# Cartesian Symmetry Classes associated with certain subgroups of $S_{m}$

Seyyed Sadegh Gholami and Yousef Zamani∗ Department of Mathematics, Faculty of
Basic Sciences, Sahand University of Technology, Tabriz, Iran
<EMAIL_ADDRESS>Department of Mathematics, Faculty of Basic Sciences,
Sahand University of Technology, Tabriz, Iran<EMAIL_ADDRESS>
###### Abstract.

Let $V$ be an $n$-dimensional inner product space. Assume $G$ is a subgroup of
the symmetric group of degree $m$, and $\lambda$ is an irreducible character
of $G$. Consider the _Cartesian symmetrizer_ $C_{\lambda}$ on the Cartesian
space $\times^{m}V$ defined by

$C_{\lambda}=\frac{\lambda(1)}{|G|}\sum_{\tau\in G}\lambda(\tau)Q(\tau).$

The vector space $V^{\lambda}(G)=C_{\lambda}(\times^{m}V)$ is called the
Cartesian symmetry class associated with $G$ and $\lambda$.

In this paper, we give a formula for the dimension of the cyclic subspace
$V^{\lambda}_{ij}$. Then we discuss the problem existing an $O$-basis for the
Cartesian symmetry class $V^{\lambda}(G)$. Also, we compute the dimension of
the symmetry class $V^{\lambda}(G)$ when
$G=\langle\sigma_{1}\sigma_{2}\cdots\sigma_{p}\rangle$ or
$G=<\sigma_{1}><\sigma_{2}>\cdots<\sigma_{k}>$, where $\sigma_{i}$ are
disjoint cycles in $S_{m}$. The dimensions are expressed in terms of the
Ramanujan sum.

Additionally, we provide a necessary and sufficient condition for the
existence of an $O$-basis for Cartesian symmetry classes associated with the
irreducible characters of the dihedral group $D_{2m}$. The dimensions of these
classes are also computed.

###### Key words and phrases:

Irreducible characters, Cartesian symmetry classes, orthogonal basis, cyclic
group, Dihedral group, Ramanujan sum

###### 2020 Mathematics Subject Classification:

20C15, 15A69

∗ Corresponding author

##  1\. Introduction and Preliminaries

The Cartesian symmetry class associated with a subgroup $G$ of the symmetric
group $S_{m}$ and a mapping $f:G\rightarrow\mathbb{C}$ was initially explored
by Tian-Gang Lei in [5]. The dimension of this Cartesian symmetry class,
corresponding to an irreducible character of the subgroup $G$ in $S_{m}$, has
been computed in terms of the fixed point character of $S_{m}$ in [7].
Additionally, a formula expressing the dimension of the Cartesian symmetry
class in terms of the rank of an idempotent matrix. The article also deduces
certain properties of generalized trace functions of a matrix.

In this research, we give a formula for the dimension of the cyclic subspace
$V^{\lambda}_{ij}$. Then we discuss the problem existing an $O$-basis for the
Cartesian symmetry class $V^{\lambda}(G)$. Also, we compute the dimension of
the symmetry class $V^{\lambda}(G)$ in terms of the Ramanujan sum when
$G=\langle\sigma_{1}\sigma_{2}\cdots\sigma_{p}\rangle$ or
$G=<\sigma_{1}><\sigma_{2}>\cdots<\sigma_{k}>$, where $\sigma_{i}$ are
disjoint cycles in $S_{m}$. Additionally, we provide a necessary and
sufficient condition for the existence of an $O$-basis for Cartesian symmetry
classes associated with the irreducible characters of the Dihedral group
$D_{2m}$. The dimensions of these classes are also computed.

We commence with a review of Cartesian symmetry classes. For more details, we
refer the reader to [5, 7].

Let $V$ be a complex inner product space of dimension $n\geq 2$. Consider
$\mathbb{E}=\\{f_{1},\ldots,f_{n}\\}$ as an orthonormal basis of $V$. Let
$\times^{m}V$ be the Cartesian product of $m$-copies of $V$. We have an
induced inner product of $\times^{m}V$, which is defined by

$<z^{\times},w^{\times}>=\sum_{i=1}^{m}<z_{i},w_{i}>,$

where

$z^{\times}=(z_{1},\ldots,z_{m}),\ w^{\times}=(w_{1},\ldots,w_{m}).$

For every $1\leq i\leq n,\ 1\leq j\leq m$, we define

$f_{ij}=(\delta_{1j}f_{i},\delta_{2j}f_{i},\ldots,\delta_{mj}f_{i})\in\times^{m}V.$

Then

$\mathbb{E}^{\times}=\\{f_{ij}~{}|~{}i=1,2,\ldots,n,j=1,2,\ldots,m\\}$

is an orthonormal basis of $\times^{m}V$.

Let $G$ be a subgroup of the symmetric group $S_{m}$. For any $\tau\in G$, we
define the Cartesian permutation linear operator

$Q(\tau):\times^{m}V\longrightarrow\times^{m}V$

by

$Q(\tau)(u_{1},\ldots,u_{m})=(u_{\tau^{-1}(1)},\ldots,u_{\tau^{-1}(m)}).$

It is easy to see that $Q:\tau\rightarrow Q(\tau)$ defines a faithful unitary
representation of $G$ over $\times^{m}V$.

Let $\lambda$ be a complex irreducible character of $G$. Define the _Cartesian
symmetrizer_ $C_{\lambda}$ as follows:

$C_{\lambda}=\dfrac{\lambda(1)}{|G|}\sum_{\tau\in G}\lambda(\tau)Q(\tau).$

It has been shown [7] that $C_{\lambda}$ is an orthogonal projection on
$\times^{m}V$, i.e., $C_{\lambda}^{2}=C_{\lambda}^{\ast}=C_{\lambda}$.

The range of the linear operator $C_{\lambda}$ is called _the Cartesian
symmetry class associated with $G$ and $\lambda$_, denoted by
$V^{\lambda}(G)$. It is easy to see that

$\times^{m}V=\sum_{\lambda\in
Irr(G)}^{\bot}{V^{\lambda}(G)}~{}(\text{orthogonal direct sum}).$

The elements

$u^{\lambda}=C_{\lambda}(u_{1},\ldots,u_{m})$

of $V^{\lambda}(G)$ are called _$\lambda$ -symmetrized vectors_. Note that
$V^{\lambda}(G)$ is spanned by the standard $\lambda$-symmetrized vectors

$f^{\lambda}_{ij}=C_{\lambda}(f_{ij})~{}(i=1,2,\ldots,n,j=1,2,\ldots,m).$

######  Theorem 1.1.

[7] Let $G$ be a subgroup of the symmetric group $S_{m}$ and
$\lambda\in\text{Irr}(G)$. Then for each $1\leq i,r\leq n,1\leq j,s\leq m$, we
have

$\displaystyle\langle f_{ij}^{\lambda},f_{rs}^{\lambda}\rangle$
$\displaystyle=$
$\displaystyle\dfrac{\lambda(1)^{2}}{|G|^{2}}\sum_{\sigma,\tau\in
G}\lambda(\sigma){\overline{\lambda(\tau)}}{\delta_{ir}}\delta_{\sigma^{-1}(j),\tau^{-1}(s)}$
$\displaystyle=$ $\displaystyle\begin{cases}0&\text{if $i\neq r$ or $j\nsim s$
}\\\ \frac{\lambda(1)}{|G|}\sum_{g\in G_{j}}\lambda(g\tau^{-1})&\text{if
$s=\tau(j)$}\end{cases}.$

In particular

(1) $\displaystyle\parallel
f_{ij}^{\lambda}\parallel^{2}=\dfrac{\lambda(1)[\lambda,1_{G_{j}}]}{{[G:G_{j}]}},$

where $G_{j}$ is the stabilizer of $j$ in $G$ and $[\ ,\ ]$ the inner product
of characters [3].

Let $\mathcal{D}$ be a set of representatives of orbits of the action of $G$
on the set $\mathbf{I}_{m}=\\{1,2,\cdots,m\\}$ and
$\mathcal{O}=\\{j~{}|~{}1\leq j\leq m,\ [\lambda,1_{G_{j}}]\neq 0\\}.$ We put
$\bar{\mathcal{D}}=\mathcal{D}\cap\mathcal{O}$. It is easy to see that the set

$\\{f^{\lambda}_{ij}~{}\mid~{}1\leq i\leq n,j\in\bar{D}\\}$

is an orthogonal set of non-zero vectors in $V^{\lambda}(G)$. Notice that the
above set may not be a basis for $V^{\lambda}(G)$.
For any $1\leq i\leq n$ and $j\in\bar{\mathcal{D}}$, define the cyclic
subspace

$V^{\lambda}_{ij}=\langle f^{\lambda}_{i\sigma(j)}\mid\sigma\in G\rangle.$

Clearly

(2) $\displaystyle V^{\lambda}(G)=\sum_{1\leq i\leq
n,~{}j\in\bar{\mathcal{D}}}^{\bot}{V^{\lambda}_{ij}}$

the orthogonal direct sum of the cyclic subspaces $V^{\lambda}_{ij}\ (1\leq
i\leq n,\ j\in\bar{\mathcal{D}})$. If $\lambda$ is a linear character of $G$
and $j\in\bar{\mathcal{D}}$, then it is easy to see that
$f^{\lambda}_{i\sigma(j)}=\lambda(\sigma^{-1})f^{\lambda}_{ij}$, so $\dim
V^{\lambda}_{ij}=1$ and the set

$\\{f^{\lambda}_{ij}\mid 1\leq i\leq n,j\in\bar{\mathcal{D}}\\}$

is an orthogonal basis of $V^{\lambda}(G)$. Therefore, we have $\dim
V^{\lambda}(G)=(\dim V)|\bar{\mathcal{D}}|$. For example we have

$\dim V^{1}(S_{m})=\dim V,\ \ \ \dim V^{\varepsilon}(S_{m})=\begin{cases}\dim
V&\text{if $m=2$}\\\ 0&\text{if $m\geq 3$}\end{cases},$

where $1$ and $\varepsilon$ are the principal and the alternating character of
$S_{m}$, respectively.

Suppose $\lambda$ is a non-linear irreducible character of $G$. We will now
proceed to construct a basis for $V^{\lambda}(G)$. For each
$j\in\bar{\mathcal{D}}$, we choose the set $\\{j_{1},\ldots,j_{s_{j}}\\}$ from
the orbit of $j$ such that
$\\{f^{\lambda}_{ij_{1}},\ldots,f^{\lambda}_{ij_{s_{j}}}\\}$ forms a basis for
the cyclic subspace $V^{\lambda}_{ij}$. This procedure is executed for each
$k\in\bar{\mathcal{D}}$. If $\bar{\mathcal{D}}=\\{j,k,l,\ldots\\}$ with
$j<k<l<\ldots$, then we define

$\hat{\mathcal{D}}=\\{j_{1},\ldots,j_{s_{j}};k_{1},\ldots,k_{s_{k}},\ldots\\}$

with an indicated order. Consequently,

$\mathbb{E}^{\lambda}=\\{f^{\lambda}_{ij}~{}|~{}1\leq i\leq n,\
j\in\hat{\mathcal{D}}\\}$

forms a basis for $V^{\lambda}(G)$. Note that this basis may not be
orthogonal. Therefore,

$\dim V^{\lambda}(G)=(\dim V)|\hat{\mathcal{D}}|.$

If the subspace $W$ of $V^{\lambda}(G)$ possesses a basis consisting of
orthogonal standard $\lambda$-symmetrized vectors, we will refer to $W$ as
having an $O$-basis.

In section 2 we give a formula for the dimension of the cyclic subspace
$V^{\lambda}_{ij}$. Then we discuss the problem existing $O$-basis for
$V^{\lambda}(G)$. In section 3, we determine the dimensions of Cartesian
symmetry classes associated with a cyclic subgroup of the symmetric group
$S_{m}$ (generated by a product of disjoint cycles) and the product of cyclic
subgroups of $S_{m}$. These dimensions are expressed in terms of the Ramanujan
sum.
In Section 4, we establish a necessary and sufficient condition for the
existence of an $O$-basis for Cartesian symmetry classes associated with the
irreducible characters of the dihedral group. Additionally, we compute the
dimensions of these classes.

##  2\. Orthogonal bases of Cartesian symmetry classes

In this section we discuss the problem existing $O$-basis for $V^{\lambda}(G)$
consisting of standard $\lambda$-symmetrized vectors. We first prove the
following theorem.

######  Theorem 2.1.

For every $1\leq i\leq n$ and $j\in\bar{\mathcal{D}}$, we have

$\dim V^{\lambda}_{ij}=\lambda(1)[\lambda,1_{G_{j}}].$

###### Proof.

Let $[G:G_{j}]=k$ , $G=\bigcup_{i=1}^{k}\tau_{i}G_{j}$ be the left coset
decomposition of $G_{j}$ in $G$ and

$Orb~{}(j)=\\{\tau_{1}(j),\ldots,\tau_{k}(j)\\}.$

We put

$\displaystyle V_{ij}=$ $\displaystyle\langle f_{i\sigma_{(}j)}\mid\sigma\in
G\rangle.$

Then

$\displaystyle\mathbb{E}_{ij}=\\{f_{i\tau_{1}(j)},\ldots,f_{i\tau_{k}(j)}\\}$

is a basis for $V_{ij}$ but the set

$\displaystyle\\{f_{i\tau_{1}(j)}^{\lambda},\ldots,f_{i\tau_{k}(j)}^{\lambda}\\},$

may not be a basis for $V_{ij}^{\lambda}$. Notice that
$V_{ij}^{\lambda}=C_{\lambda}(V_{ij})$. Since $V_{ij}$ is invariant under
$C_{\lambda}$, so we put

$C_{ij}(G,\lambda)=C_{\lambda}\mid_{V_{ij}}.$

This restriction is a linear operator on $V_{ij}$. Since $C_{\lambda}$ is an
orthogonal projection on $\times^{m}V$, so $C_{ij}(G,\lambda)$ is also an
orthogonal projection on $V_{ij}$.

