text:
\begin{document}

\title {\bf Periodicity related to a sieve method of producing primes}

\author{\it Haifeng Xu\thanks{Project supported by NSFC(Grant No. 11401515), the University Science Research Project of Jiangsu Province (14KJB110027) and the Foundation of Yangzhou University 2014CXJ004.},\quad Zuyi Zhang,\quad Jiuru Zhou}
\date{\small\today}

\maketitle

\begin{abstract}
In this paper we consider a slightly different sieve method from Eratosthenes' to get primes. We find the periodicity and mirror symmetry of the pattern.
\end{abstract}

\noindent{\bf MSC2010:} 11A41.\\
{\bf Keywords:} sieve method, periodicity of pattern, mirror symmetry of pattern.

\section{Definitions and Notations}

Let $\{p_1,p_2,\ldots,p_n,\ldots\}$ be the set of all primes, where $p_n$ denotes the $n$-th prime number. $\pi(x)$ is the number of primes which are less than or equal to $x$. $\mathbb{N}$ is the set of positive numbers. Let
\[
M_{p_n}=\mathbb{N}-\{2k,3k,5k,\ldots,p_n k\mid k\in\mathbb{N}\},
\]
and $D_{p_n}$ be the set of difference of two consecutive numbers in $M_{p_n}$, that is,
\[
D_{p_n}=\{d_k\mid d_k=x_{k+1}-x_k, x_i\in M_{p_n}\},
\]
where $x_i$ is the $i$-th number in $M_{p_n}$.

For example, we list the first few numbers of $M_3$ and $D_3$.

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$M_3$ & 1 & 5 & 7 & 11 & 13 & 17 & 19 & 23 & 25 & 29 & 31 & 35 & 37 & $\ldots$\\
\hline
$D_3$ & 4 & 2 & 4 & 2  & 4  & 2  & 4  & 2  & 4  & 2  & 4  & 2  & 4  & $\ldots$\\
\hline
\end{tabular}

We call the minimum subset $\mathcal{P}_3:=\{4,2\}$ of $D_3$ as the pattern of $D_3$ since it occurs periodically. We will prove this fact in the next section and also for the general situation. Hence, we put the general definitions of $P_{p_n}$ and $T_{p_n}$ in the proof of Theorem \ref{thm:periodicity} due to the induction.

The number of the elements in the pattern $\mathcal{P}_3$ for $D_3$ is called the period of $D_3$. We write it as $T_3=2$.

The length of the pattern $\mathcal{P}_{p_n}$ is defined as the sum of the elements in the pattern. We write it as $L(\mathcal{P}_{p_n})$.

\section{Periodicity of the pattern}

\begin{prop}\label{prop:1}
$M_3=\{u_n\}_{n=1}^{\infty}$ has the following property:
\[
\begin{cases}
u_{2k}=u_{2k-1}+4,\\
u_{2k+1}=u_{2k}+2,\\
\end{cases}
\]
where $k=1,2,\ldots$. In other words, $\{4,2\}$ is the pattern of $D_3$, so the period is $2$.
\end{prop}
\begin{proof}
$u_n$ satisfies the equations:
\[
\begin{cases}
x\equiv 1(\text{mod}\ 2),\\
x\equiv 1(\text{mod}\ 3),\\
\end{cases}
\quad\text{or}\quad
\begin{cases}
x\equiv 1(\text{mod}\ 2),\\
x\equiv 2(\text{mod}\ 3).\\
\end{cases}
\]
The left infers that $x\equiv 1(\text{mod}\ 6)$. The right is:
\[
\begin{cases}
x=2k+1,\\
x=3\ell+2.\\
\end{cases}
\]
If $u_n=2k+1=3\ell+2$, then $2k=3\ell+1$, which infers that $\ell$ must be odd. Let $\ell=2h+1$, then $2k=3(2h+1)+1=6h+4$. Thus $2k+1=6h+5=3\ell+2$. Hence, $u_n\equiv 5(\text{mod}\ 6)$.
\end{proof}

\begin{thm}\label{thm:periodicity}
The period of $D_p$ is
\[
T_p=(2-1)(3-1)(5-1)\cdots(p-1).
\]
\end{thm}
\begin{proof}
We prove this by induction. The case for $D_3$ has been proved. For simplicity, we try to explain the procedure by proving the case for $D_5$. First, we observe that
\[
M_{p_{n+1}}=M_{p_n}-\{p_{n+1}h\mid h\in M_{p_n}\}.
\]
We show this procedure (getting $M_5$ from $M_3$) in the following table (see Figure \ref{fig:M5}).\\

\begin{figure}[htbp]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
1& {\bf\color{red}\sout{5}} & 7 & 11 & 13 & 17 & 19 & 23 & {\bf\color{blue}\sout{25}} & 29 & 31 \\
 & {\bf\color{red}\sout{35}} & 37 & 41 & 43 & 47& 49 & 53 & {\bf\color{blue}\sout{55}} & 59 & 61 \\
 & {\bf\color{red}\sout{65}} & 67 & 71 & 73 & 77 & 79 & 83 & {\bf\color{blue}\sout{85}} & 89 & 91 \\
 & {\bf\color{red}\sout{95}} & 97 & 101 & 103 & 107 & 109 & 113 & {\bf\color{blue}\sout{115}} & 119 & 121 \\
 & {\bf\color{red}\sout{125}} & 127 & 131 & 133 & 137& 139 & 143 & {\bf\color{blue}\sout{145}} & 149 & 151 \\
 & {\bf\color{red}\sout{155}} & 157 & 161 & 163 & 167 & 169 & 173 & {\bf\color{blue}\sout{175}} & 179 & 181\\
 & {\bf\color{red}\sout{185}} & 187 & 191 & 193 & 197 & 199 & 203 & {\bf\color{blue}\sout{205}} & 209 & 211\\
\hline
 & {\bf\color{red}\sout{215}} & 217 & 221 & 223 & 227 & 229 & 233 & {\bf\color{blue}\sout{235}} & 239 & 241\\
 & {\bf\color{red}\sout{245}} & 247 & 251 & 253 & 257 & 259 & 263 & {\bf\color{blue}\sout{265}} & 269 & 271\\
 & {\bf\color{red}\sout{275}} & 277 & 281 & 283 & 287 & 289 & 293 & {\bf\color{blue}\sout{295}} & 299 & 301\\
 & {\bf\color{red}\sout{305}} & 307 & 311 & 313 & 317 & 319 & 323 & {\bf\color{blue}\sout{325}} & 329 & 331\\
 & {\bf\color{red}\sout{335}} & 337 & 341 & 343 & 347 & 349 & 353 & {\bf\color{blue}\sout{355}} & 359 & 361\\
 & {\bf\color{red}\sout{365}} & 367 & 371 & 373 & 377 & 379 & 383 & {\bf\color{blue}\sout{385}} & 389 & 391\\
 & {\bf\color{red}\sout{395}} & 397 & 401 & 403 & 407 & 409 & 413 & {\bf\color{blue}\sout{415}} & 419 & 421\\
 \hline
 & $\cdots$ & & & & & & & & & \\
\end{tabular}
\caption{$M_5$}
\label{fig:M5}
\end{figure}

We need to delete the multiples of $5$ in $M_3$, i.e.,
\[
\begin{split}
&{\bf\color{red}5\cdot 1},\quad 5\cdot 5,\\
&{\bf\color{red}5\cdot 7},\quad 5\cdot 11,\\
&{\bf\color{red}5\cdot 13},\quad 5\cdot 17,\\
&{\bf\color{red}5\cdot 19},\quad \ldots
\end{split}
\]
By Proposition \ref{prop:1}, $D_3$ has pattern $\{4,2\}$, hence the positions of the elements needed to delete occur periodically in $M_3$. The period is $5\times 6=30$, where $6$ is the length of the previous patter $\{4,2\}$.

From another view, we make $5$ copies of the pattern $\{4,2\}$. Say,
\[
4, 2, 4, 2, 4, 2, 4, 2, 4, 2.
\]
After deleting $5$ and $25$, we get the string:
\[
(4+2), 4, 2, 4, 2, 4, (2+4), 2.
\]
It is the pattern of $D_5$, we denote it by $\mathcal{P}_{5}=\{6,4,2,4,2,4,6,2\}$. Hence the period is $T_5=8=2\cdot 5-2=(5-1)\cdot T_3$.

Hence, we have seen that we give the definition of $\mathcal{P}_{p_n}$ and $T_{p_n}$ inductively. When we prove that $D_{p_n}$ contains a minimum subset $\mathcal{P}_{p_n}$ which occurs periodically, we will prove that $D_{p_{n+1}}$ also contains a minimum subset which is denoted by $\mathcal{P}_{p_{n+1}}$ that occurs periodically. Then, we can define the period $T_{p_{n+1}}$.

To see this fact more clearly, we now show that the pattern of $D_7$ is
\begin{equation}\label{pattern:D_7}
\begin{split}
&10=(6,4),2,4,2,4,6,2;\\
&6,4,2,4,6=(2,4),6,2;\\
&6,4,2,6=(4,2),4,6,8=(2,6);\\
&\ 4,2,4,2,4,8=(6,2);\\
&6,4,6=(2,4),2,4,6,2;\\
&6,6=(4,2),4,2,4,6,2;\\
&6,4,2,4,2,10=(4,6),2.
\end{split}
\end{equation}
The period is $T_7=48=(7-1)\cdot(5-1)\cdot(3-1)\cdot(2-1)$.

In fact, $M_7$ is obtained by deleting the multiples of $7$ in $M_5$. We can rearrange the numbers in Figure \ref{fig:M5} as in Figure \ref{fig:M5-arrange}.

\begin{figure}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
1& 7 & 11 & 13 & 17 & 19 & 23 & 29 & 31 & {\it 37} & {\it 41} \\
\hline
 & {\it 43} & {\it 47} & {\it 49} & {\it 53} & {\it 59} & {\it 61}& {\bf 67} & {\bf 71} & {\bf 73} & {\bf 77} \\
 & {\bf 79} & {\bf 83} & {\bf 89} & {\bf 91} & {\it 97} & {\it 101}&{\it 103} &{\it 107}&{\it 109} &{\it 113} \\
 &{\it 119} &{\it 121} &{\bf 127} &{\bf 131} &{\bf 133} &{\bf 137} &{\bf 139} &{\bf 143}&{\bf 149} & {\bf 151} \\
 &{\it 157} &{\it 161} &{\it 163} &{\it 167} &{\it 169} &{\it 173} &{\it 179} &{\it 181}&{\bf 187} & {\bf 191} \\
 &{\bf 193} &{\bf 197} &{\bf 199} &{\bf 203} & {\bf 209} & {\bf 211}& & & & \\
\hline
\hline
 & 217 & 221 & 223 & 227 & 229 & 233 & 239 & 241 & 247 & 251 \\
 & 253 & 257 & 259 & 263 & 269 & 271 & 277 & 281 & 283 & 287 \\
 & 289 & 293 & 299 & 301 & 307 & 311 & 313 & 317 & 319 & 323 \\
 & 329 & 331 & 337 & 341 & 343 & 347 & 349 & 353 & 359 & 361 \\
 & 367 & 371 & 373 & 377 & 379 & 383 & 389 & 391 & 397 & 401 \\
 & 403 & 407 & 409 & 413 & 419 & 421 & & & & \\
\hline
\hline
 & 427 & 431 & 433 & 437 & 439 & 443 & 449 & 451 & 457 & 461 \\
 & 463 & 467 & 469 & 473 & 479 & 481 & 487 & 491 & 493 & 497 \\
 & 499 & 503 & 509 & 511 & 517 & 521 & 523 & 527 & 529 & 533 \\
 & 539 & 541 & 547 & 551 & 553 & 557 & 559 & 563 & 569 & 571 \\
 & 577 & 581 & 583 & 587 & 589 & 593 & 599 & 601 & 607 & 611 \\
 & 613 & 617 & 619 & 623 & 629 & 631 & & & & \\
\hline
\hline
 & $\cdots$ & & & & & & & & & \\
\end{tabular}
\caption{rearranging $M_5$}
\label{fig:M5-arrange}
\end{figure}

We need to delete the multiples of $7$ in $M_5$ (See Figure \ref{fig:M7}). They are
\[
\begin{split}
&{\bf\color{red}7\cdot 1},\quad 7\cdot 7,\quad 7\cdot 11,\quad 7\cdot 13,\quad 7\cdot 17,\quad 7\cdot 19,\quad 7\cdot 23,\quad 7\cdot 29,\\
&{\bf\color{red}7\cdot 31},\quad 7\cdot 37,\quad 7\cdot 41,\quad 7\cdot 43,\quad 7\cdot 47,\quad 7\cdot 49,\quad 7\cdot 53,\quad 7\cdot 59,\\
&{\bf\color{red}7\cdot 61},\cdots
\end{split}
\]

Since we have proved that $D_5$ has period pattern $\{6,4,2,4,2,4,6,2\}$, we can assert that the elements to be deleted above are periodic. Every first item in the period has gap $7\times 30$, where $30$ is exactly the length of previous pattern $\{6,4,2,4,2,4,6,2\}$. Therefore, the pattern is obtained in the following way. First copy the previous pattern $7$ times, then combine some pairs of consecutive numbers to get a new pattern.
\[
\begin{aligned}
\underline{6,4},2,4,2,4,6,2\\
6,4,2,4,\underline{2,4},6,2\\
6,4,2,\underline{4,2},4,6,\underline{2}\\
\underline{6},4,2,4,2,4,\underline{6,2}\\
6,4,\underline{2,4},2,4,6,2\\
6,\underline{4,2},4,2,4,6,2\\
6,4,2,4,2,\underline{4,6},2\\
\end{aligned}
\]
Thus, we get the pattern of $D_7$, see \eqref{pattern:D_7}.

\begin{figure}[htbp]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
1& {\bf\color{red}\sout{7}} & 11 & 13 & 17 & 19 & 23 & 29 & 31 & {\it 37} & {\it 41} \\
\hline
 & {\it 43} & {\it 47} & {\bf\color{blue}\sout{49}} & {\it 53} & {\it 59} & {\it 61}& {\bf 67} & {\bf 71} & {\bf 73} & {\bf\color{blue}\sout{77}} \\
 & {\bf 79} & {\bf 83} & {\bf 89} & {\bf\color{blue}\sout{91}} & {\it 97} & {\it 101}&{\it 103} &{\it 107}&{\it 109} &{\it 113} \\
$M_7^{(0)}$ &{\bf\color{blue}\sout{119}} &{\it 121} &{\bf 127} &{\bf 131} &{\bf\color{blue}\sout{133}} &{\bf 137} &{\bf 139} &{\bf 143}&{\bf 149} & {\bf 151} \\
 &{\it 157} &{\bf\color{blue}\sout{161}} &{\it 163} &{\it 167} &{\it 169} &{\it 173} &{\it 179} &{\it 181}&{\bf 187} & {\bf 191} \\
 &{\bf 193} &{\bf 197} &{\bf 199} &{\bf\color{blue}\sout{203}} & {\bf 209} & {\bf 211}& & & & \\
\hline
\hline
 & {\bf\color{red}\sout{217}} & 221 & 223 & 227 & 229 & 233 & 239 & 241 & 247 & 251 \\
 & 253 & 257 & {\bf\color{blue}\sout{259}} & 263 & 269 & 271 & 277 & 281 & 283 & {\bf\color{blue}\sout{287}} \\
 & 289 & 293 & 299 & {\bf\color{blue}\sout{301}} & 307 & 311 & 313 & 317 & 319 & 323 \\
$M_7^{(1)}$ & {\bf\color{blue}\sout{329}} & 331 & 337 & 341 & {\bf\color{blue}\sout{343}} & 347 & 349 & 353 & 359 & 361 \\
 & 367 & {\bf\color{blue}\sout{371}} & 373 & 377 & 379 & 383 & 389 & 391 & 397 & 401 \\
 & 403 & 407 & 409 & {\bf\color{blue}\sout{413}} & 419 & 421 & & & & \\
\hline
\hline
 & {\bf\color{red}\sout{427}} & 431 & 433 & 437 & 439 & 443 & 449 & 451 & 457 & 461 \\
 & 463 & 467 & {\bf\color{blue}\sout{469}} & 473 & 479 & 481 & 487 & 491 & 493 & {\bf\color{blue}\sout{497}} \\
 & 499 & 503 & 509 & {\bf\color{blue}\sout{511}} & 517 & 521 & 523 & 527 & 529 & 533 \\
$M_7^{(2)}$ & {\bf\color{blue}\sout{539}} & 541 & 547 & 551 & {\bf\color{blue}\sout{553}} & 557 & 559 & 563 & 569 & 571 \\
 & 577 & {\bf\color{blue}\sout{581}} & 583 & 587 & 589 & 593 & 599 & 601 & 607 & 611 \\
 & 613 & 617 & 619 & {\bf\color{blue}\sout{623}} & 629 & 631 & & & & \\
\hline
\hline
 & $\cdots$ & & & & & & & & & \\
\end{tabular}
\caption{Obtain $M_{7}$ from $M_5$ by deleting the $7h$, $h\in M_5$}
\label{fig:M7}
\end{figure}

If we construct $M_{11}$, then we will consider the following eleven blocks:
\[
M_7^{(0)},M_7^{(1)},M_7^{(2)},\ldots,M_7^{(10)},
\]
where we use $M_{7}^{(0)}$ to denote the first block in $M_7$ corresponding to the first period of $D_{7}$.

