# Symbolic Dynamic Formulation for the Collatz Conjecture: I. Local and Quasi-
global Behavior

Eric Sakk
<EMAIL_ADDRESS>
Department of Computer Science
Morgan State University,
Baltimore, USA.

###### Abstract

In this work, a symbolic dynamical formulation based upon discrete iterative
mappings derived from the Collatz conjecture is introduced. It is demonstrated
that this formulation naturally induces a ternary alphabet useful for
characterizing the expansive and dissipative behavior of generated
itineraries. Furthermore, local and quasi-global analyses indicate cyclic
behaviors that should prove useful for describing global stability properties
of itineraries. Additionally, techniques for generating arbitrarily long
divergent itineraries are presented. Finally, this symbolic formulation allows
itineraries to be grouped into sequence families that retain equivalent
dynamical behavior.

##  1 Introduction

In this work, we present a symbolic dynamical formulation for the Collatz
function

$f(x)=\left\\{\begin{array}[]{ll}x/2&x\;even\\\ 3x+1&x\;odd\\\
\end{array}\right.$ (1)

for any $x\in\mathbb{Z}^{+}$. Much has been written about this mapping [5, 6,
7, 8] and it has been observed (but, not rigorously proved) that all initial
conditions tested so far converge to the periodic solution $1\rightarrow
2\rightarrow 4\rightarrow 1\rightarrow\cdots$ [1]. Intuitively, the above
mapping should be describable using some type of shift map using integers
represented in, for example, base 2 or base 3. Obviously, the $x/2$ portion of
the mapping represents a right shift in base 2. The $3x+1$ portion represents
a left shift plus one in base 3; or, using $x+2x+1$, the $3x+1$ portion could
be described using a shift left plus $x$ plus $1$ in base 2. Since the Collatz
function is a discrete dynamical system, we adopt terminology consistent with
the analysis of this class of systems. For example. given an initial
condition, the term ’itinerary ’ is used to describe the set of points visited
by iterating the Collatz function.

The goal of this paper is to introduce a symbolic formalism that enables the
characterization of all possible itineraries in order to determine if any
given trajectory is unstable (i.e. divergent). Given that the Collatz function
can be viewed as a piecewise series of left and right shifts, it seems
sensible to suggest such an approach. This work deviates from previous work
concerning Collatz graphs [2, 3] in that this formulation can be applied to
characterize the long term behavior of families of trajectories. From the
treatment presented here, it then becomes possible to identify dynamically
equivalent symbolic sequences at many different scales. This work focuses on
the local and quasi-global behavior of the proposed symbolic dynamical
formulation. The results presented here are consistent with findings regarding
the Collatz function [11, 4, 10]. Similarities and differences will be pointed
out in the upcoming sections.

##  2 Preliminary Concepts

We begin by introducing the definitions necessary for characterizing the
mapping defined in Equation (1).

###  2.1 Canonical Relation

A consequence of the Collatz function is that, given any positive, odd integer
$x$,

$f(x)=\frac{f(4x+1)}{4}.$ (2)

This relationship highlights a main feature of the formalism we introduce to
characterize the dynamical behavior of the Collatz function. It also hints at
the possibility of developing a renormalization procedure for understanding
Equation (1).

###  2.2 The C-Ladder

Our goal then is to introduce an approach that can be used to order and
collect the itineraries generated by any set of initial conditions.

######  Definition 2.1.

A node with respect to Equation (1) is any $x\in\mathbb{Z}^{+}$ such that $x$
is odd.

######  Definition 2.2.

A hub is a node that is not of the form $4k+1$ for any positive odd integer
$k=1,3,5,\ldots$.

In other words, all hubs are nodes, but not all nodes are hubs.

The hub is the starting point or ’generator’ for an object we will refer to as
a C-ladder. One side of the C-ladder is formed by starting with a hub and then
pumping the associated odd number $\bf{k}$ using successive applications of
$\bf{4k+1}$; hence, one side of the ladder consists of a sequence of nodes. In
base 2, this sequence is formed by two left shifts then adding a value of one.
The ’rungs’ of the ladder are generated by applying the Collatz function to
the pumped nodes. The other side of a C-ladder is naturally induced by
Equation (1) and the canonical relation in Equation 2. For example, the odd
integer $3$ is a hub and generates the following C-ladder

$\begin{matrix}3&\Rightarrow&13&\Rightarrow&53&\Rightarrow&213&\Rightarrow&\cdots\\\
\downarrow&&\downarrow&&\downarrow&&\downarrow&&\\\
10&\leftarrow_{4}&40&\leftarrow_{4}&160&\leftarrow_{4}&640&\leftarrow_{4}&\cdots\\\
\end{matrix}$

In this diagram, numbers connected by $\rightarrow$ indicate a node and its
image of the Collatz function. Numbers connected by $\leftarrow_{4}$ indicate
division by 4. Numbers connected by $\Rightarrow$ indicate a $4k+1$ pump.
Given, for example, $x=53$, Equation 2 tells us that $f(53)=f(4(53)+1)/4=160$.

Finally, we have the following

######  Definition 2.3.

A C-tree is the connected union of all C-ladders.

Constructions similar to the even side of the C-ladder have been previously
mentioned [4, 10]. This focal point makes sense as this side of the ladder is
responsible for the convergence to a node under successive iterates of the
Collatz function. When successive pumps are viewed in base 2, it is clear why
this is so. As successive $4k+1$ pumps of a node are applied to the C-ladder,
the original node is shifted left by two bits and a one is then inserted into
the least significant bit. After several pumps the suffix of the original
value will look like $...01010101$. When the odd portion of the Collatz
function is applied as $x+2x+1$, the pumped value $...01010101$ is added to
the shifted value $...010101010$ leading to the values $....11111111$. To
complete the $x+2x+1$ operation, simply add one to $....11111111$ and the
resulting value looks like $.....00000000$ which will ’fall’ down the even
side of the C-ladder until the original node is reached. It is clear why some
previous works have focused on this dissipative process in the hopes that some
kind of convergence will ultimately result as unions of C-ladders are
considered. While such an approach is worthy of exploration, there is no
inherent guarantee that the final node value reached at the bottom of a
particular C-ladder will be less than some previous iterate leading to one of
the pumped nodes in the ladder. If there were such a guarantee, the Collatz
conjecture would have been solved. On the other hand, C-ladders are extremely
important as they represent strongly dissipative events along the itinerary of
an initial condition.

###  2.3 Node Definitions

We can say much more about the nodes of a C-ladder. In order to do so, the
mapping in Equation (1) will require us to partition the set of all nodes into
three disjoint sets using the following definitions.

######  Definition 2.4.

A 0-node is an odd integer of the form $6m+3$ for any nonnegative integer
$m=0,1,2,\ldots$.

Since $6m+3=3(2m+1)$, is should be clear that set of all 0-nodes represents
the set of nonnegative odd integers that are divisible by three.

######  Definition 2.5.

A 1-node is an odd integer of the form $6m+5$ for any nonnegative integer
$m=0,1,2,\ldots$.

######  Definition 2.6.

A 2-node is an odd integer of the form $6m+1$ for any nonnegative integer
$m=0,1,2,\ldots$.

The set of 0-nodes, 1-nodes and 2-nodes comprise a disjoint partition of the
set of nonnegative odd integers.

Finally, we require one more definition in order to further describe an
intrinsic property of 0-nodes.

######  Definition 2.7.

Let $x$ be a $0-node$ of the form $6m+3$. The set of all 0-nodes can be
further partitioned into a set defined by $3(6n+t)$ where $t=1,3\;or\;5$ and
$n=0,1,2,3,\ldots$. Given the partition $3(6n+t)$, the character of a 0-node
is the value $t\in\\{1,3,5\\}$ associated with any 0-node.

######  Example 2.8.

We can show all partitions of each character as follows:

  * •

0-nodes of character $t=1$: $\\{3,\text{{21}},39,67,\ldots\\}$

  * •

0-nodes of character $t=3$: $\\{9,27,\text{{45}},63,\ldots\\}$

  * •

0-nodes of character $t=5$: $\\{15,33,51,\text{{69}},\ldots\\}$

In this example, we have also highlighted nodes in boldface that cannot be
hubs as they are of the form $4k+1$ where $k$ is odd. Finally, when describing
a 0-node, the character of the node is depicted with a superscript as follows:
$3^{(1)},\;51^{(5)},\;27^{(3)},\;etc$.

According to Equation (1), backward iterations of 1-nodes and 2-nodes can lead
to either even preimages or odd preimages. On the other hand, 0-nodes are
unique as their backward iterations cannot possess odd preimages. Since a
0-node is of the form $3(2m+1)$, there cannot exist an odd integer $x$ such
that $2^{-n}(3x+1)$ can map to $3y$ where $y$ is odd. Therefore, given a
0-node $x$, all Collatz preimages must be of the form $2^{n}x$. Thus, 0-nodes
are the end point of successive divisions by 2. In some sense, the C-tree is
’surrounded’ by ’threads’ of even numbered itineraries of the Collatz function
whose forward iterations ultimately converge to a 0-node on a Collatz ladder.
To see this, consider the example from the previous section where the odd
integer $3$ was a hub used to generate a C-ladder of the form

$\begin{matrix}&&\vdots&&&&&&\vdots&\\\ &&12&&&&&&852&\\\
&&\downarrow_{2}&&&&&&\downarrow_{2}&\\\ &&6&&&&&&426&\\\
&&\downarrow_{2}&&&&&&\downarrow_{2}&\\\
1&&3^{(1)}&\Rightarrow&13&\Rightarrow&53&\Rightarrow&213^{(5)}&\Rightarrow\\\
\Downarrow&&\downarrow&&\downarrow&&\downarrow&&\downarrow&\\\
5&\leftarrow_{2}&10&\leftarrow_{4}&40&\leftarrow_{4}&160&\leftarrow_{4}&640&\leftarrow_{4}\\\
\Downarrow&&&&&&&&&\\\
21^{(1)}&\leftarrow_{2}&42&\leftarrow_{2}&84&\cdots&&&&\\\
\Downarrow&&&&&&&&&\\\ \end{matrix}$

(where $\leftarrow_{4}$ indicates division by 4 and $\leftarrow_{2}$ indicates
division by 2). This diagram also begins to hint at the fact that a C-ladder
generated by a given hub can connect to another ladder generated by a
different hub. Here, we can see that the 0-nodes 3, 21 and 213 are the end
point of a series of successive divisions by 2. More importantly, we shall see
in a moment that they exist at the beginning of what we will define as a
’primitive itinerary’. Before taking this step, we will use our node
definitions to develop the symbolic formalism. Specifically, the node type
will be included as part of the C-tree.

$\begin{matrix}&&\vdots&&&&&&\vdots&\\\ &&12&&&&&&852&\\\
&&\downarrow_{2}&&&&&&\downarrow_{2}&\\\ &&6&&&&&&426&\\\
&&\downarrow_{2}&&&&&&\downarrow_{2}&\\\
1:2&&3:0^{(1)}&\Rightarrow&13:2&\Rightarrow&53:1&\Rightarrow&213:0^{(5)}&\Rightarrow\\\
\Downarrow&&\downarrow&&\downarrow&&\downarrow&&\downarrow&\\\
5:1&\leftarrow_{2}&10&\leftarrow_{4}&40&\leftarrow_{4}&160&\leftarrow_{4}&640&\leftarrow_{4}\\\
\Downarrow&&&&&&&&&\\\
21:0^{(1)}&\leftarrow_{2}&42&\leftarrow_{2}&84&\cdots&&&&\\\
\Downarrow&&&&&&&&&\\\ \end{matrix}$

According to this illustration, each node is directed to another node via
successive application of the Collatz function. When the $4k+1$ pumps are
viewed in base 2, the $f(x)=x+2x+1$ interpretation of the Collarz function
makes clear how successive divisions by four down one side of a C-ladder can
take place via a series of right shifts.

##  3 Local Behavior

Given the above definitions, we now characterize the dynamical behavior of
1-nodes and 2-nodes. Consider the following examples.

$\begin{matrix}&&3:0\\\ &&\downarrow\\\ 5:1&\leftarrow_{2}&10\\\
\end{matrix}\quad\quad\quad\quad\quad\begin{matrix}&&17:1\\\ &&\downarrow\\\
13:2&\leftarrow_{4}&52\\\ \end{matrix}$

These cases indicate that a 1-node is the image of a hub under the mapping
$(3x+1)/2$ and a 2-node is the image of a hub under the mapping $(3x+1)/4$.
Even preimages can be recovered from a C-ladder using the canonical relation,
so we will focus on hub images for a moment.

######  Definition 3.1.

The first odd preimage of a 1-node $y$ is formed from the inverse mapping
$(2y-1)/3$.

######  Definition 3.2.

The first odd preimage of a 2-node $y$ is formed from the inverse mapping
$(4y-1)/3$.

To establish the sensibility of these definitions requires the analyses
presented in the following sections.

###  3.1 Characterization of 1-nodes

Given a node $x$, in order for $(3x+1)/2$ to map to a 1-node, the equation
$(3(2m+1)+1)/2=6p+5$ must have a solution. This observation implies that there
exists a solution to $6m+4=12p+10$ or $m=2p+1$ for $p=0,1,2,\ldots$. For
example, $p=0$ leads to $6p+5=5$ and $m=2p+1=1\Rightarrow 2m+1=3$; hence 3
maps to 5 which is a 1-node (see above diagram). It should also be clear that
neither nodes of the form $6p+1$ (2-node) nor $6p+3$ (0-node) can be the image
of $(3x+1)/2$. We have already dealt with the 0-node case. To understand the
2-node case, the equation $(3(2m+1)+1)/2=6p+1$ has no solution. Such a
condition reduces to $6m+4=12p+2$ or $3m+1=6p$ which obviously cannot be
satisfied for any integer $p$.

If we invert the 1-node value $6p+5$ to find its first odd preimage, we must
solve $6p+5=(3x+1)/2$; or $x=4p+3$. Since $x$ cannot be of the form $4k+1$
with $k$ odd, it must therefore, by definition, be a hub.

######  Lemma 3.3.

The first odd preimage of a 1-node must always be a hub.