Let $M=(m_{ij})=[C_{ij}(G,\lambda)]_{\mathbb{E}_{ij}}$. Then

$\dim
V_{ij}^{\lambda}=rank~{}C_{ij}(G,\lambda)=tr~{}C_{ij}(G,\lambda)=tr~{}M.$

Now we compute the entries of the matrix $M$. We have

$\displaystyle C_{ij}(G,\lambda)f_{i\tau_{q}(j)}=$ $\displaystyle
C_{\lambda}(f_{i\tau_{q}(j)})$ $\displaystyle=$
$\displaystyle\frac{\lambda(1)}{|G|}\sum_{\sigma\in
G}{\lambda(\sigma)Q(\sigma)(f_{i\tau_{q}(j)})}$ $\displaystyle=$
$\displaystyle\frac{\lambda(1)}{|G|}\sum_{\sigma\in
G}{\lambda(\sigma)f_{i\sigma\tau_{q}(j)}}$ $\displaystyle=$
$\displaystyle\frac{\lambda(1)}{|G|}\sum_{\pi\in
G}{\lambda(\pi\tau_{q}^{-1})f_{i\pi(j)}}\hskip 28.45274pt(\sigma\tau_{q}=\pi)$
$\displaystyle=$
$\displaystyle\frac{\lambda(1)}{|G|}\sum_{p=1}^{k}\sum_{\pi\in{G_{j}}}\lambda(\tau_{p}\pi\tau^{-1}_{q})f_{i\tau_{p}\pi(j)}\hskip
28.45274pt$ $\displaystyle=$
$\displaystyle\frac{\lambda(1)}{|G|}\sum_{p=1}^{k}\sum_{\pi\in{G_{j}}}\lambda(\tau_{p}\pi\tau^{-1}_{q})f_{i\tau_{p}(j)}$
$\displaystyle=$
$\displaystyle\sum_{p=1}^{k}\left(\frac{\lambda(1)}{|G|}\sum_{\pi\in{G_{j}}}\lambda(\tau_{p}\pi\tau^{-1}_{q})\right)f_{i\tau_{p}(j)}.$

Thus

$m_{pq}=\frac{\lambda(1)}{|G|}\sum_{\pi\in{G_{j}}}\lambda(\tau_{p}\pi\tau_{q}^{-1}).$

Hence

$\displaystyle\dim
V^{\lambda}_{ij}=trM=\sum_{p=1}^{k}m_{pp}=\sum_{p=1}^{k}\frac{\lambda(1)}{|G|}\sum_{\pi\in{G_{j}}}\lambda(\tau_{p}\pi\tau^{-1}_{p})=\sum_{p=1}^{k}\frac{\lambda(1)}{|G|}\sum_{\pi\in{G_{j}}}\lambda(\pi)$
$\displaystyle=\frac{k}{|G|}\lambda(1)\sum_{\pi\in{G_{j}}}\lambda(\pi)=\lambda(1)\frac{1}{|G_{j}|}\sum_{\pi\in{G_{j}}}\lambda(\pi)=\lambda(1)[\lambda,1_{G_{j}}],$

so the result holds. ∎

By using Theorem 2.1 and Equation (2), we have the following corollary.

######  Corollary 2.2.

Let $G$ be a subgroup of $S_{m}$. Then for any $\lambda\in\text{Irr}(G)$, we
have

$\dim V^{\lambda}(G)=(\dim
V)\lambda(1)\sum_{j\in\bar{\mathcal{D}}}[\lambda,1_{G_{j}}].$

Similar to the proof of Theorem 1 in [6], we can prove the following important
theorem.

######  Theorem 2.3.

Let $\lambda$ be a non-linear irreducible character of $G$ and suppose there
exists $j\in\bar{\mathcal{D}}$ such that

$\frac{\sqrt{2}}{2}<\parallel f_{ij}^{\lambda}\parallel<1.$

Then $V^{\lambda}(G)$ has no $O$-basis.

We will show that $\frac{\sqrt{2}}{2}$ is the best lower bound for $\parallel
f_{ij}^{\lambda}\parallel$ in Theorem 2.3. Also, we will give a counter-
example that the converse of this theorem is not true.

Recall that $G$ is $2$-transitive if $m\geq 2$ and $G_{i}$ is transitive on
$\mathbf{I}_{m}\setminus\\{i\\}$ for some $1\leq i\leq m$. Let
$\theta(\sigma)=\mid\\{i~{}|~{}1\leq i\leq m,\sigma(i)=i\\}\mid$. By [3,
Corollary 5.17], $G$ is $2$-transitive if and only if
$\lambda=\theta-1\in\text{Irr}(G)$. Now we have the following corollary.

######  Corollary 2.4.

Let $G\leq S_{m}~{}(m\geq 3)$ be $2$-transitive and $\lambda=\theta-1$. Then
$V^{\lambda}(G)$ has no $O$-basis. In particular, if $G=S_{m}\ (m\geq 3)$ or
$G=A_{m}\ (m\geq 4)$, then $V^{\lambda}(G)$ does not have an $O$-basis.

###### Proof.

Clearly $G_{m}=G\cap S_{m-1}$ so $[\lambda,1_{G_{m}}]=1$ (Burnside’s Lemma)
and $[G:G_{m}]=[S_{m}:S_{m-1}]=m$ . By Equation (1), we have

$\parallel f_{im}^{\lambda}\parallel^{2}=\frac{m-1}{m},$

so the result holds by the previous theorem. ∎

We will say that $\times^{m}V$ has $O$-basis if for each irreducible character
$\lambda$ of $G$, the Cartesian symmetry class $V^{\lambda}(G)$ has an
$O$-basis. We know that
$\times^{m}V=\sum_{\lambda\in\text{Irr}(G)}^{\bot}{V^{\lambda}(G)}$
(orthogonal direct sum), so the following result follows immediately from the
above corollary.

######  Corollary 2.5.

Assume $m\geq 3$. If $G\leq S_{m}$ is 2-transitive, then $\times^{m}V$ does
not have an $O$-basis.

##  3\. Cyclic Cartesian symmetry classes

The well-known Ramanujan sum is given by

$\mathit{c}_{m}(q)=\sum_{s=0,\\\ (s,m)=1}^{m-1}\exp(\frac{2\pi iqs}{m}),$

where $m$ is a positive integer, and $q$ is a non-negative integer (see [1]).

######  Theorem 3.1.

Let $G$ be the cyclic subgroup of $S_{m}$, generated by the $m$-cycle
$(1~{}2~{}\cdots~{}m)$. Then for any $\lambda\in\text{Irr}(G)$, we have

$\dim V^{\lambda}(G)=\dim V.$

###### Proof.

By [3, Corollary 2.6], a finite group $G$ is abelian if and only if every
irreducible character of it is linear. Let $\lambda\in\text{Irr}(G)$. Then
$\lambda(1)=1$. It is easy to see that the action of $G$ on $\mathbf{I}_{m}$
is transitive, so $|\mathcal{D}|=1$. Assume that $\mathcal{D}=\\{1\\}$.
Clearly $G_{1}=\\{1\\}$ and $[\lambda,1_{G_{1}}]=1$. Therefore $\dim
V^{\lambda}(G)=\dim V$, by Corollary 2.2. ∎

Now, let’s consider the cyclic group
$G=\langle\sigma_{1}\sigma_{2}\cdots\sigma_{p}\rangle$, where $\sigma_{i}\
(1\leq i\leq p)$ are disjoint cycles in $S_{m}$. Assume that the order of
$\sigma_{i}\ (1\leq i\leq p)$ is $m_{i}$. The irreducible characters of $G$
are given by

$\lambda_{q}\left((\sigma_{1}\cdots\sigma_{p})^{s}\right)=\exp\left(\frac{2\pi
iqs}{M}\right),\quad s=0,\ldots,M-1,$

where $M$ denotes the least common multiple of the integers
$m_{1},\ldots,m_{p}$.

In the following theorem we obtain $\dim V^{\lambda_{q}}(G)$ in terms of
Ramanujan sum.

######  Theorem 3.2.

Let $G=\langle\sigma_{1}\sigma_{2}\cdots\sigma_{p}\rangle$, where $\sigma_{i}\
(1\leq i\leq p)$ are disjoint cycles in $S_{m}$ of orders
$m_{1},\cdots,m_{p}$, respectively. Assume $m=\sum_{i=1}^{p}m_{i}$ and let
$\lambda_{q},~{}0\leq q\leq M-1$, be an irreducible complex character of $G$.
Then

$\dim V^{\lambda_{q}}(G)=(\dim
V)\sum_{i=1}^{p}\frac{1}{m_{i}^{\prime}}\sum_{d\mid
m_{i}^{\prime}}\mathit{c}_{m_{i}^{\prime}/d}(q),$

where $m_{i}^{\prime}=M/m_{i}$. In particular, if $G$ is generated by an
$m$-cycle in $S_{m}$, then $\dim V^{\lambda_{q}}(G)=\dim V$.

###### Proof.

To compute the dimension $\dim V^{\lambda_{q}}(G)$ we use Corollary 2.2. For
every $1\leq i\leq p$, suppose

$\mathcal{O}_{i}=\\{1\leq j\leq m~{}\mid~{}\sigma_{i}(j)\neq j\\}.$

Then $\mathcal{O}_{1},\cdots,\mathcal{O}_{p}$ are distinct orbits of $G$. If
$j\in\mathcal{O}_{i}$ then

$\displaystyle G_{j}=$
$\displaystyle\\{(\sigma_{1}\sigma_{2}\cdots\sigma_{p})^{s}~{}|~{}0\leq
s\leq{M-1},\sigma_{i}^{s}(j)=j\\}$ $\displaystyle=$
$\displaystyle\\{\sigma_{1}^{s}\sigma_{2}^{s}\cdots\sigma_{p}^{s}~{}|~{}0\leq
s\leq{M-1},\sigma_{i}^{s}\in\langle\sigma_{i}\rangle_{j}=1\\}$
$\displaystyle=$
$\displaystyle\\{\sigma_{1}^{s}\cdots\sigma_{i-1}^{s}\sigma_{i+1}^{s}\cdots\sigma_{p}^{s}~{}|~{}0\leq
s\leq{M-1},m_{i}|s\\}.$

In this case, it is easy to see that $|G_{j}|=\frac{M}{m_{i}}$. Now for
$j\in\mathcal{O}_{i}$ we have

$\displaystyle[\lambda_{q},1_{G_{j}}]=$
$\displaystyle\frac{1}{|G_{j}|}\sum_{g\in G_{j}}\lambda_{q}(g)\hskip
28.45274pt$ $\displaystyle=$ $\displaystyle\frac{1}{|G_{j}|}\sum_{g\in
G_{j}}\lambda_{q}((\sigma_{1}\sigma_{2}\cdots\sigma_{p})^{s})$
$\displaystyle=$ $\displaystyle\frac{m_{i}}{M}\sum_{s=0,\ m_{i}\mid
s}^{M-1}\exp\left(\dfrac{2\pi iqs}{M}\right)$

For any $1\leq i\leq p$ we choose $j_{i}\in\mathcal{O}_{i}$. Then
$\mathcal{D}=\\{j_{1},\cdots,j_{p}\\}$. Therefore

$\displaystyle\dim V^{\lambda_{q}}(G)$ $\displaystyle=$ $\displaystyle(\dim
V)\lambda_{q}(1)\sum_{j\in\bar{\mathcal{D}}}[\lambda_{q},1_{G_{j}}]$
$\displaystyle=$ $\displaystyle(\dim
V)\sum_{j\in\mathcal{D}}[\lambda_{q},1_{G_{j}}]$ $\displaystyle=$
$\displaystyle(\dim V)\sum_{i=1}^{p}[\lambda_{q},1_{G_{j_{i}}}]$
$\displaystyle=$ $\displaystyle(\dim
V)\sum_{i=1}^{p}\frac{m_{i}}{M}\sum_{s=0,m_{i}\mid
s}^{M-1}\exp\left(\dfrac{2\pi iqs}{M}\right)$

Now letting $s_{i}^{\prime}=s/m_{i}$ and $m_{i}^{\prime}=M/m_{i}$, we obtain

$\displaystyle\dim V^{\lambda_{q}}(G)$ $\displaystyle=$ $\displaystyle(\dim
V)\sum_{i=1}^{p}\frac{1}{m_{i}^{\prime}}\sum_{s_{i}^{\prime}=0}^{m_{i}^{\prime}-1}\exp\left(\dfrac{2\pi
iqs_{i}^{\prime}}{m_{i}^{\prime}}\right)$ $\displaystyle=$ $\displaystyle(\dim
V)\sum_{i=1}^{p}\frac{1}{m_{i}^{\prime}}\sum_{d\mid
m_{i}^{\prime}}~{}~{}~{}\sum_{(s_{i}^{\prime},m_{i}^{\prime})=d}\exp\left(\dfrac{2\pi
iqs_{i}^{\prime}}{m_{i}^{\prime}}\right)$ $\displaystyle=$ $\displaystyle(\dim
V)\sum_{i=1}^{p}\frac{1}{m_{i}^{\prime}}\sum_{d\mid
m_{i}^{\prime}}\mathit{c}_{m_{i}^{\prime}/d}(q).$

∎

Our the other goal is to obtain the dimension of $V^{\lambda}(G)$, when $G\leq
S_{m}$ has the structure

$G=<\sigma_{1}><\sigma_{2}>\cdots<\sigma_{k}>,$

with $\sigma_{\ell}$ $(1\leq\ell\leq k)$ representing disjoint cycles in
$S_{m}$ of specific orders, denoted as $m_{1},\ldots,m_{k}$, respectively. The
irreducible characters of $G$ are exclusively linear and can be expressed as
products of the irreducible characters of the cyclic groups
$\langle\sigma_{\ell}\rangle$, where $\ell=1,\ldots,k$:

$\lambda_{(q_{1},\ldots,q_{k})}=\lambda_{q_{1}}\cdots\lambda_{q_{k}},~{}q_{\ell}=0,\ldots,m_{\ell}-1,$

and the character $\lambda_{q_{\ell}}$ is defined as
$\lambda_{q_{\ell}}(\sigma_{\ell}^{j_{\ell}})=\exp\left(\dfrac{2\pi
i{q_{\ell}}{j_{\ell}}}{m_{\ell}}\right)$.