Generally, when constructing the new pattern for $D_{p_{n+1}}$ from $D_{p_n}$, we first take $p_{n+1}$ copies of previous pattern for $D_{p_n}$. Then the period obeys the recursive formula:
\[
T_{p_{n+1}}=p_{n+1}\cdot T_{p_n}-T_{p_n}=(p_{n+1}-1)\cdot T_{p_n}.
\]
According to the induction hypothesis,
\[
T_{p_n}=\prod_{i=1}^{n}(p_i-1),
\]
we have
\[
T_{p_{n+1}}=\prod_{i=1}^{n+1}(p_i-1).
\]
\end{proof}

\begin{cor}
$L(\mathcal{P}_{p_n})=\prod_{i=1}^n p_i$.
\end{cor}
\begin{proof}
Using $L(\mathcal{P}_{p_n})=p_nL(\mathcal{P}_{p_{n-1}})$.
\end{proof}

\begin{lem}
If $p_n\geqslant 11$ (i.e., $n\geqslant 5$), $p_n^2$ is contained in the first block of $M_{p_{n-1}}$. (Here the block means the subset of $M_{p_{n-1}}$ corresponding to the $D_{p_{n-1}}$.)
\end{lem}
\begin{proof}
First,
\[
\begin{split}
&p_n^2< p_n+L(\mathcal{P}_{p_{n-1}}) \\
\Leftrightarrow\  & p_n(p_n-1)< p_{n-1}\cdot p_{n-2}\cdots 5\cdot 3\cdot 2.\\
\end{split}
\]
It is easy to check that $p_k=11$ satisfies the inequality:
\[
p_k(p_k-1)< p_{k-1}\cdot p_{k-2}\cdots 5\cdot 3\cdot 2.
\]
By Bertrand's postulate $p_{k+1}<2p_k$, then it is easy to show that
\[
p_{k+1}(p_{k+1}-1)< p_{k}\cdot p_{k-1}\cdot p_{k-2}\cdots 5\cdot 3\cdot 2
\]
also holds if $k\geqslant 5$.

Then we have completed the proof.
\end{proof}

Hence, if $n\geqslant 5$, then the numbers between $p_n$ and $p_n^2$ which belong to the first block of $M_{p_{n-1}}$ are all primes. In fact, we observe that for any $n\geqslant 1$, for any element $q(>1)$ in the the first block of $M_{p_{n-1}}$, if $q$ satisfies $p_n<q<p_n^2$, then it is a prime.

\begin{thm}\label{thm:mirror-symmetry}
The gap sequence except the last element in the pattern is mirror symmetric.
\end{thm}
\begin{proof}
For example, when constructing $D_{11}$ from $D_7$, we delete the numbers $\{11k\mid k\in M_{7}\}$. We list the divisors $k$ of the first period. They are
\[
\begin{matrix}
 1 &     11&    13&    17&        19&    23&    29&    31&    37&    41\\
 43&     47&    53&    59&        61&    67&    71&    73&    79&    83\\
 89&     97&  101&  {\bf 103} &  *  {\bf 107} &  109&  113&  121&  127&  131\\
 137&   139&  143&  149&      151&  157&  163&  167&  169&  173\\
 179&   181&  187&  191&      193&  197&  199&  209&     &     \\
\end{matrix}
\]
These $48$ numbers are mirror complemental. The sum of every two symmetric numbers about $*$ are equal to $210$. More precisely, every block in $M_7$ are mirror complemental about the center of the block. It infers that the mirror symmetry of the gap sequence except the last element in the pattern $\mathcal{P}_{11}$. Therefore, when constructing $D_{p_{n+2}}$ from $D_{p_{n+1}}$, we conclude that the numbers which to be deleted $\{p_{n+2}k\mid k\in M_{p_{n+1}}\}$ are mirror complemental because the numbers $k\in M_{p_{n+1}}$ corresponding the first period are mirror complemental by the inductive hypothesis. And by inductive hypothesis, the first $p_{n+2}$ blocks in $M_{p_{n+1}}$ are mirror complemental. Therefore, the gap sequence in the pattern $\mathcal{P}_{p_{n+2}}$ except the last number is mirror symmetric.
\end{proof}

By the argument above, we easily have
\begin{cor}\label{cor:1}
The last element in every pattern $\mathcal{P}_{p_n}$ is $2$.
\end{cor}
\begin{proof}
We still illustrate this fact first by the above example. Where the length of the period is $210$. Thus the next number of $209$ in the second block of $M_7$ is $211$. Thus the last element in $\mathcal{P}_7$ is $2=211-209$.

Generally, if we use $x$ to denote the last element in the first block of $M_{p_n}$, then $1+x=L(\mathcal{P}_{p_n})$ by the mirror complementary. On the other hand, the first element denoted by $y$ in the second block of $M_{p_n}$ satisfies that $y-1=L(\mathcal{P}_{p_n})$. Thus, $y-x=2$ which is just the last element of the patter $\mathcal{P}_{p_n}$.
\end{proof}

\begin{thm}\label{thm:about-skips}
(i) The multiplicities of skips $2$ and $4$ are always same and odd.\\
(ii) The central gap is always $4$, and the multiplicities of all gaps except $2$ and $4$ are even.
\end{thm}
\begin{proof}
(i) In fact, these multiplicities for $k$-prime basis, $k>1$, are both precisely equal to
\begin{equation}\label{eqn:tpk}
(3-2)\cdot(5-2)\cdots(p_k-2),
\end{equation}
which is always odd as well. For example, there are 15 skips $2$ in the pattern $\mathcal{P}_7$ (see \eqref{pattern:D_7}).

For element $x$ in $M_{p_k}$, we consider the residues of $2,3,5,7,\ldots,p_k$ respectively, i.e.,
\[
x\equiv(x_1,x_2,x_3,\ldots,x_k)\mod (p_1,p_2,p_3,\ldots,p_k),
\]
and for simplicity we write it as a vector $x=(x_1,x_2,x_3,\ldots,x_k)$.

The multiplicity of skip $2$ is easily obtained by noting that some integer $x = (x_1, x_2,\ldots, x_k)$ is the first relatively prime (r-prime) before skip $2$, iff $x$ has residues of the form $(1, x_2, x_3,\ldots, x_k)$ where no $x_i$ is $0$ or $-2$. (If $x_i=0$, then $x$ is a multiple of $p_i$ which will not in $M_{p_k}$ for $i\leqslant k$. Similarly, if $x_i=-2$, then $x_i+2$ is a multiple of $p_i$.) Since number of such $k$-tuples with $2$ forbidden values on the upper $k-1$ residues is $(3-2)\cdot(5-2)\cdots(p_k-2)$, that is the multiplicity of skip $2$.

Similarly, for skip $4$, the integer $x$ is the first r-prime before skip $4$ iff it has residues of the form $(1, x_2, x_3,\ldots,x_k)$ where no $x_i$ is $0$ or $-4$, yielding the above multiplicity with the two forbidden values on $x_2, x_3,\ldots, x_k$.

Note that the above simple method doesn't work for skips larger than $4$ since these can be obtained both via replication (by increasing k) and via merging of adjacent skips which follows replication. In contrast, for $k>1$, skips $2$ and $4$ can be obtained only via replication (with growing $k$), but not via skip merging during filtering, since skip subsequences $2, 2$ cannot occur for $k>1$ (the replicated residue pattern for $k>1$ is $2,4,2,4,\ldots$).

(ii) Regarding the formula above \eqref{eqn:tpk} for multiplicities of gaps $2$ and $4$, the second statement of (ii) (the multiplicities of all gaps except $2$ and $4$ are even) is true. This follows easily from the mirror symmetry (which is easiest seen as symmetry gaps with respect to change of signs of residues) and the fact that the central gap (which could straddle the symmetry center and mirror into itself) is always 4. (End gap is always 2, and these are the only 2 exceptions.) Regarding the latter clause, for a k-prime basis $p_1=2, p_2=3, \ldots, p_k$, the central integer (symmetry axis) is at $C=3\cdot 5\cdots p_k$ and its residue pattern is $C=(1, 0, 0,\ldots, 0)$. The integers around $C$ are then in residue form as follows:
\[
\begin{array}{rrrrrcrl}
C-2: & -1 & -2 & -2 & -2 & \ldots & -2  & \text{r-prime}\\
C-1: &  0 & -1 & -1 & -1 & \ldots & -1  & \text{r-composite}\\
  C: &  1 & 0  & 0  & 0  & \ldots & 0   & \text{r-composite}\\
C+1: &  0 & 1  & 1  & 1  & \ldots & 1   & \text{r-composite}\\
C+2: &  1 & 2  & 2  & 2  & \ldots & 2   & \text{r-prime}\\
\end{array}
\]

which always has $2$ r-primes at $C\pm 2$, and $3$ r-composites at $C, C\pm 1$, yielding thus gap $4$ at the symmetry axis. The mirror symmetry around $C$ then implies the second statement.
\end{proof}

\begin{remark}\label{rem:largest-skip}
The largest skip comes in pairs only up to first $8$ basis primes (i.e. up to $\mathcal{P}_{19}$), but already for the $9$ prime basis ($\mathcal{P}_{23}$) the largest skip is $40$ which occurs $12$ times. Beyond the $9$ prime basis, the longest pattern occurs in pairs only for $3$ more cases: $10$, $11$ and $13$ basis primes i.e. $\mathcal{P}_{29}$, $\mathcal{P}_{31}$ and $\mathcal{P}_{41}$. All other cases have larger multiplicities of the longest skip.

Note also that for the basis with first $k$ primes $p_1=2, p_2=3, \ldots, p_k$, the trivial low bound for the max skip is $2p_{k-1}$, e.g. for the set $\mathcal{P}_{13} (k=6)$, the low bound is $2\cdot 11=22$ which happens to be the actual longest skip for $\mathcal{P}_{13}$.

The general residue pattern for solution meeting this low bound with $k$ prime basis is $( 0, 0,  0, ... 0, \mp 1, \pm 1 )$, where first $k-2$ residues are $0$, and last two are $+1$ and $-1$ or $-1$ and $+1$ (which corresponds to the two solutions). The two longest patterns occur only when this low bound $2p_{k-1}$ is met, otherwise there are more than $2$ maxima (again there is even number of solutions due to mirror symmetry).

For the $9$ prime basis ($\mathcal{P}_{23}$), the above low bound is $2\cdot 19=38$, while the max skip is $40$, hence the low bound defect is $40-38=2$. The max skip is $100$ for $\mathcal{P}_{47}$, while the low bound is $2\cdot 43=86$, yielding defect $14$. The max skip is $200$ for $\mathcal{P}_{79}$ while the low bound is $2\cdot 73=146$, yielding defect $54$. The max skip is $300$ for $\mathcal{P}_{107}$ ($28$ primes basis), while low bound is $2\cdot 103=206$, defect $94$. The max skip is $414$ (i.e. the first $\geqslant 400$) for $\mathcal{P}_{139}$ ($34$ primes basis), while low bound is $2\cdot 137=274$, defect $140$. The max skip is $510$ (the first $\geqslant 500$) for $\mathcal{P}_{167}$  ($39$ primes basis), defect $184$, etc.
\end{remark}

\begin{conj}
$6$ is the number that occurs the most times in any pattern $\mathcal{P}_{p_n}$ when $n\geqslant 11$.
\end{conj}

\begin{remark}
Let $t_{p_{n+1}}$ denote the multiplicity of $2$ in the pattern for $D_{p_{n+1}}$, then we get a recurrence inequality,
\[
t_{p_{n+1}}\geqslant t_{p_{n}}\cdot p_{n+1}-T_{p_n}.
\]
But it is a rough estimate. In fact, by equation \eqref{eqn:tpk}, we have
\[
t_{p_{n+1}}>T_{p_n}.
\]
\end{remark}

\section{Application to get consecutive primes}

Based on the above discussion, to obtain the pattern for $D_{p_{n}}$, we first take $p_{n}$ copies of the previous pattern for $D_{p_{n-1}}$. Then, delete the corresponding elements $p_n h$ in $M_{p_{n-1}}$, where $h\in M_{p_{n-1}}$. These elements to be deleted are distributed in the following way. The number of them is $T_{p_{n-1}}$.

\begin{center}
\begin{tikzpicture}[scale=1]
\draw[gray,dashed] (0,0) -- (9,0);
\fill [red] ($(0,0)$) circle (2pt);
\node  at (-0.5,-0.7)[anchor=south] {$1$};
\node  at (0,-0.7)[anchor=south] {$p_n$};

\node  at (0.7,0.5)[anchor=south] {$p_nd_{p_{n-1}}^1$};

\fill [red] ($(1.5,0)$) circle (2pt);
\node  at (1.5,-0.7)[anchor=south] {$p_n^2$};

\fill [red] ($(2.5,0)$) circle (2pt);
\node  at (2.7,-0.7)[anchor=south] {$p_n p_{n+1}$};
\node  at (2.1,0.5)[anchor=south] {$p_nd_{p_{n-1}}^2$};
\node  at (3.4,0.5)[anchor=south] {$p_nd_{p_{n-1}}^3$};
\node  at (4.75,0.5)[anchor=south] {$p_nd_{p_{n-1}}^4$};

\node  at (8,0.5)[anchor=south] {$p_nd_{p_{n-1}}^{T_{p_{n-1}}}$};

\fill [red] ($(4.0,0)$) circle (2pt);
\fill [red] ($(5.5,0)$) circle (2pt);
\fill [red] ($(7.0,0)$) circle (2pt);
\fill [green] ($(9.0,0)$) circle (2pt);