The hub itself can be a 0-node,, a 1-node or a 2-node, but the first odd
preimage of a 1-node must be a hub. Given the preceding analysis, we can also
conclude that

######  Lemma 3.4.

The first odd preimage of a 1-node of the form $6p+5$ must always be a hub of
the form $4p+3$ where $p=0,1,2,3,\ldots$.

Given these observations, we have the following

######  Theorem 3.5.

A hub $x$ of the form $4p+3$ under the image of the mapping $(3x+1)/2$ always
maps to a 1-node.

Observe, the 1-node under this image of the hub need not also be a hub, but,
as we shall see, it can be.

We can collect these results in the form of a table to exemplify the behavior
of 1-nodes.

Table 1: 1-node behavior $p$ | (first odd preimage) $4p+3$: node type |  $6p+5$ 1-node
---|---|---
0 | $3:0^{(1)}$ | 5
1 | $7:2$ | 11
2 | $11:1$ | 17
3 | $15:0^{(5)}$ | 23
4 | $19:2$ | 29
5 | $23:1$ | 35
6 | $27:0^{(3)}$ | 41
7 | $31:2$ | 47
8 | $35:1$ | 53
9 | $39:0^{(1)}$ | 59
10 | $43:2$ | 65
11 | $47:1$ | 71
12 | $51:0^{(5)}$ | 77
13 | $55:2$ | 83
14 | $59:1$ | 89
15 | $63:0^{(3)}$ | 95

Table 1 provides the first hint that, when viewed from the proper reference
frame, there exist a number of interesting period behaviors in the set of
generated 1-nodes. In addition to the preimage 0-node, 2-node, 1-node pattern
as $p$ increases, for the set of 0-nodes, the type is periodic in the sequence
$1,5,3$. Furthermore, 1-nodes that are of the form $4k+1$ where $k$ is odd are
indicated in boldface. Given their structure, there should be no surprise that
these occur periodically as well.

The periodic nature of these values allows us to work backward from a 1-node
value $y$ to determine node type $s$ of its first odd preimage using the
formula

$s=\frac{y+4}{3}\quad(mod\;3)$ (3)

which is the first hint at generating a symbolic dynamical description for the
Collatz function.

###  3.2 Characterization of 2-nodes

Given a node $x$, in order for $(3x+1)/4$ to map to a 2-node, the equation
$(3(2m+1)+1)/4=6p+1$ must have a solution. This observation implies that there
exists a solution to $6m+4=24p+4$ or $m=4p$ for $p=0,1,2,\ldots$. For example,
$p=0$ leads to $6p+1=1$ and $m=2p+1=1\Rightarrow 2m+1=1$; hence 1 maps to 1
which is a 2-node. It should also be clear that neither nodes of the form
$6p+5$ (1-node) nor $6p+3$ (0-node) can be the image of $(3x+1)/4$. We have
already dealt with the 0-node case. To understand the 1-node case, the
equation $(3(2m+1)+1)/4=6p+5$ has no solution. Such a condition reduces to
$6m+4=24p+20$ or $6m=24p+16$ which cannot be satisfied for any integer $p$.

If we invert the 2-node value $6p+1$ to find its first odd preimage, we must
solve $6p+1=(3x+1)/4$; or $x=8p+1$. Since $x$ cannot be of the form $4k+1$
with $k$ odd, it must therefore, by definition, be a hub.

######  Lemma 3.6.

The first odd preimage of a 2-node must always be a hub.

The hub itself can be a 0-node,, a 1-node or a 2-node, but the first odd
preimage of a 2-node must itself be a hub. Given the preceding analysis, we
can also conclude that

######  Lemma 3.7.

The first odd preimage of a 2-node of the form $6p+1$ must always be a hub of
the form $8p+1$ where $p=0,1,2,3,\ldots$.

Given these observations, we have the following

######  Theorem 3.8.

A hub $x$ of the form $8p+1$ under the image of the mapping $(3x+1)/4$ always
maps to a 2-node.

Observe, the 2-node under this image of the hub need not also be a hub, but,
as we shall see, it can be.

We can collect these results in the form of a table to exemplify the behavior
of 2-nodes.

Table 2: 2-node behavior $p$ | (first odd preimage) $8p+1$: node type |  $6p+1$ 2-node
---|---|---
0 | $1:2$ | 1
1 | $9:0^{(3)}$ | 7
2 | $17:1$ | 13
3 | $25:2$ | 19
4 | $33:0^{(5)}$ | 25
5 | $41:1$ | 31
6 | $49:2$ | 37
7 | $57:0^{(1)}$ | 43
8 | $65:1$ | 49
9 | $73:2$ | 55
10 | $81:0^{(3)}$ | 61
11 | $89:1$ | 67
12 | $97:2$ | 73
13 | $105:0^{(5)}$ | 79
14 | $113:1$ | 85
15 | $121:2$ | 91

Table 2 provides the first hint that, when viewed from the proper reference
frame, there exist a number of interesting period behaviors in the set of
generated 2-nodes. In addition to the preimage 2-node, 0-node, 1-node pattern
as $p$ increases, for the set of 0-nodes, the type is periodic in the sequence
$3,5,1$. Furthermore, 2-nodes that are of the form $4k+1$ where $k$ is odd are
indicated in boldface. Given their structure, there should be no surprise that
these occur periodically as well.

The periodic nature of these values allows us to work backward from a 2-node
value $y$ to determine node type $s$ of its first odd preimage using the
formula

$s=\frac{2y+4}{3}\quad(mod\;3)$ (4)

which is the another hint at generating a symbolic dynamical description for
the Collatz function

###  3.3 Characterization of the set of all nodes

As pointed out above, even preimages of a hub image can be recovered from a
C-ladder using the canonical relation by applying successive pumps of the
mapping $4k+1$ to the hub. Therefore, a C-ladder can be generated using
0-node, 1-node, 2-node as a hub and then pumping that value using $4k+1$. From
the previous sections, all hubs are either of the form $4p+3$ (which maps to a
1-node) or $8p+1$ (which maps to a 2-node). Given these possibilities, if we
partition of all odd numbers using the rule $8p+t$ where $t\in\\{1,3,5,7\\}$,
we can see that the set of all odd numbers consists of only hubs and ’pumps’.
The value $8p+1$ is a hub whose image corresponds to a 2-node. If we define,
$p=2q$ is even and $p+1=2q+1$ is odd, then $4p+3$ (a hub that maps to a
1-node) corresponds to $8q+3$ and $4(2q+1)+3$ corresponds to $8p+7$; while
$4(2p+1)+1$ pumps correspond to $8p+5$. Therefore, using the partition of all
odd numbers using the rule $8p+t$ where $t\in\\{1,3,5,7\\}$ leads to 1-node
hubs when $t\in\\{3,7\\}$, 2-node hubs when $t=1$ and pumps when $t=5$.

This characterization is significant because it tells us the how dissipative
and expansive nodes are distributed. A hub that transitions to a 1-node is
expansive as it is of the form $(3x+1)/2$. A hub that transitions to a 2-node
is dissipative as it is of the form $(3x+1)/4$. A pumped $4k+1$ node (where
$k$ is odd) is dissipative by at least a factor of $(3x+1)/8$ if it
transitions to a 1-node and at least a factor of $(3x+1)/16$ if it transitions
to a 2-node. These observations will allow us to begin entering into a
discussion regarding the stability of trajectories.

##  4 Quasi-global Behavior

###  4.1 Symbolic construction and the stability of an itinerary

In order to begin a discussion about stability, we require the definition

######  Definition 4.1.

A primitive itinerary of Equation (1) is one that begins at a 0-node and ends
at a pumped $4k+1$ node where $k$ is odd.

######  Example 4.2.

$21:0$ is a primitive itinerary of length one as it is both a 0-node and a
pumped $4k+1$ node where $k=5$.

######  Example 4.3.

$3:0\rightarrow_{2}5:1$ is a primitive itinerary of length two.

######  Example 4.4.

$81:0\rightarrow_{4}61:2$ is a primitive itinerary of length two.

######  Example 4.5.

$15:0\rightarrow_{2}23:1\rightarrow_{2}35:1\rightarrow_{2}53:1$ is a primitive
itinerary of length four.

More directly, a primitive itinerary begins at a 0-node and traverses a
sequence of hubs that are either 1-nodes or 2-nodes until a pumped node is
encountered. A primitive itinerary will naturally induce a sequence of 1’s and
2’s.

######  Example 4.6.

$15:0\rightarrow_{2}23:1\rightarrow_{2}35:1\rightarrow_{2}53:1$ is a primitive
itinerary that generates the sequence $\\{0111\\}$.

######  Example 4.7.

$9:0\rightarrow_{4}7:2\rightarrow_{2}11:1\rightarrow_{2}17:1\rightarrow_{4}13:2$
is a primitive itinerary that generates the sequence $\\{02112\\}$.

If the value of the pumped node is less than that of the 0-node, then we say
the primitive itinerary is dissipative. If the value of the pumped node is
greater than that of the 0-node, then we say the primitive itinerary is
expansive. We can make an asymptotic statement regarding the dissipative or
expansive behavior of a primitive itinerary. After the 0-node, a primitive
itinerary will consist only of 1-nodes and 2-nodes (i.e a sequence of 1’s and
2’s). It is therefore possible to asymptotically characterize the stability of
such an itinerary by counting the occurrences of 1’s and 2’s in a sequence
generated by the itinerary.

######  Observation 4.8.

Letting $n$ represent the number of 2’s and $m$ represent the number of 1’s, a
primitive itinerary will be expansive if the following condition holds:

$\left(\frac{3}{2}\right)^{m}\left(\frac{3}{4}\right)^{n}>1\Rightarrow
m>n\frac{\log{4/3}}{\log{3/2}}\approx 0.7095n\equiv\gamma n$ (5)

This bound is consistent results regarding the Collatz function (see for
example, [9]).

After the 0-node, a primitive itinerary is simply a binary sequence of length
$N$ and the binomial theorem can be used to count the number of expansive
sequences $N_{E}$ generated by a primitive itinerary:

$N_{E}=\sum_{k=0}^{\hat{n}}{N\choose k}$ (6)

where $\hat{m}>\gamma\hat{n}$ and $\hat{m}=N-\hat{n}$. This, in turn, yields
the condition that

$N-\hat{n}>\gamma\hat{n}\Rightarrow\hat{n}<\frac{N}{1+\gamma}$ (7)

meaning that the number of 2’s in an expansive primitive sequence (i.e. a
sequence generated by a primitive itinerary) of length $\mathbf{N}$
asymptotically must be less than the quantity  $\mathbf{N/(1+\gamma)}$.

######  Example 4.9.

Consider primitive sequences of length $N=5$ with the first symbol equal $0$.
Then, $\hat{n}<\frac{N}{1+\gamma}\approx 2.3$. Therefore, expansive sequences
can visit at most $2$ 2-nodes.

A complete itinerary of an initial condition is simply the intersection of a
set of primitive itineraries. Once a primitive itinerary lands at a $4k+1$
node, it will necessarily drop by at least a factor of $8$ into the midst of
another primitive itinerary and so on. This behavior is readily observable if
one plots each value as function of its iteration index. For complex
trajectories, it is common to see an expansive portion that suddenly
dissipates and intersects another primitive itinerary and then recommences the
expansion-dissipation process. Obviously, in order to encompass the
possibility of returning to the $1\rightarrow 2\rightarrow 4$ cycle, there
must ultimately be more dissipative events than expansive events.

####  4.1.1 An expansive construction

One major goal of Section 4 will be to systematically characterize exactly how
to generate expansive sequences as this would deal directly with the question
regarding the existence of an asymptotically expansive sequence. Before
presenting a more comprehensive development, we can deal more directly with
the worst case scenario, a sequence consisting purely of 1-nodes that are
hubs.

Table 1 indicates that it should be possible generate a sequence consisting
purely of hubs that are of 1-nodes since 1-nodes that map to 1-nodes are
separated by a value of 18. Therefore, given a 1-node $y$, there should exist
an $n$ such that $y+18n=(3y+1)/2$. We can begin exploring this goal by
checking that a hub can be mapped to a 1-node using the condition

$[3(4p+3)+1]/2=6p+5$ (8)

To ensure this condition can be iterated to generate a sequence of hubs that
are 1-nodes, the following observation will prove useful.

######  Observation 4.10.

Given a node $y_{i}$ and the mapping $y_{i+1}=(3y_{i}+1)/2$

$y_{i+1}+1=\frac{3y_{i}+1}{2}+1=\frac{3}{2}(y_{i}+1)$ (9)

The procedure is to start with an $x$ that maps to 1-node $y_{0}$ that is a
hub. Given Equation (8), it follows that

$y_{0}+1=6p+6=6(p+1).$ (10)

Observation 4.10 tells us that 1-node iterations can be constructed simply by
successively multiplying $y_{i}+1$ by an expansive factor of $3/2$. In order
to ensure that $y_{i}+1$ continues to be an integer after successive divisions
by $2$, we must choose $p=s2^{M}-1$, for some $s$ an odd integer, in Equation
(10) resulting in

$y_{0}+1=6s2^{M}=3s2^{M+1}.$ (11)

For this prescription, it should be clear that $p$ must be an odd integer in
order to create the initial condition. It should also be clear that $\bf{M}$
can be chosen arbitrarily large in order to create an arbitrarily long
subsequence of $\bf{M+1}$ 1-nodes within a primitive sequence by applying
Observation 4.10 to Equation (11) until all powers of two have been exhausted

$\begin{split}y_{0}+1&=3s2^{M+1}\\\
y_{1}+1&=\left(\frac{3}{2}\right)(3s2^{M+1})=s3^{2}2^{M}\\\
y_{2}+1&=\left(\frac{3}{2}\right)(s3^{2}2^{M})=s3^{3}2^{M-1}\\\ &\vdots\\\
y_{M+1}+1&=\left(\frac{3}{2}\right)(s3^{M+1}2)=s3^{M+2}\\\ \end{split}$ (12)

The value $y_{M+1}$ will necessarily be an even integer indicating the last
iteration can map to a 2-node or a pumped node.

