectors of $L^2\left(R^n\right)$ defined by

(7) $Amb\left(\psi ,\phi \right)\left(x,p\right)={\left(\frac{1}{2\pi \eta }\right)}^n{⟨\psi |T_{\eta }\left(x,p\right)\phi ⟩}_{L^2}$(7)

Such defined cross-ambiguity function depends on the scalar parameter $\eta$

Applying the definition of the Weyl-Heisenberg translation $T_{\eta }\left(x,p\right)$, we have

(8) $Amb\left(\psi ,\phi \right)\left(x,p\right)={\left(\frac{1}{2\pi \eta }\right)}^n{⟨exp\left(-\frac{i}{\eta }p\cdot z\right)\text{break}\psi \left(z+\frac{1}{2}x\right)\bar{\phi }\left(z-\frac{1}{2}x\right)⟩}_z$(8)

So, the cross-ambiguity function is a Fourier transform of the cross Wigner function, thus we have

(9) $Amb\left(\psi ,\phi \right)=F_{\sigma ,\eta }{W}_{\eta }\left(\psi ,\phi \right)$(9)

Definition 1. A function $a\in C^{\mathrm{\infty }}\left(R^n\times R^n\right)$ belongs to class $S^m$ if $a$ satisfies the differential inequalities

(10) $\left|{\mathrm{\partial }}_x^{\beta }{\mathrm{\partial }}_p^{\alpha }a\left(x,p\right)\right|\le M_{\alpha ,\beta }{\left(1+\left|p\right|\right)}^{m-\left|\alpha \right|}$(10)

For all multi-indices $\alpha$ and$\beta$

Let $a$ be a classical symbol of the tempered distributions space $S{ }^{\mathrm{\prime }}\left(R^n\oplus R^n\right)$. A Weyl pseudodifferential operator ${\hat{A}}^W$ can be defined as a mapping ${\hat{A}}^W:S\left(R^n\right)\to S\left(R^n\right)$ such that

(11) $⟨{\hat{A}}^W\psi ,\bar{\phi }⟩=⟨a,W_{\eta }\left(\psi ,\phi \right)⟩$(11)

defined for all $\psi ,\phi \in S\left(R^n\right)$.

The pseudo-differential operator has an integral representation

(12) $\hat{A}\psi \left(x\right)={\left(\frac{1}{2\pi \eta }\right)}^n{⟨exp\left(\frac{i}{\eta }p\cdot \left(x-z\right)\right)\text{break}{a}_{\tau }\left(\left(1-\tau \right)x+\tau z,p\right)\psi \left(z\right)⟩}_{\left(z,p\right)}$(12)

if we take $\tau =0$ then

(13) $\hat{A}\psi \left(x\right)={\left(\frac{1}{2\pi \eta }\right)}^n{⟨exp\left(\frac{i}{\eta }p\cdot \left(x-z\right)\right)a\left(x,p\right)\psi \left(z\right)⟩}_{\left(z,p\right)}.$(13)

Assume $\tau =\frac{1}{2}$ and $\tau =\frac{1}{2}\sqrt{\eta }$, we denote operators

(14) ${\hat{A}}_{1/2}\psi \left(x\right)={\left(\frac{1}{2\pi \eta }\right)}^n{⟨exp\left(\frac{i}{\eta }p\cdot \left(x-z\right)\right)a\left(\frac{1}{2}\left(x+z\right),p\right)\psi \left(z\right)⟩}_{\left(z,p\right)}$(14)

and

(15) ${\hat{A}}_{1/2\sqrt{\eta }}\psi \left(x\right)={\left(\frac{1}{2\pi }\right)}^n{⟨exp\left(i\xi \cdot \left(x-z\right)\right)\text{break}a\left(\frac{1}{2}\sqrt{\eta }\left(x+z\right),\sqrt{\eta }\xi \right)\psi \left(z\right)⟩}_{\left(z,\xi \right)}$(15)

then if we denote $M_{\sqrt{\eta }}\psi \left(x\right)={\eta }^{\frac{n}{4}}\psi \left(x\sqrt{\eta }\right)$ operators ${\hat{A}}_{1/2}$ and ${\hat{A}}_{1/2\sqrt{\eta }}$ are connected according to formula ${\hat{A}}_{1/2}={M_{\sqrt{\eta }}}^{-1}{\hat{A}}_{1/2\sqrt{\eta }}{M}_{\sqrt{\eta }}$.