######  Theorem 3.3.

Suppose $G=<\sigma_{1}><\sigma_{2}>\cdots<\sigma_{k}>,k\geq 2$, where
$\sigma_{\ell},\ell=1,\ldots,k$, are disjoint cycles in $S_{m}$ of orders
$m_{1},\cdots,m_{k}$. Let $\lambda=\lambda_{(q_{1},\ldots,q_{k})}$, where
$0\leq q_{\ell}\leq m_{\ell}-1$, $1\leq\ell\leq k$. Then

$\dim V^{\lambda}(G)=(\dim
V)\left[(m-\sum_{\ell=1}^{k}m_{\ell})[\lambda,1_{G}]+\prod_{l\neq
j}\frac{1}{m_{\ell}}\sum_{d\mid
m_{\ell}}\mathit{c}_{m_{\ell}/d}(q_{\ell})\right].$

In the case $k=1$, $\dim V^{\lambda}(G)=(\dim
V)\left[(m-m_{1})[\lambda,1_{G}]+1\right]$.

###### Proof.

For any $j\in\mathcal{D}$, if $\sigma_{i}(j)=j$ for all $1\leq i\leq k$, then
$G_{j}=G$ and so $[\lambda,1_{G_{j}}]=[\lambda,1_{G}]$. Otherwise there exists
$1\leq i\leq k$, such that $\sigma_{i}(j)\neq j$. Then $G_{j}=\prod_{i\neq
j}<\sigma_{i}>$. Thus we have

$\displaystyle[\lambda,1_{G_{j}}]=$ $\displaystyle\frac{1}{|G_{j}|}\sum_{g\in
G_{j}}\lambda(g)\hskip 28.45274pt$ $\displaystyle=$
$\displaystyle\prod_{\ell\neq
j}\frac{1}{m_{\ell}}\sum_{j_{\ell}=0}^{m_{\ell}-1}\lambda_{q_{\ell}}(\sigma_{\ell}^{j_{\ell}})$
$\displaystyle=$ $\displaystyle\prod_{\ell\neq
j}\frac{1}{m_{\ell}}\sum_{j_{\ell}=0}^{m_{\ell}-1}\exp\left(\dfrac{2\pi
i{q_{\ell}}{j_{\ell}}}{m_{\ell}}\right)$ $\displaystyle=$
$\displaystyle\prod_{\ell\neq j}\frac{1}{m_{\ell}}\sum_{d\mid
m_{\ell}}\sum_{d=(j_{\ell},m_{\ell})}\exp\left(\dfrac{2\pi
i{q_{\ell}}{j_{\ell}}}{m_{\ell}}\right)$ $\displaystyle=$
$\displaystyle\prod_{\ell\neq j}\frac{1}{m_{\ell}}\sum_{d\mid
m_{\ell}}\mathit{c}_{m_{\ell}/d}(q_{\ell}).$

Now, by letting $\mathcal{A}=\\{j~{}|~{}\sigma_{i}(j)=j~{}\text{for
all}~{}1\leq i\leq k\\}$, we have

$\displaystyle\dim V^{\lambda}(G)$ $\displaystyle=$ $\displaystyle(\dim
V)\lambda(1)\sum_{j\in{\mathcal{D}}}[\lambda,1_{G_{j}}]$ $\displaystyle=$
$\displaystyle(\dim
V)\left[\sum_{j\in\mathcal{A}}[\lambda,1_{G}]+\sum_{j=1}^{k}[\lambda,1_{G_{j}}]\right],$
$\displaystyle=$ $\displaystyle(\dim
V)\left[(m-\sum_{\ell=1}^{k}m_{\ell})[\lambda,1_{G}]+\prod_{\ell\neq
j}\frac{1}{m_{\ell}}\sum_{d\mid
m_{\ell}}\mathit{c}_{m_{\ell}/d}(q_{\ell})\right],$

so the result holds. ∎

##  4\. Cartesian symmetry classes associated with dihedral group

In this section, we first obtain the dimensions of Cartesian symmetry classes
associated with the irreducible characters of the dihedral group $D_{2m}$.

The subgroup $D_{2m}$ of $S_{m}~{}(m\geq 3)$ generated by the elements

$\ r=(1\ 2\ \cdots\ m)\ \ \text{and}\ \
s=\begin{pmatrix}{}_{1}&{}_{2}&{}_{3}&{}_{\cdots}&{}_{m-1}&{}_{m}\\\
{}_{1}&{}_{m}&{}_{m-1}&{}_{\cdots}&{}_{3}&{}_{2}\end{pmatrix}$

is the dihedral group of degree $m$ (see [2, page 50]). The generators $r$ and
$s$ satisfy

$r^{m}=1=s^{2}\ \text{and}\ s^{-1}rs=r^{-1}.$

In particular,

$D_{2m}=\\{r^{k},\ sr^{k}|\ 0\leq k<m\\}.$

If $m$ is even, i.e., $m=2k$ ($k\geq 2$), then $D_{2m}$ has $k+3$ conjugacy
classes. If $m$ is odd, i.e., $m=2k+1$ ($k\geq 1$), then $D_{2m}$ has $k+2$
conjugacy classes.

For each integer $h$ with $0<h<m/2$, $D_{2m}$ has an irreducible character
$\psi_{h}$ of degree $2$ given by

$\psi_{h}(r^{k})=2\cos\frac{2kh\pi}{m},\ \ \psi_{h}(sr^{k})=0,\ \ \ 0\leq
k<m.$

The other characters of $D_{2m}$ are of degree $1$, namely $\lambda_{j}$. The
character table of $D_{2m}$ is shown in Table 1 (see [4, page 182]).

Clearly, the action of $G=D_{2m}$ on $\mathbf{I}_{m}=\\{1,2,\cdots,m\\}$ is
transitive, so $\mathcal{D}=\\{1\\}$. On the other hand, $G_{1}=\\{1,s\\}$.
Thus for any $\lambda\in\text{Irr}(G)$, we have

(3) $\displaystyle\dim V^{\lambda}(G)=(\dim
V)~{}\lambda(1)~{}[\lambda,1_{G_{1}}]=(\dim
V)~{}\frac{\lambda(1)}{2}~{}[\lambda(1)+\lambda(s)].$

Now, by using Table 1 and Equation (3), we have the following result.

######  Theorem 4.1.

Let $G=D_{2m}~{}(m\geq 3)$. Assume $n=\dim V\geq 2$.

Case (i) If $m$ is even, then

  * (a)

$\dim V^{\lambda_{1}}(G)=\dim V^{\lambda_{3}}(G)=n$,

  * (b)

$\dim V^{\lambda_{2}}(G)=\dim V^{\lambda_{4}}(G)=0$,

  * (c)

$\dim V^{\psi_{h}}(G)=2n~{}(0<h<\frac{m}{2})$.

Case (ii) If $m$ is odd, then

  * (a)

$\dim V^{\lambda_{1}}(G)=n$,

  * (b)

$\dim V^{\lambda_{2}}(G)=0$,

  * (c)

$\dim V^{\psi_{h}}(G)=2n~{}(0<h<\frac{m}{2})$.

In the following theorem we give a necessary and sufficient condition for the
existence an $O$ -basis for Cartesian symmetry classes
$V^{\psi_{h}}(G)~{}(0<h<\frac{m}{2})$.

######  Theorem 4.2.

Let $G=D_{2m}~{}(m\geq 3)$ and $\psi=\psi_{h}~{}(0<h<\frac{m}{2})$. Then
$V^{\psi}(G)$ has $O$-basis if and only if $m\equiv
0~{}(\text{mod}~{}4h_{2})$, where $h_{2}$ is the $2$-part of $h$.

###### Proof.

Since $G$ acts on $\mathbf{I}_{m}$ transitively, so $\mathcal{D}=\\{1\\}$.
Clearly, $G_{1}=\\{1,s\\}$. So
$[\psi,1_{G_{1}}]=\frac{1}{2}[\psi(1)+\psi(s)]=1\neq 0$. This implies that
$1\in\bar{\mathcal{D}}$. But $V^{\psi}(G)=\sum_{1\leq i\leq
n}^{\bot}V^{\psi}_{i1}$. Notice that that $\dim V^{\psi}_{i1}=2$ and the set
$\\{f^{\psi}_{ir(1)},f^{\psi}_{is(1)}\\}$ is a basis for it. By using Theorem
1.1, we have

$\langle
f^{\psi}_{ir(1)},f^{\psi}_{is(1)}\rangle=\frac{\psi(1)}{|G|}\sum_{g\in
G_{1}}\psi(gsr^{-1})=\frac{1}{m}[\psi(sr^{-1})+\psi(r^{-1})]=\frac{2}{m}\cos~{}\frac{2\pi
h}{m}.$

It is observe that $\langle f^{\psi}_{ir(1)},f^{\psi}_{is(1)}\rangle=0$ if and
only if $\cos~{}\frac{2\pi h}{m}=0$, or equivalently $m\equiv
0~{}(\text{mod}~{}4h_{2})$, $h_{2}$ is the $2$-part of $h$. This completes the
proof. ∎

######  Corollary 4.3.

Let $G=D_{2m}$ and assume $\dim V\geq 2$. Then $\times_{1}^{m}V$ has an
$O$-basis if and only if $m$ is a power of $2$.

###### Proof.

Suppose $m_{2}$ is the $2$-part of $m$. Assume that $m_{2}<m$. Then
$0<m_{2}<\frac{m}{2}$ and $m\not\equiv 0~{}(\text{mod}~{}4h_{2})$. Hence, if
$\psi=\psi_{m_{2}}$, then Theorem 4.2 implies that $V^{\psi}(G)$ has no
$O$-basis, and so $\times_{1}^{m}V$.
Conversely, assume $m$ is a power of $2$. If $0<h<\frac{m}{2}$, then
$h_{2}\leq\frac{m}{4}$ and $m\equiv 0~{}(\text{mod}~{}4h_{2})$, where $h_{2}$
is the $2$-part of $h$. Now Theorem 4.2 implies that $V^{\psi_{h}}(G)$ has an
$O$-basis, and so $\times_{1}^{m}V$. ∎

######  Remark 4.4.

Let $G=D_{8}\leq S_{4}$ . Then $G$ has only one non-linear irreducible
character $\psi(r^{k})=2\cos\frac{k\pi}{2},~{}\psi(sr^{k})=0,~{}0\leq k<4$. By
Theorem 4.2, $V^{\psi}(G)$ has an $O$-basis. Again, by Theorem 4.2, we have

$\parallel
f^{\psi}_{ij}\parallel^{2}=\frac{\psi(1)}{[G:G_{1}]}[\psi,1_{G_{1}}]=\frac{1}{4}[\psi(1)+\psi(s)]=\frac{1}{2}.$

So $\parallel f^{\psi}_{ij}\parallel=\frac{\sqrt{2}}{2}$. This examle show
that $\frac{\sqrt{2}}{2}$ is the best lower bound for $\parallel
f^{\lambda}_{ij}\parallel$ in Theorem 2.3.

Now we give a counter-example that the converse of Theorem 2.3 is not true.

######  Example 4.5.

Let $q>3$ be a prime and

$G=D_{2q}=<r^{q}=1=s^{2}\ \text{and}\ s^{-1}rs=r^{-1}>.$

Consider $G$ as a subgroup of $S_{q}$. Then the non-linear irreducible
characters of $D_{2q}$ are $\psi_{h}$, where $0<h<q/2$ and
$\psi_{h}(r^{k})=2\cos\frac{2k\pi h}{q},~{}\psi_{h}(sr^{k})=0,~{}0\leq k<q$.
Let $\psi=\psi_{h},~{}0<h<\frac{q}{2}$. By Theorem 4.2, $V^{\psi}(G)$ has no
$O$-basis. But we have

$\parallel
f^{\psi}_{i1}\parallel^{2}=\frac{\psi(1)}{[G:G_{1}]}[\psi,1_{G_{1}}]=\frac{1}{q}[\psi(1)+\psi(s)]=\frac{2}{q}<\frac{1}{2}.$

## References

  * [1] T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.
  * [2] T. W. Hungerford, Algebra, New York: Holt, Rinehart and Wilson, 1974.
  * [3] M. Isaacs, Character Theory of Finite Groups, Academic Press, 1976.
  * [4] G. James and M. Liebeck, Representations and Characters of Groups, Cambridge University press, 1993.
  * [5] T. G. Lei, Notes on Cartesian symmetry classes and generalized trace functions, Linear Algebra Appl. 292 (1999) 281-288.
  * [6] M. Shahryari, On the orthogonal bases of symmetry classes, J. Algebra 220 (1999) 327-332.
  * [7] Y. Zamani and M. Shahryari, On the dimensions of Cartesian symmetry classes, Asian-Eur. J. Math. 5 (3) (2012) Article ID 1250046 (7 pages).


# Spectral splitting method for nonlinear Schrödinger equations with quadratic
potential

Andrea Sacchetti  Department of Physics, Informatics and Mathematics,
University of Modena and Reggio Emilia, Modena, Italy.
<EMAIL_ADDRESS>
###### Abstract.

In this paper we propose a modified Lie-type spectral splitting approximation
where the external potential is of quadratic type. It is proved that we can
approximate the solution to a one-dimensional nonlinear Schrödinger equation
by solving the linear problem and treating the nonlinear term separately, with
a rigorous estimate of the remainder term. Furthermore, we show by means of
numerical experiments that such a modified approximation is more efficient
than the standard one.

###### Key words and phrases:

Spectral splitting approximation; nonlinear Schrödinger equation; harmonic and
inverted potential; evolution operator.

This work is partially supported by GNFM-INdAM and by the UniMoRe-FIM project
“Modelli e metodi della Fisica Matematica”.