######  Example 4.11.

Consider the case where $s=1$, $M=5$ and $p=2^{5}-1=31$; hence,

$\begin{split}y_{0}+1&=3s2^{M+1}=192\\\
y_{1}+1&=\left(\frac{3}{2}\right)(192)=288\\\
y_{2}+1&=\left(\frac{3}{2}\right)(288)=432\\\
y_{3}+1&=\left(\frac{3}{2}\right)(432)=648\\\
y_{4}+1&=\left(\frac{3}{2}\right)(648)=972\\\
y_{5}+1&=\left(\frac{3}{2}\right)(972)=1458\\\
y_{6}+1&=\left(\frac{3}{2}\right)(1458)=2187\\\ \end{split}$

This implies the following sequence containing $M+1=6$ successive 1-nodes that
are hubs (and, hence, a subsequence within a primitive sequence),

$191:1\rightarrow_{2}287:1\rightarrow_{2}287:1\rightarrow_{2}431:1\rightarrow_{2}647:1\rightarrow_{2}971:1\rightarrow_{2}1457:1\rightarrow_{2}2188$

The end point of this sequence is an even integer that can still be further
divided by 2 (this is because the end of the sequence $1457$ maps to a
2-node).

To check that the prescribed 1-node to 1-node mapping truly results in a new
1-node we first check the following. Given a value $6p+5$ for some $p$, there
must exist a $q$ such that $6p+5=4q+3$, it is easily found that, when $p$ is
odd, there must exist a $q$. But, recall, the condition $p=s2^{M}-1$ will be
odd if $s$ is odd. Therefore, the initial condition must represent a hub as it
maps a 1-node to a 1-node. Next, given Observation 4.10 to Equation (11), it
is enough to check the first iteration on the initial condition

$\begin{split}y_{1}+1&=(\frac{3}{2})(y_{0}+1)\\\ &=(\frac{3}{2})(6p+6)\\\
&=9p+9\\\ &=9(s2^{M}-1)+9\\\ \Rightarrow y_{1}&=9s2^{M}-1\\\ \end{split}$ (13)

Rearranging and regrouping $y_{1}$ as follows yields the desired result

$\begin{split}y_{1}&=9s2^{M}-1\\\ &=3\cdot 2s2^{M-1}-6+5\\\
&=6(s2^{M-1}-1)+5\\\ &=6q+5\\\ \end{split}$ (14)

which is of the form of a 1-node. Hence, the initial prescription $p=s2^{M}-1$
does enable the mapping of 1-nodes to 1-nodes a total of $M$ times.

Given the possibility of generating an expansive subsequence of 1-nodes of
arbitrary length, it is only natural to attempt to enumerate and categorize
the set of all possible expansive primitive itineraries. This is the overall
goal of the following subsections.

###  4.2 Cyclic Behavior

Given the characterization of nodes into the partition $\\{6p+1,6p+3,6p+5\\}$,
it should not be surprising that the C-ladder exhibits a substantial amount of
cyclic behavior. We state many of these observations without proof as they are
fairly straightforward to demonstrate.

######  Observation 4.12.

The node sequence of $4k+1$ pumps along a C-ladder obey the periodic node
ordering

$\cdots\Rightarrow 0\Rightarrow 2\Rightarrow 1\Rightarrow 0\Rightarrow
2\Rightarrow 1\Rightarrow\cdots.$ (15)

We can say a bit more, given the character of a 0-node.

######  Observation 4.13.

The node sequence of $4k+1$ pumps along a C-ladder obey the periodic node
ordering

$\begin{split}\cdots\Rightarrow&0^{(1)}\Rightarrow 2\Rightarrow
1\Rightarrow\\\ &0^{(5)}\Rightarrow 2\Rightarrow 1\Rightarrow\\\
&0^{(3)}\Rightarrow 2\Rightarrow 1\Rightarrow\\\ &0^{(1)}\Rightarrow
2\Rightarrow 1\Rightarrow\\\ &0^{(5)}\Rightarrow 2\Rightarrow 1\Rightarrow\\\
&0^{(3)}\Rightarrow 2\Rightarrow 1\Rightarrow\cdots\\\ \end{split}$ (16)

where the character of 0-nodes is also periodic.

As 0-nodes can possess different characters, the above observation begins to
hint at the possibility of topologically equivalent behaviors between
different C-ladders in a C-tree. To explore this possibility further, it is
important to recognize that C-ladders are numerically constrained along its
sequence of $4k+1$ pumps,

######  Observation 4.14.

Consider a node sequence of $4k+1$ pumps along a C-ladder. Along the pump, the
first odd preimage of 1-nodes and 2-nodes obey the periodic node ordering

$\begin{split}\cdots\Rightarrow&0^{(1)}\Rightarrow(1\rightarrow_{4}2)\Rightarrow(1\rightarrow_{2}1)\Rightarrow\\\
&0^{(5)}\Rightarrow(0\rightarrow_{4}2)\Rightarrow(2\rightarrow_{2}1)\Rightarrow\\\
&0^{(3)}\Rightarrow(2\rightarrow_{4}2)\Rightarrow(0\rightarrow_{2}1)\Rightarrow\\\
&0^{(1)}\Rightarrow(1\rightarrow_{4}2)\Rightarrow(1\rightarrow_{2}1)\Rightarrow\\\
&0^{(5)}\Rightarrow(0\rightarrow_{4}2)\Rightarrow(2\rightarrow_{2}1)\Rightarrow\\\
&0^{(3)}\Rightarrow(2\rightarrow_{4}2)\Rightarrow(0\rightarrow_{2}1)\Rightarrow\cdots\\\
\end{split}$ (17)

Table 3: C-ladder successive $4k+1$ pump behavior starting with the hub value of 1 (*the hub with periodic itinerary $1\rightarrow 2\rightarrow 4$ is not primitive). 0-node value | 2-node value | primitive sequence | 1-node value | primitive sequence
---|---|---|---|---
– | $1^{\ast}$ | $222\ldots$ | $5$ | $10^{3(1)}$
$21^{(1)}$ | $85$ | $210^{75(1)}$ | $341$ | $1120^{201(1)}$
$1365^{(5)}$ | $5461$ | $20^{7281(3)}$ | $21845$ | $12210^{17259(3)}$
$87381^{(3)}$ | $349525$ | $2211120^{24581(5)}$ | $1398101$ | $10^{932067(3)}$
$5592405^{(1)}$ | $22369621$ | $21120^{17674761(1)}$ | $89478485$ | $1110^{26512143(5)}$
$357913941^{(5)}$ | $1431655765$ | $20^{(1)}$ | $5726623061$ | $120^{(3)}$

We can collect the above observations in the form of a table. In Table 3, in
the first column, the character of each 0-node value is included. The third
column provides the complete sequence leading to the 1-node value in column
two, all sequences therefore begin at a 0-node in the rightmost position,
travel to the left and end at the 1-node value. The fifth column provides the
complete sequence leading to the 2-node value in column four, all sequences
therefore begin at a 0-node in the rightmost position, travel to the left and
end at the 2-node value. In the sequence columns, for convenience, the
character and value of the starting 0-node is provided. Because the 1-node and
2-node columns represent $4k+1$ pumps and all sequences begin at 0-nodes, by
definition, all sequences (except the hub with periodic itinerary
$1\rightarrow 2\rightarrow 4$ ) represent primitive itineraries. Table 4 shows
a similar sequence analysis using $3$ as the hub.

Table 4: C-ladder successive $4k+1$ pump behavior starting with the hub value of 3. 0-node value | 2-node value | primitive sequence | 1-node value | primitive sequence
---|---|---|---|---
$3^{(1)}$ | $13$ | $21120^{9(3)}$ | $53$ | $1110^{15(5)}$
$213^{(5)}$ | $853$ | $\bf{20^{1137(1)}}$ | $3413$ | $\bf{120^{3033(3)}}$
$13653^{(3)}$ | $54613$ | $\bf{220^{97089(5)}}$ | $218453$ | $\bf{10^{145635(5)}}$
$873813^{(1)}$ | $3495253$ | $\bf{212220^{(5)}}$ | $13981013$ | $\bf{110^{(1)}}$
$55924053^{(5)}$ | $223696213$ | $20^{(5)}$ | $894784853$ | $1210^{(3)}$
$3579139413^{(3)}$ | $14316557653$ | $222120^{(3)}$ | $57266230613$ | $10^{(1)}$
$0^{(1)}$ |  | $210^{(3)}$ |  | $112210^{(3)}$
$0^{(5)}$ |  | $20^{(3)}$ |  | $12220^{(5)}$

In addition to confirming Observations 4.12-4.14, several other observations
are possible. For example, as one steps down the rows in the sequence column,
the symbolic structure induces a form of counting in each sequence position
over the alphabet $\\{0,1,2\\}$. There is also much more to say regarding
character periodicities. In fact, given this symbolic structure, C-trees and
C-ladders exhibit a wide spectrum of periodic behaviors and topological
consistencies that extend globally throughout the tree. Let us take another
look at these behaviors from the perspective of the character of 0-nodes.

Table 5: In each row of this table, each 0-node value generated by $6m+3$ (and starting with $m=0$) is followed by its 2-node and 1-node generated by $4k+1$ pumps. 0-node | 2-node | sequence | 1-node | sequence
---|---|---|---|---
$3^{(1)}$ | $13$ | $21120^{9(3)}$ | $53$ | $1110^{15(5)}$
$9^{(3)}$ | $37$ | $22120^{57(1)}$ | $149$ | $10^{99(3)}$
$15^{(5)}$ | $61$ | $20^{81(3)}$ | $245$ | $122220^{513(3)}$
$\bf{21^{(1)}}$ | $85$ | $210^{75(1)}$ | $341$ | $1120^{201(1)}$
$27^{(3)}$ | $109$ | $22210^{171(3)}$ | $437$ | $10^{291(1)}$
$33^{(5)}$ | $133$ | $20^{177(5)}$ | $533$ | $1210^{315(3)}$
$39^{(1)}$ | $157$ | $21210^{123(5)}$ | $629$ | $110^{279(3)}$
$\bf{45^{(3)}}$ | $181$ | $220^{321(5)}$ | $725$ | $10^{483(5)}$
$\bf{51^{(5)}}$ | $\bf{205}$ | $\bf{20^{273(1)}}$ | $\bf{821}$ | $\bf{120^{729(3)}}$
$57^{(1)}$ | $229$ | $2110^{135(3)}$ | $917$ | $111222121212220^{897(5)}$
$63^{(3)}$ | $253$ | $2211221220^{417(1)}$ | $1013$ | $10^{675(3)}$
$\bf{69^{(5)}}$ | $277$ | $20^{369(3)}$ | $1109$ | $1221120^{777(1)}$
$75^{(1)}$ | $301$ | $210^{267(5)}$ | $1205$ | $112120^{633(1)}$
$81^{(3)}$ | $325$ | $22221110^{303(5)}$ | $1301$ | $10^{867(1)}$
$87^{(5)}$ | $348$ | $20^{465(5)}$ | $1397$ | $1211120^{489(1)}$
$\bf{93^{(1)}}$ | $373$ | $2120^{441(3)}$ | $1493$ | $110^{663(5)}$
$99^{(3)}$ | $397$ | $220^{705(1)}$ | $1589$ | $10^{1059(5)}$
$105^{(5)}$ | $421$ | $20^{561(3)}$ | $1685$ | $120^{1497(1)}$
$111^{(1)}$ | $445$ | $211121121221111210^{27(3)}$ | $1781$ | $11110^{351(3)}$
$\bf{117^{(3)}}$ | $469$ | $2210^{555(5)}$ | $1877$ | $10^{1251(3)}$
$123^{(5)}$ | $493$ | $20^{657(3)}$ | $1973$ | $1220^{2357(5)}$
$129^{(1)}$ | $517$ | $210^{459(3)}$ | $2069$ | $1122211212120^{1017(3)}$

Once again, in Table 5, in addition to periodic behavior, a ordered ’counting’
process takes place in the sequences associated with each primitive itinerary
(since these sequences are constrained by Observations 4.12-Observations
4.12). Given the $\\{6m+3\\}$ ordering of the 0-nodes, the second position of
each sequence generates the periodic ordering $\\{0,1,2\\}$ for the 2-node
column and $\\{0,2,1\\}$ for the 1-node column for every change by $6$ in the
0-node value. The third position of each sequence generates the periodic
ordering $\\{0,2,1\\}$ for the 2-node column and $\\{0,1,2\\}$ for the 1-node
column for every change by $18$ in the 0-node value. The fourth position of
each sequence generates the periodic ordering $\\{0,1,2\\}$ for the 2-node
column and $\\{0,2,1\\}$ for the 1-node column for every change by $54$ in the
0-node value and so on.

These observations imply an immense amount of consistent topological structure
that is globally applicable across the C-tree. To see this, it is best to
start with a motif that generates short primitive sequences that terminate
quickly for most sequences in the motif. For example, consider the block of
sequences highlighted in Table 4 that pump across three iterations of $4k+1$.
The start of this motif can be found in Table 5 at the 0-node labeled $51$. By
simply adding 54 consecutively ($51,105,159,\ldots$), the reader can check
that a measure of topological consistency and counting periodicity is
preserved. To preserve even greater topological consistency, one can take into
account and preserve the character of the 0-node generating the primitive
sequence (as these exhibit periodic behavior as well). More will be said about
these behaviors when we present the global treatment in part II of this work.
Before tackling the global behavior, we require one more set of observations.