Definition 2. A lattice in $R^{2n}$ is a discrete subgroup $M\left(Z^{2n}\right)$ here $M\in GL\left(2n,R\right)$, then the determinant of $M$ is called the volume of the lattice.

The Weyl-Heisenberg system is called the set

(16) $\mathfrak{C}\left(\phi ,M\left(Z^{2n}\right)\right)=\left\{\hat{T}\left(x,p\right)\phi :\left(x,p\right)\in M\left(Z^{2n}\right),\text{break}\phi \ne 0,\phi \in L^2\left(R^n\right)\right\}$(16)

If the system $\mathfrak{C}\left(\phi ,M\left(Z^{2n}\right)\right)$ constitutes a frame, this frame is called a Weyl-Heisenberg frame

For all $\psi \in S\left(R^n\right)$ the Weyl-Heisenberg operator is

(17) ${\hat{T}}_{\eta }\left(x_0,p_0\right)\psi \left(x\right)=exp\left(\frac{i}{\eta }{p}_0\left(x-\frac{1}{2}{x}_0\right)\right)\psi \left(x-x_0\right)$(17)

The Wigner-Fourier transform of a function $\psi \in L^2\left(R^n\right)$ is the cross auto-ambiguity transform of $\psi$ defined by

(18) $Amb\left(\psi ,\psi \right)\left(x,p\right)={\left(\frac{1}{2\pi \eta }\right)}^n{⟨\psi |{\hat{T}}_{\eta }\left(x,p\right)\psi ⟩}_{L^2}$(18)

Of course, we can write the cross-ambiguity function in the following inverse form

(19) $Amb\left(\psi ,\phi \right)\left(-x,-p\right)={\left(\frac{1}{2\pi \eta }\right)}^n{⟨{\hat{T}}_{\eta }\left(x,p\right)\psi |\bar{\phi }⟩}_{L^2}$(19)

or

(20) $Amb\left(\psi ,\phi \right)\left(x,p\right)={\left(\frac{1}{2\pi \eta }\right)}^n{⟨{\hat{T}}_{\eta }\left(x,p\right){\psi }^{-}|{\phi }^{-}⟩}_{L^2}$(20)

where ${\psi }^{-}\left(x\right)=\psi \left(-x\right)$ and ${\phi }^{-}\left(x\right)=\phi \left(-x\right)$.

The function $\phi \ne 0,\phi \in L^2\left(R^n\right)$ is called a window of the frame.

Definition 3. The function ${W_{\eta }}^M\left(\psi ,\phi \right)\left(x,p\right)$ defined by

(21) ${W_{\eta }}^M\left(\psi ,\phi \right)\left(x,p\right)={\left(\frac{1}{2\pi \eta }\right)}^n{⟨exp\left(2\frac{i}{\eta }p\cdot \left(z-x\right)\right)\text{break}\psi \left(2x-z\right)\bar{\phi }\left(z\right)⟩}_z$(21)

is called a Wigner-Moyal function or distribution, which is correctly defined at least for $\psi ,\phi \in S\left(R^n\right)$.

An integral operator ${\hat{T}}_R$ defined by

(22) ${\hat{T}}_R\left(x_0,p_0\right)\psi \left(x\right)=exp\left(2\frac{i}{\eta }{p}_0\cdot \left(x-x_0\right)\right)\psi \left(2x_0-x\right)$(22)

Is called Royer-Grossman operator

Since

(23) $\begin{array}{rl}& {⟨{\hat{T}}_R\left(x,p\right)\psi |\phi ⟩}_{L^2}={⟨exp\left(2\frac{i}{\eta }p\cdot \left(z-x\right)\right)\psi \left(2z-x\right)|\bar{\phi }\left(z\right)⟩}_z=\\ & ={⟨2^{-n}exp\left(-\frac{i}{\eta }p\cdot 2\left(x-z\right)\right)\psi \left(x+x-z\right)|\bar{\phi }\left(x-x+z\right)⟩}_z=\\ & ={\left(2\pi \eta \right)}^n{W_{\eta }}^M\left(\psi ,\phi \right)\left(x,p\right)\end{array}$(23)

we obtained the representation of the Wigner-Moyal function as

(24) ${W_{\eta }}^M\left(\psi ,\phi \right)\left(x,p\right)=\frac{1}{{\left(2\pi \eta \right)}^n}{⟨{\hat{T}}_R\left(x,p\right)\psi |\bar{\phi }⟩}_{L^2}$(24)