\draw[gray] (0,0) -- (0,0.5);
\draw[gray] (1.5,0) -- (1.5,0.5);
\draw[gray] (2.5,0) -- (2.5,0.5);
\draw[gray] (4,0) -- (4,0.5);
\draw[gray] (5.5,0) -- (5.5,0.5);
\draw[gray] (7,0) -- (7,0.5);
\draw[gray] (9,0) -- (9,0.5);

\draw[gray][<->] (0,0.3) -- (1.5,0.3);
\draw[gray][<->] (1.5,0.3) -- (2.5,0.3);
\draw[gray][<->] (2.5,0.3) -- (4.0,0.3);
\draw[gray][<->] (4.0,0.3) -- (5.5,0.3);
\draw[gray][<->] (7,0.3) -- (9,0.3);

\fill [blue] ($(1.2,0)$) circle (2pt);
\draw[blue][->] (1.2,-0.6) -- (1.2,-0.1);
\node  at (1.2,-1.0)[anchor=south] {$q_n$};
\end{tikzpicture}
\end{center}

Here $\mathcal{P}_{p_{n-1}}=\{d_{p_{n-1}}^1, d_{p_{n-1}}^2,\ldots, d_{p_{n-1}}^{T_{p_{n-1}}}\}$ is the pattern for $D_{p_{n-1}}$. The red points are just the elements to be deleted. The blue point $q_n$ denotes the biggest prime which is less than $p_n^2$. The last green point is also deleted. But it is regarded as in the next period.

\begin{prop}\label{prop:2}
The numbers in the set
\[
\{q\in M_{p_{n-1}}\mid p_n<q<p_n p_{n+1},\quad q\neq p_n^2\}
\]
are consecutive primes.
\end{prop}

If there exist at least one gap $2$ in the pattern $\mathcal{P}_{p_{n-1}}$ located in the subset corresponding to the interval $[p_{n+1},q_n]$. Then, this gap $2$ is kept in the next pattern $\mathcal{P}_{p_n}$, and also in the subset corresponding to the interval $[p_{n+1},q_n]$.

\begin{thm}\label{thm:chi-interval}
Let $c$ be a positive integer. Define a characteristic function for each interval $I_k=[(k-1)c,kc)$, $k=1,2,\ldots$ as follows:
\[
\chi_{_{I_k}}=\begin{cases}
1, & \text{if}\ I_k\ \text{contains one or more primes},\\
0, & \text{otherwise}.\\
\end{cases}
\]
If $m$ is large enough, and if $c\geqslant 3$, then
\[
\frac{1}{c}\pi(mc)\leqslant\sum_{k=1}^{m}\chi_{_{I_k}}<\pi(mc),
\]
that is,
\[
\sum_{k=1}^{m}\chi_{_{I_k}}\asymp\frac{m}{\ln m}.
\]
\end{thm}
\begin{proof}
Let $J_{ki}, i=1,2,\ldots,c$ be the subintervals in $I_k$, each $J_{ki}$ has the form $[d,d+1)$. Then,
\[
\chi_{_{I_k}}=\max_{1\leqslant i\leqslant c}\chi_{_{J_{ki}}}.
\]
Thus,
\[
\sum_{k=1}^{m}\chi_{_{I_k}}=\sum_{k=1}^{m}\max_{i}\chi_{_{J_{ki}}}\geqslant\sum_{k=1}^{m}\frac{1}{c}\sum_{i=1}^{c}\chi_{_{J_{ki}}}=\frac{1}{c}\pi(mc).
\]
On the other hand, it is obvious that
\[
\sum_{k=1}^{m}\chi_{_{I_k}}<\pi(mc)\quad\text{for}\ c\geqslant 3.
\]

\end{proof}

Yitang Zhang \cite{ZhangYitang} proved the following result for consecutive primes based on the recent work of Goldston, Pintz and Y{\i}ld{\i}r{\i}m \cite{GPY2009A,GPY2010A} on the small gaps between consecutive primes.
\begin{equation}\label{eqn:zhang_result}
\liminf_{n\rightarrow\infty}(p_{n+1}-p_n)< H,
\end{equation}
where $H=7\times 10^7$. Then it was reduced to $4680$ by Polymath project \cite{ Granville,Polymath}. In late 2013, James Maynard and Terry Tao found a much simpler proof of Zhang's result giving $H=600$. A further progress based on this work has reduced $H$ to $252$ by Polymath project\cite{Green}. As of April 14, 2014, one year after Zhang's announcement, according to the Polymath project wiki, the bound has been reduced to $246$.

Hence, for $c\geqslant 246$, we have

\begin{prop}
There are infinitely many pairs $m_2>m_1>0$, $m_2$ and $m_1$ are large enough, such that
\[
\frac{1}{c}(\pi(m_2 c)-\pi(m_1 c))\leqslant\sum_{k=m_1}^{m_2}\chi_{_{I_k}}< \pi(m_2 c)-\pi(m_1 c).
\]
\end{prop}

\begin{conj}
For any $m_2>m_1>0$, $m_2$ and $m_1$ are large enough, and any $c>2$, we have
\[
\frac{1}{c}(\pi(m_2 c)-\pi(m_1 c))\leqslant\sum_{k=m_1}^{m_2}\chi_{_{I_k}}< \pi(m_2 c)-\pi(m_1 c).
\]
\end{conj}

\noindent{\bf Acknowledgments :}
We express our gratitude to Mr. Ratko V. Tomic (a researcher in Infinetics Technologies, Inc) who pointed out the mirror symmetry of the gap sequence which is stated in Theorem \ref{thm:mirror-symmetry}. And also he gives the proof of Theorem \ref{thm:about-skips} and makes the remark \ref{rem:largest-skip}. We thank him for discussing the question and introducing us the ideas involved.


TITLE: Rewriting H(jw) in exponential form
QUESTION [0 upvotes]: Link to equation in question
Why can H(jw) be expressed as the absolute value of H(jw) multiplied by an exponent? And what is the meaning behind the actual power (j*theta(jw))?

REPLY [0 votes]: For any complex number $z$ one can rewrite it in polar form $z = |z|e^{i \arg(z)}$ which is presumably what they did here.


TITLE: Shortest distance between skew line *segments*
QUESTION [0 upvotes]: The answer for the shortest distance between two skew lines has already been answered here and is also found on Wikipedia.
However, I need to apply a further constraint. I need to find the shortest distance between two skew line segments. Or, if it is easier, two rays, whose starting points are $P_1$ and $P_2$. How can I modify the equations for $c_1$ and $c_2$ in the wiki link to account for this? Or is this a far more complicated problem?

REPLY [0 votes]: Defining
$$
s_1(\lambda)=\lambda p_1+(1-\lambda)p_2\\
s_2(\mu) = \mu p_3+(1-\mu)p_4
$$
with $0\le \lambda\le 1$ and $0 \le \mu\le 1$ and with the help of four slack variables $\epsilon_1,\epsilon_2,\epsilon_3,\epsilon_4$ to handle the inequality restrictions, the problem can be stated using the Lagrange Multipliers as follows
$$
L(\lambda, \mu, \eta,\epsilon) = \frac{1}{2}||s_1(\lambda)-s_2(\mu)||^2+\eta_1 (\lambda-\epsilon_1^2) +\eta_2(\mu-\epsilon_2^2)+\eta_3(\lambda-1+\epsilon_3^2)+\eta_4(\mu-1+\epsilon_4^2)
$$
with the stationary conditions
$$
\nabla L = 0
$$
or
$$
\left\{
\begin{array}{rcl}
(p_2+\lambda(p_1-p_2))(p_1-p_2)+\eta_1 + \eta_3 & = & 0\\
(p_4+\mu(p_3-p_4))(p_3-p_4)+\eta_2 + \eta_4 & = & 0\\
\lambda-\epsilon_1^2 & = & 0\\
\mu-\epsilon_2^2 & = & 0\\
\lambda-1+\epsilon_3^2 & = & 0\\
\mu-1+\epsilon_4^2 & = & 0\\
\eta_1\epsilon_1 & = & 0\\
\eta_2\epsilon_2 & = & 0\\
\eta_3\epsilon_3 & = & 0\\
\eta_4\epsilon_4 & = & 0
\end{array}
\right.
$$
Solving those conditions we get the possible stationary points.


TITLE: How To Slice $Re(1/(1+z))$ Into A Cartesian Function For Any Angle?
QUESTION [2 upvotes]: The complex function $\frac{1}{1+z^2}$  may be broken into real and imaginary components:
$$Re \bigg(\frac{1}{(1+z^2)}\bigg)=\frac{1+x^2-y^2}{(1+x^2-y^2)^2+4x^2y^2}$$
$$Im \bigg(\frac{1}{(1+z^2)}\bigg)=\frac{-2xyi}{(1+x^2-y^2)^2+4x^2y^2}$$
Graphing the real components and colouring them according to the imaginary components yields:

Slicing the graph of the real components along the x-axis yeilds the graph of $\frac{1}{1+x^2}$, which is:

Slicing the graph of the real components along the y-axis yields the graph of $\frac{1}{1-x^2}$, which is:

Is it possible to write a cartesian equation equivalent to slicing the above 3D graph of $Re(1/(1+z))$ for any angle between the x and y axes?
For your convenience see the graph the $Re(1/(1+z^2))$ on Wolfram Alpha!

Note that graphing the imaginary components yields something entirely different.

REPLY [1 votes]: Recapitulating the question with just a little extra notation,
we have two real functions
$$f(z) = \mathrm{Re}\left(\frac{1}{1+z^2}\right)$$
and
$$g(z) = \mathrm{Im}\left(\frac{1}{1+z^2}\right).$$
Taking $z = x + iy,$
we can plot one of these functions (for example, $f$) in three dimensions
by labeling the three axes $x,$ $y,$ and $w$ (using $w$ for the third axis since we've already used $z$ for something else)
and plotting the equation
$$ w = f(x + iy).$$
Now we want to "slice" this plot along one of the planes through the
origin perpendicular to the $x,y$ plane,
that is, we want a slice along a plane containing the $w$-axis,
but rotated an angle $\theta$ away from the $x$-axis.
Because complex analysis is involved here,
let's measure $\theta$ in radians rather than degrees.
(This will simplify some formulas later.)
We might as well assume $\theta \in [0, \pi),$at least for now,
because every other angle will slice the three-dimensional graph
along one of the planes described by $\theta \in [0, \pi).$
This slice can then be viewed as a graph in two dimensions.
Naturally, we might expect to be able to impose a Cartesian coordinate system on this plane so that we can write an equation for the graph.
Since the $w$ axis already lies in the plane, it is an obvious candidate
for one of the coordinate axes of the slice.
In order to form a Cartesian coordinate system that describes the slice
in the usual way (mapping equations to shapes in the way we usually expect),
the other coordinate axis of the slice should be perpendicular to the
$w$-axis, and the unit of measurement along that axis should be equal to
the unit along the $w$-axis.
That is, we want to set up a coordinate axis along the line
$y = x\tan\theta$ in the $x,y$ plane;
and if we call this the $u$-axis, then we want the point
$(u,w) = (1,0)$ in the plane of the slice to be the point
$(x,y,w) = (\cos\theta, \sin\theta, 0).$
In order to evaluate $f(z)$ (or $g(z)$) at any given value of $u,$ we
can write $z = u\cos\theta + iu \sin\theta,$
or more simply, $z = u e^{i\theta}.$
Observing that $\bar z = u e^{-i\theta},$ we have
\begin{align}
\frac{1}{1+z^2}
&= \frac{1}{1+u^2 e^{i2\theta}} \\
&= \frac{1}{1+u^2 e^{i2\theta}}
      \cdot\frac{1+u^2 e^{-i2\theta}}{1+u^2 e^{-i2\theta}} \\
&= \frac{1+u^2 e^{-i2\theta}}
   {1+u^2 (e^{-i2\theta} + e^{-i2\theta}) + e^{i2\theta}e^{-i2\theta} u^4}\\
&= \frac{1+(\cos(2\theta))u^2 - i(\sin(2\theta))u^2}
        {1+(2\cos(2\theta))u^2 + u^4}\\
\end{align}
and therefore
$$
f(z) = \mathrm{Re}\left(\frac{1}{1+z^2}\right)
= \frac{1+(\cos(2\theta))u^2}{1+(2\cos(2\theta))u^2 + u^4}
$$
and
$$
g(z) = \mathrm{Im}\left(\frac{1}{1+z^2}\right)
= \frac{-(\sin(2\theta))u^2}{1+(2\cos(2\theta))u^2 + u^4}
$$
We can now relax our constraints on $\theta$; if we let
$\theta$ be any real number, the angles $\theta_1$ and
$\theta_2 = \theta_1 - \pi$
both specify the same slice of the three-dimensional graph,
except that the $u$-axis for $\theta_1 - \pi$ runs in the opposite
direction from the $u$-axis for $\theta_1.$
If you just pick angles randomly then you will almost certainly end
up with transcendental coefficients in these equations;
but if you simply set either $\cos\theta$  or $\sin\theta$
to a rational number between $-1$ and $1$ and apply the usual
trigonometric identities, you will get relatively reasonable coefficients
(rational numbers and/or square roots of rational numbers).
Examples
The table below shows the functions we obtain by “slicing” the graph of $\frac1{1+z^2}$ at a few different angles:
$\theta=0$ (slicing along the $x$ axis), $\theta=\frac\pi4$ radians ($45$ degrees, slicing along the line $y=x$), and $\theta=\frac\pi2$ ($90$ degrees, slicing along the $y$ axis).
\begin{array}{ccccccccc}
\theta&\ &\cos(2\theta)&\ &\sin(2\theta)&\ &\mathrm{Re}\left(\dfrac{1}{1+z^2}\right)&\ & \mathrm{Im}\left(\dfrac{1}{1+z^2}\right) \\
\hline
0        && 1&& 0 && \dfrac{1+u^2}{1+2u^2+u^4} = \dfrac{1}{1+u^2} && 0\\
\frac\pi4&& 0&& 1 && \dfrac{1}{1+u^4} && \dfrac{-u^2}{1+u^4} \\
\frac\pi2&&-1&& 0 && \dfrac{1-u^2}{1-2u^2+u^4} = \dfrac{1}{1-u^2} && 0\\
\end{array}
Alternatives
In general, if you write $z = x + iy,$ then $f(z)$ and $g(z)$ can also be written as functions of the real and imaginary components of $z$:
\begin{align}
f(x, y) &= f(x + iy) = \frac{1+x^2-y^2}{(1+x^2-y^2)^2+4x^2y^2}, \\
g(x, y) &= g(x + iy) = \frac{-2xy}{(1+x^2-y^2)^2+4x^2y^2}.
\end{align}
A "slice through the origin" gives a line in the complex plane with
an equation like the following in $x$ and $y$:
$$ Ax + By = 0,$$
where $A$ and $B$ are not both zero.
Take $A = 0$ and you have the equation of the $x$-axis;
take $B = 0$ and you have the $y$-axis;
take $A = \tan\theta$ and $B = -1$ and you have a line rotated by an angle
$\theta$ from the $x$-axis, with  slope $m = \tan\theta$
and equivalent equation $y = mx.$
Note that after you "split" $z$ into real and imaginary components
$x$ and $y,$ you generally require both components in order to
evaluate $f$ and $g$ at any value $z.$
If you simply erase $y$ from the formula for $f$ above, you end up with
$\frac{1+x^2}{(1+x^2)^2+4x^2},$ which you can easily verify is
not a faithful formula for $\mathrm{Re}\left(\frac{1}{1+z^2}\right).$
On the other hand, if you "slice" a function such as $f(x,y)$ along a line
through the origin, you have something that resembles a function of
one (real-valued) coordinate plotted in a plane,
that is, a single-parameter real function.
There should be a way to write $f$ as such a function.
Any resolution to this question is going to come down to choosing
one real-valued variable as an independent coordinate, and using some
functions of that variable as the $x$ and $y$ values in $f(x,y).$
In the question, one method is explored in which we set
$f(x,y) = f(x, x\tan\theta).$  That is, we use $x$ for $x$ itself and
$x\tan\theta$ for $y.$
This makes it possible to plot $f$ as a function of $x$ along along any line
whose equation can be written $y = x\tan\theta.$
It is not possible to plot $f$ in this way along the $y$ axis.
Another method is to set $f(x,y) = f(y\cot\theta, y).$
This makes it possible to plot $f$ as a function of $y$ along along any line
whose equation can be written $x = y\cot\theta,$
that is, along any line through the origin except the $x$-axis.
You can use the second method to describe the slice of $f$ along the $y$-axis
as a single-parameter real function,
and the first method to describe any other slice of $f$ through the origin.
If you apply this method along lines at angles near $90$ degrees
(but not at $90$ degrees exactly),
you can observe that the function increases from $1$ toward a maximum value,
then falls to a negative minimum value, then increases asymptotically to zero;
as the angle of the line approaches $90$ degrees, the maximum and minimum
values get more extreme and also occur at $x$-values closer and closer to zero.
At $90$ degrees exactly, the function increases to infinity and then from
negative infinity back up to zero, which is consistent with the trend of the
other functions, but rather than doing this at an input value of zero
(to which the extreme points of the other functions were converging),
it does this at the input value $1.$
Alternatively, you can use the first method just to describe the slice of $f$
along the $x$-axis, and use the second method for the other slices.
This gives quite different results (most of the time) compared to the
previous paragraph, but the results are just as valid.
Yet another alternative is to set $f(x,y) = f(x, x\tan\theta)$
when slicing along any line up to $45$ degrees from the $x$-axis
(that is, any line $y = mx$ where $-1 \leq m \leq 1)$,
and set $f(x,y) = f(y\cot\theta, y)$ when slicing at any other angle.
A nice thing about this is that the two methods match up perfectly
along the lines $y = x$ and $y = -x,$ which are the lines at exactly
the angles where we switch from one method to the other.
The method I described above is just another method for reducing $f$
to a single-parameter real function.
In that method, we set $f(x,y) = f(u\cos\theta, u\sin\theta).$
Along a line at angle $\theta$ from the $x$-axis,
where $\theta$ is a constant,
this reduces $f$ to a function of the real variable $u.$
It can also be viewed as using polar coordinates to describe points
in the complex plane, with $u$ as the radius from the origin.
The formulas I gave for $f(z)$ and $g(z)$ in terms of $u$ are just
what you get if you substitute $x = u\cos\theta$ and $y = u\sin\theta$
in the formulas for $f(x,y)$ and $g(x,y).$
(Note that then $x^2 - y^2 = u^2 \cos(2\theta)$ and $2xy = u^2 \sin(2\theta).$)
And these are merely a few of the infinite number of alternative methods.
You can take your pick or come up with another that you prefer.