Table 6: Characterization of prefixes within a sequence ending at a node. Here, it is convenient to use the 1-node and 2-node definitions given in Section 2.3 in order to generate a natural ordering. Boldface sequences are primitive sequences. $p$ | 2-node 6p+1 | sequence | 1-node 6p+5 | sequence
---|---|---|---|---
$0$ | $1_{\leftarrow_{4}}1$ | $222\ldots$ | $\bf{5_{\leftarrow_{2}}3}$ | $\bf{10^{(1)}}$
$1$ | $7_{\leftarrow_{4}}9$ | $20^{(3)}$ | $11_{\leftarrow_{2}}7_{\leftarrow_{4}}9$ | $120^{(3)}$
$2$ | $\bf{13_{\leftarrow_{4}}17_{\leftarrow_{2}}11_{\leftarrow_{2}}7_{\leftarrow_{4}}9}$ | $\bf{21120^{(3)}}$ | $17_{\leftarrow_{2}}11_{\leftarrow_{2}}7_{\leftarrow_{4}}9$ | $1120^{(3)}$
$3$ | $19_{\leftarrow_{4}}25_{\leftarrow_{4}}33$ | $220^{(5)}$ | $23_{\leftarrow_{2}}15$ | $10^{(5)}$
$4$ | $25_{\leftarrow_{4}}33$ | $20^{(5)}$ | $\bf{29_{\leftarrow_{2}}19_{\leftarrow_{4}}25_{\leftarrow_{4}}33}$ | $\bf{1220^{(5)}}$
$5$ | $31_{\leftarrow_{4}}41_{\leftarrow_{2}}27$ | $210^{(3)}$ | $35_{\leftarrow_{2}}23_{\leftarrow_{2}}15$ | $110^{(5)}$
$6$ | $\bf{37_{\leftarrow_{4}}49_{\leftarrow_{4}}65_{\leftarrow_{2}}43_{\leftarrow_{4}}57}$ | $\bf{22120^{(1)}}$ | $41_{\leftarrow_{2}}27$ | $10^{(3)}$
$7$ | $43_{\leftarrow_{4}}57$ | $20^{(1)}$ | $47_{\leftarrow_{2}}31_{\leftarrow_{4}}41_{\leftarrow_{2}}27$ | $1210^{(3)}$
$8$ | $49_{\leftarrow_{4}}65_{\leftarrow_{2}}43_{\leftarrow_{4}}57$ | $2120^{(1)}$ | $\bf{53_{\leftarrow_{2}}35_{\leftarrow_{2}}23_{\leftarrow_{2}}15}$ | $\bf{1110^{(5)}}$
$9$ | $55_{\leftarrow_{4}}73_{\leftarrow_{4}}97_{\leftarrow_{4}}129$ | $2220^{(1)}$ | $59_{\leftarrow_{2}}39$ | $10^{(1)}$
$10$ | $\bf{61_{\leftarrow_{4}}81}$ | $\bf{20^{(3)}}$ | $65_{\leftarrow_{2}}43_{\leftarrow_{4}}57$ | $120^{(1)}$
$11$ | $67_{\leftarrow_{4}}89_{\leftarrow_{2}}59_{\leftarrow_{2}}39$ | $2110^{(1)}$ | $71_{\leftarrow_{2}}47_{\leftarrow_{2}}31_{\leftarrow_{4}}41_{\leftarrow_{2}}27$ | $11210^{(5)}$
$12$ | $73_{\leftarrow_{4}}97_{\leftarrow_{4}}129$ | $220^{(1)}$ | $\bf{77_{\leftarrow_{2}}51}$ | $\bf{10^{(5)}}$
$13$ | $79_{\leftarrow_{4}}105$ | $20^{(5)}$ | $83_{\leftarrow_{2}}55_{\leftarrow_{4}}73_{\leftarrow_{4}}97_{\leftarrow_{4}}129$ | $12220^{(1)}$
$14$ | $\bf{85_{\leftarrow_{4}}113_{\leftarrow_{2}}75}$ | $\bf{210^{(1)}}$ | $89_{\leftarrow_{2}}59_{\leftarrow_{2}}39$ | $110^{(1)}$
$15$ | $91_{\leftarrow_{4}}\ldots$ | $221111210^{(3)}$ | $95_{\leftarrow_{2}}63$ | $10^{(3)}$
$16$ | $97_{\leftarrow_{4}}129$ | $20^{(1)}$ | $\bf{101_{\leftarrow_{2}}67_{\leftarrow_{4}}\cdots}$ | $\bf{12110^{(3)}}$

###  4.3 Prefix Properties

Yet another way of analyzing sequence cyclic behavior is to characterize
primitive sequences as subsequences of prefixes. Consider Table 6 where it can
be seen that there are many instances where one sequence is a prefix of
another sequence. In this section, we present a set of rules for generating
sets of all prefixes in an ordered fashion. That this is a possibility should
be clear from the cyclic nature of the sequences generated as a function of
the index $p$ in Table 6. Appendices A and B form the foundation and provide
the details details behind the prefix rules presented in this section.

######  Observation 4.15.

  1. The following steps can generate the set of all possible prefixes for all possible primitive sequences:

  2. 1.

(Initial condition) Begin with the sequence $m=0,1,2,3,\ldots$ that imply a
sequence of 0-nodes of the form $6m+3$

  3. 2.

Mark all values of $m$ whose $6m+3$ value will map to $4k+1$ pumps (these
values are periodic at every fourth value of $m$).

  4. 3.

Create new tree levels as follows.

     * i.

Create a triple by sequentially taking three values at a time (do not include
the marked values as they represent those prefixes that terminate at a $4k+1$
pump - the end of a primitive sequence).

     * ii.

Create next prefix ending in a 1-node by taking every other value in the
sequence of triples

     * iii.

Create next prefix ending in a 2-node by taking every middle value in the
sequence of triples

  5. 4.

return to step 2

The above procedure can be made clear via the following demonstration (the
numbers shown correspond to values of $m$ as developed in Appendices A and B).
Consider the first level to generate all possible prefixes of length $2$:

$\begin{matrix}&&01:\underline{0}\;2\;4\;6\;\underline{8}\;10\;\ldots\\\
&\nearrow_{1}&\\\
0-node:0\;1\;2\;\underline{3}\;4\;5\;6\;\underline{7}\;8\;\ldots&&\\\
&\searrow_{2}&\\\ &&02:1\;5\;9\;\underline{13}\;17\;21\;\ldots\\\
\end{matrix}$ (18)

Iterate the above procedure to generate all possible prefixes of length $3$:

$\begin{matrix}&\nearrow_{1}&011:2\;6\;10\;\underline{14}\;18\;22\;26\;\underline{30}\;\ldots\\\
01:\underline{0}\;2\;4\;6\;\underline{8}\;10\;\ldots&\\\
&\searrow_{2}&012:4\;\underline{12}\;20\;28\;\ldots\\\ &&\\\
&\nearrow_{1}&021:1\;9\;17\;\underline{25}\;33\;\ldots\\\
02:1\;5\;9\;\underline{13}\;17\;21\;\ldots&\\\
&\searrow_{2}&022:5\;21\;37\;\underline{53}\;\ldots\\\ \end{matrix}$ (19)

Be careful to synchronize this procedure with respect to the $4k+1$ pump.
There are various possibilities for iterating this process as the relative
position of sequences leading to $4k+1$ pumps will differ for each tree node.
The general form of the generated sequences can be of four possible forms:

$\begin{matrix}1&2&1&\underline{X}&1&2&1&\underline{X}&\ldots\\\
2&1&\underline{X}&1&2&1&\underline{X}&1&\ldots\\\
1&\underline{X}&1&2&1&\underline{X}&1&2&\ldots\\\
\underline{X}&1&2&1&\underline{X}&1&2&1&\ldots\\\ \end{matrix}$

The above procedure can be iterated to map every 0-node to its $4k+1$ pump
along a primitive sequence. Observe, the $m$ value is enough to generate this
information. The astute reader will recognize that the above decimation
process is analogous to a wavelet transformation thus, once again, hinting at
a renormalization procedure. Additionally, this symbolic formulation bears
some resemblance to encoding vectors developed for characterizing the
convergent behavior of the Collatz function [11]. However, this work is
focused primarily on the quasi-global behavior of families of primitive
itineraries having the same symbolic sequence. Along with the C-ladder,
primitive itineraries are crucial to understand because they begin at a
0-node, visit only hubs that either $1$-nodes or $2$-nodes and terminate at a
$4k+1$ pump of a C-ladder.

####  4.3.1 An Expansive Sequence Example

While there are many results to consider in light of Section 4.1, we once
again focus on applying this formulation to the most expansive case:
generating a sequence of 1-nodes of arbitrary length. Such a construction
amounts to traversing the tree outlined in Observation 4.15 along its
outermost branches (corresponding to a sequence of 1-node transitions). In
this case, it can shown that the position of $4k+1$ pump end points at each
tree level will alternate between the $0^{th}$ position and the $3^{rd}$
position (e.g. in the above tree figures, consider the position of $4k+1$
pumps for the $10$ case versus the $110$ case).

Given this construction, it is straightforward to see that there always exists
an $m$, where, starting at a 0-node value $6m+3$, it is possible to generate a
sequence of 1-nodes of arbitrary length. However, given the alternating nature
of the $4k+1$ end points, once the expansive sequence encounters the $4k+1$
pump value, the question arises as to whether or not the entry point into the
next primitive sequence will be enough to cancel the expansion. We take up
this question in part II of this work which addresses the global behavior of
the Collatz function.

##  5 Conclusions

In this work, we have presented a symbolic dynamical formalism for
characterizing all itineraries generated by the Collatz function. This
symbolic approach enables the establishment of dynamical similarities within
the presented tree structure. The demonstrated techniques allow for
understanding the local and quasi-global behavior of all trajectories. Future
work will involve applying this approach to characterize the global behavior
of itineraries.

## Disclosure statement

No potential conflict of interest was reported by the author.

## References

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##  Appendix A Prefix Analysis

###  A.1 Prefixes of length 2

####  A.1.1 The $10$ prefix

Start with a 0-node $x=6m+3$ and applying

$\frac{3x+1}{2}=\frac{3(6m+3)+1}{2}=9m+5$

to the constraint that the output must be a 1-node

$9m+5=6p+5$

leads to the condition

$p=\frac{3}{2}m.$ (20)

This generates the following progression

$\begin{matrix}k=0,1,2,3,4,\ldots&m=2k=0,2,4,6,8,\ldots&6m+3=3,15,27,39,51\ldots\\\
&p=3k=0,3,6,9,12\ldots&6p+5=\underline{5},23,41,59,\underline{77}\ldots\\\
\end{matrix}$

where each $6m+3$ 0-node maps to its corresponding $6p+5$ 1-node (4k+1 pumps
are underlined and mark the end point of a primitive sequence). The reader can
check that these results are consistent with Table 6.

####  A.1.2 The $20$ prefix

Start with a 0-node $x=6m+3$ and applying

$\frac{3x+1}{4}=\frac{3(6m+3)+1}{4}=\frac{18m+10}{4}$

to the constraint that the output must be a 2-node

$\frac{18m+10}{4}=6p+1$

leads to the condition

$p=\frac{3}{4}m+\frac{1}{4}.$ (21)

This generates the following progression

$\begin{matrix}k=0,1,2,3,4,\ldots&m=4k+1=1,5,9,13,17\ldots&6m+3=9,33,57,81,105\ldots\\\
&p=3k+1=1,4,7,10,13,\ldots&6p+1=7,25,43,\underline{61},79\ldots\\\
\end{matrix}$

where each $6m+3$ 0-node maps to its corresponding $6p+1$ 2-node (4k+1 pumps
are underlined and mark the end point of a primitive sequence). The reader can
check that these results are consistent with Table 6.

###  A.2 Prefixes of length 3

####  A.2.1 The $110$ prefix

Start with a 0-node $x=6m+3$ and applying

$\frac{3[\frac{3x+1}{2}]+1}{2}=\frac{3[9m+5]+1}{2}=\frac{27m+16}{2}$

to the constraint that the output must be a 1-node

$\frac{27m+16}{2}=6p+5$

leads to the condition (it will shortly become clear why we leave the fraction
in non-reduced form)

$p=\frac{9}{4}m+\frac{2}{4}.$ (22)

This generates the following progression

$\begin{matrix}k=0,1,2,3,4,\ldots&m=4k+2=2,6,10,14,18,\ldots&6m+3=15,39,63,87,111,\ldots\\\
&p=9k+5=5,14,23,32,41,\ldots&6p+5=35,89,143,\underline{197},251,\ldots\\\
\end{matrix}$

where each $6m+3$ 0-node maps to its corresponding $6p+5$ 1-node (4k+1 pumps
are underlined and mark the end point of a primitive sequence). The reader can
check that these results are consistent with Table 6.

####  A.2.2 The $210$ prefix

Start with a 0-node $x=6m+3$ and applying

$\frac{3[\frac{3x+1}{2}]+1}{4}=\frac{27m+16}{4}$

to the constraint that the output must be a 2-node

$\frac{27m+16}{4}=6p+1$

leads to the condition (it will shortly become clear why we leave the fraction
in non-reduced form)

$p=\frac{9}{8}m+\frac{4}{8}.$ (23)

This generates the following progression

$\begin{matrix}k=0,1,2,3,4,\ldots&m=8k+4=4,12,20,28,\ldots&6m+3=27,75,123,171,\ldots\\\
&p=9k+5=5,14,23,32,\ldots&6p+1=31,\underline{85},139,187,\ldots\\\
\end{matrix}$

where each $6m+3$ 0-node maps to its corresponding $6p+1$ 2-node (4k+1 pumps
are underlined and mark the end point of a primitive sequence).

####  A.2.3 The $120$ prefix

Start with a 0-node $x=6m+3$ and applying

$\frac{3[\frac{3x+1}{4}]+1}{2}=\frac{54m+34}{8}$

to the constraint that the output must be a 1-node

$\frac{54m+34}{8}=6p+5$

leads to the condition

$p=\frac{9}{8}m-\frac{1}{8}.$ (24)

This generates the following progression

$\begin{matrix}k=0,1,2,3,4,\ldots&m=8k+1=1,9,17,25,\ldots&6m+3=9,57,105,153,\ldots\\\
&p=9k+1=1,10,19,28,\ldots&6p+5=11,65,119,\underline{173},\ldots\\\
\end{matrix}$

where each $6m+3$ 0-node maps to its corresponding $6p+5$ 1-node (4k+1 pumps
are underlined and mark the end point of a primitive sequence).