Assume $\psi ,\phi \in L^2\left(R^n\right)$, we have the formula

$\begin{array}{rl}& {W_{\eta }}^M\left(\hat{T}\left(x_1,p_1\right)\psi ,\hat{T}\left(x_2,p_2\right)\phi \right)\left(x,p\right)=\\ & \frac{1}{{\left(2\pi \eta \right)}^n}exp\left(\frac{i}{\eta }\left(\left(p_1-p_2\right)x-\left(x_1-x_2\right)p+\frac{1}{2}\left(p_2{x}_1-x_2{p}_1\right)\right)\right)\times \\ & \times {⟨exp\left(-\frac{i}{\eta }\left(p-\frac{1}{2}z\left(p_1+p_2\right)\right)\right)\psi \left(x-\frac{1}{2}\left(\left(x_1+x_2\right)-z\right)\right)\text{break}\bar{\phi }\left(x-\frac{1}{2}\left(\left(x_1+x_2\right)+z\right)\right)⟩}_z=\\ & =exp\left(-\frac{i}{\eta }\left(\left(p_2-p_1\right)x-\left(x_2-x_1\right)p+\frac{1}{2}\left(x_2{p}_1-p_2{x}_1\right)\right)\right)\times \\ & \times {W_{\eta }}^M\left(\psi ,\phi \right)\left(\left(x,p\right)-\frac{1}{2}\left(x_1+x_2,p_1+p_2\right)\right),\end{array}$

Which consequence is that the equality

(25) ${W_{\eta }}^M\left(\hat{T}\left(x_1,p_1\right)\psi ,\hat{T}\left(x_1,p_1\right)\phi \right)\left(x,p\right)=\hat{T}\left(x_1,p_1\right){W_{\eta }}^M\left(\psi ,\phi \right)\left(x,p\right)$(25)

holds for all functions $\psi ,\phi \in L^2\left(R^n\right)$ and all $\left(x_1,p_1\right)\in R^{2n}$.

2. The connection between Cross ambiguity and cross Wigner functions

Let us calculate a Fourier transform $F_{\sigma ,\eta }$ of the Wigner function ${W_{\eta }}^M\left(\psi ,\phi \right)$ as

$\begin{array}{rl}& F_{\sigma ,\eta }{W_{\eta }}^M\left(\psi ,\phi \right)=\\ & ={\left(\frac{1}{2\pi \eta }\right)}^{2n}{⟨{⟨{⟨exp\left(-\frac{i}{\eta }\left(\sigma \left(\left(x,p\right),\left(\tilde{x},\tilde{p}\right)\right)+\tilde{p}\cdot z\right)\right)\text{break}\psi \left(\tilde{x}+\frac{1}{2}z\right)\bar{\phi }\left(\tilde{x}-\frac{1}{2}z\right)⟩}_{\tilde{p}}⟩}_{\tilde{x}}⟩}_z=\\ & ={\left(\frac{1}{2\pi \eta }\right)}^{2n}{⟨{⟨{⟨exp\left(-\frac{i}{\eta }\left(\tilde{p}\cdot \left(z-x\right)+p\cdot \tilde{x}\right)\right)\text{break}\psi \left(\tilde{x}+\frac{1}{2}z\right)\bar{\phi }\left(\tilde{x}-\frac{1}{2}z\right)⟩}_{\tilde{p}}⟩}_{\tilde{x}}⟩}_z=\\ & ={\left(\frac{1}{2\pi \eta }\right)}^n{⟨{⟨exp\left(-\frac{i}{\eta }p\cdot \tilde{x}\right)\delta \left(x-z\right)\text{break}\psi \left(\tilde{x}+\frac{1}{2}z\right)\bar{\phi }\left(\tilde{x}-\frac{1}{2}z\right)⟩}_{\tilde{x}}⟩}_z=\\ & ={\left(\frac{1}{2\pi \eta }\right)}^n{⟨exp\left(-\frac{i}{\eta }p\cdot \tilde{x}\right)\psi \left(\tilde{x}+\frac{1}{2}x\right)\text{break}\bar{\phi }\left(\tilde{x}-\frac{1}{2}x\right)⟩}_{\tilde{x}}=\\ & =Amb\left(\psi ,\phi \right)\left(x,p\right)\end{array}$

since we integrate

${⟨exp\left(-\frac{i}{\eta }\left(\tilde{p}\cdot \left(z-x\right)\right)\right)exp\left(-\frac{i}{\eta }p\cdot \tilde{x}\right)\psi \left(\tilde{x}+\frac{1}{2}z\right)\text{break}\bar{\phi }\left(\tilde{x}-\frac{1}{2}z\right)⟩}_{\tilde{p}}={\left(2\pi \eta \right)}^n\delta \left(x-z\right)$