Appendix
Here are some reasons why I personally prefer the method I described in the first part of this answer,
writing $z = u\cos\theta + i u\sin\theta.$
In general, except when $\theta = 0$ the
$x$-axis does not actually lie in the slice taken through the origin.
In effect, the method of setting $f(z) = f(x, x\tan\theta)$
is a  projection of  the $x$-axis onto the plane of the slice
(or vice versa).
The projection is parallel to the $y$-axis, so when
$\theta$ is near $\frac\pi2,$ that is, when the slice is nearly parallel
to the $y$-axis, the projection onto the slice is "stretched"
in the horizontal  direction.
This can have some weird results.
For example, consider the plot of the real function $h(z) = |z|^2.$
The graph $w = h(x+iy)$ is a paraboloid of revolution around the $w$-axis,
that is, every slice is a parabola of the same shape.
But if we take the slice at $\theta = \frac32$
(that is, rotated about $85.94$ degrees from the $x$-axis),
and plot $w$ as a function of $x$ in that slice,
we find that in this slice,
$$
w = \left|x + i\tan\left(\tfrac32\right)\right|^2
= \left(\sec^2\left(\tfrac32\right)\right) x^2
\approx 199.85 x^2,
$$
although in the slice for $\theta = 0$ the equation is $w = x^2.$
How can two slices have such different equations
when they have the same shape?
The answer is that we have used an unnatural, distorted coordinate system
for the slice at $\theta = \frac32.$
This coordinate system works weirdly in that slice.
(And as we know, it doesn't work at all in the slice $\theta = \frac\pi2.$)
If you set up a $u$-axis in the slice, however, then no matter what
value of $\theta$ you use you will always find that the equation
of the slice is $w = u^2,$ which describes the actual
shape of the slice in the way we usually expect.
And when $\theta = \frac\pi2,$ then $u = y,$
so we end up using the $y$ axis instead of the $x$ axis
just as in the other answer.


TITLE: How to create solvable equations with algebraic fractions?
QUESTION [1 upvotes]: I came across this simple algebraic fractions question:
$\frac{8}{x+3}+\frac{3}{x+8}=1$
After a bit of manipulation you are left with a difference of two squares and it is easily solvable.
I then wondered how these questions are made, so had a go at creating a new one and it has been surprisingly difficult.
My first thought was to work backwards from a factorisable quadratic, but I hit a dead end.
My attempt now is to begin with something in the form:
$\frac{a}{x+b}+\frac{b}{x+a}=1$
Which leads to:
$0=x^2+ab-a^2-b^2$
So my question is, how to pick values of a and b so that this will be factorisable, or more specifically so that it will be a difference of two squares? i.e:
$ab-a^2-b^2=-n^2$ for some integer n.

REPLY [1 votes]: To elaborate on lulu's perfect answer:
$a^2+b^2-ab=n^2\iff x^2+y^2-xy=1$, where $x=a/n, y=b/n$.
Now, to parametrize the points of this ellipse, you can start from the point $I=(1,0)$, and then intersect a line of $\mathbb{R}^2$, passing through $I$. The equation of such a line has the form $y=t(x-1)$. Plugging into the ellipse equation yields $x^2+t^2(x-1)^2- tx(x-1)-1=0$, that is $(x-1)(x+1+t^2(x-1)-tx)=0$.
The case $x=1$ will give the point $I$. If $x\neq 1$, little algebraic computations yield $x=\dfrac{t^2-1}{t^2-t+1}, y=\dfrac{t^2-2t}{t^2-t+1}$.
If you take for $t$ any rational value, you will get rational points on your ellipse (in fact, one may convince ourselves that we get all of them).
Side remark. Letting $t$ go to $+\infty$ yield the missing point $I$.
Multiplying $x,y$ by a suitable integer $n$ (for example a denominator of $x,y$), you will get integers $a,b$ as you want.
An easy infinite family of examples would be $a=m^2-1, b=m^2-2m$, where $m$ is an arbitrary integer.
A little algebra shows that  $(m^2-1)^2+(m^2-2m)^2-(m^2-1)(m^2-2m)=(m^2-m+1)^2$, so for $a=m^2-1, b=m^2-2m$, you may take $n=m^2-m+1$.
Note that the case $a=8$ and $b=3$ corresponds to $m=3$.


text:
\begin{document}
\begin{titlepage}

\title{The quantum state vector in phase space and Gabor's windowed Fourier transform}

\author{A.J. Bracken\footnote{{\em Email:} a.bracken@uq.edu.au} and P. Watson\footnote{{\em Email:} pw.cmp@optusnet.com.au}\\Centre for Mathematical Physics\\Department of Mathematics\\
University of Queensland\\Brisbane 4072\\Queensland\\Australia}

\date{}
\maketitle

\begin{abstract}
\ni
Representations of quantum state vectors by complex  phase space amplitudes, complementing the description of the density operator
by the Wigner function, have been defined by applying the Weyl-Wigner transform to dyadic operators,
linear in the state vector and  anti-linear in a fixed `window state vector'.
Here aspects of this construction are explored, and a connection is established with
Gabor's `windowed Fourier transform'.
The amplitudes that arise for simple quantum states from various choices of window are presented as illustrations.
Generalized Bargmann representations of the state vector appear as  special cases, associated with  Gaussian windows.
For every choice of window, amplitudes lie in a corresponding linear subspace of square-integrable functions on phase space.
A generalized Born interpretation of amplitudes is described, with both the  Wigner function and a generalized Husimi function
appearing as quantities linear in an amplitude and anti-linear in its complex conjugate. Schr\"odinger's time-dependent and
time-independent equations are represented on phase space amplitudes, and their solutions described in simple cases.

\end{abstract}

\ni
PACS numbers: 03.65.Ca, 03.65.Ta, 03.65.Db, 03.65.Wj, 03.65.Yz

\end{titlepage}

\section{Introduction}

\vs\ni
The phase space formulation of quantum mechanics \cite{dubin,folland,zachos,gosson2}
now plays a central role in
theoretical quantum optics \cite{schleich} and quantum tomography \cite{leonhardt}, and it has also become an important vehicle for investigations of
fundamental questions such as
the nature of quantization \cite{flato,kontsevich} and the quantum-classical interface \cite{littlejohn,bracken1,bracken2}.
New features continue to be revealed by ongoing
investigations of the underlying structures \cite{bracken3,bracken4,cassinelli}.
\vs\ni
The overwhelming majority of the very large number of studies in this field
have focussed on the Wigner function $W$, the real-valued quasiprobability density on phase space $\Gamma$ which is
the representative of the density operator
${\widehat \rho}$ under the action of the Weyl-Wigner transform $\W$.

However, in the last two decades a small number of studies \cite{torres,harriman,wlodarz,ban,oliveira,gosson,smith1}
have gradually made clear how the state vector
$|\psi\rangle$, which is after all a
more fundamental object than ${\widehat \rho}$, can also be depicted in the phase space picture,
as a complex-valued amplitude $\Psi$ on $\Gamma$. The recent work of Smith \cite{smith1} in particular
shows that this is achieved most simply by applying
$\W$ to a suitable multiple of the dyadic operator $|\psi\rangle\langle \varphi_0|$,
where $|\varphi_0\rangle$ is an
arbitrary normalized state vector that, once chosen, is held fixed while $|\psi\rangle$
ranges over all states of the given quantum system.   From this definition
it follows that $\Psi$ is a multiple of the
two-state Wigner function \cite{hillery} $W_{\psi \varphi_0}$, which is thereby
given an important role in the phase space formulation.

\vs\ni
Several questions concerning this important extension of the phase   space formulation
suggest themselves:
\begin{itemize}
\item
In what sense is $|\psi\rangle\langle\varphi_0|$, and hence its image $\Psi$, equivalent to $|\psi\rangle$
in  quantum mechanical calculations?
\item
What relationships do the Wigner function $W$ and  the Husimi function
\cite{husimi} have
with the phase space amplitude $\Psi$?
\item
What form does the amplitude
$\Psi$ take for special system states $|\psi\rangle$ such as  coherent states, or eigenstates of position or momentum, for
different choices of $|\varphi_0\rangle$?
\item For a given $|\psi\rangle$, how does
the choice of $|\varphi_0\rangle$ influence the structure of $\Psi$
and its place in the set of functions on $\Gamma$?
\item What forms do the time-dependent and time-independent
Schr\"odinger equations take on phase space amplitudes,
and how do their solutions look in simple cases, for various choices of $|\varphi_0\rangle$?
\item What is the nature of the mapping from
$|\psi\rangle$ to $\Psi$, when viewed as a transform of the coordinate space wavefunction $\psi$,
and how does it relate to other, well-known
transforms in the literature?
\end{itemize}

\vs\ni
These are  the questions that we attempt to address in what follows.  With regard to the  last question,
we shall see that the transform in question
is closely related to Gabor's `windowed Fourier transform' \cite{gabor}, widely used
in the signal processing literature \cite{cohen1,cohen2} and also called
there the `short-time Fourier transform'.
Accordingly we shall refer to the fixed state $|\varphi_0\rangle$ that appears
in the definition of
phase space amplitudes $\Psi$ as the window vector,
window state, or simply `the window' in what follows.  Other names for closely related objects in the
literature  are `probe functions' \cite{torres} and `drone states' or `fiducial states' \cite{smith1}.
We shall see also that the choice of an oscillator ground state as window gives a $\Psi$ that is, up to a
factor independent of the state $|\psi\rangle$, the well-known Bargmann wavefunction \cite{bargmann},
which is thereby seen as a precursor to the more recent efforts
\cite{torres,harriman,wlodarz,ban,oliveira,gosson,smith1}
to represent the state vector in phase space.

\vs\ni
For simplicity of presentation in what follows, we consider mainly one linear degree of freedom,
ignore spin degrees of freedom, and treat all variables as dimensionless, setting Planck's constant $\hbar$ equal to $1$.
We use hats to label operators on the usual complex Hilbert space $\Hil$ of quantum mechanics.
Variables without hats are defined on $\Gamma$, unless otherwise specified.

\section{Definition of phase space amplitudes}
\vs\ni
Our starting point is the observation that  pure
state vectors $|\psi\rangle$ can be replaced in all quantum mechanical calculations,
without any loss of generality,  by the corresponding dyadic operators
$|\psi\rangle\langle\varphi_0|$, for any {\it fixed} vector $|\varphi_0\rangle$ of unit length,
$\langle\varphi_0|\varphi_0\rangle =1$.
For example, we can superpose dyadic operators,
\be
\alpha |\psi_1\rangle +\beta |\psi_2\rangle \longleftrightarrow
\alpha |\psi_1\rangle\langle\varphi_0|  +\beta |\psi_2\rangle \langle \varphi_0|\,;
\label{superpose}
\ee
we can evolve them in time using Schr\"odinger's equation,
\be
i \partial_t |\psi(t)\rangle={\hat H}|\psi(t)\rangle
\longleftrightarrow
i\partial_t |\psi(t)\rangle\langle\varphi_0|={\hat H}
|\psi(t)\rangle\langle  \varphi_0|\,;
\label{schro1}
\ee
we can use them to calculate corresponding pure-state density operators,

\bea
\R=|\psi\rangle\langle\psi|= |\psi\rangle\langle\varphi_0|(|\psi\rangle\langle\varphi_0|)^{\dagger}\,;
\label{rho_calc}
\eea
we can use them to calculate expectation values,
\be
\langle {\hat A}\rangle = \langle \psi |{\hat A}|\psi\rangle =
{\rm Tr}\left(\left(|\psi\rangle\langle\varphi_0|\right)^\dagger {\hat A}
|\psi\rangle\langle\varphi_0|\right)\,;
\label{expec1}
\ee
and, most important, we can use them to calculate transition amplitudes,
\be
\langle\psi_1|\psi_2\rangle=
{\rm Tr} \left(\left(|\psi_1\rangle\langle \varphi_0|\right)^\dagger | \psi_2\rangle
\langle\varphi_0 | \right)\,,
\label{transition1}
\ee
and not just transition {\it probabilities} $|\langle\psi_1|\psi_2\rangle|^2$.
Transition probabilities, but not transition amplitudes, can
readily be determined from
the density matrix or, equivalently,  from the Wigner function.