##  1\. Introduction

In this paper we consider non-linear Schrödinger equation of the form

$\displaystyle\left\\{\begin{array}[]{l}i\hbar\frac{\partial\psi_{t}(x)}{\partial
t}=\left[-\frac{\hbar^{2}}{2m}\Delta+V(x)\right]\psi_{t}(x)+\nu|\psi_{t}(x)|^{2\sigma}\psi_{t}(x)\\\
\psi_{t_{0}}(x)=\psi_{0}(x)\end{array}\right.\,,\psi_{t}(\cdot)\in
L^{2}({\mathbb{R}}^{d},dx)\,,$ (3)

where $\sigma>0$, $V(x)$ is a real-valued quadratic potential and
$\nu\in{\mathbb{R}}$. Hereafter, we assume the units such that $2m=1$ and
$\hbar=1$, we simply denote by $\psi_{t}$ the wavefunction $\psi_{t}(x)$, by
$\psi_{0}$ the initial wavefunction $\psi_{0}(x)$,
$\psi^{\prime}=\frac{\partial\psi}{\partial x}$,
$\psi^{\prime\prime}=\frac{\partial^{2}\psi}{\partial x^{2}}$, etc., and
$\dot{\psi}=\frac{\partial\psi}{\partial t}$. Furthermore, we restrict our
attention, for sake of simplicity, to the one-dimensional case, i.e. $d=1$.

Nonlinear Schrödinger equations with a quadratic potential are a useful tool
in order to describe Bose-Einstein condensates in a trapping potential [10,
14], as well as in the theory of nonlinear optics [12].

An efficient numerical treatment of such an equation is based on the Lie-type
splitting approximation. The basic idea is quite simple (see, e.g., the paper
[5]): suppose to consider an evolution equation

$\displaystyle\left\\{\begin{array}[]{l}i\dot{\psi}_{t}=\left[A+B\right]\psi_{t}\\\
\psi_{t_{0}}=\psi_{0}\end{array}\right.\,,\psi_{t}\in
L^{2}({\mathbb{R}},dx)\,,$ (6)

where $A$ and $B$ are two given operators. Let us denote by
$S^{t-t_{0}}\psi_{0}$ the solution to (6) where $S^{t-t_{0}}$ is the
associated evolution operator; let us denote by $X^{t-t_{0}}$ and
$Y^{t-t_{0}}$ the evolution operators respectively associated to the equations

$\displaystyle i\dot{\psi}_{t}=A\psi_{t}\ \mbox{ and }\
i\dot{\psi}_{t}=B\psi_{t}\,.$

It is well known that, in general,

$\displaystyle S^{\delta}\psi_{0}\not=X^{\delta}Y^{\delta}\psi_{0}\,,\
\delta\in{\mathbb{R}}\,,$

but this difference may be proved, under some circumstances, to be small when
$\delta$ is small. More precisely, if one fix any $T>0$, a $\delta>0$ small
enough and a positive integer number $n$ such that $n\delta\leq T$, then the
solution $\psi_{t}=S^{t-t_{0}}\psi_{0}$ to (6), where $t=n\delta+t_{0}$, can
be approximated by

$\displaystyle\left[X^{\delta}Y^{\delta}\right]^{n}\psi_{0}\,,$ (7)

up to a remainder term that goes to zero when $\delta$ goes to zero.

In fact, a better result may be obtained by means of the Strang-type
approximation where the solution $\psi_{t}$ to (6) is approximated by

$\displaystyle\left[X^{\delta/2}Y^{\delta}X^{\delta/2}\right]^{n}\psi_{0}\,.$

However, for sake of definiteness we restrict our analysis to the Lie-type
approximation method (7).

When one applies such an approximation to the problem (3) a typical choice
consists in choosing $A=-\frac{\partial^{2}}{\partial x^{2}}$, i.e. the one-
dimensional linear Laplacian operator, and $B=V+\nu|\psi_{t}|^{2\sigma}$.
Here, we denote by $X_{1}^{\delta}$ and $Y_{1}^{\delta}$ the associated
evolution operators. Thus, with such a choice
$X_{1}^{t-t_{0}}=e^{-iA(t-t_{0})}$ is the evolution operator associated to the
Laplacian and it is an integral linear operator with well known kernel
function. For what concerns $Y_{1}^{t-t_{0}}$ it is the evolution operator
obtained by means of the solution to the ordinary differential equation

$\displaystyle\left\\{\begin{array}[]{l}i\dot{w}_{t}=Vw_{t}+\nu|w_{t}|^{2\sigma}w_{t}\\\
w_{t_{0}}=w_{0}\end{array}\right.\,.$ (10)

We observe that $|w_{t}|$ is constant with respect to $t$ since $V(x)$ is a
real-valued function; indeed, one can check that

$\displaystyle\frac{\partial|w_{t}|^{2}}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{\partial w_{t}}{\partial
t}\overline{w_{t}}+\frac{\partial\overline{w_{t}}}{\partial t}{w_{t}}$
$\displaystyle=$
$\displaystyle{-i}\left[Vw_{t}+\nu|w_{t}|^{2\sigma}w_{t}\right]\bar{w}_{t}+{i}\left[V\bar{w}_{t}+\nu|w_{t}|^{2\sigma}\bar{w}_{t}\right]w_{t}=0\,.$

Thus, equation (10) takes the form

$\displaystyle\left\\{\begin{array}[]{l}i\dot{w}_{t}=\left[V+\nu|w_{0}|^{2\sigma}\right]w_{t}\\\
w_{t_{0}}=w_{0}\end{array}\right.$ (13)

which has solution

$\displaystyle
w_{t}(x)=\left[Y_{1}^{t-t_{0}}w_{0}\right](x)=e^{-i[V(x)+\nu|w_{0}(x)|^{2\sigma}](t-t_{0})}w_{0}(x)\,,$
(14)

that is $Y_{1}^{t-t_{0}}$ is a multiplication nonlinear operator such that

$\displaystyle\|Y^{t-t_{0}}w_{0}\|_{L^{p}}=\|w_{0}\|_{L^{p}}\,,\ \forall
p\in[1,+\infty]\,.$

Therefore, both evolution operators $X_{1}^{t-t_{0}}$ and $Y_{1}^{t-t_{0}}$
have an explicit expression.

The crucial point is to give a rigorous estimate of the remaining term

$\displaystyle{\mathcal{R}}_{1}\psi_{0}:=S^{t-t_{0}}\psi_{0}-\left[X_{1}^{\delta}Y_{1}^{\delta}\right]^{n}\psi_{0}\,.$
(15)

Let us recall here some rigorous results concerning the estimate of
${\mathcal{R}}_{1}$.

In the case where the external potential is absent, i.e. $V\equiv 0$, and
under some assumption on the initial state $\psi_{0}$ then the estimate

$\displaystyle\|{\mathcal{R}}_{1}\psi_{0}\|_{L^{2}}\leq C{\delta}\,,$ (16)

for some positive constant $C=C(\psi_{0},T)$, has been proved by [5, 9].

If the external potential $V$ is not identically zero then a similar estimate
of the remainder term holds true provided that the Schrödinger equation is
restricted to a bounded domain $U\subset{\mathbb{R}}^{d}$ and provided that
its solution $\psi_{t}$ is such that (see, e.g. Thm. 4.3 [3])

$\displaystyle\psi\in C\left([0,T];H^{m}(U)\cap H_{0}^{1}(U)\right)\,,$

for some $m\geq 5$.

We should also mention that a purely formal (not completely rigorous) argument
(see [1]) suggests that

$\displaystyle\|{\mathcal{R}}_{1}\psi_{0}\|_{L^{2}({\mathbb{R}})}\leq
C\delta^{2}e^{C\delta}$

for some positive constant $C=C(\psi_{0},T)$, provided that the potential
$V(x)$ is a bounded function and $\psi_{0}\in H^{2}({\mathbb{R}})$.

We must remark that such an approach does not properly work when the potential
$V(x)$ is singular, e.g. $V$ is a Dirac’s delta. In such a case the method
should be modified by choosing $A=H=-\frac{\partial^{2}}{\partial x^{2}}+V$,
where $H$ is the linear Schrödinger operator, and where
$B=\nu|\psi|^{2\sigma}$ is the nonlinear term [13].

In this paper we prove the validity of the Lie-type approximation for a
nonlinear Schrödinger equation with quadratic potential following the approach
introduced by [13] in the case of singular potential. Let
$X_{2}^{\delta}:=e^{-iH\delta}$ be the evolution operator associated to the
linear Schrödinger operator and

$\displaystyle\left[Y_{2}^{\delta}w\right](x):=e^{-i\nu|w(x)|^{2\sigma}\delta}w(x)\,.$
(17)

If we denote by

$\displaystyle{\mathcal{R}}_{2}\psi_{0}:=S^{t-t_{0}}\psi_{0}-\left[X_{2}^{\delta}Y_{2}^{\delta}\right]^{n}\psi_{0}$
(18)

the remainder term, we are going to prove that it goes to zero when $\delta$
goes to zero and $n\delta\leq T$ for any fixed $T>0$ (see Theorem 1). We can
thus show that this method has at least as solid a theoretical basis as the
one based on the approximation (15).

One must remark that approximation (15) can be implemented by means of a quite
simple numerical algorithm basically independent on the shape of the potential
$V(x)$; in contrast, approximation (18) is substantially useful when the
evolution operator $X_{2}^{\delta}$, associated to the linear Schrödinger
operator, can be efficiently computed, like in the case of a quadratic
potential. On the other side, by means of numerical experiments, the
approximation (18) turns out to be more accurate than the usual one (15).

The paper is organized as follows. In Section 2 we state our main result
(Theorem 1); Section 3 is devoted to the proof of Theorem 1; in Section 4 we
compare the approximations (15) and (18) on test models; is Section 5 we draw
the conclusions; a short Section A appendix is devoted to the Mehler’s
formulas, that is to the kernel of the evolution operator $X_{2}^{\delta}$ of
the linear Schrödinger operator with harmonic or inverted oscillator
potential.

Hereafter $C$ denotes any positive constant which may change from line to
line.

##  2\. Main result

Let us consider the one-dimensional (i.e. $d=1$) nonlinear Schrödinger
equation of the form

$\displaystyle\left\\{\begin{array}[]{l}i\frac{\partial\psi_{t}}{\partial
t}=H\psi_{t}+\nu|\psi_{t}|^{2\sigma}\psi_{t}\\\
\psi_{t_{0}}=\psi_{0}\end{array}\right.\,,\psi_{t}\in
L^{2}({\mathbb{R}},dx)\,,\ H=-\frac{\partial^{2}}{\partial x^{2}}+V(x)\,,$
(21)

where

$\displaystyle V(x)=\alpha x^{2}$

is a real-valued quadratic potential for some
$\alpha\in{\mathbb{R}}\setminus\\{0\\}$. Let $t_{0}=0$ for the sake of
definiteness.

Solutions to (21) are usually studied in the space

$\displaystyle\Sigma:=\left\\{\psi\in{\mathcal{S}}^{\prime}\ :\
\|\psi\|_{\Sigma}:=\|\psi\|_{L^{2}}+\left\|\psi^{\prime}\right\|_{L^{2}}+\|x\psi\|_{L^{2}}<+\infty\right\\}$

and the existence of a local solution to (21), with the conservation of the
norm

$\displaystyle{\mathcal{N}}(\psi_{t})={\mathcal{N}}(\psi_{0})\,,\ \mbox{ where
}{\mathcal{N}}(\psi):=\|\psi\|_{L^{2}}\,,$

and of the energy

$\displaystyle{\mathcal{E}}(\psi_{t})={\mathcal{E}}(\psi_{0})\,,\ \mbox{ where
}\
{\mathcal{E}}(\psi):=\left\|\psi^{\prime}\right\|_{L^{2}}^{2}+\alpha\|x\psi\|_{L^{2}}^{2}+\frac{\nu}{\sigma+1}\|\psi\|_{L^{2\sigma+2}}^{2\sigma+2}\,.$

has been proved (see [7, 8]).

Solution to (21) globally exists when $\sigma<\frac{2}{d}$ and the map
$t\in{\mathbb{R}}\to\psi_{t}\in\Sigma$ is continuous provided that
$\psi_{0}\in\Sigma$. On the other hand, blow-up may occur as proved by [8]
under some circumstances for some $\nu<0$ and $\alpha>0$ when
$\sigma\geq\frac{d}{2}$.

Let ${{\Gamma}}$ be the vector space

$\displaystyle{{\Gamma}}=\left\\{\psi\in{\mathcal{S}}^{\prime}\ :\
\|\psi\|_{{\Gamma}}:=\|\psi\|_{H^{2}}+\|x^{2}\psi\|_{L^{2}}<+\infty\right\\}\subset\Sigma\,.$

Let $X^{\delta}_{2}=e^{-iH\delta}$ be the evolution operator associated to the
linear Schrödinger operator and let $Y^{\delta}_{2}$ be the multiplication
operator defined by (17).

In order to compare the approximate solution
$(X_{2}^{\delta}Y_{2}^{\delta})^{n}\psi_{0}$ with the solution $S^{t}\psi_{0}$
for any $t=n\delta\leq T$, where $T>0$ is any fixed positive real number, we
have to assume that the solution $S^{t}\psi_{0}$ does not blow up. We remark
that the approximate solution $(X_{2}^{\delta}Y_{2}^{\delta})^{n}\psi_{0}$
always exists; however, we have to introduce the following technical
assumption: we assume that

$\displaystyle\max_{j=0,1,\ldots,n-1}\|(X_{2}^{\delta}Y_{2}^{\delta})^{n-j-1}S^{(j+1)\delta}\psi_{0}\|_{L^{\infty}}\leq
C$ (22)

for some positive constant $C$ depending on $\psi_{0}$ and $T$, but
independent of $t$ and $n$. We should remark that for each index $j$ the
vector $(X_{2}^{\delta}Y_{2}^{\delta})^{n-j-1}S^{(j+1)\delta}\psi_{0}$ belongs
to $L^{\infty}$ because of Lemma 3 and Lemma 4, the technical assumption
concerns the uniformity of the bound with respect to $n$. Assumption (22) is
necessary when we make use of the estimate obtained in Lemma 5 and its is a
rather usual kind of assumption in such a contest (see, e.g., equation (2.4c)
in Lemma 2.3 by [5] and its application in equation (16) of the same paper,
see also the assumptions of Theorem 1 by [13]).