####  A.2.4 The $220$ prefix

Start with a 0-node $x=6m+3$ and applying

$\frac{3[\frac{3x+1}{4}]+1}{4}=\frac{54m+34}{16}$

to the constraint that the output must be a 2-node

$\frac{27m+17}{8}=6p+1$

leads to the condition

$p=\frac{9}{16}m+\frac{3}{16}.$ (25)

This generates the following progression

$\begin{matrix}k=0,1,2,3,4,\ldots&m=16k+5=5,21,37,\ldots&6m+3=33,129,225,\ldots\\\
&p=9k+3=3,12,21,\ldots&6p+1=19,\underline{73},127,\ldots\\\ \end{matrix}$

where each $6m+3$ 0-node maps to its corresponding $6p+1$ 2-node (4k+1 pumps
are underlined and mark the end point of a primitive sequence).

###  A.3 Prefixes of length 4

A similar set of basic algebraic steps can be used to generate formulae for
higher order sequences. After a certain point, this exercise becomes
unnecessary as a reproducible pattern arises that can be used to unveil the
cyclic nature of the symbolic sequences.

####  A.3.1 The $1110$ prefix

$p=\frac{27}{8}m+\frac{10}{8}.$ (26)

This generates the following progression

$\begin{matrix}k=0,1,2,3,4,\ldots&m=8k+2=2,10,18,\ldots&6m+3=15,63,111,\ldots\\\
&p=27k+8=8,35,62,\ldots&6p+5=\underline{53},215,377,\ldots\\\ \end{matrix}$

####  A.3.2 The $2110$ prefix

$p=\frac{27}{16}m+\frac{14}{16}.$ (27)

This generates the following progression

$\begin{matrix}k=0,1,2,3,4,\ldots&m=16k+6=6,22,38,\ldots&6m+3=39,135,231,\ldots\\\
&p=27k+11=11,38,65,\ldots&6p+1=67,\underline{229},391,\ldots\\\ \end{matrix}$

####  A.3.3 The $1210$ prefix

$p=\frac{27}{16}m+\frac{4}{16}.$ (28)

This generates the following progression

$\begin{matrix}k=0,1,2,3,4,\ldots&m=16k+4=4,20,36,\ldots&6m+3=27,123,219,\ldots\\\
&p=27k+7=7,34,61,\ldots&6p+5=47,209,371,\ldots\\\ \end{matrix}$

####  A.3.4 The $2210$ prefix

$p=\frac{27}{32}m+\frac{12}{32}.$ (29)

This generates the following progression

$\begin{matrix}k=0,1,2,3,4,\ldots&m=32k+28=28,60,92,\ldots&6m+3=171,363,555,\ldots\\\
&p=27k+24=24,51,78,\ldots&6p+1=145,307,469,\ldots\\\ \end{matrix}$

####  A.3.5 The $1120$ prefix

$p=\frac{27}{16}m+\frac{5}{16}.$ (30)

This generates the following progression

$\begin{matrix}k=0,1,2,3,4,\ldots&m=16k+1=1,17,33,\ldots&6m+3=9,105,291,\ldots\\\
&p=27k+2=2,29,56,\ldots&6p+5=17,179,341,\ldots\\\ \end{matrix}$

####  A.3.6 The $2120$ prefix

$p=\frac{27}{32}m+\frac{13}{32}.$ (31)

This generates the following progression

$\begin{matrix}k=0,1,2,3,4,\ldots&m=32k+9=9,41,73,\ldots&6m+3=57,249,441,\ldots\\\
&p=27k+8=8,35,62,\ldots&6p+1=49,211,\underline{373},\ldots\\\ \end{matrix}$

####  A.3.7 The $1220$ prefix

$p=\frac{27}{32}m-\frac{7}{32}.$ (32)

This generates the following progression

$\begin{matrix}k=0,1,2,3,4,\ldots&m=32k+5=5,37,69,\ldots&6m+3=33,225,417,\ldots\\\
&p=27k+4=4,31,58,\ldots&6p+5=\underline{29},191,353,\ldots\\\ \end{matrix}$

####  A.3.8 The $2220$ prefix

$p=\frac{27}{64}m+\frac{9}{64}.$ (33)

This generates the following progression

$\begin{matrix}k=0,1,2,3,4,\ldots&m=64k+21=21,85,149,\ldots&6m+3=129,513,897,\ldots\\\
&p=27k+9=9,36,63,\ldots&6p+1=55,217,379,\ldots\\\ \end{matrix}$

##  Appendix B Prefix Equations

Once a sequence begins from a 0-node, it can thereafter be viewed as a binary
sequence consisting of the symbols$\\{1,2\\}$. The results of Appendix A can
therefore be phrased in terms of binary tree. The head node on the binary tree
is a 0-node which can transition to one of two possibilities

$\begin{matrix}&&01:p=\frac{3}{2}m\\\ &\nearrow_{1}&\\\ 0-node&&\\\
&\searrow_{2}&\\\ &&02:p=\frac{3}{4}m+\frac{1}{4}\\\ \end{matrix}$ (34)

which is the initial condition for the tree.

Transitioning from a given node to a 1-node necessarily introduces a scale
factor of $3/2$ which must be applied to the $p$ equation. Transitioning from
a given node to a 2-node necessarily introduces a scale factor of $3/4$ which
must be applied to the $p$ equation. After applying the appropriate scale
factor, for any node, a shift value must also be added in order to obtain the
$p$ equation for the next node. After the initial condition, there are four
possible shift values from one tree node to the next.

$\begin{matrix}1\rightarrow 1&+\frac{1}{2}\\\ 1\rightarrow 2&+\frac{1}{2}\\\
2\rightarrow 1&-\frac{1}{2}\\\ 2\rightarrow 2&0\\\ \end{matrix}$ (35)

For example, to build the next level in the tree representing prefixes of
length $3$, apply these scaling and shift rules to obtain

$\begin{matrix}&&&&011:p=\frac{9}{4}m+\frac{2}{4}\\\
&&&\nearrow_{1}(+\frac{2}{4})&\\\ &&01:p=\frac{3}{2}m&&\\\
&&&\searrow_{2}(+\frac{4}{8})&\\\ &&&&012:p=\frac{9}{8}m+\frac{4}{8}\\\
&\nearrow_{1}&&&\\\ &&&&\\\ 0-node&&&&\\\ &&&&\\\ &\searrow_{2}&&&\\\
&&&&021:p=\frac{9}{8}m-\frac{1}{8}\\\ &&&\nearrow_{1}(-\frac{2}{4})&\\\
&&02:p=\frac{3}{4}m+\frac{1}{4}&&\\\ &&&\searrow_{2}(+0)&\\\
&&&&022:p=\frac{9}{16}m+\frac{3}{16}\\\ \end{matrix}$ (36)

where the length $3$ prefix equations are recovered from the tree. This
process can be iterated to recover any $p$ equation for any primitive sequence
emanating from a $0-node$.

As pointed out in the main text of Section 4.3, there are important questions
regarding divergent sequences that can be answered without the need for the
complete set of $p$ equations for all possible binary sequences. One important
point to consider for the $p$ values would be to ensure that the complete set
would recover all possible odd integers $\bigcup_{p}\\{6p+1,6p+5\\}$. If it
can be shown that every nonnegative integer can be generated by the complete
set of $p$ values, then it can be concluded that every 1-node and 2-node
exists in the path of some primitive itinerary.


# Erfonium: A Hooke Atom with Soft Interaction Potential

Jacek Karwowski  Institute of Physics, Faculty of Physics, Astronomy and
Informatics,
Nicolaus Copernicus University, Grudzia̧dzka 5, 87-100 Toruń, Poland
email<EMAIL_ADDRESS>Andreas Savin  Laboratoire de Chimie Théorique, CNRS and
Sorbonne University, 4 place Jussieu, 75252 Paris cedex 05, France
email<EMAIL_ADDRESS>
###### Abstract

Properties of erfonium, a Hooke atom with the Coulomb interaction potential
$1/\/r$ replaced by a non-singular $\mathrm{erf}(\mu\,r)/\/r$ potential are
investigated. The structure of the Hooke atom potential and properties of its
energy spectrum, relative to the ones of the spherical harmonic oscillator and
of harmonium, are analyzed. It is shown, that at a certain value of $\mu$ the
system changes its behavior from a harmonium-like regime to a harmonic-
oscillator-like regime.

Schrödinger equation, electron interaction, Hooke atoms, erf potential, range
separation.

††thanks: This is a preprint of the following chapter: Jacek Karwowski and
Andreas Savin, A Hooke Atom with Soft Interaction Potential, published in
Recent Progress in Methods and Applications of Quantum Systems, edited by
Ireneusz Grabowski, Karolina Słowik, Erkki J. Brändas, and Jean Maruani, 2023,
Springer Nature, reproduced with permission of Springer Nature Switzerland AG
Gewerbestrasse 11, 6330 Cham, Switzerland. The final authenticated version is
available online at: https://doi.org/10.1007/978-3-031-52078-5_5.

##  I Motivation

The interaction between electrons can be split to two parts:

$\frac{1}{r}=\frac{\mathrm{erf}(\mu\,r)}{r}+\frac{\mathrm{erfc}(\mu\,r)}{r}.$
(1)

The first, long-range, term is smooth at $r\,\sim\,0$ and behaves as the
Coulomb potential at $r\rightarrow\infty$. The second, short-range one,
correctly represents the Coulomb potential at small $r$ and decays
exponentially as $r\,\rightarrow\,\infty$. From this observation stems the
concept of range separation Gori-Giorgi and Savin (2006); Savin (2011, 2020),
where the long-range section and the short-range section are treated in
different ways.

In theories of many-electron systems, the replacement of the singular Coulomb
potential by a smooth long-range potential, signifficantly reduces the
computational effort. On the other hand, the short-range behavior of the wave
function is mostly defined by the universal properties of the Coulomb
singularity. Therefore the idea of improving models which utilize the long-
range interactions only, by some universal corrections describing the short-
range properties is very tempting. Indeed, correcting solutions of the
Schrödinger equation with the long-range part of the interaction potential, by
using the generalized cusp conditions to represent the wave functions in the
small $r$ area, was recently shown not only feasible, but also very accurate
Savin and Karwowski (2023).

The long-range Coulomb interaction potentials,

$\mathsf{w}(r;\mu)=\frac{\mathrm{erf}(\mu\,r)}{r},$ (2)

have been introduced in several areas of chemistry and physics. For example, a
model with the Coulomb interaction replaced by $\mathsf{w}(r;\mu)$ correctly
reproduces some selected properties of the hydrogen atom, harmonium, and
electron gas González-Espinoza  _et al._ (2016). In quantum chemistry, the
replacement of the singular Coulomb potential by a smooth one represented by
the error function is particularly convenient with the Gaussian basis sets:
the integrals with this potential are easy to calculate. Therefore, the
applications of the range separation concept, though most common in the
density functional theory Savin (2011); Pernal and Hapka (2022), extend beyond
this field Savin (2020).

Also in theory of complex systems, as crystals, liquids, or plasmas, using
$\mathrm{erf}(\mu\,r)/r$ potential is motivated by its mathematical
properties. Probably for the first time it was used by Ewald in 1921, well
before the formulation of quantum mechanics, to replace the Coulomb
interaction in order to secure the convergence of some series describing
infinite systems of particles Ewald (1921). A reader interested in these areas
of applications is referred to a recent paper Demyanov and Levashov (2022),
where also references to the earlier works are supplied. Due to its convenient
mathematical form and simple relation to the Coulomb potential,
$\mathrm{erf}(\mu\,r)/r$ potential is also used in many other sections of
physics, just to mention so distant fields as interactions between quarks
Lucha  _et al._ (1992), or general theory of relativity Plamondon (2018).

Two electrons with interaction represented by $\mathsf{v}(r)$, in an external
potential
$\mathsf{W}(\mbox{\boldsymbol{$r$}}_{1})+\mathsf{W}(\mbox{\boldsymbol{$r$}}_{2})$,
are described by the Hamiltonian

$\mathsf{H}(\mbox{\boldsymbol{$r$}}_{1},\mbox{\boldsymbol{$r$}}_{2})=-{\textstyle\frac{1}{2}}\bigtriangleup_{\,\mbox{\boldsymbol{$\textstyle{r}$}}_{1}}-{\textstyle\frac{1}{2}}\bigtriangleup_{\,\mbox{\boldsymbol{$\textstyle{r}$}}_{2}}+\mathsf{W}(\mbox{\boldsymbol{$r$}}_{1})+\mathsf{W}(\mbox{\boldsymbol{$r$}}_{2})+\mathsf{v}(r),$
(3)

where
$r=|\mbox{\boldsymbol{$r$}}|\equiv\left|\mbox{\boldsymbol{$r$}}_{1}-\mbox{\boldsymbol{$r$}}_{2}\right|$.
Introducing
$\mbox{\boldsymbol{$R$}}=\left(\mbox{\boldsymbol{$r$}}_{1}+\mbox{\boldsymbol{$r$}}_{2}\right)/2$
we get

$\mbox{\boldsymbol{$r$}}_{1}=\mbox{\boldsymbol{$R$}}+{\textstyle\frac{1}{2}}\,\mbox{\boldsymbol{$r$}},\;\;\;\
\mbox{\boldsymbol{$r$}}_{2}=\mbox{\boldsymbol{$R$}}-{\textstyle\frac{1}{2}}\,\mbox{\boldsymbol{$r$}}.$
(4)

If $\mathsf{W}(\mbox{\boldsymbol{$r$}}_{i})\,\propto\,r_{i}^{2}$, $i=1,2$,
then the Hamiltonian is separable and the two-electron eigenvalue equation
splits to two spherically-symmetric equations Kestner and Sinanoglu (1962).
The first one, in $R$, is independent of the interaction potential, and
describes the motion of the center of mass of two electrons in the parabolic
confining potential. The second equation, in $r$, describes the relative
motion of two interacting electrons in the confining potential
$(\omega\,r/2)^{2}$, where parameter $\omega$ defines the strength of the
confinement. We assume that the interaction potential is equal to
$\mathsf{w}(r;\mu)$, defined in (2). The resulting model potential reads

$\mathsf{V}(r)\,\equiv\,\mathsf{V}(r;\omega,\mu)\,=\,\frac{\mathrm{erf}(\mu\,r)}{r}+\left(\frac{\omega\,r}{2}\right)^{2}.$
(5)

The elimination of the angular dependence from the second equation gives an
infinite set of radial equations labelled by the angular momentum quantum
number $\ell$

$\mathsf{H}(r;\ell,\omega,\mu)\,\phi_{n,\ell}(r)=E_{n,\ell}(\omega,\mu)\,\phi_{n,\ell}(r),\;\;\;n,\ell=0,1,2,\ldots,$
(6)

where $\phi_{n,\ell}(r)=r\,\psi_{n,\ell}(r)$ is the reduced radial function,
$\psi_{n,\ell}(r)$ is the radial part of the one-particle wave function, and

$\mathsf{H}(r;\ell,\omega,\mu)=\left(-\frac{d^{2}}{dr^{2}}+\frac{\ell(\ell+1)}{r^{2}}+\mathsf{V}(r;\omega,\mu)\right)$
(7)

is the radial Hamiltonian.