\vs\ni
It is now a simple matter \cite{smith1} to combine  two ideas --- the notion of replacing state vectors $|\psi\rangle$
by dyadic operators $|\psi\rangle\langle \varphi_0 |$, and the notion of  mapping operators into functions on phase space
using the Weyl-Wigner transform  --- in order to define phase space amplitudes $\Psi(q,p)$.  We simply set

\bea
\Psi(q,p) =\frac{1}{\sqrt{2\pi}}\W \left(\,|\psi\rangle\langle\varphi_0|\,\right)(q,p)
\equiv \frac{1}{\sqrt{2\pi}}{\rm Tr }\left(|\psi\rangle\langle \varphi_0|\Del (q,p)\right) \,.
\label{Psi_def}
\eea
Here $\Del (q,p)$ is the Weyl-Wigner-Stratonovich kernel operator
\cite{weyl,wigner,stratonovich,grossmann,royer,brif,lobo,ellinas}, given
in terms of $q$, $p$
and corresponding (dimensionless) canonical operators $\qO$, $\pO$, by

\bea
\Del (q,p)=2\,e^{2i(p\,\,\qO -q\,\pO)}\,\Par\,,
\label{Del_def}
\eea
where $\Par$ is the parity operator, satisfying

\bea
\qO\,\Par=-\Par\,\qO\,,\quad \pO\,\Par=-\Par\,\pO\,,\quad \Par^2=1.
\label{parity}
\eea
\vs\ni
We recall at this point that the star product \cite{neumann,groenewold,moyal} of phase space functions is
defined through the Weyl-Wigner transform by

\bea
&\,& \W(\A)=A\,,\,\, \W(\B)=B\,\,\Rightarrow\,\,\W(\A\B)=A\star B\,,
\label{star1}
\eea
leading to
\bea
(A\star B)(q,p)
&=&A(q,p)\,e^{iJ/2}\,B(q,p)\,,\quad (J=\leftder_q\rightder_p-\leftder_p\rightder_q )
\mea\mea
&=& e^{i(\partial_q\partial_{p'}-\partial_p\partial_{q'})/2}\,A(q,p)B(q',p')|_{(q',p')=(q,p)}\,.
\label{star2}
\eea
The expressions involving differential operators are well-defined if
$A$ and $B$ are polynomial functions.  In more general cases, they define asymptotic expansions,

\bea
A(q,p)\star B(q,p)=A(q,p)B(q,p)+\half iA(q,p)\,J\,B(q,p)+{\rm O}(2)\,,
\label{asymptotic}
\eea
provided $A$ and $B$ represent observables that are asymptotically regular at $\hbar =0$ \cite{osborn}.
In (\ref{asymptotic}),  O(2) denotes terms of second order in Planck's constant (here implicit).

\vs\ni
A string of results now follows from (\ref{Psi_def}) and (\ref{star1}-\ref{asymptotic}):

\vs\ni
\begin{enumerate}
 \item {\em Complex phase space amplitudes}: For each choice of window vector $|\varphi_0\rangle$,
a (distinct) set of amplitudes $\Psi$
is defined by (\ref{Psi_def}), corresponding to the set of all state vectors $|\psi\rangle$.
Each $\Psi$ therefore carries implicitly  a label $\varphi_0$,
which we suppress.
For any choice of $|\varphi_0\rangle$, the amplitudes $\Psi$ are complex-valued in general, and carry the same phases as the
corresponding state vectors $|\psi\rangle$, up to a constant phase shift determined by $|\varphi_0\rangle$.

\vs\ni \item
{\em Expectation values}: Corresponding to (\ref{expec1}), expectation values can be expressed in the form

\bea
\langle{\widehat A}\rangle &=&\int \overline{\Psi(q,p)}\star A(q,p)\star \Psi(q,p)\,d\Gamma
\mea\mea
&=&\int \left\{\Psi(q,p)\star \overline{\Psi(q,p)}\right\} A(q,p)\,d\Gamma
\mea\mea
&=&  \int \overline{\Psi(q,p)}\left\{A(q,p)\star \Psi(q,p)\right\}\,d\Gamma\,.
\label{expec2}
\eea
Here we have used the result \cite{kubo,fairlie}

\bea
{\rm Tr}(\A\B)\longrightarrow \int A(q,p)\star B(q,p)\,d \Gamma =\int A(q,p)B(q,p)\,d\Gamma\,,
\label{trace_formula}
\eea
for suitably smooth functions $A$ and $B$.
\vs\ni
\item
 {\em Transition probabilities}: Corresponding to (\ref{transition1}), transition amplitudes and subsequently transition probabilities
can be calculated from

\bea
\langle\psi_1|\psi_2\rangle=\int \overline{\Psi_1(q,p)}\Psi_2(q,p)\,d\Gamma\,.
\label{transition3}
\eea
In particular,

\bea
\langle\psi|\psi\rangle=\int \overline{\Psi(q,p)}\Psi(q,p)\,d\Gamma=1\,
\label{Psi_normalization1}
\eea
which suggests that $|\Psi(q,p)|^2$, like $W(q,p)$, can be regarded as a quasi--
probability distribution over $\Gamma$ in its own right.  We shall see below that in fact
$|\Psi(q/2,p/2)/2|^2$ is a generalized Husimi distribution.

\vs\ni
\item {\em Generalized Born interpretation}: Corresponding to (\ref{rho_calc}), we have

\bea
W(q,p)=\Psi(q,p)\star \overline{\Psi(q,p)}\,,
\label{pseudoBorn}
\eea
which, bearing (\ref{expec2}) in mind, we refer to as the Born interpretation of $\Psi$,
by analogy with the relation between the
wavefunction and probability density in configuration space.
Then (\ref{asymptotic}) suggests the expansion

\bea
W(q,p)=\Psi(q,p)\overline{\Psi(q,p)}+\half i \Psi(q,p)\,J\,\overline{\Psi(q,p)}+ \dots\,,
\label{approx1}
\eea
which in turn suggests a non-negative approximation to the Wigner function,

\bea
W(q,p)\approx |\Psi(q,p)|^2\geq 0\,.
\label{approx2}
\eea
However, (\ref{approx1}) and (\ref{approx2}) must be treated with caution because $\Psi$ and $W$ may
not be asymptotically regular  at $\hbar=0$ \cite{osborn,berry}.
Comparisons of $|\Psi|^2$ with $W$ in examples that follow illustrate the difficulty  --- see for example
(\ref{cs12}) and (\ref{cs13}).

\vs\ni
 \item {\em Superposition property}: For each choice of $|\varphi_0\rangle$, the mapping from $|\psi\rangle$ to $\Psi$ is linear and phase space amplitudes can be superposed,

\bea
|\psi_{12}\rangle=c_1|\psi_1\rangle+c_2|\psi_2\rangle \longleftrightarrow \Psi_{12}(q,p)=c_1\Psi_1(q,p)+c_2\Psi_2(q,p)\,,
\label{linearity}
\eea
preserving the phase relations between state vectors.

\vs\ni
Note from (\ref{pseudoBorn}) that if $W_1$, $W_2$ and $W_{12}$ are the Wigner functions
corresponding to the
states and phase space amplitudes in (\ref{linearity}),  then

\bea
W_{12}= \left(c_1\Psi_1 +c_2\Psi_2\right)\star \overline{\left(c_1\Psi_1 +c_2\Psi_2\right)}
\mea\mea
= |c_1|^2 W_1 +|c_2|^2 W_2 \qquad\qquad\qquad\qquad
\mea\mea
+c_1\overline{c_2}\,\Psi_1\star\overline{\Psi_2}+c_2\overline{c_1}\,\Psi_2\star\overline{\Psi_1}\,.
\label{wigner_superpose}
\eea
In contrast, it is possible \cite{bracken5} but not straightforward to express $W_{12}$, for example,
directly in terms of $W_1$ and $W_2$.
\vs\ni
Similarly for the generalized Husimi distribution, we have

\bea
|\Psi_{12}|^2= \left(c_1\Psi_1 +c_2\Psi_2\right)\, \overline{\left(c_1\Psi_1 +c_2\Psi_2\right)}
\mea\mea
= |c_1|^2 |\Psi_1|^2 +|c_2|^2 |\Psi_2|^2 \qquad\qquad\qquad\qquad
\mea\mea
+c_1\overline{c_2}\,\Psi_1\overline{\Psi_2}+c_2\overline{c_1}\,\Psi_2\overline{\Psi_1}\,.
\label{husimi_superpose}
\eea

\vs\ni
 \item {\em Normalization and subspace of amplitudes}: The results (\ref{transition3})
and (\ref{Psi_normalization1}) show that, for whatever choice of $|\varphi_0\rangle$,
the complex amplitude $\Psi$ belongs to the Hilbert space  of  square-integrable
functions on phase space, and is normalized when $|\psi\rangle$ is normalized.
In fact all such amplitudes for a given window state
lie in a proper closed subspace $\subspace$ of that Hilbert space,
characterized by the property that

\bea
\Psi(q,p)\star W_{\varphi_0}(q,p)=\Psi(q,p)/2\pi\,,
\label{Psi_charac1}
\eea
where $W_{\varphi_0}$ is the Wigner function corresponding to the state $|\varphi_0\rangle$.  This
follows from the identity

\bea
\psistate\langle\varphi_0|\varphi_0\rangle\langle\varphi_0|=\psistate\langle\varphi_0|
\label{Psi_charac2}
\eea
under the action of $\W$.
Note also the identity

\bea
W_{\psi}(q,p)\star\Psi(q,p)=\Psi(q,p)/2\pi\,,
\label{Psi_charac3}
\eea
which follows from

\bea
|\psi\rangle\langle\psi|\psi\rangle\langle\varphi_0|=|\psi\rangle\langle\varphi_0|\,.
\label{Psi_charac4}
\eea
\vs\ni
It can be seen from (\ref{transition3}) that if $\{|\psi_n\rangle\}$ is a complete
orthonormal set of vectors in $\Hil$, then the corresponding
set of phase space amplitudes $\{\Psi_n\}$ provides a complete orthonormal set in
$\subspace$ under the scalar product given by the RHS of (\ref{transition3}).
Thus $\subspace$ is a Hilbert space in its own right, which for each choice of window
provides an image of $\Hil$ within the space of
square-integrable functions on $\Gamma$.  It is  clear \cite{ascensi} that $\Hil$,
which after all is commonly realized in the coordinate representation
as the space of square-integrable functions on the real {\em line},
has uncountably many images in
the space of square-integrable functions on the phase {\em plane}.

\end{enumerate}

\section{A class of integral transforms}
In the coordinate representation, formula (\ref{Psi_def}) takes the form

\bea
\Psi(q,p)&=&\frac{1}{\sqrt{2\pi}}\int \psi (q-\half y)\overline{\varphi_0(q+\half y)}\, e^{ipy} \,dy
\mea\mea
&=&\sqrt{2/\pi}\int \psi (u)\overline{\varphi_0(2q-u)}\, e^{2ip(q-u)} \,du
\mea\mea
&=&\sqrt{2/\pi}\int \psi (2q-u)\overline{\varphi_0(u)}\, e^{2ip(u-q)} \,du\,,
\label{Psi_def2}
\eea
where $\psi$ and $\varphi_0$ are the wavefunctions corresponding to $\psistate$ and $|\varphi_0\rangle$ respectively.
\vs\ni
The first of these formulas shows that,
apart from a normalization factor, $\Psi$ is the `two-state' or `two-sided' Wigner function \cite{zachos,hillery} that is sometimes denoted
$W_{\psi\varphi_0}(q,p)$.  But as the notation $\Psi$ suggests,
we consider it now as varying with, and determined by
$|\psi\rangle$, with $|\varphi_0\rangle$ held fixed once and for all. We emphasize in particular
that when $|\psi(t)\rangle$ evolves in time,  $\Psi(t)$ evolves also
with {\em only the LH member} of $|\psi(t)\rangle\langle\varphi_0|$ changing (see below).

\vs\ni
The second formula (\ref{Psi_def2}) shows that the  transform from $\psi$ to $\Psi$, obtained with any fixed choice
of the normalized wavefunction $\varphi_0$, is closely related to Gabor's `windowed Fourier transform' \cite{gabor,cohen1,cohen2},
which is

\bea
\Phi(q,p)=\frac{1}{\sqrt{2\pi}}\int \psi(u) w(u-q)\,e^{-ipu}\,du\,,
\label{window_trasform}
\eea
with $w(x)$ the window function.  Choosing $w(x)=\overline{\varphi_0(-x)}$ then leads to

\bea
\Psi(q/2,p/2)=2\,e^{iqp/2}\,\Phi(q,p)\,.
\label{transform_relation}
\eea
It is easily checked that $\Psi$ and $e^{iqp/2}\,\Phi$ are also related by
a `symplectic Fourier transform',

\bea
e^{i qp/2}\,\Phi(q,p)=\frac{1}{2\pi}\,\int \Psi(q',p')\,e^{i(p'q-q'p)}\,d\Gamma'\,.
\label{transform_relation1}
\eea
Gabor  initially chose a simple Gaussian centred on the origin for the window function $\varphi_0(x)$.
In the present context, this choice leads to Bargmann's representation \cite{bargmann} of the wavefunction as an entire function on $\Gamma$,
regarded as the complex plane (see below).

\vs\ni
Note also that the second formula (\ref{Psi_def2}) shows each transform in the general form

\bea
\Psi(q,p)=\int T_{\varphi_0}(q,p,u)\psi(u)\,du\,,
\mea\mea
 T_{\varphi_0}(q,p,u)= \sqrt{2/\pi}\,\overline{\varphi_0(2q-u)}e^{2ip(q-u)}\,.
\label{transform1}
\eea

\vs\ni
From (\ref{Psi_def}) we can also write

\bea
\Psi (q,p) ={\rm Tr} (\psistate\langle\varphi_0|\Del(q,p))&=&\langle\varphi_0|\Del (q,p)|\psi\rangle =\langle {\mathcal T}_{\varphi_0}(q,p)\psistate\,,
\mea\mea
|{\mathcal T}_{\varphi_0}(q,p)\rangle&=&\Del(q,p)|\varphi_0\rangle\,.
\label{transform2}
\eea
\vs\ni
In a similar way it can be seen that

\bea
\Phi(q,p)&=&\langle {\mathcal Q}_{\varphi_0}(q,p)\psistate\,,
\mea\mea
|{\mathcal Q}_{\varphi_0}(q,p)\rangle&=&e^{i(p\,\,\qO -q\,\pO)}\,|\varphi_0\rangle\,.
\label{transform_relation2}
\eea
The transform inverse to (\ref{Psi_def2}) is obtained by considering $\W^{-1}\left(\Psi\right)$
in the coordinate representation, which leads to

\bea
\frac{1}{2\pi}\int\Psi((x+y)/2, p)\,e^{ip(x-y)}\,dp
&=&\langle x|\psi\rangle\langle\varphi_0|y\rangle \,,
\mea\mea
&=& \psi(x) \overline{\varphi_0(y)}\,.
\label{inverse_transform1}
\eea
Hence

\bea
\psi(x)=\frac{1}{2\pi \overline{\varphi_0(y)}}\int\Psi((x+y)/2, p)\,e^{ip(x-y)}\,dp
\label{inverse_transform2}
\eea
wherever $\varphi_0(y)\neq 0$, and also

\bea
\psi(x)= \frac{1}{2\pi}\int\!\!\!\int \Psi((x+y)/2, p)\varphi_0(y)\,e^{ip(x-y)}\,dy\,dp\,.
\label{inverse_transform3}
\eea
Using the definition of $\W^{-1}$ in terms of $\Del(q,p)$ \cite{ellinas},
the result (\ref{inverse_transform3}) can also be written as