and since

(26) $Amb\left(\psi ,\phi \right)\left(x,p\right)={\left(\frac{1}{2\pi \eta }\right)}^n{⟨exp\left(-\frac{i}{\eta }p\cdot z\right)\text{break}\psi \left(z+\frac{1}{2}x\right)\bar{\phi }\left(z-\frac{1}{2}x\right)⟩}_z$(26)

thus, we proved that $F_{\sigma ,\eta }{W_{\eta }}^M\left(\psi ,\phi \right)=Amb\left(\psi ,\phi \right)\left(x,p\right)$.

3. The general Moyal identity

The correspondence between the ambiguity and Wigner functions appears as the integral identities so that both types of mappings are satisfying the orthogonality relationship in the form of the Moyal identities.

Theorem 1. Let ${\psi }_1,{\phi }_1,{\psi }_2,{\phi }_2\in L^2\left(R^n\right)$ then we have identities

(27) ${⟨Amb\left({\psi }_1,{\phi }_1\right)|Amb\left({\psi }_2,{\phi }_2\right)⟩}_{L^2\left(R^{2n}\right)}={\left(\frac{1}{2\pi \eta }\right)}^n⟨{\psi }_1,{\bar{\psi }}_2⟩⟨{\bar{\phi }}_1,{\phi }_2⟩$(27)

and

(28) ${⟨W_{\eta }\left({\psi }_1,{\phi }_1\right)|W_{\eta }\left({\psi }_2,{\phi }_2\right)⟩}_{L^2\left(R^{2n}\right)}={\left(\frac{1}{2\pi \eta }\right)}^n⟨{\psi }_1,{\bar{\psi }}_2⟩⟨{\bar{\phi }}_1,{\phi }_2⟩$(28)

thus, we have

(29) ${⟨Amb\left({\psi }_1,{\phi }_1\right)|Amb\left({\psi }_2,{\phi }_2\right)⟩}_{L^2\left(R^{2n}\right)}={⟨W_{\eta }\left({\psi }_1,{\phi }_1\right)|W_{\eta }\left({\psi }_2,{\phi }_2\right)⟩}_{L^2\left(R^{2n}\right)}$(29)

Proof. We have already established $Amb\left(\psi ,\phi \right)=F_{\sigma ,\eta }{W}_{\eta }\left(\psi ,\phi \right)$, which means that the ambiguity function is the Fourier transform of the cross-Wigner function and since the Fourier transform is unitary so the last equality is proven. Next, let us consider an integral

$\begin{array}{rl}& {⟨W_{\eta }\left({\psi }_1,{\phi }_1\right)|W_{\eta }\left({\psi }_2,{\phi }_2\right)⟩}_{L^2\left(R^{2n}\right)}=\\ & ={\left(\frac{1}{2\pi \eta }\right)}^{2n}{⟨{⟨{⟨{⟨\begin{array}{c}exp\left(-\frac{i}{\eta }p\left(z-\tilde{z}\right)\right){\psi }_1\left(x+\frac{1}{2}z\right){\bar{\phi }}_1\left(x-\frac{1}{2}z\right)\times \\ \times {\bar{\psi }}_2\left(x+\frac{1}{2}\tilde{z}\right){\phi }_2\left(x-\frac{1}{2}\tilde{z}\right)\end{array}⟩}_z⟩}_{\tilde{z}}⟩}_x⟩}_p=\\ & ={\left(\frac{1}{2\pi \eta }\right)}^n{⟨{⟨{⟨\begin{array}{c}\delta \left(z-\tilde{z}\right){\psi }_1\left(x+\frac{1}{2}z\right){\bar{\phi }}_1\left(x-\frac{1}{2}z\right)\times \\ \times {\bar{\psi }}_2\left(x+\frac{1}{2}\tilde{z}\right){\phi }_2\left(x-\frac{1}{2}\tilde{z}\right)\end{array}⟩}_z⟩}_{\tilde{z}}⟩}_x=\\ & ={\left(\frac{1}{2\pi \eta }\right)}^n{⟨{⟨{\psi }_1\left(x+\frac{1}{2}z\right){\bar{\phi }}_1\left(x-\frac{1}{2}z\right){\bar{\psi }}_2\left(x+\frac{1}{2}z\right){\phi }_2\left(x-\frac{1}{2}z\right)⟩}_z⟩}_x,\end{array}$