If the interaction potential $\mathsf{v}(r)$ is repulsive then its energy
spectrum is continuous. The introduction of a parabolic confinement leads to
the discretization of the spectrum. Two confined electrons form a bound system
which resembles a two-electron atom. It is referred to as a Hooke atom (due to
the confining Hooke force). If the two particles interact by the Coulomb
potential, the system is called harmonium. We propose here the name erfonium
for a Hooke atom with the Coulomb interatcion between electrons replaced by
$\mathsf{w}(r;\mu)$. Erfonium is a generalization and unification of two well-
studied systems: the spherical harmonic oscillator and harmonium. Its radial
Hamiltonian (7) transforms to the Hamiltonian of the spherical harmonic
oscillator if $\mu=0$, and to the Hamiltonian of harmonium, if
$\mu\rightarrow\infty$ (10). Accordingly, at these two limits the eigenvalues
$E_{n,\ell}(\omega,\mu)$, and other quantities characterizing the system,
approach the corresponding quantities of the harmonic oscillator and of
harmonium. They are marked hereafter by superscripts o and h, respectively.

The close relation between the Coulomb and $\mathsf{w}(r;\mu)$ potentials can
be seen by comparig their integral representations. We have

$\frac{1}{r}=\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}\,\mathrm{e}^{-(x\,r)^{2}}\,dx\;\;\;\;\;\;\mbox{and}\;\;\;\;\;\;\frac{\mathrm{erf}(\mu\,r)}{r}=\frac{2}{\sqrt{\pi}}\int_{0}^{\mu}\,\mathrm{e}^{-(x\,r)^{2}}\,dx.$
(8)

Then,

$\frac{1}{r}\,=\,\lim_{\mu\rightarrow\infty}\,\frac{\mathrm{erf}(\mu\,r)}{r}$
(9)

and at the limit of large $\mu$ the model potential (5) transforms to the
potential of harmonium:

$\mathsf{V}(r;\omega,\mu)\,\mathrel{\raisebox{-6.0pt}{$\stackrel{{\scriptstyle\displaystyle{\rightarrow}}}{{\scriptstyle{\mu\to\infty}}}$}}\,\mathsf{V}^{\mathrm{h}}(r;\omega)=\frac{1}{r}+\left(\frac{\omega\,r}{2}\right)^{2}.$
(10)

The $\ell$-dependent term in equation (7) describes the centrifugal force, and
together with the model potential is called the effective radial potential:

$\mathcal{V}(r;\ell,\omega,\mu)\,=\,\frac{\ell(\ell+1)}{r^{2}}+\mathsf{V}(r;\omega,\mu).$
(11)

Notice, that the parameter $\mu$ in $\mathcal{V}(r;\ell,\omega,\mu)$ describes
the adiabatic connection between spherical harmonic oscillator
$\left[\mathcal{V}(r;\ell,\omega,0)\right]$, and harmonium
$\left[\mathcal{V}(r;\ell,\omega,\infty)\right]$.

For $\ell=0$, $\mathcal{V}(r;\ell,\omega,\mu)$ is equal to the model potential
(5):

$\mathcal{V}(r;0,\omega,\mu)\,=\,\mathsf{V}(r;\omega,\mu).$ (12)

But, for $l>0$ these two potentials are notably different. The most
significant difference is in the area of small $r$. At $r=0$,

$\lim_{r\rightarrow\,0}\mathcal{V}(r;0,\omega,\mu)\,=\,\lim_{r\rightarrow\,0}\mathsf{V}(r;\omega,\mu)=\frac{2\,\mu}{\sqrt{\pi}},$
(13)

while

$\mathcal{V}(r;\ell,\omega,\mu)\mathrel{\raisebox{-6.0pt}{$\stackrel{{\scriptstyle\displaystyle{\sim}}}{{\scriptstyle{r\to
0}}}$}}\frac{\ell\,(\ell+1)}{r^{2}}$ (14)

i.e. it is singular if $\ell\,\neq\,0$.

This paper is aimed at the exploration of some general properties of erfonium.
General characteristics of the model potential (5) are presented in the next
section. The effective radial potential (11) is discussed in section 3. The
energy spectrum of erfomium is analyzed in section 4. Final remarks complete
the work. All equations and numerical values are here expressed in the Hartree
atomic units.

##  II The model potential

The radial potential of erfonium (5) can be expressed as

$\displaystyle\mathsf{V}(r;\omega,\mu)$ $\displaystyle=$
$\displaystyle\mu\,\left[\frac{\mathrm{erf}(\mu\,r)}{(\mu\,r)}+\frac{2\,q}{3\,\sqrt{\pi}}(\mu\,r)^{2}\right]$
(15) $\displaystyle=$
$\displaystyle\frac{2\mu}{\sqrt{\pi}}\left[1-\frac{(1-q)}{3\cdot
1!}(\mu\,r)^{2}+\frac{(\mu\,r)^{4}}{5\cdot 2!}-\frac{(\mu\,r)^{6}}{7\cdot
3!}+\cdots\right]$ (16)

where

$q=\frac{3\omega^{2}\sqrt{\pi}}{8\,\mu^{3}}.$ (17)

Potential $\mathsf{V}(r;\omega,\mu)$ has interesting scaling properties.
According to equation (15)

$\mathsf{V}(r/\mu;\omega,\mu)=\mu\,\left(\frac{\mathrm{erf}(r)}{r}+\frac{2\,q}{3\,\sqrt{\pi}}\,r^{2}\right).$
(18)

As we can see, all potentials with the same value of $q$ have exactly the same
shape - they only differ by units in which $\mathsf{V}$ and $r$ are expressed.
Therefore, for an arbitrary pair $\left(\mu_{1},\mu_{2}\right)$

$\frac{\mathsf{V}(r/\mu_{1};\omega_{\,1},\mu_{1})}{\mathsf{V}(r/\mu_{2};\omega_{\,2},\mu_{2})}=\frac{\mu_{1}}{\mu_{2}}$
(19)

if

$\omega_{\,2}=\omega_{\,1}\,\left(\frac{\mu_{2}}{\mu_{1}}\right)^{3/2},$ (20)

i.e. if the cofinement parameters $\omega_{\,1}$ and $\omega_{\,2}$ are
related to $\mu_{1}$ and $\mu_{2}$ as

$\left(\frac{\omega_{\,1}}{\omega_{\,2}}\right)^{2}=\left(\frac{\mu_{1}}{\mu_{2}}\right)^{3}.$
(21)

For example, the potential energy curve corresponding to ($\mu=1$, $\omega=1$)
is, except for scaling, the same as the curve for
($\mu=10^{-2/3}\,\approx\,0.2154$, $\omega\,=\,0.1$) or for
($\mu=100^{-2/3}\,\approx\,0.0464$, $\omega\,=\,0.01$). From equation (18) we
can get similar scaling relations for the derivatives of $\mathsf{V}$ with
respect to $r$:

$\frac{\mathsf{V}^{(n)}(r/\mu_{1};\omega_{\,1},\mu_{1})}{\mathsf{V}^{(n)}(r/\mu_{2};\omega_{\,2},\mu_{2})}=\left(\frac{\mu_{1}}{\mu_{2}}\right)^{n+1},$
(22)

where $\mathsf{V}^{(n)}$ is the $n$-th derivative of $\mathsf{V}$ with respect
to $r$.

The first and the second derivatives of $\mathsf{V}(r;\omega,\mu)$ are
expressed as

$\displaystyle\mathsf{V}^{\prime}(r;\omega,\mu)$ $\displaystyle=$
$\displaystyle\mu^{2}\,\left[-\frac{\mathrm{erf}(\mu\,r)}{(\mu\,r)^{2}}+\frac{2}{\sqrt{\pi}}\frac{\mathrm{e}^{-(\mu\,r)^{2}}}{(\mu\,r)}+\frac{4\,q}{3\,\sqrt{\pi}}(\mu\,r)\right]$
(23) $\displaystyle=$
$\displaystyle\frac{2\,\mu^{2}}{\sqrt{\pi}}\left[-\frac{2\,(1-q)}{3\cdot
1!}(\mu\,r)+\frac{4}{5\cdot 2!}(\mu\,r)^{3}-\frac{6}{7\cdot
3!}(\mu\,r)^{5}+\cdots\right];$ (24)
$\displaystyle\mathsf{V}^{\prime\prime}(r;\omega,\mu)$ $\displaystyle=$
$\displaystyle-\frac{2}{r}\,\frac{d\mathsf{V}(r;\omega,\mu)}{dr}-\mu^{3}\,\frac{4}{\sqrt{\pi}}\left(\mathrm{e}^{-(\mu\,r)^{2}}-q\right)$
(25) $\displaystyle=$
$\displaystyle\frac{2\,\mu^{3}}{\sqrt{\pi}}\left[-\frac{2\,(1-q)}{3\cdot
1!}+\frac{3\cdot 4}{5\cdot 2!}(\mu\,r)^{2}-\frac{5\cdot 6}{7\cdot
3!}(\mu\,r)^{4}+\cdots\right].$ (26)

For finite $\mu$, $\mathsf{V}^{\prime}(0;\omega,\mu)=0$, and the potential has
an extremum at $r=0$ – a minimum if $q>1$ and a maximum if $q<1$ [cf.
equations (24) and (26)]. If $q\neq{1}$ then for small $r$, the potential is a
quadratic function of $r$. If $q=1$ then, at $r\,\sim\,0$, the dependence on
$r$ is quartic. The threshold value of $\mu$ corresponding to $q=1$ is equal
to

$\mu_{0}(\omega)=\frac{1}{2}\left(3\sqrt{\pi}\omega^{2}\right)^{1/3}.$ (27)

Then, $\mathsf{V}(r;\omega,\mu)$ at $r=0$ has a minimum if
$\mu<\mu_{0}(\omega)$ and a maximum if $\mu>\mu_{0}(\omega)$. If
$\mu\rightarrow\infty$ then, for $r\,\rightarrow\,0$, $\mathsf{V}\,\sim\,1/r$.
Potential $\mathsf{V}(r;\omega,\mu)$ grows up to infinity with increasing $r$.
Therefore, a maximum at $r=0$ implies the existence of a minimum at
$r=r_{\mathrm{e}}(\omega,\mu)\equiv r_{\mathrm{e}}>0$. In the case of
harmonium,

$\displaystyle
r_{\mathrm{e}}^{\mathrm{h}}\,\equiv\,r_{\mathrm{e}}^{\mathrm{h}}(\omega)$
$\displaystyle=$
$\displaystyle\lim_{\mu\rightarrow\infty}r_{\mathrm{e}}(\omega,\mu)=\left(\frac{2}{\omega^{2}}\right)^{1/3},$
(28)
$\displaystyle\mathsf{V}_{\mathrm{e}}^{\mathrm{h}}\,\equiv\,\mathsf{V}_{\mathrm{e}}^{\mathrm{h}}(\omega)$
$\displaystyle=$
$\displaystyle\lim_{\mu\rightarrow\infty}\mathsf{V}(r_{\mathrm{e}}^{\mathrm{h}};\omega,\mu)\,=\,\frac{3}{4}\,\left[\omega\,r_{\mathrm{e}}^{\mathrm{h}}(\omega)\right]^{2}\,=\,\frac{3}{4}\,\left(2\,\omega\right)^{2/3},$
(29)

Taut (1993); Mandal  _et al._ (2003). Notice that

$\mu_{0}(\omega)\,r_{\mathrm{e}}^{\mathrm{h}}(\omega)=\left(\frac{3\sqrt{\pi}}{4}\right)^{1/3}\equiv\alpha$
(30)

is independent of $\omega$.

For a finite $\mu$, the requirement that
$\mathsf{V}^{\prime}(r_{\mathrm{e}};\omega,\mu)=0$ yields:

${\widetilde{r}_{\mathrm{e}}\,}^{3}=\mathrm{erf}(\mu\,r_{\mathrm{e}})-\frac{2\,\mu\,r_{\mathrm{e}}}{\sqrt{\pi}}\,\mathrm{e}^{-(\mu\,r_{\mathrm{e}})^{2}},$
(31)

where

$\widetilde{r}_{\mathrm{e}}\,\equiv\,\widetilde{r}_{\mathrm{e}}(\omega,\mu)=\frac{r_{\mathrm{e}}(\omega,\mu)}{r_{\mathrm{e}}^{\mathrm{h}}(\omega)}=r_{\mathrm{e}}(\omega,\mu)\,\left(\frac{\omega^{2}}{2}\right)^{1/3}$

is the normalized coordinate of the minimum. Notice, that rhs of equation (31)
is equal to $0$ if $\mu=0$ and approaches $1$ if $\mu\rightarrow\infty$. Its
derivative with respect to $x=\mu\,r_{\mathrm{e}}$ is
$(4\,x^{2}/\sqrt{\pi})\,\mathrm{e}^{-x^{2}}\,>\,0$. Therefore, with increasing
$\mu$, $r_{\mathrm{e}}$ monotonically increases from $0$ to
$r_{\mathrm{e}}^{\mathrm{h}}$. The dependence of $\widetilde{r}_{\mathrm{e}}$
on $\mu$, obtained by numerical solving equation (31), is presented, for
several values of omega, in the upper left panel of Fig. 1.