Here we state our main result.

######  Theorem 1.

Let $\sigma\geq\frac{1}{2}$; let $T>0$ be any fixed positive real number and
let $\psi_{0}\in{\Gamma}$ be such that $S^{t}\psi_{0}\in{\Gamma}$ for any
$t\in[0,T]$. Let $\delta>0$ and $n\in{\mathbb{N}}$ such that $t=n\delta\leq
T$. Let (22) holds true. Then, there exists a positive constant
$C:=C(\psi_{0},T)$ depending on $\psi_{0}$ and $T$ such that

$\displaystyle\left\|\left[X^{\delta}_{2}Y^{\delta}_{2}\right]^{n}\psi_{0}-S^{n\delta}\psi_{0}\right\|_{L^{2}}\leq
C\delta|\nu|\,.$ (23)

######  Remark 1.

In fact, we expect that such a result may be extended to subquadratic
potentials $V(x)\in C^{\infty}({\mathbb{R}})$ such that
$\left\|\frac{\partial^{r}V(x)}{\partial x^{r}}\right\|_{L^{\infty}}\leq C$ as
soon as $r\geq 2$. Also the extension to an higher dimension $d\geq 2$ could
be considered, too. However, in both two cases we have to face some problems:
e.g. the proof of Lemma 3 is based on the explicit expression of the
propagator $X^{t}$ of the linear Schrödinger operator. Some results [6]
concerning the generalized Mehler’s formula could be the basis for such an
extension. However, we don’t dwell here on the details concerning these two
generalizations.

##  3\. Proof of Theorem 1

Hereafter, in this Section we simply denote $X_{2}$ and $Y_{2}$ respectively
by $X$ and $Y$.

###  3.1. Preliminary results

We require some preliminary Lemmas and Remarks.

######  Lemma 1.

${{\Gamma}}\subseteq L^{p}$ for any $p\in[1,+\infty]$. In particular

$\displaystyle\|w\|_{L^{1}({\mathbb{R}})}\leq
C\left[\|x^{2}w\|_{L^{2}({\mathbb{R}})}+\|w\|_{L^{2}({\mathbb{R}})}\right]\,,$
(24)

where $C=(2^{5}/3)^{1/8}$.

###### Proof.

The statement ${{\Gamma}}\subseteq L^{p}$ holds true for $p=+\infty$ by making
use of the Gagliardo-Nirenberg inequality:

$\displaystyle\|w\|_{L^{\infty}}\leq
C\|w\|_{L^{2}}^{\frac{1}{2}}\|w^{\prime}\|_{L^{2}}^{\frac{1}{2}}\leq
C\|w\|_{{{\Gamma}}}.$

If we are able to prove that the statement holds true for $p=+1$ too, then the
Riesz-Thorin interpolation Theorem prove the statement for any
$p\in[+1,+\infty]$. In order to prove the statement when $p=+1$ we observe
that for any $R>0$

$\displaystyle\|w\|_{L^{1}({\mathbb{R}})}$ $\displaystyle=$
$\displaystyle\left[\int_{-\infty}^{-R}|w(x)|dx+\int_{+R}^{+\infty}|w(x)|dx+\int_{-R}^{+R}|w(x)|dx\right]$
$\displaystyle=$
$\displaystyle\left[\int_{-\infty}^{-R}\frac{1}{x^{2}}x^{2}|w(x)|dx+\int_{+R}^{+\infty}\frac{1}{x^{2}}x^{2}|w(x)|dx+\int_{-R}^{+R}|w(x)|dx\right]$
$\displaystyle=$ $\displaystyle\langle
x^{-2},x^{2}w\rangle_{L^{2}(-\infty,-R)}+\langle
x^{-2},x^{2}w\rangle_{L^{2}(R,\infty)}+\langle 1,|w|\rangle_{L^{2}(-R,+R)}$
$\displaystyle\leq$
$\displaystyle\frac{2}{\sqrt{3}}R^{-3/2}\|x^{2}w\|_{L^{2}({\mathbb{R}})}+\sqrt{2R}\|w\|_{L^{2}({\mathbb{R}})}<+\infty$

from the Hölder’s inequality. Hence, (24) follows for $R=(2/3)^{1/4}$. ∎

######  Remark 2.

From Lemma 1 it follows that

$\displaystyle\|w\|_{L^{1}}\leq C\|w\|_{{\Gamma}}\,.$

The following result holds true

######  Lemma 2.

Let $w\in{{\Gamma}}$ then

$\displaystyle\|e^{-iHt}w-w\|_{L^{2}}\leq C|t|\|w\|_{{\Gamma}}$

where $C=\max[1,|\alpha|]$.

###### Proof.

Indeed, since $w\in{{\Gamma}}\subset{\mathcal{D}}$ where ${\mathcal{D}}$ is
the self-adjointness domain of $H$, then the evolution
$v_{t}(x):=\left[e^{-itH}w\right](x)\in{\mathcal{D}}$ is such that

$\displaystyle\|e^{-itH}w-w\|_{L^{2}}$ $\displaystyle=$
$\displaystyle\|v_{t}-v_{0}\|_{L^{2}}=\left\|\int_{0}^{t}\dot{v}_{\tau}d\tau\right\|_{L^{2}}=\left\|\int_{0}^{t}iHv_{\tau}d\tau\right\|_{L^{2}}$
$\displaystyle=$ $\displaystyle\left\|\int_{0}^{t}iHe^{-i\tau
H}wd\tau\right\|_{L^{2}}=\left\|\int_{0}^{t}e^{-i\tau
H}Hwd\tau\right\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle|t|\,\left\|Hw\right\|_{L^{2}}\leq|t|\,\left[\|w^{\prime\prime}\|_{L^{2}}+|\alpha|\,\|x^{2}w\|_{L^{2}}\right]\,,$

since the two operators $H$ and $e^{-itH}$ commute: $[H,e^{-itH}]=0$. ∎

Furthermore, we have that

######  Lemma 3.

Let $w\in{{\Gamma}}$, then $X^{t}w\in{{\Gamma}}$ for any $t\in[0,T]$. In
particular:

$\displaystyle\left\|X^{t}w\right\|_{{{\Gamma}}}\leq
C\left\|w\right\|_{{{\Gamma}}}\,,$ (25)

for some positive constant $C>0$ independent of $t$ and $w$.

###### Proof.

Assume, for argument’s sake, that $\alpha=+\frac{1}{4}\omega^{2}$. Now, let
$a>0$ be fixed and small enough, and let us consider, at first, the case where
$a\leq\left|t-n\frac{\pi}{\omega}\right|\leq\frac{\pi}{\omega}-a$,
$n\in{\mathbb{Z}}$. Let us recall that

$\displaystyle\left(X^{t}w\right)(x)$ $\displaystyle:=$
$\displaystyle\left[e^{-itH}w\right](x)=\int_{{\mathbb{R}}}K_{HO}(x,y,t)w(y)dy$
$\displaystyle=$ $\displaystyle\sqrt{\frac{\omega}{4\pi i\sin(\omega
t)}}\int_{{\mathbb{R}}}e^{i\frac{\omega}{4\sin(\omega
t)}\left[(x^{2}+y^{2})\cos(\omega t)-2xy\right]}w(y)dy$

from the Mehler’s formula (42). Hence, for any positive integer $n$

$\displaystyle x^{n}\left[e^{-itH}w\right](x)=\sqrt{\frac{\omega}{4\pi
i\sin(\omega t)}}\int_{{\mathbb{R}}}x^{n}e^{i\frac{\omega}{4\sin(\omega
t)}\left[(x^{2}+y^{2})\cos(\omega t)-2xy\right]}w(y)dy$ $\displaystyle\ \
=\frac{1}{\sqrt{2\pi}}\left[\frac{i{2\sin(\omega
t)}}{\omega}\right]^{n-\frac{1}{2}}e^{i\frac{\omega x^{2}\cos(\omega
t)}{4\sin(\omega t)}}\int_{{\mathbb{R}}}e^{-i\frac{\omega xy}{2\sin(\omega
t)}}\frac{\partial^{n}\left[e^{i\frac{\omega y^{2}\cos(\omega t)}{4\sin(\omega
t)}}w(y)\right]}{\partial y^{n}}dy$

In particular, for $n=1$ and $n=2$ it turns out that

$\displaystyle x\left[e^{-itH}w\right](x)$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}}K_{HO}(x,y,t)\left[a_{1}(t)yw(y)+b_{1}(t)w^{\prime}(y)\right]dy$
$\displaystyle=$
$\displaystyle\left\\{e^{-itH}\left[a_{1}(t)yw(y)+b_{1}(t)w^{\prime}(y)\right]\right\\}(x)$
$\displaystyle x^{2}\left[e^{-itH}w\right](x)$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}}K_{HO}(x,y,t)\left[a_{2}(t)y^{2}w(y)+b_{2}(t)yw^{\prime}(y)+c_{2}(t)w^{\prime\prime}(y)\right]dy$
$\displaystyle=$
$\displaystyle\left\\{e^{-itH}\left[a_{2}(t)y^{2}w(y)+b_{2}(t)yw^{\prime}(y)+c_{2}(t)w^{\prime\prime}(y)\right]\right\\}(x)$

for some bounded functions $a_{1}(t)$, $b_{1}(t)$, $a_{2}(t)$, $b_{2}(t)$ and
$c_{2}(t)$ since
$a\leq\left|t-n\frac{\pi}{\omega}\right|\leq\frac{\pi}{\omega}-a$. Then, we
can conclude that

$\displaystyle\left\|x\left[e^{-itH}w\right]\right\|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle|a_{1}(t)|\,\|yw\|_{L^{2}}+|b_{1}(t)|\,\|w^{\prime}\|_{L^{2}}\leq
C\|w\|_{{\Gamma}}$
$\displaystyle\left\|x^{2}\left[e^{-itH}w\right]\right\|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle|a_{2}(t)|\,\|y^{2}w\|_{L^{2}}+|b_{2}(t)|\,\|yw^{\prime}\|_{L^{2}}+|c_{2}(t)|\|w^{\prime\prime}\|_{L^{2}}\leq
C\|w\|_{{\Gamma}}$

for some $C$ since

$\displaystyle\|yw\|_{L^{2}}\leq\|w\|_{L^{2}}^{1/2}\|y^{2}w\|_{L^{2}}^{1/2}$

and

$\displaystyle\|yw^{\prime}\|_{L^{2}}\leq\frac{1}{2}\left[\|y^{2}w\|_{L^{2}}+\|w^{\prime\prime}\|_{L^{2}}\right]\,.$

Indeed, the last inequality follows by observing that

$\displaystyle\left\|yw^{\prime}\right\|_{L^{2}}^{2}=\langle
yw^{\prime},yw^{\prime}\rangle=-\langle 2yw^{\prime},w\rangle-\langle
y^{2}w^{\prime\prime},w\rangle$

and thus

$\displaystyle\left\|yw^{\prime}\right\|_{L^{2}}^{2}\leq
2\|yw^{\prime}\|_{L^{2}}\|w\|_{L^{2}}+\|w^{\prime\prime}\|_{L^{2}}\|y^{2}w\|_{L^{2}}\,.$

Similarly, if one notices that

$\displaystyle\frac{\partial}{\partial x}\left[e^{-itH}w\right](x)$
$\displaystyle=$ $\displaystyle
a_{3}(t)x\left[e^{-itH}w\right](x)+b_{3}(t)\left[e^{-itH}xw\right](x)$

for some bounded functions $a_{3}(t)$ and $b_{3}(t)$, then the same arguments
as above prove that

$\displaystyle\left\|\frac{\partial^{2}}{\partial
x^{2}}\left[e^{-itH}w\right]\right\|_{L^{2}}\leq C\|w\|_{{\Gamma}}$

for some $C>0$.

Now, one can check that (25) holds true for any $t$; indeed if $t$ is such
that $|t|<a$ then we observe that

$\displaystyle e^{-itH}w=e^{iaH}e^{-i(t+a)H}w$

from which, since $a\leq|t+a|\leq\frac{\pi}{\omega}-a$ if $0<t<a$ and $a$ is
small enough,

$\displaystyle\left\|e^{-itH}w\right\|_{{\Gamma}}=\left\|e^{iaH}e^{-i(t+a)H}w\right\|_{{\Gamma}}\leq
C\left\|e^{-i(t+a)H}w\right\|_{{\Gamma}}\leq
C^{2}\left\|w\right\|_{{\Gamma}}\,.$

The case $\left|t-n\frac{\pi}{\omega}\right|<a$, $n\in{\mathbb{Z}}$, follows
in the same way, too.

Eventually, the case $\alpha=-\frac{1}{4}\omega^{2}<0$ is similarly treated by
making use of (43). ∎

Concerning the evolution operator

$\displaystyle\left(Y^{t}w\right)(x):=e^{-i\nu|w(x)|^{2\sigma}t}w(x)\,,$

we recall that

$\displaystyle\|Y^{t}w\|_{L^{p}}=\|w\|_{L^{p}}\,,\ \forall p\in[1,+\infty]\,.$

Furthermore:

######  Lemma 4.

Let $w\in{{\Gamma}}$, then $Y^{t}w\in{{\Gamma}}$ for any $t$; in particular

$\displaystyle\|Y^{t}w\|_{{\Gamma}}\leq\left[1+C|\nu
t|\|w\|_{L^{\infty}}^{2\sigma}\right]^{2}\,\|w\|_{{\Gamma}}\,.$

for some positive constant $C>0$ independent of $t$ and $w$.