let us change variables in accordance with the formulae

$\alpha =x+\frac{1}{2}z$

$\beta =x-\frac{1}{2}z$

so that

$\begin{array}{rl}& {⟨W_{\eta }\left({\psi }_1,{\phi }_1\right)|W_{\eta }\left({\psi }_2,{\phi }_2\right)⟩}_{L^2\left(R^{2n}\right)}=\\ & ={\left(\frac{1}{2\pi \eta }\right)}^n{⟨{⟨{\psi }_1\left(\alpha \right){\bar{\phi }}_1\left(\beta \right){\bar{\psi }}_2\left(\alpha \right){\phi }_2\left(\beta \right)⟩}_{\alpha }⟩}_{\beta }=\\ & ={\left(\frac{1}{2\pi \eta }\right)}^n{⟨{\psi }_1\left(\alpha \right){\bar{\psi }}_2\left(\alpha \right)⟩}_{\alpha }{⟨{\bar{\phi }}_1\left(\beta \right){\phi }_2\left(\beta \right)⟩}_{\beta },\end{array}$

which proves the theorem.

The following statements are the consequences of the theorem. The cross-ambiguity $Amb$ and cross-Wigner $W_{\eta }$ maps can be extended as follows:

(30) $Amb:L^2\left(R^n\right)\times L^2\left(R^n\right)\to L^2\left(R^{2n}\right)\cap C_0\left(R^{2n}\right)$(30)

(31) $W_{\eta }:L^2\left(R^n\right)\times L^2\left(R^n\right)\to L^2\left(R^{2n}\right)\cap C_0\left(R^{2n}\right)$(31)

4. Generalization of the Lieb theory

Definition 4. Let $A,B,C$ be complex numbers and $Re\left(A\right)>0$. The function $f\left(x\right)=exp\left(-A{x}^2+Bx+C\right)$ is called a Gaussian. Two Gaussians $f$ and $g$ are called a matched Gaussian pair if they have the same coefficient before $x^2$.

Theorem (General Lieb inequality) 2. Let $\frac{1}{r}+\frac{1}{q}=1$ and $\frac{1}{\alpha }+\frac{1}{\beta }=1$, and let $\psi \in L^{\alpha }\left(R^n\right)$ and $\phi \in L^{\beta }\left(R^n\right)$ then inequality

(32) ${⟨{\left|Amb\left(\psi ,\phi \right)\left(x,p\right)\right|}^r⟩}_{\left(x,p\right)}\le Const\left(r,\alpha ,\beta \right){⟨{\left|\psi \right|}^{\alpha }⟩}^{\frac{r}{\alpha }}{⟨{\left|\phi \right|}^{\beta }⟩}^{\frac{r}{\beta }}$(32)

Holds for all $\frac{r}{r-1}\le \alpha \le r$ and$\frac{r}{r-1}\le \beta \le r,1<r$

Proof. This statement follows from

$\begin{array}{rl}& {⟨{\left|Amb\left(\psi ,\phi \right)\left(x,p\right)\right|}^r⟩}_{\left(x,p\right)}=\\ & ={\left(\frac{1}{2\pi \eta }\right)}^{2n}{⟨{⟨{⟨{⟨{\left|exp\left(-\frac{i}{\eta }p\left(z-\tilde{z}\right)\right)\psi \left(x+\frac{1}{2}z\right)\bar{\phi }\left(x-\frac{1}{2}z\right)\right|}^r⟩}_z⟩}_{\tilde{z}}⟩}_x⟩}_p=\\ & ={\left(\frac{1}{2\pi \eta }\right)}^n{r}^n{⟨{⟨{⟨\delta \left(z-\tilde{z}\right){\left|\psi \left(x+\frac{1}{2}z\right)\bar{\phi }\left(x-\frac{1}{2}z\right)\right|}^r⟩}_z⟩}_{\tilde{z}}⟩}_x=\\ & ={\left(\frac{1}{2\pi \eta }\right)}^n{r}^n{⟨{⟨{\left|\psi \left(x+\frac{1}{2}z\right)\bar{\phi }\left(x-\frac{1}{2}z\right)\right|}^r⟩}_z⟩}_x=\\ & ={\left(\frac{1}{2\pi \eta }\right)}^n{r}^n{⟨{\left|\psi \left(\tilde{z}\right)\right|}^r{\left|\bar{\phi }\left(\tilde{x}\right)\right|}^r⟩}_{\left(\tilde{x},\tilde{z}\right)},\end{array}$

where $\tilde{z}=x+\frac{1}{2}z$ and $\tilde{x}=x-\frac{1}{2}z$. Next, one can use Holder inequality and prove Lieb’s inequality.