Using equation (30) we can rewrite equation (31) as

${\widetilde{r}_{\mathrm{e}}\,}^{3}\,=\,\mathrm{erf}(\widetilde{\mu}\,\widetilde{r_{\mathrm{e}}}\,\alpha)-\frac{2\,\widetilde{\mu}\,\widetilde{r_{\mathrm{e}}}\,\alpha}{\sqrt{\pi}}\,\mathrm{e}^{-(\widetilde{\mu}\,\widetilde{r_{\mathrm{e}}}\,\alpha)^{2}},$
(32)

where

$\widetilde{\mu}\equiv\widetilde{\mu}(\omega)=\frac{\mu}{\mu_{0}(\omega)}.$

Equation (32) shows, that the relation between $\widetilde{r}_{\mathrm{e}}$
and $\widetilde{\mu}$ is independent of $\omega$. It is plotted in the upper
right panel of Fig. 1.

Figure 1: Plots of $\widetilde{r}_{\mathrm{e}}(\omega,\mu)$ (top row) and
$\widetilde{\mathsf{V}}_{\mathrm{e}}(\omega,\mu)$ (bottom row) versus $\mu$
(left column) and versus $\widetilde{\mu}$ (right column). Curves $1$, $2$,
$3$, correspond, respectively, to $\omega=1.0,0.5,0.1$. The vertical lines
mark $\mu_{0}(1.0)\approx 0.8727$, $\mu_{0}(0.5)\approx 0.5498$, and
$\mu_{0}(0.1)\approx 0.1880$. In the right panels the curves overlap:
$\widetilde{r}_{\mathrm{e}}(\omega,\widetilde{\mu})$ and
$\widetilde{\mathsf{V}}_{\mathrm{e}}(\omega;\widetilde{\mu})$ are the same for
all values of $\omega$. The parameter $\mu$ is expressed in
$\mathrm{bohrs}^{-1}$.

As one can see consulting equations (15) and (23),

$\mathsf{V}_{\mathrm{e}}\equiv\mathsf{V}_{\mathrm{e}}(\omega,\mu)\equiv\mathsf{V}(r_{\mathrm{e}};\omega,\mu)=\frac{3}{4}\,(\omega\,r_{\mathrm{e}})^{2}+\frac{2\mu}{\sqrt{\pi}}\,\mathrm{e}^{-(\mu\,r_{\mathrm{e}})^{2}}.$
(33)

The normalized potential at its minimum is defined as

$\widetilde{\mathsf{V}}_{\mathrm{e}}\equiv\widetilde{\mathsf{V}}_{\mathrm{e}}(\omega,\mu)=\frac{\mathsf{V}_{\mathrm{e}}(\omega,\mu)}{\mathsf{V}^{\mathrm{h}}(\omega)}.$
(34)

Using equations (29), (30), and (33), we get

$\widetilde{\mathsf{V}}_{\mathrm{e}}={\widetilde{r}_{\mathrm{e}}\,}^{2}+\frac{\widetilde{\mu}}{\alpha^{2}}\mathrm{e}^{-(\widetilde{\mu}\,\widetilde{r}_{\mathrm{e}})^{2}}.$
(35)

Since $\widetilde{r}_{\mathrm{e}}$, as a function of $\widetilde{\mu}$, is
$\omega$-independent, equation (35) implies that also the relation between
$\widetilde{\mathsf{V}}_{\mathrm{e}}$ and $\widetilde{\mu}$ does not depend on
$\omega$. Plots of $\widetilde{\mathsf{V}}_{\mathrm{e}}$ versus $\mu$, for
three values of $\omega$ are presented in the lower left panel of Fig. 1. The
three curves collapse into one in the lower right panel where they are plotted
against $\widetilde{\mu}(\omega)$.

Figure 2: Potentials $\mathsf{V}(r)\equiv\mathsf{V}(r;\omega,\mu)$, in
$\mathrm{hartrees}$, versus $r$, in $\mathrm{bohrs}$. Left panel: $\omega=1.0$
[$\mu_{0}(1.0)=0.8727$]; right panel $\omega=0.1$ [$\mu_{0}(0.1)=0.1880$].
Thick lines: spherical harmonic oscillator ($\mu=0$), and harmonium
($\mu=\infty$). Thin lines: $\mathsf{V}(r;\omega,\mu_{n})$,
$\mu_{n}=\mu_{0}(\omega)\cdot(n/4)$, $n=1,\ldots,12$. The horizontal lines
correspond to $q=1$, i.e. to $\mathsf{V}(0)=2\,\mu_{0}(\omega)/\sqrt{\pi}$.
Figure 3: Ratio of potentials of erfonium and harmonium versus $r$ (in
$\mathrm{bohrs}$) for $\ell=0$ (left panel) and for $\ell=1$ (right panel).
Thick lines - $\omega=1.0$; thin lines - $\omega=0.5$. The lines are marked by
the values of $\mu=\,0.1,\,0.5,\,1.5,\,5.0$. If $\mu>1$ then the lines
corresponding to different $\omega$ overlap.

Plots of $\mathsf{V}(r;\omega,\mu)$ for $\omega=1.0$ and for $\omega=0.1$ are
shown in Fig. 2. As one can see, the curves correspnding to large $\mu$ are
close to the potential of harmonium and the ones corresponding to small $\mu$
to the potential of the spherical harmonic oscillator. The shapes of the left-
and of the right-panel potentials ($\omega=1$ and $\omega=0.1$, respectively),
are the same, but the ranges of units differ by the scaling factor
$100^{1/3}=4.6416$.

Similarities and differences between erfonium and harmonium potentials are
well reflected by their ratio:

$R_{0}(r;\omega,\mu)=\frac{\mathsf{V}(r;\omega,\mu)}{\mathsf{V}^{\mathrm{h}}(r;\omega)}=\frac{\mathrm{erf}(\mu\,r)+\omega^{2}\,r^{3}/4}{1+\omega^{2}\,r^{3}/4}.$
(36)

If $\ell=0$ then, for given $\omega$ and $\mu$, the ratio is an increasing
function of $r$. At the limit of $r\rightarrow\infty$, or
$\mu\rightarrow\infty$, or $\omega\rightarrow\infty$, the ratio is equal to
$1$. If $r\,=\,0$, or $\mu=\omega=0$, then $R_{0}$ = 0. Plots of
$R_{0}(r;\omega,\mu)$ versus $r$, for several values of $\omega$ and $\mu$ are
shown in the left panel of Fig. 3. As one can see, the $\omega$-dependence is
essential for small values of $\mu$ only. One can also notice a fast
convergence of $\;\mathsf{V}(r;\omega,\mu)$ to
$\mathsf{V}^{\mathrm{h}}(r;\omega)$ with increasing $\mu$.

##  III The effective potential

The shape of $\mathcal{V}(r;\ell,\omega,\mu)$, similarly as the shape of
$\mathsf{V}(r;\omega,\mu)$, is conserved under some specific transformations.
Equation (11) can be rewritten as

$\mathcal{V}(r/\mu;\ell,\omega,\mu)=\mu\left[\frac{\Lambda}{r^{2}}+\frac{\mathrm{erf}(r)}{r}+\frac{\omega^{2}\,r^{2}}{4\,\mu^{3}}\right],$
(37)

where $\Lambda=\ell\,(\ell+1)\,\mu$. Then,

$\frac{\mathcal{V}(r/\mu_{1};\ell_{1},\omega_{\,1},\mu_{1})}{\mathcal{V}(r/\mu_{2};\ell_{2},\omega_{\,2},\mu_{2})}=\frac{\mu_{1}}{\mu_{2}}$
(38)

if

$\left(\frac{\omega_{\,1}}{\omega_{\,2}}\right)^{2}=\left(\frac{\mu_{1}}{\mu_{2}}\right)^{3},\;\;\;\;\mbox{and}\;\;\;\;\Lambda_{1}=\Lambda_{2},$
(39)

[compare eq. (21)]. This means, that two effective potentials have the same
shape if,

$\mu_{2}=\mu_{1}\,\frac{\ell_{1}(\ell_{1}+1)}{\ell_{2}(\ell_{2}+1)},\;\;\;\;\mbox{and}\;\;\;\;\omega_{2}=\omega_{1}\,\left[\frac{\ell_{1}(\ell_{1}+1)}{\ell_{2}(\ell_{2}+1)}\right]^{3/2}$
(40)

Figure 4: The effective potential
$\mathcal{V}(r)\equiv\mathcal{V}(r;\ell,\omega,\mu)$ (in $\mathrm{hartrees}$)
for $\ell=1$, as functions of the inter-electronic distance $r$ (in
$\mathrm{bohrs}$). Left panel - $\omega=1.0$; right panel - $\omega=0.1$. The
lowest curves represent the effective potentials of the spherical harmonic
oscillator ($\mu=0$), and the upper ones of harmonium ($\mu=\infty$). The
curves between these two limits correspond to $\mu=n/10$ (left), and
$\mu=n/35$ (right), for $n=1,\ldots,8$.

As it follows from equations (10) and (11)

$\mathcal{V}(r;\ell,\omega,\mu)\,\mathrel{\raisebox{-6.0pt}{$\stackrel{{\scriptstyle\displaystyle{\rightarrow}}}{{\scriptstyle{\mu\to\infty}}}$}}\mathcal{V}^{\mathrm{h}}(r;\ell,\omega)\;\;\;\;\;\mbox{and}\;\;\;\;\;\mathcal{V}(r;\ell,\omega,\mu)\,\mathrel{\raisebox{-6.0pt}{$\stackrel{{\scriptstyle\displaystyle{\rightarrow}}}{{\scriptstyle{\mu\to
0}}}$}}\mathcal{V}^{\mathrm{o}}(r;\ell,\omega),$ (41)

where

$\displaystyle\mathcal{V}^{\mathrm{h}}(r;\ell,\omega)$ $\displaystyle=$
$\displaystyle\frac{\ell(\ell+1)}{r^{2}}+\frac{1}{r}+\left(\frac{\omega\,r}{2}\right)^{2},$
(42) $\displaystyle\mathcal{V}^{\mathrm{o}}(r;\ell,\omega)$ $\displaystyle=$
$\displaystyle\frac{\ell(\ell+1)}{r^{2}}+\left(\frac{\omega\,r}{2}\right)^{2},$
(43)

are, respectively, the effective potentials of harmonium and of the spherical
harmonic oscillator. Since $0\,\leq\,\mathsf{w}(r,\mu)\,\leq\,1/r$, we have

$\mathcal{V}^{\mathrm{o}}(r;\ell,\omega)\,\leq\,\mathcal{V}(r;\ell,\omega,\mu)\,\leq\,\mathcal{V}^{\mathrm{h}}(r;\ell,\omega).$
(44)

The general shapes of $\mathcal{V}^{\mathrm{o}}(r;\ell,\omega)$ and
$\mathcal{V}^{\mathrm{h}}(r;\ell,\omega)$ are similar: regular potential wells
approaching infinity in the same way if $r\rightarrow{0}$ and
$r\rightarrow\infty$. Because $\mathsf{w}(r,\mu)$ is regular and monotonic,
the shapes of $\mathcal{V}(r;\ell,\omega,\mu)$ are similar. In Fig. 4 plots of
$\mathcal{V}(r;\ell,\omega,\mu)$ for $\ell=1$, $\omega=1$ (left panel),
$\ell=1$, $\omega=0.1$ (right panel), and for several values of $\mu$ are
presented. The ranges of $r$ and $\mathcal{V}$ in the two panels are selected
so that the shapes of the curves are similar [the shapes cannot be the same,
because $\ell_{1}=\ell_{2}=1$, cf. eq. (39)].

Figure 5: Plots of the normalized coordinates of the minimum of the effective
potential for $\ell=1$ and $\omega=1.0,0.5,0.1$ – curves $1$, $2$, $3$,
respectively, as functions of $\widetilde{\mu}$. In the left panel
$\widetilde{r}_{\mathrm{e}}\,\equiv\,\widetilde{r}_{\mathrm{e}}(\ell,\omega,\mu)$;
in the right one
$\widetilde{\mathcal{V}}_{\mathrm{e}}\,\equiv\,\widetilde{\mathcal{V}}_{\mathrm{e}}(\ell,\omega,\mu)$.

If $\ell>0$ then the effective potential $\mathcal{V}(r;\ell,\omega,\mu)$ has
only one extremum: a minimum, at $r=r_{\mathrm{e}}(\ell,\omega,\mu)$. The
values of $r_{\mathrm{e}}$ can be obtained from the condition

$\left.\frac{\mathcal{V}(r;\ell,\omega,\mu)}{dr}\right|_{r=r_{\mathrm{e}}}\,=\,0,$

transformed to

$(r_{\mathrm{e}})^{4}=\frac{2}{\omega^{2}}\left[2\ell(\ell+1)+\mu\,r_{\mathrm{e}}^{2}\,\left(\frac{\mathrm{erf}(\mu\,r_{\mathrm{e}})}{\mu\,r_{\mathrm{e}}}-\frac{2}{\sqrt{\pi}}\,\mathrm{e}^{-(\mu\,r_{\mathrm{e}})^{2}}\right)\right],$
(45)

where $r_{\mathrm{e}}$ stands here for $r_{\mathrm{e}}(\ell,\omega,\mu$).
Equation (45) has to be solved numerically, but one can see that
$r_{\mathrm{e}}(\ell,\omega,\mu_{1})\,\geq\,r_{\mathrm{e}}(\ell,\omega,\mu_{2}$),
if $\mu_{1}\,>\,\mu_{2}$. At the limit of $\mu=0$ (harmonic oscillator),

$r^{\mathrm{o}}_{\mathrm{e}}(\ell,\omega)^{4}=\frac{4\ell(\ell+1)}{\omega^{2}}.$
(46)

If $\mu\rightarrow\infty$ (harmonium), then $r_{\mathrm{e}}^{\mathrm{h}}$ is a
root of a fourth-order polynomial:

$r^{\mathrm{h}}_{\mathrm{e}}(\ell,\omega)^{4}=\frac{2}{\omega^{2}}\left[2\ell(\ell+1)+r^{\mathrm{h}}_{\mathrm{e}}(\ell,\omega)\right].$
(47)

The explicit expression is given in Mandal  _et al._ (2003). The second
coordinate of this minimum is
$\mathcal{V}_{\mathrm{e}}\,\equiv\,\mathcal{V}(r_{\mathrm{e}};\ell,\omega,\mu)$,
The normalized coordinates of the minimum,

$\widetilde{r}_{\mathrm{e}}\,\equiv\,\widetilde{r}_{\mathrm{e}}(\ell,\omega,\mu)=\frac{r_{\mathrm{e}}(\ell,\omega,\mu)}{r_{\mathrm{e}}^{\mathrm{h}}(\ell,\omega)},\;\;\;\;\;\mbox{and}\;\;\;\;\;\widetilde{\mathcal{V}}_{\mathrm{e}}\,\equiv\,\widetilde{\mathcal{V}}_{\mathrm{e}}(\ell,\omega,\mu)=\frac{\mathcal{V}(r_{\mathrm{e}};\ell,\omega,\mu)}{\mathcal{V}^{\mathrm{h}}(r_{\mathrm{e}};\ell,\omega)},$
(48)

are plotted versus $\widetilde{\mu}$ in, respectively, left and right panels
of Fig. 5. Similarly as in the case of the model potential (cf. Fig. 1) the
minimum of the effective potential is approximately located at its asymptotic
position already for $\mu\,\approx\,2\,\mu_{0}(\omega)$. But, unlike the case
of $\ell=0$, here both
$\widetilde{r}_{\mathrm{e}}(\ell,\omega,\widetilde{\mu})$ and
$\widetilde{\mathcal{V}}_{\mathrm{e}}(\ell,\omega,\widetilde{\mu})$ depend on
$\omega$.