###### Proof.

A straightforward calculation proves that

$\displaystyle\|x^{2}Y^{t}w\|_{L^{2}}=\|x^{2}w\|_{L^{2}}$

and that

$\displaystyle\left\|\frac{\partial^{2}Y^{t}w}{\partial
x^{2}}\right\|_{L^{2}}\leq\left[1+C|\nu
t|\,\|w\|_{L^{\infty}}^{2\sigma}\right]^{2}\|w\|_{H^{2}}\,.$ (26)

Indeed,

$\displaystyle\left\|\frac{\partial^{2}Y^{t}w}{\partial
x^{2}}\right\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\|w^{\prime\prime}\|_{L^{2}}+C\left[|t\nu|\,\|w^{2\sigma}w^{\prime\prime}\|_{L^{2}}+\nu^{2}t^{2}\|w^{4\sigma-1}\left(w^{\prime}\right)^{2}\|_{L^{2}}+|\nu
t|\|w^{2\sigma-1}\left(w^{\prime}\right)^{2}\|_{L^{2}}\right]$
$\displaystyle\leq$ $\displaystyle\|w^{\prime\prime}\|_{L^{2}}+C\left[|\nu
t|\|w\|_{L^{\infty}}^{2\sigma}\|w^{\prime\prime}\|_{L^{2}}+\nu^{2}t^{2}\|w\|_{L^{\infty}}^{4\sigma-1}\|\left(w^{\prime}\right)^{2}\|_{L^{2}}+|\nu
t|\,\|w\|_{L^{\infty}}^{2\sigma-1}\|\left(w^{\prime}\right)^{2}\|_{L^{2}}\right]$

since $\sigma\geq 1/2$. Concerning the term
$\|\left(w^{\prime}\right)^{2}\|_{L^{2}}$ we have that

$\displaystyle\|\left(w^{\prime}\right)^{2}\|_{L^{2}}^{2}$ $\displaystyle=$
$\displaystyle\left|\int_{{\mathbb{R}}}\left({\bar{w}^{\prime}}\right)^{2}\left({w^{\prime}}\right)^{2}dx\right|=\left|-\int_{{\mathbb{R}}}w\left[2w^{\prime}\bar{w}^{\prime}\bar{w}^{\prime\prime}+w^{\prime\prime}\left({\bar{w}^{\prime}}\right)^{2}\right]dx\right|$
$\displaystyle\leq$ $\displaystyle
3\|w\|_{L^{\infty}}\int_{{\mathbb{R}}}|w^{\prime\prime}|\,\left|{w^{\prime}}\right|^{2}dx\leq
C\|w\|_{L^{\infty}}\|w^{\prime\prime}\|_{L^{2}}\|(w^{\prime})^{2}\|_{L^{2}}\,;$

hence

$\displaystyle\|\left(w^{\prime}\right)^{2}\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle C\|w\|_{L^{\infty}}\|w^{\prime\prime}\|_{L^{2}}\,.$ (27)

Thus, we conclude that

$\displaystyle\left\|\frac{\partial^{2}Y^{t}w}{\partial
x^{2}}\right\|_{L^{2}}\leq\|w^{\prime\prime}\|_{L^{2}}+C\left[2|\nu
t|\,\|w\|_{L^{\infty}}^{2\sigma}+\nu^{2}t^{2}\|w\|_{L^{\infty}}^{4\sigma}\right]\|w^{\prime\prime}\|_{L^{2}}$

from which (26) follows. ∎

The evolution operator $Y^{t}$ satisfies to the Lipschitz condition, too (see
Lemmas 2 and 3 [13]).

######  Lemma 5.

Let $w_{1},w_{2}\in L^{2}\cap L^{\infty}$ and let

$\displaystyle
M:=\max\left[\|w_{1}\|_{L^{\infty}},\|w_{2}\|_{L^{\infty}}\right]\,.$

Then,

$\displaystyle\|Y^{t}w_{1}-Y^{t}w_{2}\|_{L^{2}}\leq\left[1+2\sigma|\nu
t|M^{2\sigma-1}\right]\|w_{1}-w_{2}\|_{L^{2}}\,.$

######  Remark 3.

Since the linear operator $X^{t}:=e^{-itH}$ is unitary from $L^{2}$ to $L^{2}$
then the Lipschitz condition

$\displaystyle\|X^{t}Y^{t}w_{1}-X^{t}Y^{t}w_{2}\|_{L^{2}}\leq\left[1+2\sigma|\nu
t|M^{2\sigma-1}\right]\|w_{1}-w_{2}\|_{L^{2}}$

holds true.

Finally.

######  Lemma 6.

Let $F(w):=|w|^{2\sigma}w$, $w\in{{\Gamma}}$, then $F(w)\in{{\Gamma}}$; in
particular

$\displaystyle\|F(w)\|_{{\Gamma}}\leq
C\|w\|_{L^{\infty}}^{2\sigma}\|w\|_{{\Gamma}}\,,$

for some positive constant $C>0$ independent of $w$.

###### Proof.

At first we consider

$\displaystyle\|x^{2}F(w)\|_{L^{2}}=\|x^{2}|w|^{2\sigma}w\|_{L^{2}}\leq\|w\|_{L^{\infty}}^{2\sigma}\|x^{2}w\|_{L^{2}}\leq\|w\|_{L^{\infty}}^{2\sigma}\|w\|_{{\Gamma}}$

and then, similarly,

$\displaystyle\left\|\frac{\partial^{2}F(w)}{\partial
x^{2}}\right\|_{L^{2}}\leq
C\left[\|w\|_{L^{\infty}}^{2\sigma}\,\|w^{\prime\prime}\|_{L^{2}}+\|w\|_{L^{\infty}}^{2\sigma-1}\|(w^{\prime})^{2}\|_{L^{2}}\right]\leq
C\|w\|_{L^{\infty}}^{2\sigma}\,\|w^{\prime\prime}\|_{L^{2}}$

since (27). ∎

######  Remark 4.

Finally, we recall here some previous technical results. In particular in
Lemma 4 by [13] we proved that

$\displaystyle\|F(w_{1})-F(w_{2})\|_{L^{2}}\leq(2\sigma+1)M^{2\sigma}\|w_{1}-w_{2}\|_{L^{2}}\,,$
(28)

where $M=\max\left[\|w_{1}\|_{L^{\infty}},\|w_{2}\|_{L^{\infty}}\right]$.

###  3.2. Estimate of the remainder term

Now, let $S^{t}$ be the evolution operator associated to the Cauchy problem
(21); it satisfies to the mild equation

$\displaystyle\psi_{t}$ $\displaystyle=$ $\displaystyle
S^{t}\psi_{0}=X^{t}\psi_{0}-i\nu\int_{0}^{t}X^{t-s}|\psi_{s}|^{2\sigma}\psi_{s}ds$
$\displaystyle=$ $\displaystyle
X^{t}\psi_{0}-i\nu\int_{0}^{t}X^{t-s}F[S^{s}(\psi_{0})]ds\,.$

Now, we are going to compare $S^{t}\psi_{0}$ with $X^{t}Y^{t}\psi_{0}$ where
$Y^{t}$ satisfies to the mild equation

$\displaystyle
Y^{t}\psi_{0}=\psi_{0}-i\nu\int_{0}^{t}F\left[Y^{s}(\psi_{0})\right]ds\,.$

Then, we prove that

######  Theorem 2.

Let $w\in{{\Gamma}}$ and let $T>0$ be fixed, then

$\displaystyle\|S^{t}w-X^{t}Y^{t}w\|_{L^{2}}\leq|\nu|C_{2}t^{2}e^{C_{1}t}\,,\
\forall t\in[0,T]\,,$

where $C_{1}$ and $C_{2}$ are two positive constants given by:

$\displaystyle
C_{1}:=C_{1}(w,t)=|\nu|(2\sigma+1)\max_{s\in[0,t]}\left\\{\max\left[\|S^{s}w\|_{L^{\infty}},\|X^{s}Y^{s}w\|_{L^{\infty}}\right]\right\\}^{2\sigma+1}\,,$
(29)

and

$\displaystyle
C_{2}:=C_{2}(w)=C\|w\|_{{\Gamma}}^{2\sigma+1}\max\left[1,T^{2}\nu^{2}\|w\|_{{\Gamma}}^{4\sigma}\right]^{2\sigma+1}\,,$
(30)

where $C>0$ is a positive constant independent of $w$, $t$, $\nu$ and $T$.

######  Remark 5.

Indeed, if we assume that $S^{s}(w)$ does not blow up for $s\in[0,T]$ then
$\|S^{s}(w)\|_{L^{\infty}}$ is uniformly bounded on time; furthermore, from
Lemmas 3 and 4, we already known that

$\displaystyle\|X^{s}Y^{s}w\|_{L^{\infty}}$ $\displaystyle\leq$
$\displaystyle\|X^{s}Y^{s}w\|_{{\Gamma}}\leq C\|Y^{s}w\|_{{\Gamma}}$
$\displaystyle\leq$ $\displaystyle
C\left[1+C|s\nu|\|w\|_{L^{\infty}}^{2\sigma}\right]^{2}\|w\|_{{\Gamma}}\,.$

Hence $C_{1}(w)<+\infty$.

###### Proof.

Let $w\in{{\Gamma}}$, then we have that

$\displaystyle
S^{t}w-X^{t}Y^{t}w=-i\nu\left[\int_{0}^{t}X^{t-s}F\left[S^{s}(w)\right]ds-\int_{0}^{t}X^{t}F\left[Y^{s}(w)\right]ds\right]$
$\displaystyle\ \
=-i\nu\int_{0}^{t}X^{t-s}\left\\{F\left[S^{s}(w)\right]-F\left[X^{s}Y^{s}(w)\right]\right\\}ds+{\mathcal{R}}(t,w)$
(31)

where

$\displaystyle{\mathcal{R}}(t,w)=-i\nu\int_{0}^{t}X^{t-s}{\mathcal{R}}_{I}(s,w)ds$

and

$\displaystyle{\mathcal{R}}_{I}(s,w)=F\left[X^{s}Y^{s}w\right]-X^{s}F\left[Y^{s}(w)\right]\,.$

######  Lemma 7.

Let

$\displaystyle
M_{s}:=\max\left[\|X^{s}Y^{s}w\|_{L^{\infty}},\|Y^{s}w\|_{L^{\infty}}\right]\,.$

Then

$\displaystyle\|{\mathcal{R}}_{I}(s,w)\|_{L^{2}}\leq
C|s|M_{s}^{2\sigma}\max\left[1,s^{2}\nu^{2}M_{s}^{4\sigma}\right]\|w\|_{{\Gamma}}$

for some positive constant $C>0$ independent of $s$, $\nu$ and $w$.

###### Proof.

Indeed,

$\displaystyle\|{\mathcal{R}}_{I}(s,w)\|_{L^{2}}$ $\displaystyle=$
$\displaystyle\|F\left[X^{s}Y^{s}w\right]-X^{s}F\left[Y^{s}(w)\right]\|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle\|F\left[X^{s}Y^{s}w\right]-F\left[Y^{s}w\right]\|_{L^{2}}+\|X^{s}F\left[Y^{s}(w)\right]-F\left[Y^{s}w\right]\|_{L^{2}}$

where from (28) and from Lemma 2 it follows that

$\displaystyle\|F\left[X^{s}Y^{s}w\right]-F\left[Y^{s}w\right]\|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle(2\sigma+1)M_{s}^{2\sigma}\|X^{s}Y^{s}w-Y^{s}w\|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle(2\sigma+1)M_{s}^{2\sigma}C|s|\|Y^{s}w\|_{{\Gamma}}\,.$

Concerning the other term we apply, at first, Lemma 2 and then Lemma 6
obtaining that

$\displaystyle\|X^{s}F\left[Y^{s}(w)\right]-F\left[Y^{s}w\right]\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle C|s|\|F\left[Y^{s}w\right]\|_{{\Gamma}}\leq
C|s|\|Y^{s}w\|_{L^{\infty}}^{2\sigma}\|Y^{s}w\|_{{\Gamma}}$
$\displaystyle\leq$ $\displaystyle C|s|M_{s}^{2\sigma}\|Y^{s}w\|_{{\Gamma}}$

Hence, we have proved that

$\displaystyle\|{\mathcal{R}}_{I}\|_{L^{2}}\leq
C|s|M_{s}^{2\sigma}\|Y^{s}w\|_{{\Gamma}}\,.$

From this result and from Lemma 4 the proof follows. ∎

######  Remark 6.

From Remark 5 it follows that

$\displaystyle
M_{s}\leq\max\left[1+C|s\nu|\|w\|_{L^{\infty}}^{2\sigma}\right]^{2}\|w\|_{{\Gamma}}\,.$

Thus

$\displaystyle\|{\mathcal{R}}_{I}(s,w)\|_{L^{2}}\leq
C|s|\max\left[1,s^{2}\nu^{2}\|w\|_{{\Gamma}}^{4\sigma}\right]^{2\sigma+1}\|w\|_{{\Gamma}}^{2\sigma+1}$

for some positive constant $C>0$ independent of $s$ and $w$.