Theorem 3. Let functions $\psi \in L^{\alpha }\left(R^n\right)$ and $\phi \in L^{\beta }\left(R^n\right)$, $\frac{1}{\alpha }+\frac{1}{\beta }=1$. Assume the following estimation $0<{⟨{\left|Amb\left(\psi ,\phi \right)\left(x,p\right)\right|}^r⟩}_{\left(x,p\right)}<\mathrm{\infty }$ holds for $1\le r<2$, then functions $\psi$ and $\phi$ belong to $L^r\left(R^n\right)$ and the inequality

(33) ${⟨{\left|Amb\left(\psi ,\phi \right)\left(x,p\right)\right|}^r⟩}_{\left(x,p\right)}\ge \tilde{C}onst\left(r,\alpha ,\beta \right){⟨{\left|\psi \right|}^{\alpha }⟩}^{\frac{r}{\alpha }}{⟨{\left|\phi \right|}^{\beta }⟩}^{\frac{r}{\beta }}$(33)

holds for $r\le \alpha$ and $\beta \le q$, $\frac{1}{r}+\frac{1}{q}=1$.

Remark. The equalities can be achieved for a pair of the Gaussians with some additional conditions.

5. Pseudo-differential operators in the light of the quantum theory

Let function $a$ be a symbol belonging to $S\left(R^{2n}\right)$ .

The linear pseudo-differential operator $\hat{A}$ with symbol $a$ mapping $S\left(R^n\right)\to S\left(R^n\right)$ by formula

(34) $\hat{A}\psi \left(x\right)={\left(\frac{1}{2\pi \eta }\right)}^n{⟨a\left(z,p\right)exp\left(2\frac{i}{\eta }p\cdot \left(x-z\right)\right)\psi \left(2z-x\right)⟩}_{\left(z,p\right)}$(34)

is called Weyl operator.

Since ${\hat{T}}_R\left(x_0,p_0\right)\left(\psi \left(x\right)\right)=exp\left(2\frac{i}{\eta }{p}_0\cdot \left(x-x_0\right)\right)\psi \left(2x_0-x\right)$, the Weyl operator can be written in the form

(35) $\hat{A}\psi \left(x\right)={\left(\frac{1}{2\pi \eta }\right)}^n{⟨a\left(z,p\right){\hat{T}}_R\left(z,p\right)\left(\psi \left(x\right)\right)⟩}_{\left(z,p\right)}$(35)

Statement. The Weyl operator extends to the continuous operator $\hat{A}:S{ }^{\mathrm{\prime }}\left(R^n\right)\to S{ }^{\mathrm{\prime }}\left(R^n\right)$.

Indeed, since $a\in S\left(R^{2n}\right)$ the function $a\left(z,p\right)x^{\alpha }{\mathrm{\partial }}_x^{\alpha }{\hat{T}}_R\left(z,p\right)\psi \in S\left(R^{2n}\right)$ for all functions $\psi \in S\left(R^n\right)$ and all multi-indices $\alpha \in N^n$ therefore $\left|x^{\alpha }{\mathrm{\partial }}_x^{\alpha }\hat{A}\psi \left(x\right)\right|<\mathrm{\infty }$.

Weyl established that correspondence between symbols $a$ and Weyl operators $A$ is one-to-one and linear, unit symbol corresponds to the identity operator on $S{ }^{\mathrm{\prime }}\left(R^n\right)$. Thus, the set of all Weyl operators coincides with the set of all symbols on $S\left(R^n\oplus R^n\right)$. The Weyl operators are pseudo-differential operators with rapidly decreasing kernels.

Since the Weyl operator can be rewritten as

(36) $\hat{A}\psi \left(x\right)={\left(\frac{1}{2\pi \eta }\right)}^n{⟨a\left(\frac{1}{2}\left(x+z\right),p\right)exp\left(\frac{i}{\eta }p\cdot \left(x-z\right)\right)\psi \left(z\right)⟩}_{\left(z,p\right)},$(36)

so that the kernel of the Weyl operator $A$ can be calculated by the formula

(37) $K_{\hat{A}}\left(x,y\right)={\left(\frac{1}{2\pi \eta }\right)}^n{⟨exp\left(\frac{i}{\eta }p\cdot \left(x-y\right)\right)a\left(\frac{1}{2}\left(x+y\right),p\right)⟩}_p$(37)

then, the symbol can be represented as

(38) $a\left(x,p\right)={⟨exp\left(-\frac{i}{\eta }p\cdot z\right)K_{\hat{A}}\left(x+\frac{1}{2}z,x-\frac{1}{2}z\right)⟩}_z$(38)

The last three formulae are circular.