Finally, the ratio

$R_{\ell}(r;\ell,\omega,\mu)=\frac{\mathcal{V}(r;\ell,\omega,\mu)}{\mathcal{V}^{\mathrm{h}}(\ell,r;\omega)},$
(49)

for $\ell=1$, and for several values of $\omega$ and $\mu$ is plotted in the
right panel of Fig. 3. As in the case of $\ell=0$, the $\omega$-dependence is
noticeable for small values of $\mu$ only. The differences between the
effective potentials for finite values of $\mu$, and their asymptotic forms
is, in the case of $\ell>0$, much smaller than for $\ell=0$. This can be seen
in Fig. 3: the maximum difference between $1$ and $R_{1}$ is about three times
smaller than between 1 and $R_{0}$.

##  IV Spectrum

Figure 6: Plots of $\Delta_{0}^{\mathrm{o}}$ [eq. (53)] expressed in
$\mathrm{hartrees}$ versus $\mu$ (in $\mathrm{bohrs^{-1}}$) for $\omega=0.5$
(top panels) and $\omega=0.001$ (bottom panels); $\ell=0$ (left panels) and
$\ell=1$ (right panels). The vertical lines mark $\mu_{0}(\omega)$; the
tangent lines $t=t(\mu)$ in the bottom-left corners of the panels and the
horizontal lines in the upper parts of the panels show the asymptotic behavior
of $\Delta_{0}^{\mathrm{o}}(\mu)$ for $\mu\rightarrow{0}$ and
$\mu\rightarrow\infty$, respectively. The upper curves (marked $1$) correspond
to the ground states and the lowest ones (marked $6$) - to the fifth excited
states.

At $\mu=0$ the energy spectrum is given by the well known analytic formula

$E_{n,\ell}(\omega,0)\,=\,E_{n,\ell}^{\mathrm{o}}(\omega)\,=\,\omega\,\left(2\,n+\ell+\frac{3}{2}\right).$
(50)

According to the Hellmann-Feynman theorem,

$\frac{\partial\,E_{n,\ell}(\omega,\mu)}{\partial\,\mu}\,=\,\frac{2}{\sqrt{\pi}}\langle\phi_{n,\ell}\left|\mathrm{e}^{-(\mu\,r)^{2}}\right|\phi_{n,\ell}\rangle\,\longrightarrow\,\left\\{\begin{array}[]{cl}2/\sqrt{\pi},\;\;\mbox{if}\;\;\mu=0,\\\
0,\;\;\mbox{if}\;\;\mu\rightarrow\infty.\end{array}\right.$ (51)

Therefore, for a fixed $\omega$, and $\mu\,\ll\,1$,

$E_{n,\ell}(\omega,\mu)\,\;\mathrel{\raisebox{-6.0pt}{$\stackrel{{\scriptstyle\displaystyle{\rightarrow}}}{{\scriptstyle{\mu\to
0}}}$}}\,\;\frac{2\mu}{\sqrt{\pi}}+E_{n,\ell}(\omega,0)\,=\,\frac{2\mu}{\sqrt{\pi}}+\omega\,\left(2\,n+\ell+\frac{3}{2}\right).$
(52)

Differences

$\Delta_{0}^{\mathrm{o}}\,\equiv\,\Delta_{0}^{\mathrm{o}}(n,\ell;\omega,\mu)=E_{n,\ell}(\omega,\mu)-E^{\mathrm{o}}_{n,\ell}(\omega),$
(53)

for $n=0,1,\ldots,6$, $\ell=0,1$, and for two values of $\omega$ differing by
factor $500$ ($\omega=0.5$, and $\omega=0.001$) are plotted, as functions of
$\mu$, in Fig. 6. Though the scaling invariance discussed in previous sections
is not valid exactly, one can see that the spectra for different values of the
confinement parameter have similar structure.

The average distance between electrons, $\langle\,r\,\rangle$, in the case of
the harmonic oscillator, is proportional to $1/\sqrt{\omega}$. This kind of
dependence is approximately valid for a broad range of $\omega$. The
increasing repulsion between electrons (increasing $\mu$) leads to an increase
of $\langle\,r\,\rangle$. In the case of $\omega=0.001$ the average distance
between electrons is of the order of $100\;\mathrm{bohrs}$, increasing from
about $50\;\mathrm{bohrs}$ at the limit of the harmonic oscillator to
$130\;\mathrm{bohrs}$ at the limit of harmonium. In such cases the influence
of the centrifugal term, proportional to $1/r^{2}$, is small – for small $r$
the wave function is close to $0$. This can be seen by comparing the left and
the right panels in Fig. 6: a difference is visible if $\omega=0.5$, but if
$\omega=0.001$, then the plots for $\ell=0$ and for $\ell=1$ are nearly the
same – the relative difference between the ground state energies of $\ell=0$
and $\ell=1$ is, in the case of $\omega=0.001$, about $1\%$. The range of the
validity of the linear approximation (52) can be estimated for small $\mu$ by
comparing the tangent lines

$t(\mu)\,=\,\frac{2\,\mu}{\sqrt{\pi}}$ (54)

with the curves representing the exact energies.

Figure 7: Plots of $\Delta_{0}^{\mathrm{h}}$ [eq. (55)] expressed in
$\mathrm{hartrees}$ versus $\mu$ (in $\mathrm{bohrs}^{-1}$) for $\ell=0$. Left
panel: $\omega=0.5$; right panel: $\omega=0.001$. The vertical lines mark
$\mu_{0}(\omega)$. The curves (marked $1$) correspond to the ground state and
the ones marked $6$ \- to the fifth excited state.

The energy of harmonium can be expressed analytically only in some special
cases, but the structure of its spectrum is well known (see e.g. Karwowski and
Cyrnek (2004)). As a function of $\omega$, the spectrum can be divided to
three parts: (1) $\omega\,>\,1$ – strong confinement dominates the Coulomb
interaction, the electron correlation is weak and we have the harmonic
oscillator regime; (2) $\omega\,\in\,(10^{-5}\,,\,1)$ – the intermediate
regime, the relation between the confinement and the Coulomb interaction is,
in a sense, similar as in atoms; (3) $\omega\,<\,10^{-5}$ – very weak
confinement, the electrons are far apart and their motion is strongly
correlated. Differences

$\Delta_{0}^{\mathrm{h}}\,\equiv\,\Delta_{0}^{\mathrm{h}}(n,\ell;\omega,\mu)=E_{n,\ell}(\omega,\mu)-E^{\mathrm{h}}_{n,\ell}(\omega),$
(55)

for $n=0,1,\ldots,6$, $\ell=0$, and $\omega=0.5,\,0.001$ are plotted, as
functions of $\mu$, in Fig. 7. Similarly as in the diagrams presented in Fig.
6, also here one can see a qualitative change of the spectrum when
$\mu\,\approx\,\mu_{0}(\omega)$. In the case of the harmonic oscillator the
distances between the neighboring energy levels, for given $\ell$ and
$\omega$, are constant (independent of $n$). In the case of harmonium, these
distances increase with increasing $n$ Karwowski and Cyrnek (2004). When $\mu$
changes from the large to the small values, the distances between the energy
levels gradually change from the mode of harmonium to the mode of harmonic
oscillator. This can be seen in Fig. 7 – in the area
$\mu\,\approx\,\mu_{0}(\omega)$, the order of the values of
$\Delta_{0}^{\mathrm{h}}$ in terms of $n$, changes to the reverse.

Figure 8: Plots of
$\Delta_{\mathrm{H}}^{\mathrm{h}}\,\equiv\,\Delta_{\mathrm{H}}^{\mathrm{h}}(n,\ell;\omega,\mu)$
[eq. (56)] expressed in $\mathrm{hartrees}$ versus $\mu$ (in
$\mathrm{bohrs}^{-1}$), for $\omega=0.5$ (left panel) and $\omega=0.001$
(right panel), and $\ell=0$. The vertical lines mark $\mu_{0}(\omega)$. The
curves marked $1$ correspond to the ground state and the ones marked $6$ \- to
the fifth excited state.

As it was already mentioned, using the smooth long-range interaction potential
instead of the singular Coulomb one, reduces the computational effort in the
approximate solving the Schrödinger equation for many-electron systems Gori-
Giorgi and Savin (2006); Savin (2011, 2020). Therefore, one of the objectives
of studying properties of erfonium, is an exploration of the possibility of an
extrapolation of the energy derived from a model with a finite $\mu$ to the
limit of electrons interacting by the Coulomb force, i.e. the transition from
the spectrum of a finite-$\mu$ erfonium to the spectrum of harmonium. The
first step in such a procedure is the approximation of the energies of
harmonium by the expectation values of the harmonium Hamiltonian calculated
using the wave functions of erfonium Savin and Karwowski (2023). The resulting
correction to the energy, equal to

$\Delta_{\mathrm{H}}^{\mathrm{h}}\,\equiv\,\Delta_{\mathrm{H}}^{\mathrm{h}}(n,\ell;\omega,\mu)=\langle\,\phi_{n,\ell}(\mu)|\mathsf{H}^{\mathrm{h}}|\phi_{n,\ell}(\mu)\,\rangle-
E_{n,\ell}^{\mathrm{h}}(\omega),$ (56)

where

$\langle\,\phi_{n,\ell}(\mu)|\mathsf{H}^{\mathrm{h}}|\phi_{n,\ell}(\mu)\,\rangle\,\equiv\,\langle\,\phi_{n,\ell}(r;\omega,\mu)|\mathsf{H}(r;\ell,\omega,\infty)|\phi_{n,\ell}(r;\omega,\mu)\,\rangle,$

is plotted in Fig. 8. As one can see by comparing with Fig. 7, the difference
between $\left|\Delta_{0}^{\mathrm{h}}\right|$ and
$\left|\Delta_{\mathrm{H}}^{\mathrm{h}}\right|$ depends on the value of
$\omega$. If $\omega$ is large (left panels), then
$\left|\Delta_{\mathrm{H}}^{\mathrm{h}}\right|\ll\left|\Delta_{0}^{\mathrm{h}}\right|$
(for $\omega=0.5$ by factor $\approx\,10$). But if $\omega$ is small (right
panels), then both differences are of the same order of magnitude.

The difference between $E_{n,\ell}(\omega,\mu)$ and
$E_{n,\ell}^{\mathrm{h}}(\omega)$ can be dramatically reduced if the
generalized cusp conditions are used to represent the wave function at small
$r$. But, if $\mu$ is smaller than a certain threshold, a further reduction of
the difference using this method proved to be impossible Savin and Karwowski
(2023). From the perspective of the present analysis it is understandable that
the threshold value is close to $\mu_{0}(\omega)$, where the character of the
potential (and of the spectrum) changes from a harmonium-like to an
oscillator-like.

##  V Final remarks

Erfonium, a Hooke atom with a soft interaction potential represented by
$\mathrm{erf}(\mu\,r)/r$, makes a bridge between harmonium
($\mu\,\rightarrow\,\infty$) and the spherical harmonic oscillator ($\mu=0$).
The parameter $\mu$ establishes an adiabatic connection between these two
limits. An interplay between the effects specific for harmonium and for the
harmonic oscillator creates a rich and interesting pattern which can be
observed in the structure of the potential and of the spectrum. The parameter
$\mu_{0}(\omega)$ determined by the strength of the confinement (27), marks a
fuzzy boundary between these two ”spheres of influence”. If
$\mu>\mu_{0}(\omega)$ then the regime of harmonium dominates, if
$\mu<\mu_{0}(\omega)$, then the features of the harmonic oscillator prevail.

A vast field for further studies remains open. Particularly interesting (and
difficult) are issues related to the behavior of erfonium at the limit of
$\omega\,\rightarrow\,0$. Notice that at this limit the spectrum abruptly
changes from entirely discrete, to entirely continuous. If simultaneously
$\mu\,\rightarrow\,0$, then at the limit we get two unconfined and non-
interacting ’electrons’.

Last, but not least, a broad class of semi-analytic (cf. Taut (1993); Mandal
_et al._ (2003)) and variational approaches may result in precise analytic
approximations of the wave functions and of the energies of erfonium.

Acknowledgement

We thank Prof. Henryk A. Witek (National Chiao Tung University, Hsinchu,
Taiwan) for his constructive remarks and for useful discussions.

## ORCID

Jacek Karwowski: https://orcid.org/0000-0003-1508-2929
Andreas Savin: https://orcid.org/0000-0001-8401-8037

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