Then, an estimate of the term ${\mathcal{R}}$ will follow

######  Lemma 8.

Let $w\in{{\Gamma}}$, then

$\displaystyle\|{\mathcal{R}}(t,w)\|_{L^{2}}\leq|\nu|C_{2}(w)t^{2}\,.$

###### Proof.

Indeed, let $t\geq 0$ for argument’s sake; then:

$\displaystyle\|{\mathcal{R}}(t,w)\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle|\nu|\int_{0}^{t}\|X^{t-s}{\mathcal{R}}_{I}(s,w)\|_{L^{2}}ds$
$\displaystyle\leq$
$\displaystyle|\nu|\int_{0}^{t}\|{\mathcal{R}}_{I}(s,w)\|_{L^{2}}ds$
$\displaystyle\leq$
$\displaystyle|\nu|\int_{0}^{t}Cs\max\left[1,s^{2}\nu^{2}\|w\|_{{\Gamma}}^{4\sigma}\right]^{2\sigma+1}\|w\|_{{\Gamma}}^{2\sigma+1}ds$

from which the Lemma follows. ∎

Now, we are ready to estimate the difference (31):

$\displaystyle\|S^{t}w-X^{t}Y^{t}w\|_{L^{2}}\leq|\nu|\int_{0}^{t}\left\|X^{t-s}\left\\{F\left[S^{s}w\right]-F\left[X^{s}Y^{s}w\right]\right\\}\right\|_{L^{2}}ds+\|{\mathcal{R}}(t,w)\|_{L^{2}}$
$\displaystyle\
\leq|\nu|\int_{0}^{t}\left\|F\left[S^{s}w\right]-F\left[X^{s}Y^{s}w\right]\right\|_{L^{2}}ds+\|{\mathcal{R}}(t,w)\|_{L^{2}}$
$\displaystyle\ \
\leq|\nu|(2\sigma+1)\int_{0}^{t}\max\left[\|S^{s}w\|_{L^{\infty}},\|X^{s}Y^{s}w\|_{L^{\infty}}\right]^{2\sigma}\left\|S^{s}w-X^{s}Y^{s}w\right\|_{L^{2}}ds+\|{\mathcal{R}}(t,w)\|_{L^{2}}$
$\displaystyle\ \ \leq
C_{1}(w,t)\int_{0}^{t}\left\|S^{s}w-X^{s}Y^{s}w\right\|_{L^{2}}ds+|\nu|C_{2}(w)t^{2}$

from Remark 4, recalling that $X^{t}$ is an unitary operator on $L^{2}$ and
where $C_{1}(w,t)$ and $C_{2}(w)$ are respectively defined by (29) and (30).
That is

$\displaystyle y(t)\leq C_{1}\int_{0}^{t}y(s)ds+|\nu|C_{2}t^{2}\,,\
t\in[0,T]\,,$

where we set

$\displaystyle y(t):=\|S^{t}w-X^{t}Y^{t}w\|_{L^{2}}\,.$

Thus, the Gronwall’s Lemma implies that

$\displaystyle y(t)\leq|\nu|C_{2}t^{2}e^{C_{1}t}\,,\ \forall t\in[0,T]\,.$

and Theorem 2 is so proved. ∎

Finally, we can conclude the proof of Theorem 1. Let us fix $t\leq T$, let
$\delta>0$ small enough and let $n\in{\mathbb{N}}$ such that $t=n\delta$, let
$Z^{\delta}=X^{\delta}Y^{\delta}$; then, the triangle inequality yields to

$\displaystyle\left\|\left(Z^{\delta}\right)^{n}\psi_{0}-S^{n\delta}\psi_{0}\right\|_{L^{2}}$
$\displaystyle=$
$\displaystyle\left\|\sum_{j=0}^{n-1}\left[\left(Z^{\delta}\right)^{(n-j-1)}Z^{\delta}S^{j\delta}\psi_{0}-\left(Z^{\delta}\right)^{(n-j-1)}S^{(j+1)\delta}\psi_{0}\right]\right\|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle\sum_{j=0}^{n-1}\left\|Z^{\delta}\left(Z^{\delta}\right)^{(n-j-1)}S^{j\delta}\psi_{0}-Z^{\delta}\left(Z^{\delta}\right)^{(n-j-2)}S^{(j+1)\delta}\psi_{0}\right\|_{L^{2}}\,.$

From this inequality, by making use of Lemma 5, Remark 3 and Theorem 2 it
follows that

$\displaystyle c_{n-j-1,j}$ $\displaystyle:=$
$\displaystyle\left\|Z^{\delta}\left(Z^{\delta}\right)^{(n-j-1)}S^{j\delta}\psi_{0}-Z^{\delta}\left(Z^{\delta}\right)^{(n-j-2)}S^{(j+1)\delta}\psi_{0}\right\|_{L^{2}}$
(32) $\displaystyle\leq$
$\displaystyle\left[1+2\sigma|\nu\delta|C^{2\sigma-1}\right]c_{n-j-2,j}$
$\displaystyle\leq$
$\displaystyle\left[1+2\sigma|\nu\delta|C^{2\sigma-1}\right]^{n-j-1}c_{0,j}$

for some positive constant $C$ independent of $n$ since (22). Therefore, we
have proved that

$\displaystyle\left\|\left(Z^{\delta}\right)^{n}\psi_{0}-S^{n\delta}\psi_{0}\right\|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle\sum_{j=0}^{n-1}\left[1+2\sigma|\nu\delta|C^{2\sigma-1}\right]^{n-j-1}\left\|Z^{\delta}S^{j\delta}\psi_{0}-S^{(j+1)\delta}\psi_{0}\right\|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle\sum_{j=0}^{n-1}\left[1+2\sigma|\nu\delta|C^{2\sigma-1}\right]^{n-j-1}|\nu|C_{2,j}\delta^{2}e^{C_{1,j}\delta}$

where

$\displaystyle C_{2,j}$ $\displaystyle:=$ $\displaystyle
C_{2}(S^{j\delta}\psi_{0})=C\|S^{j\delta}\psi_{0}\|_{{\Gamma}}^{2\sigma+1}\max\left[1,T^{2}\nu^{2}\left\|S^{j\delta}\psi_{0}\right\|_{{{\Gamma}}}^{4\sigma}\right]^{2\sigma+1}$
$\displaystyle C_{1,j}$ $\displaystyle:=$ $\displaystyle
C_{1}(S^{j\delta}\psi_{0},\delta)=|\nu|(2\sigma+1)\max_{s\in[0,\delta]}\left\\{\max\left[\left\|S^{(j+1)\delta}\psi_{0}\right\|_{L^{\infty}},\left\|Z^{s}S^{j\delta}\psi_{0}\right\|_{L^{\infty}}\right]\right\\}\,.$

From Lemma 3 and Lemma 4 then it follows that

$\displaystyle\left\|Z^{s}S^{j\delta}\psi_{0}\right\|_{L^{\infty}}\leq\left\|Z^{s}S^{j\delta}\psi_{0}\right\|_{{{\Gamma}}}\leq
C\max\left[1,\nu^{2}\delta^{2}\left\|S^{j\delta}\psi_{0}\right\|_{L^{\infty}}^{4\sigma}\right]\left\|S^{j\delta}\psi_{0}\right\|_{{{\Gamma}}}$

and that

$\displaystyle\left\|S^{(j+1)\delta}\psi_{0}\right\|_{L^{\infty}}\leq
C\left\|S^{(j+1)\delta}\psi_{0}\right\|_{{\Gamma}}\,.$

Since we assume that the solution $S^{t}\psi_{0}\in{{\Gamma}}$ for any $t\leq
T$ then we can conclude that

$\displaystyle C_{1,j}\,,\ C_{2,j}\leq C_{3}\,,\ \forall j=0,1,\ldots,n-1\,,$

for some positive constant $C_{3}:=C_{3}(T,\psi_{0})>0$. Hence,

$\displaystyle\left\|\left(Z^{\delta}\right)^{n}\psi_{0}-S^{n\delta}\psi_{0}\right\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
C|\nu|\delta^{2}\sum_{j=0}^{n-1}\left[1+2\sigma|\nu\delta|C^{2\sigma-1}\right]^{n-j-1}C_{3}e^{C_{3}\delta}$
$\displaystyle\leq$ $\displaystyle C\delta|\nu t|,\ t=n\delta<T\,,$

from which the proof of Theorem 1 follows.

##  4\. Numerical Experiments

For any fixed $t$ we numerically compute the approximate solutions

$\displaystyle\psi_{t,j}=[X_{j}^{\delta}Y^{\delta}_{j}]^{n}\psi_{0}$

for different values of $n$ where $\delta=\frac{t}{n}$;
$\psi_{j}=\left[X_{j}^{\delta}Y_{j}^{\delta}\right]^{n}\psi_{0}$, $j=1,2$,
where $X_{1}^{\delta}$ is the evolution operator associated to
$-\frac{\partial^{2}}{\partial x^{2}}$, $X_{2}^{\delta}$ is the evolution
operator associated to $-\frac{\partial^{2}}{\partial x^{2}}+V$,
$Y_{1}^{\delta}$ is the evolution operator associated to the differential
equation $i\dot{\psi}=V\psi+\nu|\psi|^{2\sigma}\psi$ and $Y_{2}^{\delta}$ is
the evolution operator associated to the differential equation
$i\dot{\psi}_{t}=\nu|\psi_{t}|^{2\sigma}\psi_{t}$. We consider the harmonic
oscillator potential where $V(x)=+\frac{1}{4}\omega^{2}x^{2}$ and the inverted
oscillator potential where $V(x)=-\frac{1}{4}\omega^{2}x^{2}$. In both cases
we consider the focusing (where $\nu<0$) and defocusing (where $\nu>0$)
nonlinearity.

In this Section, for simplicity’s sake, let us drop out the index $t$, i.e.
$\psi_{t}=\psi$, $\psi_{t,j}=\psi_{j}$, and so on. For argument’s sake the
initial wavefunction is a Gaussian function

$\displaystyle\psi_{0}(x)=\frac{1}{\sqrt[4]{2\pi
s^{2}}}e^{-(x-x_{0})^{2}/4s^{2}+iv_{0}x}$

where

$\displaystyle x_{0}=-3\,,\ v_{0}=2\ \mbox{ and }\ s=0.5\,.$

We compare in numerical experiments the rate of convergence of the numerical
solutions $\psi_{j}$. More precisely, we compare the probability densities

$\displaystyle\rho_{j}(x)=|\psi_{j}(x)|^{2}\,,\ j=1,2\,,$

and the expectation value of the position observable

$\displaystyle\langle
x\rangle_{j}:=\langle\psi_{j},x\psi_{j}\rangle_{L^{2}}\,,\ j=1,2\,,$

for a fixed value of $t$.

We recall that the evolution operators $Y_{1}^{\delta}$ and $Y_{2}^{\delta}$
are the multiplication operators (14) and (17); $X_{1}^{\delta}$ and
$X_{2}^{\delta}$ are the integral operators (39), (41) and (43). Since the
evolution operators $X_{j}^{\delta}$ are integral operators then we
numerically compute the integral on a large enough fixed interval
$[x_{min},x_{max}]$ by dividing it in $m$ intervals with the same amplitude
$\frac{x_{max}-x_{min}}{m}$, that is $m$ is the number of points of the mesh.
Let $\psi_{j}^{n,m}$ be the numerical solutions given by the vector
$\left(\left[X_{j}^{\delta}Y_{j}^{\delta}\right]^{n}\psi_{0}\right)(x_{\ell})$,
where $x_{\ell}=x_{min}+\ell\frac{x_{max}-x_{min}}{m}$ for
$\ell=0,1,\ldots,m$.

If we denote by $\psi_{j}^{\infty}$ and $\rho_{j}^{\infty}$ the values of
$\psi_{j}^{n,m}$ and $\rho_{j}^{n,m}$, $j=1,2$, where $n$ and $m$ are the
largest values considered in the numerical experiment, then we are going to
estimate the quantities

$\displaystyle\Delta_{j}^{n,m}=\max_{\ell=0,1,\ldots,m}|\rho_{j}^{\infty}(x_{\ell})-\rho_{j}^{n,m}(x_{\ell})|\,,\
j=1,2\,,$

for different values of $n$ and $m$. Furthermore, we consider also the
difference

$\displaystyle\delta^{n,m}:=\max_{\ell=0,1,\ldots,m}|\rho_{1}^{n,m}(x_{\ell})-\rho_{2}^{n,m}(x_{\ell})|\,.$

Finally, we compare also the exact expected value of the position observable
$\langle x\rangle^{t}=\langle\psi_{t},x\psi_{t}\rangle_{L^{2}}$ with the ones
$\langle x\rangle^{t}_{j}=\langle\psi_{j,t},x\psi_{j,}t\rangle_{L^{2}}$,
$j=1,2$, obtained with the two approximate solutions. In fact, Ehrenfest’s
Theorem for nonlinear Schödinger equations does not generically hold true in
the usual form, but when one cosniders the position $x$ and momentum $p$
observables we still have that

$\displaystyle\frac{d\langle x\rangle^{t}}{dt}=\frac{1}{m}\langle
p\rangle^{t}\ \mbox{ and }\ \frac{d\langle
p\rangle^{t}}{dt}=-\left\langle\frac{dV}{dx}\right\rangle^{t}$

where

$\displaystyle\langle
p\rangle^{t}=-i\left\langle\psi_{t},\frac{\partial\psi_{t}}{\partial
x}\right\rangle_{L^{2}}\ \mbox{ and }\
\left\langle\frac{dV}{dx}\right\rangle^{t}=\left\langle\psi_{t},\frac{dV}{dx}\psi_{t}\right\rangle_{L^{2}}\,.$

In particular, since $m=\frac{1}{2}$ and $V(x)=\alpha x^{2}$ then $\langle
x\rangle^{t}$ is solution to the differential equation

$\displaystyle\left\\{\begin{array}[]{l}\frac{d^{2}\langle
x\rangle^{t}}{dt^{2}}+\alpha\langle x\rangle^{t}=0\\\ \langle
x\rangle^{0}=x_{0}\ \mbox{ and }\ \left.\frac{d\langle
x\rangle^{t}}{dt}\right|_{t=0}=2\langle p\rangle^{0}=2v_{0}\end{array}\right.$

Thus

$\displaystyle\langle x\rangle^{t}=\left\\{\begin{array}[]{ll}x_{0}\cos(\omega
t)+\frac{2v_{0}}{\omega}\sin(\omega t)&\mbox{ if
}\alpha=+\frac{1}{4}\omega^{2}\\\ &\\\ x_{0}\cosh(\omega
t)+\frac{2v_{0}}{\omega}\sinh(\omega t)&\mbox{ if
}\alpha=-\frac{1}{4}\omega^{2}\end{array}\right.\,.$ (37)

###  4.1. Harmonic oscillator

In such an experiment let

$\displaystyle\omega=1\,,\ \sigma=1\ \mbox{ and }\ t=10\,.$

We numerically compute the integral operators $X_{1}^{\delta}$ and
$X_{2}^{\delta}$ where the integral domain is restricted to the interval
$[x_{min},x_{max}]$ where

$\displaystyle x_{min}=-50\ \mbox{ and }\ x_{max}=+50.$

The indexes $n$ and $m$ respectively run from $60$ to $240$ and from $2000$ to
$8000$; we denote by $\psi_{j}^{\infty}=\psi_{j}^{300,10000}$ the
corresponding numerical solution obtained when $n=300$, and thus
$\delta=\frac{1}{30}$, and $m=10000$.