Theorem 4. Let $\hat{A}\stackrel{Weyl}{\leftrightarrow }a$ be the Weyl correspondence then

1. for $a\in S\left(R^n\oplus R^n\right)$ it is necessary and sufficient $K_{\hat{A}}\left(x,y\right)\in S\left(R^n\oplus R^n\right)$ and $\hat{A}\left(\psi \left(x\right)\right)={⟨K_{\hat{A}}\left(x,z\right)\psi \left(z\right)⟩}_z$;

2. the map $a\mapsto \hat{A}$ extends to an isomorphism $S{ }^{\mathrm{\prime }}\left(R^n\oplus R^n\right)\to L\left(S\left(R^n\right),S{ }^{\mathrm{\prime }}\left(R^n\right)\right)$, where $L\left(S\left(R^n\right),S{ }^{\mathrm{\prime }}\left(R^n\right)\right)$ is the space of continuous linear operators from $S\left(R^n\right)$ to $S{ }^{\mathrm{\prime }}\left(R^n\right)$.

Proof. The theorem follows from the Schwartz kernel theorem.

Theorem 5. Let the Weyl operator $\hat{A}$ corresponds to symbol $a\in L^r\left(R^n\oplus R^n\right),1\le r<2$ so $\hat{A}\stackrel{Weyl}{\leftrightarrow }a$, then there is constant $Const\left(r\right)$ such that the inequality

(39) ${∥\hat{A}\psi ∥}_{L^2\left(R^n\right)}\le Const\left(r\right){∥\psi ∥}_{L^2\left(R^n\right)}{∥a∥}_{L^r\left(R^{2n}\right)}$(39)

holds for all $\psi \in L^2\left(R^n\right)$.

From this theorem follows that for all symbols $a\in L^2\left(R^n\oplus R^n\right)$ corresponding Weyl operators are $L^2$-bounded. However, there are examples of the symbols $a\in L^r\left(R^n\oplus R^n\right),2<r$ on which $L^2$ boundness is ruined so that Weyl operators $\hat{A}$ are not $L^2$- bounded for these symbols $a\in L^r\left(R^n\oplus R^n\right),2<r$.

The complete analysis of $L^2$- regularity for Weyl operators can be made in terms of the Calderon -Zygmund theory.

Theorem 6. Let $\hat{A}$ be trace-class Weyl operator on $L^2\left(R^n\right)$ corresponded to symbol $a\in L^r\left(R^n\oplus R^n\right),1\le r<2$. Then for $\hat{A}\ge 0$ it is necessary and sufficient that

$F_{\sigma }a\left(x,p\right)={⟨exp\left(i\sigma \left(\left(x,p\right),\left(\tilde{x},\tilde{p}\right)\right)\right)a\left(\tilde{x},\tilde{p}\right)⟩}_{\left(\tilde{x},\tilde{p}\right)}$

is continuous and such that the matrix with entries

$exp\left(-\frac{i}{2}\eta \sigma \left(\left(x_j,p_j\right),\left(x_k,p_k\right)\right)\right)F_{\sigma }a\left(\left(x_j,p_j\right)-\left(x_k,p_k\right)\right)$

is positive semidefinite for all possible sets of $\left(x_1,p_1\right),\left(x_2,p_2\right),\dots .,\left(x_N,p_N\right)\in {\left(R^{2n}\right)}^N$.

Proof. Since the operator $F_{\sigma }$ is a symplectic Fourier transform so $F_{\sigma }$ is continuous. Let the system $\left\{{\psi }_j\right\}$ be an orthonormal basis in $L^2\left(R^n\right)$ then symbol $a$ can be rewritten

$a={\left(2\pi \eta \right)}^n\sum _j{\lambda }_j{W}_{\eta }\left({\psi }_j,{\psi }_j\right)$

the coefficients $\left\{{\lambda }_j\right\}\subset l^1\left(N\right)$. Next, we have already established $Amb\left(\psi ,\phi \right)=F_{\sigma ,\eta }{W}_{\eta }\left(\psi ,\phi \right)$ for all $\eta \ne 0$ so we have