\bea
|\psi\rangle=\frac{1}{2\pi}\int \Psi(q,p)|{\mathcal T}_{\varphi_0}(q,p)\rangle\,d\Gamma\,.
\label{inverse_transform4}
\eea
Similarly

\bea
|\psi\rangle=\frac{1}{2\pi}\int \Phi(q,p)|{\mathcal Q}_{\varphi_0}(q,p)\rangle\,d\Gamma\,.
\label{inverse_transform5}
\eea

\section{Generalized Husimi functions}

\vs\ni
The first of formulas (\ref{Psi_def2}) says that $\Psi(q,p)$ is the Fourier transform of

\bea
A_q(y)=\psi(q-\half y)\,\overline{\varphi_0(q+\half y)}\,.
\label{husimi1}
\eea
Similarly, $\overline{\Psi(q,p)}$ is the Fourier transform of

\bea
B_q(y)=\overline{A_q(-y)}\,,
\label{husimi2}
\eea
and so $|\Psi(q,p)|^2$ is the product of these transforms. By the convolution theorem for Fourier transforms \cite{koerner}, it follows that

\bea
|\Psi(q,p)|^2
=\frac{1}{2\pi}\int A_q(y-z)B_q(z)\,e^{ipy}\,dz\,dy&\,&
\mea\mea
=\frac{1}{2\pi}\int \psi(q-\half y+\half z)\overline{\psi(q+\half z)}\qquad\qquad\qquad&\,&
\mea\mea
\times \,\overline{\varphi_0(q+\half y -\half z)}
\varphi_0(q-\half z)\,e^{ipy}\,dz\,dy
\mea\mea
=\frac{2}{\pi}\int\psi(u+\half v)\overline{\psi(u-\half v)}\qquad\qquad\qquad&\,&
\mea\mea
\times\,\overline{\varphi_0(2q-u-\half v)}\varphi_0(2q-u+\half v)\,e^{-2ipv}\,du\,dv\,.
\label{husimi3}
\eea
Setting

\bea
f(r,v)=\int\overline{\varphi_0(\theta-\half v)}\,\varphi_0(\theta +\half v)\,e^{ir\theta }\,d\theta\,,
\label{husimi4}
\eea
so that

\bea
\overline{\varphi_0(\theta-\half v)}\,\varphi_0(\theta +\half v)=
\frac{1}{2\pi}\int f(r,v)\,e^{-ir\theta}\,dr\,,
\label{husimi5}
\eea
we then have

\bea
|\Psi(q/2,p/2)|^2=\frac{1}{\pi^2}\int \psi(u+\half v)\overline{\psi(u-\half v)}
f(r,v)e^{-ir(q-u)}\,e^{-ipv}\,dr\,du\,dv\,,
\label{husimi6}
\eea
showing that $|\Psi(q/2, p/2)/2|^2$ is a (nonnegative) distribution function
from Cohen's general class \cite{cohen1,claasen}. Note from (\ref{transform_relation}) that

\bea
|\Psi(q/2, p/2)/2|^2 \, d(q/2)\, d(p/2)=|\Phi(q,p)|^2\,dq\,dp\,.
\label{husimi6a}
\eea
In the signals literature,  $|\Phi(q,p)|^2$ is known as the spectrogram of the signal $\psi$.
It can be regarded as a generalized Husimi function \cite{smith2} (see below).
In particular, when the window function is the simple Gaussian
corresponding to the oscillator ground state (see below),

\bea
\varphi_0(x)=e^{-x^2/2}/\pi^{1/4}\,,
\label{husimi7}
\eea
we get from (\ref{husimi4})
\bea
f(r,v)=e^{-(r^2+v^2)/4}\,,
\label{husimi8}
\eea
so that

\bea
|\Psi(q/2, p/2)|^2\, d(q/2)\, d(p/2 \left[=|\Phi(q,p)|^2\,dq\,dp\right]=Q(q,p)\,dq\,dp\,,
\label{husimi9}
\eea
where $Q$ is the original Husimi function \cite{husimi}.
\vs\ni
It is remarkable, especially in view of (\ref{approx1}) and (\ref{approx2}),
that $\Psi(q,p)\star \overline{\Psi(q,p)}$ equals the Wigner function for the state $|\psi\rangle$,
while $\Psi(q/2, p/2)\,\overline{\Psi(q/2, p/2)}$ defines a generalized Husimi function for that same state, whatever the choice of window.
In this way $\Psi\star \overline{\Psi}$ and $\Psi\, \overline{\Psi}$ define different aspects of the generalized Born interpretation of $\Psi$ (see above).

\section{Gaussian windows}

 \vs\ni
Amongst the simplest choices for a window state is a squeezed state,  described by a
Gaussian function in the coordinate representation,

\bea
\varphi_0(x)=(\beta^2/\pi)^{1/4}\,e^{-\beta^2(x-x_W)^2/2+ik_W(x-x_W/2)}
\mea\mea
=(\beta^2/\pi)^{1/4}\,e^{\beta^2\lambda(\lambda-{\bar \lambda})/4}\,e^{-\beta^2(x-\lambda)^2/2}\,,\quad \lambda=x_W+ik_W/\beta^2\,,
\label{gaussian5}
\eea
where $\beta$, $x_W$ and $k_W$ are  real constants.  This has
momentum space representation (Fourier transform)

\bea
\tilde{\varphi}_0(k)=(1/\beta^2\pi)^{1/4}\,e^{-(k-k_W)^2/2\beta^2-i(k-k_W/2)x_W}
\mea\mea
= (1/\beta^2\pi)^{1/4}\,e^{-\beta^2\lambda(\lambda+{\bar \lambda})/4}\,e^{-(k+i\beta^2\lambda)^2/2\beta^2}\,.
\label{gaussian6}
\eea
 Then

\bea
|\varphi_0(x)|^2= e^{-\beta^2(x-x_W)^2}\,,\quad |\tilde{\varphi_0}(k)|^2= e^{-(k-k_W)^2/\beta^2}\,,
\label{gaussian7}
\eea
which are peaked at $x=x_W$ and $k=k_W$ respectively.
Note also that $\varphi_0(x)$ is the coordinate space representation of a solution of

\bea
\frac{1}{\sqrt{2}}(\beta  \qO+i\pO/\beta)|\varphi_0\rangle=\frac{\beta\lambda}{\sqrt{2}} |\varphi_0 \rangle\,.
\label{cs1}
\eea
Before proceding, we recall that
\bea
\W(\qO^n)=q\star q\star q\dots \star q =q^n\,,\quad \W(\pO^n)=p^n\,,
\label{qp_powers}
\eea
and that, following Bopp \cite{bopp,kubo}

\bea
\W\left(\A\,\qO^n\right)(q,p)=A(q,p)\star q^n=[q-\half i\partial_p]^n A(q,p)\equiv[\qBR]^n A(q,p)\,,\mea\mea
\W\left(\A\,\pO^n\right)(q,p)=A(q,p)\star p^n= [p+\half i\partial_q]^n A(q,p)\equiv[\pBR]^n A(q,p)\,,
\mea\mea
\W\left(\qO^n\,\A\right)(q,p)=q^n\star A(q,p)= [q+\half i\partial_p]^n A(q,p)\equiv[\qBL ]^n A(q,p)\,,
\mea\mea
\W\left(\pO^n\,\A\right)(q,p)=p^n\star A(q,p)= [p-\half i\partial_q]^n A(q,p)\equiv[\pBL]^n A(q,p)\,.
\label{star3}
\eea
Applying the first two of these results to $\A=|\psi\rangle\langle\varphi_0|$ with $n=1$,
it follows from (\ref{cs1}) that every phase space amplitude $\Psi$ constructed with the Gaussian window state
(\ref{gaussian5}) must satisfy

\bea
\frac{1}{\sqrt{2}}(\beta\qBR -i\pBR/\beta)\Psi(q,p)=\frac{\beta\bar{\lambda}}{\sqrt{2}}\Psi(q,p)\,,
\label{cs2}
\eea
and it is easily seen that the general solution of this equation has the form

\bea
\Psi(q,p)=\sqrt{2/\pi}\,e^{-\bar{z}z/2+\beta(z+{\bar z})\bar{\lambda}/\sqrt{2}}\,G_{\psi}(z)\,,\quad\!\!\!
z=\sqrt{2}(\beta q-ip/\beta)\,,
\label{cs3}
\eea
where $G_{\psi}$ is arbitrary and   we have included an extra  $z$-dependent prefactor to $G_{\psi}$ for later convenience.
The form of  $G_{\psi}(z)$ is determined by $\psi$ from (\ref{gaussian5}) and the second formula in (\ref{Psi_def2}), leading to

\bea
G_{\psi}(z)=(\beta^2/\pi)^{1/4}\,e^{\beta^2\bar{\lambda}(\bar{\lambda}-\lambda)/4}\qquad\qquad\qquad
\mea\mea
\times \int e^{-(z^2+\beta^2(u+\bar{\lambda})^2-2\sqrt{2}\beta z u)/2}\,\psi(u)\,du\,.
\label{G_transform1}
\eea

\section{Generalized Bargmann representation}
\vs\ni
Because the exponential factor in (\ref{cs3}) is independent of $\psi$, this result establishes a
generalized Bargmann representation \cite{bargmann},

\bea
|\psi\rangle \longleftrightarrow G_{\psi}(z)\,,
\label{gen_bargmann1}
\eea
with $G_{\psi}$ an analytic function of the complex variable $z$ (no $\bar{z}$-dependence),
or equivalently of $p+i\beta^2 q$,
in a Hilbert space $\Hil_G$ with weighted scalar product, from (\ref{transition3}) and (\ref{cs3}),

\bea
\left(G_{\psi_1}\,,\,G_{\psi_2}\right)=

(2/\pi)\,\int \overline{G_{\psi_1}(z)}\,G_{\psi_2}(z)\,e^{-\bar{z}z+\beta (\lambda+\bar{\lambda})(z+\bar{z})/\sqrt{2}}\,d^2z\,.
\label{gen_bargmann2}
\eea
Here $d^2z=dq\,dp$.
The creation and  annihilation  operators  $(\qO \mp i\pO)/\sqrt{2}$ acting on each $|\psi\rangle$ in $\Hil$, are represented on $\subspace$ as

\bea
(\qBL\mp i\pBL)/\sqrt{2} \,\equiv \,\pm \half(\beta\pm 1/\beta)(\half z-\partial_{\bar{z}})\mp\half(\beta\mp 1/\beta)(\half \bar{z}+\partial_z)\,,
\nonumber\\
\label{cs4}
\eea
 and therefore, taking into account the exponential factor in (\ref{cs3}), they are represented on $\Hil_G$ as

\bea
(\qO-i\pO)/\sqrt{2}\longleftrightarrow \sigma z -\tau\partial_z-\beta^2\bar{\lambda}/\sqrt{2}
\mea\mea
(\qO+i\pO)/\sqrt{2}\longleftrightarrow \sigma\partial_z-\tau z+\beta^2\bar{\lambda}/\sqrt{2}\,,
\mea\mea
\sigma=(\beta^2+1)/2\beta\,,\quad \tau = (\beta^2-1)/2\beta\,.
\label{gen_bargmann3}
\eea
Then

\bea
\qO&\longleftrightarrow &(z+\partial_z)/\beta\sqrt{2}
\mea
\mea
\pO&\longleftrightarrow &i\beta(z-\partial_z)/\sqrt{2} - i \beta^2{\bar \lambda}\,.
\label{gen_bargmann4}
\eea
Note that the RHSs of (\ref{gen_bargmann3}) and (\ref{gen_bargmann4})
do not involve $\bar{z}$ and so preserve the analyticity of
any $G_{\psi}$ upon which
they act.  They satisfy the canonical commutation relations and have the appropriate
hermiticity properties with respect to the scalar product (\ref{gen_bargmann2}).
These expressions generalize those of the usual
Bargmann representation, which corresponds to the case $\beta=1$ and $\lambda=0$,
that is, the case of a Gaussian window
(\ref{gaussian5}) with the same scaling as that used for $\qO$ and $\pO$, and centered on $x=0$.
Then $\sigma =1$ and $\tau =0$, and the expressions on the RHS of
(\ref{gen_bargmann3}) reduce to the familiar $z$ and $\partial_z$, respectively.
Furthermore, it can be seen that (\ref{G_transform1})
reduces when $\beta=1$ and $\lambda =0$ to Definition (2.3) of the Bargmann transform in \cite{bargmann}.
\vs\ni
The intimate connection between the usual Bargmann wavefunction
and the Wigner function, and with the phase space formulation of quantum mechanics
more generally, has been discussed previously \cite{janssen,wunsche,parisio} from various
other points of view.  Furthermore,
it has recently been shown \cite{ascensi} in the context of the Gabor transform,
that only Gaussian windows give rise to spaces of  analytic functions of the (generalized) Bargmann type.

\section{A test state and its phase space amplitude}
\vs\ni
Consider  a  normalized test wavefunction

\bea
&\,&\psi (x)= \frac{1}{\sqrt{\sqrt{\pi}(1+2\sqrt{2})}} \,\left( e^{-(x-1)^2/2}+4ix \,e^{-x^2}\right)
\mea
\longleftrightarrow&\,&
\mea
&\,&\tilde{\psi}(k)=\frac{1}{\sqrt{\sqrt{\pi}(1+2\sqrt{2})}} \,\left(e^{-k^2/2-ik}+\sqrt{2}k\,e^{-k^2/4}\right)\,.
\label{test1}
\eea
which has as Wigner function

\bea
W(q,p)= \frac{1}{\pi(1+2\sqrt{2})}\,\left\{\frac{}{}e^{-(q-1)^2-p^2}\qquad\qquad\qquad\qquad\right.
\mea\mea
\left. +2\sqrt{2}(4q^2+p^2-1)\,e^{-2q^2-p^2/2}
-\frac{8\sqrt{2}}{3\sqrt{3}}\,\left[(2q-1)\sin(2p\,[q+1]/3)\right.\right.
\mea\mea
\left. \left.-2p\,\cos(2p\,[q+1]/3)\right.]\,e^{-(4q^2-4q+2p^2+1)/3}\,\,\right\}
\label{test2}
\eea
and which, for the choice (\ref{gaussian5}) as window with $x_W=\langle \qO\rangle$ and $k_W=\langle \pO\rangle$, leads to the phase space amplitude

\bea
\Psi(q,p)=N\,\left\{e^{-[4\beta^2q^2+4p^2+4i(\beta^2-1)qp-4\beta^2(\bar{\lambda}+1)q]/2(\beta^2+1)}\right.
\mea\mea
\times e^{[4i(1-\beta^2\bar{\lambda})p+\beta^2(\bar{\lambda}+1)^2]/2(\beta^2+1)}/(\beta^2+1)^{1/2}
\mea\mea
+4i(2\beta^2q-2ip-\beta^2\bar{\lambda})\,e^{-[4\beta^2q^2+2p^2+2i(\beta^2-2)qp]/(\beta^2+2)}
\mea\mea
\left.\times e^{[4\beta^2\bar{\lambda}q+2i\beta^2\bar{\lambda}p-\beta^2\bar{\lambda}^2]/(\beta^2+2)}/(\beta^2+2)^{3/2}\,\,\right\}\,.
\label{test3}
\eea
Here

\bea
N=\sqrt{4\beta/\pi(1+2\sqrt{2})}\,\,e^{\beta^2\bar{\lambda}(\bar{\lambda}-\lambda)/4}\,,\qquad \lambda=x_W+ik_W/\beta^2\,,
\mea\mea
x_W=\langle \qO\rangle = 1/(1+2\sqrt{2}) \,,\qquad k_W=\langle \pO\rangle =8\sqrt{2}/9\,e^{1/3}\sqrt{3}(1+2\sqrt{2})\,.
\label{window_means1}
\eea
The values of $x_W$ and $k_W$ in the definition (\ref{gaussian5}) of
the window function have been equated to the expectation values $\langle \qO\rangle$, $\langle\pO\rangle$ of position and
momentum for the state (\ref{test1}),
so that $|\varphi_0(x)|$ and $|\tilde{\varphi}_0(k)|$ are  localized near these values in position and momentum space, respectively, in accordance with (\ref{gaussian7}).

\vs\ni
Fig. 1 shows the real and imaginary parts of $\Psi$ as in (\ref{test3}) when $\beta=1$.  Fig. 2 shows the Wigner function $W$ of (\ref{test2}),
which is not everywhere positive, and
the
distribution $|\Psi|^2$  in this case, for $\Psi$ as in (\ref{test3}),
with $\beta =1$.  Fig. 3 shows $|\Psi|^2$ for $\beta=0.5$  and $2$.  Comparison of the subplots shows the influence of  the uncertainty principle:
when $\beta^2\ll 1$, the structure of $|\Psi |^2$  better delineates the $p$-dependence of $W$,
while when $\beta^2 \gg 1$, it better delineates the $q$-dependence.

\vs\ni
[Figs. 1, 2 and 3 near here.]
\vs\ni

\section{Coherent states in phase space}

\vs\ni
The formulas (\ref{transform2}) and (\ref{transform_relation2}) show that
$\Psi$ and $\Phi$ can always be regarded as
generalized coherent states in Perelemov's sense \cite{perelemov}.  However,
we now ask what phase space amplitudes $\Psi$ correspond to the familiar
coherent state in $\Hil$  defined up to a constant phase by

\bea
\frac{1}{\sqrt{2}}(\qO+i\pO)|\psi_{\mu}\rangle=\frac{\mu}{\sqrt{2}}|\psi_{\mu}\rangle\,,
\quad \mu=x_C+ik_C\in {\mathbb C}
\label{cs5}
\eea
and normalization.  Here $x_C$ and $k_C$ are arbitrary real numbers.
It follows from (\ref{star3}) that each such state has a corresponding
phase space amplitude $\Psi_{\mu}$ satisfying

\bea
\frac{1}{\sqrt{2}}(\qBL+i\pBL)\Psi_{\mu}(q,p)&\equiv &\frac{1}{\sqrt{2}}(q+\half i\partial_p+ip+\half\partial_q)\Psi_{\mu}(q,p)
\mea\mea
&=&\frac{\mu}{\sqrt{2}}\Psi_{\mu}(q,p)\,,
\label{cs6}
\eea
and hence having the general form

\bea
\Psi_{\mu}(q,p)=K_{\mu}({\bar w})\,e^{-{\bar w}w/2}\,e^{\mu w/\sqrt{2}}\,,\quad w=\sqrt{2}(q-ip)\,.
\label{cs7}
\eea
Here the precise form of $K_{\mu}$ is determined by the phase and normalization of $|\psi_{\mu}\rangle$
and the choice of window $|\varphi_0\rangle$.
Each choice of $|\varphi_0\rangle$
leads also to a  corresponding subspace $\subspace$ of square integrable functions on $\Gamma$ through
(\ref{Psi_charac1}) and (\ref{Psi_charac2}),
and it is remarkable that the phase space amplitudes (\ref{cs7}) ---
which we may call coherent states in phase space --- must, for every choice of window state,
form an overcomplete set in the corresponding $\subspace$ for varying $\mu$,
just as the coherent states  $|\psi_{\mu}\rangle$ form an overcomplete set in $\Hil$.
\vs\ni
If the window  is a Gaussian as in (\ref{gaussian5}), then the amplitude $\Psi_{\mu}$ must also satisfy (\ref{cs2}),
leading from (\ref{cs7}) to

\bea
K_{\mu}({\bar w})={\rm const.}\,e^{-\tau {\bar w}^2/2\sigma}\,e^{(\tau\mu +\beta{\bar \lambda}){\bar w}/\sqrt{2}\sigma}\,,
\label{cs8}
\eea
where $\sigma$ and $\tau$ are as in (\ref{gen_bargmann3}).
In this case we know also that $\Psi_{\mu}$ must be of the form (\ref{cs3}), and we find

\bea
\Psi_{\mu}(q,p)=V_{\mu}(z)\,e^{-{\bar z}z/2}\,e^{\beta {\bar \lambda}{\bar z}/\sqrt{2}}\,,\quad z=\sqrt{2}(\beta q-ip/\beta)\,,
\mea\mea
V_{\mu}(z)={\rm const.}\,e^{\tau z^2/2\sigma}\,e^{(\mu-\tau\beta{\bar \lambda})z/\sigma\sqrt{2}}\,.\qquad\qquad\qquad
\label{cs9}
\eea
Consistency of (\ref{cs7}), (\ref{cs8}) and (\ref{cs9}) is easily checked, leading to

\bea
\Psi_{\mu}(q,p)={\rm const.} \,e^{-2[\beta^2q^2+p^2+i(\beta^2-1)qp-\beta^2
({\bar \lambda}+\mu)q-i(\beta^2{\bar\lambda}-\mu)p]/(\beta^2+1)}\,,
\label{cs10}
\eea
and hence

\bea
\overline{\Psi_{\mu}(q,p)} \,\Psi_{\mu}(q,p)=\frac{4\beta}{\pi(\beta^2+1)}\,e^{-4[\beta^2(q-\eta)^2+(p-\zeta)^2]/(\beta^2+1)}\,,
\mea\mea
\eta=(x_W+x_C)/2\,,
\quad
\zeta=(k_W+k_C)/2\,,
\label{cs11}
\eea
with $x_W$, $k_W$ as in (\ref{gaussian5}) and $x_C$, $k_C$ as in (\ref{cs5}).
In (\ref{cs11}), we have taken the normalization condition (\ref{Psi_normalization1}) into account.
On the other hand, the Wigner function corresponding to the coherent state $|\psi_{\mu}\rangle$ is \cite{hillery}

\bea
W_{\mu}(q,p)=\frac{1}{\pi}\,e^{-(q-x_C)^2-(p-k_C)^2}\,.
\label{cs12}
\eea
If we choose the Gaussian window to be centered on the same coordinate and momentum values $x_C$, $k_C$
as the Wigner function, with the same choice of length scale --- that
is to say, if we choose $\lambda=\mu$ and $\beta =1$ --- then

\bea
\overline{\Psi_{\mu}(q,p)} \,\Psi_{\mu}(q,p)= \frac{2}{\pi}\,e^{-2(q-x_C)^2-2(p-k_C)^2}\,
\label{cs13}
\eea
which is also centered on the same values as the Wigner function.  Note however that  these two distributions
(\ref{cs12}) and (\ref{cs13}) are  not equal,
even in this case when both are positive.
\vs\ni
Comparing (\ref{cs11}) with (\ref{cs12}) and (\ref{cs13}), we see also the effect of choosing a window that is {\em not}
centred on the key features
of the Wigner function.

\section{Schr\"odinger's equation in phase space and evolution of amplitudes}
\vs\ni
From (\ref{schro1}), corresponding to  an
evolving state vector $|\psi(t)\rangle$ in $\Hil$,
we have  for a time-dependent amplitude in phase space,  Schr\"odinger's
equation in the form

\bea
i\,\partial_t \Psi(q,p,t)=H(q,p)\star \Psi(q,p,t)\,,
\label{schro2}
\eea
where $H=\W(\H)$. Supposing $H$ is not explicitly time-dependent, this equation integrates to give

\bea
\Psi(q,p,t)=U(t-t_0)\star\Psi(q,p,t_0)\,,\quad U(t)=e^{\star [-iH(q,p)t]}\,,
\label{evolution1}
\eea
where the star exponential is defined formally by

\bea
e^{\star [A]}= \W\left(e^{{\widehat A}}\right)=1+A+A\star A/2! + A\star A\star A/3!+\dots\,.
\label{star_exponential}
\eea
When $H$ is a polynomial in $q$ and $p$, a more explicit form for the time-evolution may be available. For example,
if $H$ describes a non-relativistic particle and has the form

\bea
H(q,p)=\half p^2+V(q)\,,
\label{non_rel_Ham1}
\eea
with $V$ a polynomial in $q$, then from (\ref{star3}) we can rewrite (\ref{schro2}) as

\bea
i \,\partial_t \Psi(q,p,t)=H(\qBL,\pBL) \Psi(q,p,t)\,,
\label{schro3}
\eea
leading to

\bea
\Psi(q,p,t)=e^{-iH(\qBL,\,\pBL)(t-t_0)}\,\Psi(q,p,t_0)\,.
\label{evolution2}
\eea
\vs\ni
An efficient way to solve  (\ref{schro2}) or (\ref{schro3}) explicitly, when this is possible, is to solve Schr\"odinger's equation
in the coordinate representation to get the wavefunction $\psi(x,t)$  and then use
that in  (\ref{Psi_def2})
to construct $\Psi(q,p,t)$.
\vs\ni
For example, consider the initial-value problem for a free particle

\bea
i\partial_t \psi(x,t)=-\half \partial^2\psi(x,t)/\partial x^2\,,\quad \psi(x,0)=\sqrt{\gamma}\,e^{- \gamma^2x^2/2}/\pi^{1/4}
\label{IV_problem1}
\eea
where $\gamma$ is a positive constant, with solution

\bea
\psi(x,t)=\sqrt{\gamma}\,e^{-\gamma^2x^2/2g(t)^2}/g(t)\pi^{1/4}\,,\quad g(t)=[1 +i\gamma^2t]^{1/2}\,,\quad t\geq 0\,.
\label{evolution3}
\eea
Here the branch of the complex square-root is chosen so that $g(0)=1$.
With the Gaussian (\ref{gaussian5}) as window, (\ref{evolution3}) gives the corresponding  time-dependent phase space amplitude

\bea
\Psi(q,p,t)= [4\beta\gamma/\pi(\beta^2g(t)^2+\gamma^2)]^{1/2}\,
e^{\beta^2{\bar\lambda}({\bar\lambda}-\lambda)/4}\,
e^{-\beta^2{\bar\lambda}^2\gamma^2/2(\beta^2g(t)^2+\gamma^2)}
\mea\mea
\times e^{-2[\beta^2 \gamma^2 q^2+p^2 g(t)^2 +i(\beta^2 g(t)^2-\gamma^2 )qp-\beta^2\gamma^2
{\bar \lambda}q-i\beta^2{\bar \lambda}p g(t)^2]/(\beta^2 g(t)^2+\gamma^2)}\,,
\label{IV_problem3}
\eea
as the solution of

\bea
i\,\partial_t \Psi(q,p,t)=\half \pBL^2 \Psi(q,p,t)\,,
\label{IV_problem4}
\eea
corresponding to the initial value

\bea
\Psi(q,p,0)= [4\beta\gamma/\pi(\beta^2+\gamma^2)]^{1/2}\,
e^{\beta^2{\bar\lambda}({\bar\lambda}-\lambda)/4}\,
e^{-\beta^2{\bar\lambda}^2\gamma^2/2(\beta^2+\gamma^2)}
\mea\mea
\times e^{-2[\beta^2 \gamma^2 q^2+p^2  +i(\beta^2 -\gamma^2 )qp-\beta^2\gamma^2
{\bar \lambda}q-i\beta^2{\bar \lambda}p ]/(\beta^2+\gamma^2)}\,,
\label{IV_problem5}
\eea
which may be compared with (\ref{cs10}) in the case $\gamma=1$, $\mu=0$.
\vs\ni
If the eigenvalue problem

\bea
H(q,p)\star\Psi(q,p)=E\Psi(q,p)
\label{schro4}
\eea
can be solved to find a complete set of phase space eigenfunctions $\Psi_{E_n}(q,p)$ --- some will be generalized
eigenfunctions if $\widehat{H}$ has a (partly) continuous spectrum --- then a more explicit form of solution to (\ref{schro2}) is

\bea
\Psi(q,p,t)=\Sigma_n c_n \,e^{-iE_n (t-t_0)}\,\Psi_{E_n}(q,p)\,,
\mea\mea
\Psi(q,p,t_0)= \Sigma_n c_n \,\Psi_{E_n}(q,p)\,.\qquad
\label{schro5}
\eea
Here each sum must be extended to include an integral over the continuous spectrum, when appropriate.
The eigenvalue problem (\ref{schro4}) is distinct from the `$\star$-genvalue'
problem discussed in the context of the (one-state) Wigner function \cite{curtright,gosson3},
which can be expressed in terms of the solution of (\ref{schro4}) using (\ref{pseudoBorn}).
\vs\ni
The expressions (\ref{schro5}) are the images under $\W$ of the coordinate space formulas

\bea
\psi(x,t)=\Sigma_n c_n \,e^{-iE_n (t-t_0)}\,\psi_n(x)\,,\quad \psi(x,t_0)=\Sigma_n c_n \,\psi_n(x)\,,
\mea\mea
{\rm where} \quad {\widehat H}\psi_n=E_n\psi_n\,.\qquad\qquad\qquad\qquad
\label{schro6}
\eea
Similarly, orthogonality of the coordinate space eigenfunctions and determination of the expansion coefficients,

\bea
\int {\overline \psi_m(x)}\,\psi_n(x)\,dx =\delta_{mn}\,,\quad
c_n=\int {\overline \psi_n(x)}\,\psi(x,t_0)\,dx
\label{schro6A}
\eea
have images

\bea
\int {\overline \Psi_m(q,p)}\,\Psi_n(q,p)\,d\Gamma =\delta_{mn}\,,\quad
c_n=\int {\overline \Psi_n(q,p)}\,\Psi(q,p,t_0)\,d\Gamma\,.
\label{schro7}
\eea
If the $\psi_n$ form a complete orthonormal set in the coordinate space representation of Hilbert space,
so the $\Psi_n$ form a complete orthonormal set in $\subspace$.
\vs\ni
The simple harmonic oscillator with $V(q)=\half  q^2$ provides the simplest illustration.  In this case, (\ref{star2}) shows that
(\ref{schro4}) becomes

\bea
[\half (q^2+p^2)+\half i(q\partial_p -p\partial_q)-\textstyle{\frac{1}{8}}(\partial_q^2+\partial_p^2)]\Psi(q,p)=E\Psi(q,p)
\eea
which is easily solved using a phase space variant of the boson calculus.  Set

\bea
{\check A}=[(q+\half i \partial_p)+i(p-\half i\partial_q)]/\sqrt{2}=\half \bar{w}+\partial_w\,,
\mea\mea
{\check A} ^{\dagger}=[(q+\half i \partial_p)-i(p-\half i\partial_q)]/\sqrt{2}=\half w -\partial_{\bar{w}}\,,
\label{phase_space_bosons}
\eea
where we have used checks to distinguish phase space operators, and again introduced $w=\sqrt{2}(q-ip)$ as in (\ref{cs7}).
\vs\ni
Solving ${\check A}\Psi_0=0$
for the phase space `vacuum state', normalizing it, and then setting $\Psi_n={\check A}^{\dagger n}\Psi_0/\sqrt{n!}$ for $n=1,\,2,\,\dots$,
we find