$\begin{array}{rl}& \sum _{j,k}\begin{array}{c}{\theta }_j{\bar{\theta }}_{k}exp\left(-\frac{i}{2\eta }\sigma \left(\left(x_j,p_j\right),\left(x_k,p_k\right)\right)\right)\times \\ F_{\sigma ,\eta }{W}_{\eta }\left({\psi }_j\left(\left(x_j,p_j\right)-\left(x_k,p_k\right)\right),{\psi }_j\left(\left(x_j,p_j\right)-\left(x_k,p_k\right)\right)\right)\end{array}=\\ & =\sum _{j,k}\begin{array}{c}{\theta }_j{\bar{\theta }}_{k}exp\left(-\frac{i}{2\eta }\sigma \left(\left(x_j,p_j\right),\left(x_k,p_k\right)\right)\right)\times \\ Amb\left({\psi }_j\left(\left(x_j,p_j\right)-\left(x_k,p_k\right)\right),{\psi }_j\left(\left(x_j,p_j\right)-\left(x_k,p_k\right)\right)\right)\end{array}=\\ & ={\left(2\pi \eta \right)}^{-n}\sum _{j,k}{\theta }_j{\bar{\theta }}_{k}exp\left(-\frac{i}{2\eta }\sigma \left(\left(x_j,p_j\right),\left(x_k,p_k\right)\right)\right)\text{break}{⟨T\left(\left(x_k,p_k\right)-\left(x_j,p_j\right)\right)\psi ,\psi ⟩}_{L^2}=\text{break}\\ & ={\left(2\pi \eta \right)}^{-n}\sum _{j,k}{\theta }_j{\bar{\theta }}_k{⟨T\left(x_k,p_k\right)T\left(-x_j,-p_j\right)\psi ,\psi ⟩}_{L^2}=\\ & ={\left(2\pi \eta \right)}^{-n}{{∥\sum _{j,k}{\theta }_{j}T\left(-x_j,-p_j\right)\psi ∥}_{L^2}}^2\end{array}$

for all $\left({\theta }_j\right)\in C^N$ and all $\left(\left(x_j,p_j\right)\right)\in {\left(R^{2n}\right)}^N$. So, the necessity is proven.

From ${⟨\hat{A}\psi ,\psi ⟩}_{L^2}\ge 0$ follows ${⟨a\left(x,p\right)W_{\eta }\left(\psi \left(x,p\right),\psi \left(x,p\right)\right)⟩}_{\left(x,p\right)}\ge 0$ for all functions $\psi \in L^2\left(R^n\right)$. Since $Amb\left(\psi ,\phi \right)=F_{\sigma ,\eta }{W}_{\eta }\left(\psi ,\phi \right)$ we can consider a matrix with the entries

${\left(2\pi \eta \right)}^{n}Amb\left(\psi \left(\left(x_j,p_j\right)-\left(x_k,p_k\right)\right),\psi \left(\left(x_j,p_j\right)-\text{break}\left(x_k,p_k\right)\right)\right)\sum _j{\lambda }_{j}Amb\left({\psi }_j,{\psi }_j\right)$

the matrix with these entries is an entrywise product of the two positively defined matrices thus this matrix is positively defined.

We integrate ${\left(2\pi \eta \right)}^{n}Amb\left(\psi \left(x,p\right),\psi \text{break}\left(x,p\right)\right)\sum _j{\lambda }_{j}Amb\left({\psi }_j,{\psi }_j\right)\left(-x,-p\right)$ by $\left(x,p\right)$ over whole space so we have

$\begin{array}{rl}& {\left(2\pi \eta \right)}^n{⟨Amb\left(\psi \left(x,p\right),\psi \left(x,p\right)\right)\sum _j{\lambda }_{j}Amb\left({\psi }_j,{\psi }_j\right)\left(-x,-p\right)⟩}_{\left(x,p\right)}=\\ & ={\left(2\pi \eta \right)}^n{⟨W_{\eta }\left(\psi \left(x,p\right),\psi \left(x,p\right)\right)a\left(x,p\right)⟩}_{\left(x,p\right)}\ge 0\end{array}$

that proves sufficiency.

PUBLIC OF INTEREST STATEMENT

The connection between cross-ambiguity and cross-Wigner functions is considered. These types of studies are interesting due to their wide applications in quantum mechanics and signal processing, for example, Weyl correspondence provides us a mathematical formalism for the description of physical phenomena in terms of functions and operators on linear spaces. The open important question is how to extend the results of quantum theory from Hilbert spaces to general Banach structures. The author is interested in functional analysis and its application to the problems of quantum physics and the theory of relativity. The author is working as a researcher at NTUU KPI, the author received no direct funding for this research.

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