\bea
\Psi_n(q,p)=\frac{1}{\sqrt{n!}} \sum_{m=0}^n C^{\,n}_m(-1)^{n-m}\,w^mF^{(n-m)}(\bar{w})\,e^{-\bar{w}w/2}\,,
\label{oscillator_state}
\eea
corresponding to the familiar eigenvalue $E_n=n+\half$, for $n=0,\,1,\,2,\,\dots$.  In (\ref{oscillator_state}),
$F(\bar{w})$ and its derivatives are
determined by the choice of window and the normalization of $\Psi_0$, which requires

\bea
\int \overline{F(\bar{w})} F({\bar w})\,e^{-{\bar w}w}\,d^2w=1\,.
\label{F_normalization}
\eea
The obvious choice in the present context, $\varphi_0(x)$ as in (\ref{gaussian5})
with $\beta=1$, $\lambda=0$, leads to $F=\sqrt{2/\pi}$, and (\ref{oscillator_state}) is then the Bargmann wavefunction
for the $n$-th oscillator eigenstate, apart from the exponential factor.
But again we emphasize that any convenient window function can be chosen,
and every choice leads through (\ref{pseudoBorn}) to the same Wigner functions, which in this case are \cite{hillery}

\bea
W_n(q,p)=  (-1)^n\,L_n(2[q^2+p^2])\,e^{-(q^2+p^2)}/\pi \,,
\label{SHO_wigners}
\eea
where $L_n$ is the Laguerre polynomial \cite{abramowitz}.
Different choices of $\varphi_0$ lead to different phase space amplitudes $\Psi_n$ and hence to different
distributions $|\Psi_n|^2$ but the same Wigner function (\ref{SHO_wigners}).
For example, in the case $n=1$,  the choice (\ref{gaussian5}) with $\beta=1$, $\lambda=0$
gives from (\ref{oscillator_state})

\bea
\Psi_1(q,p)=(\sqrt{2/\pi}\,\,)\,w\,e^{-\bar{w} w/2}
\label{bargmann_approx1}
\eea
and hence
\bea
|\Psi_1(q,p)|^2 =(2/\pi)\,(\bar{w}w)\,e^{-\bar{w}w}\,,
\label{bargmann_approx2}
\eea
whereas the choice of a `square' window

\bea
\varphi_0(x)=\left\{\begin{array}{r@{\quad:\quad}l}
1/\sqrt{2a}&|x|<a\\
0&{\rm otherwise}
\end{array}\right.
\label{square_window1}
\eea
leads to
\bea
F(\bar{w})= (1/\pi^{1/4}\sqrt{a})\,e^{\bar{w}^2/2}\,\left[{\rm erf}(a/\sqrt{2}-\bar{w})+{\rm erf}(a/\sqrt{2}+\bar{w})\right]
\label{square_window2}
\eea
in (\ref{oscillator_state}) and from there to

\bea
\Psi_1(q,p) = \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad&&
\mea\mea
(1/\pi^{1/4}\sqrt{2a})\,e^{-\bar{w}w/2}\,\left\{(w-\bar{w})\,e^{\bar{w}^2/2}
\left[{\rm erf}(a/\sqrt{2}-\bar{w})+{\rm erf}(a/\sqrt{2}+\bar{w})\right]\right.&&
\mea\mea
\left. +4/\sqrt{\pi}\,e^{-(a^2+\bar{w}^2)/2}\,\sinh(\sqrt{2}a\bar{w})\right\}\,.& &
\label{square_window3}
\eea
Fig. 4 shows the Wigner function $W_1$ of (\ref{SHO_wigners}), and    $|\Psi_1|^2$ for
the Gaussian window with $\beta=1$, $\lambda=0$ as in (\ref{bargmann_approx2}).  Fig. 5 shows
$|\Psi_1|^2$  for $\Psi_1$ as in (\ref{square_window3}) in the case of a
square window with $a=1$.

\vs\ni
[Figs. 4 and 5 near here.]
\vs\ni

\section{Eigenstates of momentum and position in phase space}

\vs\ni
In view of (\ref{star3}), a (generalized) eigenstate of momentum in phase space is defined  by

\bea
\pBL\,\Psi_{k_0}(q,p)=k_0\Psi_{k_0}(q,p)\,,
\label{momentum_state1}
\eea
giving

\bea
\Psi_{k_0}(q,p)= F_{k_0}(p)\,e^{-2i(p-k_0)q}\,,
\label{momentum_state2}
\eea
with $F_{k_0}(p)$ undetermined.  The form of $F_{k_0}$ depends on the choice of $|\varphi_0\rangle$.
In fact, when the generalized momentum eigenfunction (plane wave)

\bea
\psi(x)=e^{ik_0x}/\sqrt{2\pi}
\label{config_momentum_state}
\eea
is inserted into (\ref{Psi_def2}), it is revealed that

\bea
\Psi_{k_0}(q,p)=\,\sqrt{2/\pi}\,\,\overline{\tilde{\varphi}_0(2p-k_0)}\,e^{-2i(p-k_0)q}\,,
\label{momentum_state4}
\eea
where $\tilde{\varphi}_0 (k)=\langle k|\varphi_0\rangle$ is the function in momentum space that corresponds to
($=$ Fourier transform of) $\varphi_0(x)$.
It can be checked directly using the integral form of the star product
\cite{neumann,groenewold,moyal,fairlie},

\bea
&\, & (A\star B)(q,p)
\mea\mea
&=&\frac{1}{\pi^2}\int_{\Gamma_2\times \Gamma_3} A(q+q_2,p+p_2)B(q+q_3,p+p_3)\,e^{2i(q_2 p_3-q_3 p_2)}
\mea\mea
&\,&\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\times \,d\,\Gamma_2\,d\,\Gamma_3\qquad\qquad\qquad
\label{star_integral}
\eea
that (\ref{pseudoBorn})
does indeed hold in this case whatever choice is made for $|\varphi_0\rangle$,
in every case yielding the (singular) Wigner function

\bea
W_{k_0}(q,p)=\delta(p-k_0)/2\pi\,.
\label{momentum_wigner}
\eea
\vs\ni
In a similar way, the generalized phase space position eigenfunction is defined by

\bea
\qBL\,\Psi_{x_0}(q,p)=x_0\Psi_{x_0}(q,p)
\label{position_state1}
\eea
and we find

\bea
\Psi_{x_0}(q,p)=\sqrt{2/\pi}\,\,\overline{\varphi_0(2q-x_0)}\,e^{2ip(q-x_0)}\,,
\label{position_state2}
\eea
corresponding to $W_{x_0}(q,p)=\delta(q-x_0)/2\pi$. Noting that $\Psi_{x_0}$ in  (\ref{position_state2})
is $\W(|x_0\rangle\langle\varphi_0|)/\sqrt{2\pi}$,
we then see that the second of formulas (\ref{Psi_def2}) is simply the
image under the Weyl-Wigner transform of the formula

\bea
|\psi\rangle\langle\varphi_0|=\int |u\rangle\langle u|\psi\rangle\langle\varphi_0|\,du=\int \psi(u)\,|u\rangle\langle\varphi_0|\,du\,.
\label{position_state3}
\eea
\vs\ni
Formulas (\ref{momentum_state4}) and (\ref{position_state2}) show again the benefit of choosing a
window state that is centered on a region of interest in phase space. For example
if we choose a Gaussian window centered on coordinate and
momentum values $x_W$ and $k_W$ as in (\ref{gaussian5}),
(\ref{gaussian6}), (\ref{gaussian7}),
then for the phase space momentum eigenfunction (\ref{momentum_state4}) we get

\bea
\Psi_{k_0}(q,p)=(4/\pi^3 \beta^2)^{1/4}\,

e^{-(2p-k_0-k_W)^2/2\beta^2}
\mea\mea
\times e^{-i(2qp-2qk_0-2x_Wp+x_Wk_0+x_Wk_W/2)}\,.
\label{momentum_state5}
\eea
If the window state has $k_W=k_0$ and so is centred on the momentum value $k_0$ of interest, then

\bea
|\Psi_{k_0}(q,p)|^2 = (4/\pi^3 \beta^2)^{1/2}\,e^{-4(p\,- k_0)^2/\beta^2}\,,
\label{modPsisquared_1}
\eea
which is also centred on $p=k_0$ for each value of $q$, and may be compared with the Wigner function (\ref{momentum_wigner}).

\vs\ni
Similarly,
for the phase space position eigenstate (\ref{position_state2}) with a Gaussian window we get

\bea
\Psi_{x_0}(q,p)=(4\beta^2/\pi^3)^{1/4}\,

e^{-\beta^2(2q-x_0-x_W)^2/2}
\mea\mea
\times e^{i(2qp-2x_0p-2qk_W+x_0k_W+x_Wk_W/2)}\,,
\label{position_state4}
\eea
so that if the window state has $x_W=x_0$, then

\bea
|\Psi_{x_0}(q,p)|^2 = (2\beta^2/\pi^3 )^{1/2}\,e^{-2\beta^2(q\,- x_0)^2}\,,
\label{modPsisquared2}
\eea
centered on  $q=x_0$ for each value of $p$.  Fig. 6 shows the real part of $\Psi_{k_0}$

for $k_0=-2$,

with
the parameter choices $k_W=k_0$, $x_W=4$ and $\beta=1$ in the Gaussian window (\ref{gaussian5}).  The imaginary part is similar in form.

\vs\ni
[Fig. 6 near here.]
\vs\ni

\section{Oscillator states as windows}
\vs\ni
An obvious generalization of the Gaussian window state with wavefunction $\varphi_0(x)$ as in (\ref{gaussian5}),
is  the $n$-th excited state of the oscillator, for some nonnegative integer $n$,
more precisely, the state obtained by applying a suitable `creation operator' $n$ times to the
Gaussian `ground state', to give the normalized wavefunction

\bea
{\check \varphi}_0(x)&=&(1/\sqrt{n!})\,[(\beta (x-{\bar \lambda}) -\partial_x/\beta)/\sqrt{2}]^n\varphi_0(x)\,.
\label{osc_window1}
\eea
Windows of this type have been considered recently \cite{abreu} in the context
of the Gabor transform; just as the Gaussian choice
leads to the (generalized) Bargmann transform as noted above,
so these oscillator windows lead to further generalizations of the Bargmann transform and
Bargmann representation.

\vs\ni
It follows at once from (\ref{osc_window1}) that if $\Psi (q,p)$ is the
amplitude corresponding to a system state $|\psi\rangle$
and window state $|\varphi_0\rangle$, then the amplitude corresponding to the state $|\psi\rangle$
and window state $|{\check \varphi}_0\rangle$
is

\bea
{\check \Psi}(q,p)=(1/\sqrt{n!})\,[(\beta (\qBR -\lambda)+i\pBR/\beta)/\sqrt{2}]^n\,\Psi(q,p)\,,
\label{osc_window2}
\eea
with $\qBR$, $\pBR$ as in (\ref{star3}).

\vs\ni
For example, the coherent state $\psi_{\mu}$ defined by (\ref{cs5}) leads to the amplitude $\Psi_{\mu}$ as in (\ref{cs10})
when the Gaussian window (\ref{gaussian5}) is chosen, whereas the choice of the $n=1$ (`Mexican Hat') excited state
in (\ref{osc_window1}) as window leads through (\ref{osc_window2}) to the amplitude

\bea
{\check \Psi}_{\mu}(q,p)=\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\mea\mea
\sqrt{2}\beta [(2q-x_W-x_C)+i(2p-k_W-k_C)]\,\Psi_{\mu}(q,p)/(\beta^2+1)\,,
\label{osc_window3}
\eea
and hence from (\ref{cs11}) to the non-negative expression

\bea
\overline{{\check \Psi}_{\mu}(q,p)}\,{\check \Psi}_{\mu}(q,p)
\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\qquad\qquad\qquad
\mea\mea
=\frac{32\beta^3}{\pi(\beta^2 +1)^3}[(q-\eta)^2+(p-\zeta)^2]\,e^{-4[\beta^2(q-\eta)^2+(p-\zeta)^2]/(\beta^2+1)}\,,
\label{osc_window4}
\eea
with $\eta$, $\zeta$ as in (\ref{cs11}).
This expression should be compared with $|\Psi_{\mu}(q,p)|^2$ as in (\ref{cs11}), and with $W_{\mu}(q,p)$ as in (\ref{cs12}).

\section{Concluding remarks}

\vs\ni
The introduction of phase space amplitudes extends the phase space formulation of
quantum mechanics, and may be considered
to complete that formulation by providing images of not only density operators but also state vectors.

\vs\ni
The degree of arbitrariness in the definition of the amplitude chosen to represent
a given quantum state vector reflects the freedom to choose any normalized state as window state,
and may seem surprising at first, notwithstanding our final remarks in Section 2.
Further study is needed to see how to optimize the choice of window for a given quantum system in
a given state, for purposes of computation and visualization.
Selecting a Gaussian window
with adjustable
width and location in coordinate space and momentum space
is simple and natural, and often leads to analytic expressions
for amplitudes representing simple quantum states,
as we have seen.  This has been useful for our purpose here, which was to illustrate basic ideas,
but it may not always be optimal.  A natural choice of window for an evolving
quantum system with a well-defined ground state and excited states might
be its ground state, for example.

\vs\ni
Some of the analysis of amplitudes corresponding to quantum states at a given instant,  mirrors studies
of signals in the time-frequency domain done over many years using Gabor's
windowed Fourier transform \cite{cohen2,ascensi,abreu}.
A feature of the situation in quantum mechanics that is absent in the case of signal
processing arises from the very different
role that the time variable plays in the two cases.  In the quantum case, the
time is not one of the phase space variables,
and the description of quantum phase space amplitudes evolving in time is a
feature absent in the case of signals.
Given that the window state once chosen is fixed, and does not evolve in time,
the choice of an optimal window is
therefore a more complicated problem in the quantum case.

\vs\ni
Other questions that seem worthy of further study in terms of phase space amplitudes
include the formulation of the uncertainty principle, the description of
interference effects, and the description of quantum symmetries.
We hope to return to some of these questions.

\vs\ni
{\it Acknowledgements:}    Thanks to D.R. Stevens for a stimulating conversation,
and to referees for helpful suggestions.

config: arxiv


TITLE: Convergence of $e^{ix_nt}$ implying convergence of $x_n$
QUESTION [1 upvotes]: Suppose $f_n:\mathbb{R} \to S^1$ are given by $f_n(t) = \mathrm e^{\mathrm i x_nt}$, where $x_n$ is some fixed real sequence.
It is clear that whenever $x_n$ converges, $f_n$ must converge to
$\mathrm e^{\mathrm i t \lim_{n\to \infty}x_n}$.
If $f_n$ converges pointwise to a continuous function, is it true that $x_n$ must converge?
Intuitively i think it is true, but i couldn't find a way to prove it.

REPLY [1 votes]: Your conjecture is true, and you can prove it without assuming the limit is continuous, and relaxing pointwise convergence to almost sure convergence on some open interval.
Call the limit $f$.
Case 1: $x_n$ is unbounded. For any interval $I$ we have $\int_I f=\int_I \lim f_n=\lim \int_I f_n=0$ by dominated convergence. Therefore $f=0$. This contradicts the fact that we don't have convergence in probability to $f=0$.
Case 2: $x_n$ is bounded. Then it has a convergent subsequence, with limit $x^*$. If this wasn't the limit of the original sequence, there'd be another subsequence limit $x_+$. Then you'd have $f_n$ converge to both $f_*$ and $f_+$, which contradicts the uniqueness of almost sure limits.
PS: you cannot relax your assumptions to convergence on $\mathbb{Q}$. In fact, enumerate the rationals as $q_n$ and pick $x_n$ such that $f_n(q_i)=1, i\leq n$.