####  4.1.1. Defocusing nonlinearity

We fix

$\displaystyle\nu=+10\,.$

The numerical experiment shows that the following upper bound of the absolute
value of the difference between the two probability densities
$\rho_{1}^{\infty}$ and $\rho_{2}^{\infty}$ holds true

$\displaystyle\max_{x}|\rho_{1}^{\infty}-\rho_{2}^{\infty}|=0.025\,,$ (38)

and in Figure 1 \- left hand side - we plot the graph of the function
$\rho_{2}^{\infty}$. In Table 1 we collect the difference $\Delta_{j}^{n,m}$
between $\rho_{j}^{n,m}$ and $\rho_{j}^{\infty}$, the ratio
$\Delta_{2}^{n,m}/\Delta_{1}^{n,m}$, the difference $\delta_{j}^{n,m}$ between
$\rho_{1}^{n,m}$ and $\rho_{2}^{n,m}$ and, finally, the expectation values
$\langle x\rangle_{1}^{t}$ and $\langle x\rangle_{2}^{t}$ for different values
of $n$ and $m$ and for $t=3$. It turns out that the values obtained in
correspondence of the approximation $\psi_{2}^{t}$ become rapidly stable even
for $n$ and $m$ not particularly large; in particular the expectation value
$\langle x\rangle_{2}^{10}$ is practically constant, while the expectation
value $\langle x\rangle_{1}^{10}$ slowly converges to its final value.

$\nu=+10$
---
$n$ | $m$ | $\Delta_{1}^{n,m}$ | $\Delta_{2}^{n,m}$ | ${\Delta_{2}^{n,m}}/{\Delta_{1}^{n,m}}$ | $\delta^{n,m}$ | $\langle x\rangle_{1}^{10}$ | $\langle x\rangle_{2}^{10}$
$60$ | $2000$ | $0.31$ | $0.18$ | $0.58$ | $0.16$ | $0.14$ | $0.34$
$90$ | $3000$ | $0.12$ | $0.12$ | $0.94$ | $0.08$ | $0.22$ | $0.34$
$120$ | $4000$ | $0.077$ | $0.046$ | $0.60$ | $0.070$ | $0.26$ | $0.34$
$150$ | $5000$ | $0.048$ | $0.027$ | $0.56$ | $0.055$ | $0.28$ | $0.34$
$180$ | $6000$ | $0.032$ | $0.017$ | $0.53$ | $0.044$ | $0.29$ | $0.34$
$210$ | $7000$ | $0.020$ | $0.010$ | $0.51$ | $0.037$ | $0.30$ | $0.34$
$240$ | $8000$ | $0.011$ | $0.0060$ | $0.53$ | $0.032$ | $0.30$ | $0.34$
$270$ | $9000$ | $0.0049$ | $0.0027$ | $0.55$ | $0.028$ | $0.31$ | $0.34$
Table 1. Table of values corresponding to the case of defocusing nonlinearity
$\nu=+10$ with harmonic oscillator potential
$V(x)=+\frac{1}{4}\omega^{2}x^{2}$. The exact expectation value is $\langle
x\rangle^{10}=0.3411$ from (37).

Figure 1. Harmonic oscillator. We plot the graph of the probability density
$\rho_{2}^{\infty}$ at $t=10$ of the numerical solution $\psi_{2}^{\infty}$ in
the defocusing case for $\nu=+10$ (left hand side picture) and in the focusing
case for $\nu=-10$ (right hand side picture).

####  4.1.2. Focusing nonlinearity

We fix

$\displaystyle\nu=-10\,.$

In this case the numerical experiment shows that the upper bound concerning
the difference between the two probability densities $\rho_{1}^{\infty}$ and
$\rho_{2}^{\infty}$ is of the same order of (38) since

$\displaystyle\max_{x}|\rho_{1}^{\infty}-\rho_{2}^{\infty}|=0.040\,,$

and in Figure 1 \- right hand side - we plot the graph of the function
$\rho_{2}^{\infty}$. In Table 2 we collect the difference $\Delta_{j}^{n,m}$
between $\rho_{j}^{n,m}$ and $\rho_{j}^{\infty}$, the ratio
$\Delta_{2}^{n,m}/\Delta_{1}^{n,m}$, the difference $\delta_{j}^{n,m}$ between
$\rho_{1}^{n,m}$ and $\rho_{2}^{n,m}$ and, finally, the expectation values
$\langle x\rangle_{1}^{t}$ and $\langle x\rangle_{2}^{t}$ for different values
of $n$ and $m$ and for $t=3$. It turns out that the values for the expectation
values coincide with the ones obtained in defocusing case; even in this case
the approximation $\psi_{2}^{t}$ become rapidly stable even for $n$ and $m$
not particularly large and we can observe the same behaviour of $\langle
x\rangle_{1}^{10}$ and $\langle x\rangle_{2}^{10}$ already observed in the
defocusing case (in fact, the expectation values are exactly the same of the
previous experiment).

$\nu=-10$
---
$n$ | $m$ | $\Delta_{1}^{n,m}$ | $\Delta_{2}^{n,m}$ | ${\Delta_{2}^{n,m}}/{\Delta_{1}^{n,m}}$ | $\delta^{n,m}$ | $\langle x\rangle_{1}^{10}$ | $\langle x\rangle_{2}^{10}$
$60$ | $2000$ | $0.43$ | $0.29$ | $0.68$ | $0.19$ | $0.14$ | $0.34$
$90$ | $3000$ | $0.34$ | $0.34$ | $1.00$ | $0.12$ | $0.22$ | $0.34$
$120$ | $4000$ | $0.17$ | $0.17$ | $1.00$ | $0.099$ | $0.26$ | $0.34$
$150$ | $5000$ | $0.076$ | $0.054$ | $0.71$ | $0.082$ | $0.28$ | $0.34$
$180$ | $6000$ | $0.086$ | $0.083$ | $0.96$ | $0.075$ | $0.29$ | $0.34$
$210$ | $7000$ | $0.081$ | $0.077$ | $0.95$ | $0.064$ | $0.30$ | $0.34$
$240$ | $8000$ | $0.038$ | $0.036$ | $0.94$ | $0.057$ | $0.30$ | $0.34$
$270$ | $9000$ | $0.023$ | $0.022$ | $0.94$ | $0.045$ | $0.31$ | $0.34$
Table 2. Table of values corresponding to the case of focusing nonlinearity
$\nu=-10$ with harmonic oscillator potential
$V(x)=+\frac{1}{4}\omega^{2}x^{2}$. The exact expectation value is $\langle
x\rangle^{10}=0.3411$ from (37).

###  4.2. Inverted oscillator

In such an experiment let

$\displaystyle\omega=1\,,\ \sigma=1\ \mbox{ and }\ t=3\,.$

We numerically compute the integral operators $X_{1}^{\delta}$ and
$X_{2}^{\delta}$ where the integral domain is restricted to the interval
$[x_{min},x_{max}]$ where

$\displaystyle x_{min}=-200\ \mbox{ and }\ x_{max}=+200.$

The indexes $n$ and $m$ respectively run from $30$ to $135$ and from $10000$
to $45000$; thus we denote by $\psi_{j}^{\infty}=\psi_{j}^{150,50000}$ the
corresponding numerical solution obtained when $n=150$, and thus
$\delta=\frac{1}{50}$, and $m=50000$.

####  4.2.1. Defocusing nonlinearity

We fix

$\displaystyle\nu=+10\,.$

In this case the two probability densities $\rho_{1}^{\infty}$ and
$\rho_{2}^{\infty}$ practically coincides since the numerical experiment shows
that the upper bound of their difference in much smaller than (38); indeed, it
takes the value

$\displaystyle\max_{x}|\rho_{1}^{\infty}-\rho_{2}^{\infty}|=0.00046\,,$

and in Figure 2 \- left hand side - we plot the graph of the function
$\rho_{2}^{\infty}$. In Table 3 we collect the difference $\Delta_{j}^{n,m}$
between $\rho_{j}^{n,m}$ and $\rho_{j}^{\infty}$, the ratio
$\Delta_{2}^{n,m}/\Delta_{1}^{n,m}$, the difference $\delta_{j}^{n,m}$ between
$\rho_{1}^{n,m}$ and $\rho_{2}^{n,m}$ and, finally, the expectation values
$\langle x\rangle_{1}^{t}$ and $\langle x\rangle_{2}^{t}$ for different values
of $n$ and $m$ and for $t=10$. It turns out that, as well as in the previous
experiments, the values obtained in correspondence of the approximation
$\psi_{2}^{t}$ become rapidly stable even for $n$ and $m$ not particularly
large.

$\nu=+10$
---
$n$ | $m$ | $\Delta_{1}^{n,m}$ | $\Delta_{2}^{n,m}$ | ${\Delta_{2}^{n,m}}/{\Delta_{1}^{n,m}}$ | $\delta^{n,m}$ | $\langle x\rangle_{1}^{3}$ | $\langle x\rangle_{2}^{3}$
$30$ | $10000$ | $0.017$ | $0.0050$ | $0.29$ | $0.017$ | $8.10$ | $9.87$
$45$ | $15000$ | $0.0024$ | $0.0017$ | $0.70$ | $0.0015$ | $8.84$ | $9.87$
$60$ | $20000$ | $0.0015$ | $0.0011$ | $0.69$ | $0.0011$ | $9.10$ | $9.87$
$75$ | $25000$ | $0.0010$ | $0.00069$ | $0.68$ | $0.0009$ | $9.26$ | $9.87$
$90$ | $30000$ | $0.00067$ | $0.00045$ | $0.67$ | $0.00075$ | $9.36$ | $9.87$
$105$ | $35000$ | $0.00043$ | $0.00029$ | $0.67$ | $0.00065$ | $9.43$ | $9.87$
$120$ | $40000$ | $0.00025$ | $0.00017$ | $0.66$ | $0.00057$ | $9.49$ | $9.87$
$135$ | $45000$ | $0.00011$ | $0.000073$ | $0.66$ | $0.00051$ | $9.53$ | $9.87$
Table 3. Table of values corresponding to the case of defocusing nonlinearity
$\nu=+10$ with inverted oscillator potential
$V(x)=-\frac{1}{4}\omega^{2}x^{2}$. The exact expectation value is $\langle
x\rangle^{3}=9.8685$ from (37).

Figure 2. Inverted oscillator. We plot the graph of the probability density
$\rho_{2}^{\infty}$ at $t=3$ of the numerical solution $\psi_{2}^{\infty}$ in
the defocusing case for $\nu=+10$ (left hand side picture) and in the focusing
case for $\nu=-10$ (right hand side picture).

####  4.2.2. Focusing nonlinearity

We fix

$\displaystyle\nu=-10\,.$

In this experiment we have that a significant difference between
$\rho_{1}^{\infty}$ and $\rho_{2}^{\infty}$ occurs because of a shift, as
shown in Figure 3, since

$\displaystyle\max_{x}|\rho_{1}^{\infty}-\rho_{2}^{\infty}|=0.37\,,$

that slowly decreases for increasing values of $n$ and $m$. Such a shift is
due to the fact that in the usual spectral splitting approximation
$(X_{1}^{\delta}Y_{1}^{\delta})^{n}\psi_{0}$ the linear part is approximated
as well as the nonlinear one; in contrast, in the proposed here spectral
splitting approximation $(X_{2}^{\delta}Y_{2}^{\delta})^{n}\psi_{0}$ the
linear part is exactly solved and the approximation only concerns the
nonlinear one. In order to reduce such a shift one has to significantly
increases the number $m$ of the points of the mesh and the number $n$ of
iterations in the usual spectral splitting approximation.

In Table 4 we collect the difference $\Delta_{j}^{n,m}$ between
$\rho_{j}^{n,m}$ and $\rho_{j}^{\infty}$, the ratio
$\Delta_{2}^{n,m}/\Delta_{1}^{n,m}$, the difference $\delta_{j}^{n,m}$ between
$\rho_{1}^{n,m}$ and $\rho_{2}^{n,m}$ and, finally, the expectation values
$\langle x\rangle_{1}^{t}$ and $\langle x\rangle_{2}^{t}$ for different values
of $n$ and $m$ and for $t=10$. Concerning the velocity of convergence of the
approximate solutions we can draw the same kind of conclusions of the previous
numerical experiments.

$\nu=-10$
---
$n$ | $m$ | $\Delta_{1}^{n,m}$ | $\Delta_{2}^{n,m}$ | ${\Delta_{2}^{n,m}}/{\Delta_{1}^{n,m}}$ | $\delta^{n,m}$ | $\langle x\rangle_{1}^{3}$ | $\langle x\rangle_{2}^{3}$
$30$ | $10000$ | $0.93$ | $0.12$ | $0.13$ | $0.89$ | $8.23$ | $9.86$
$45$ | $15000$ | $0.72$ | $0.063$ | $0.087$ | $0.82$ | $8.84$ | $9.87$
$60$ | $20000$ | $0.52$ | $0.037$ | $0.071$ | $0.73$ | $9.10$ | $9.87$
$75$ | $25000$ | $0.37$ | $0.023$ | $0.061$ | $0.64$ | $9.25$ | $9.87$
$90$ | $30000$ | $0.26$ | $0.014$ | $0.055$ | $0.56$ | $9.36$ | $9.87$
$105$ | $35000$ | $0.17$ | $0.0086$ | $0.051$ | $0.50$ | $9.43$ | $9.87$
$120$ | $40000$ | $0.099$ | $0.0048$ | $0.048$ | $0.45$ | $9.49$ | $9.87$
$135$ | $45000$ | $0.044$ | $0.0020$ | $0.046$ | $0.40$ | $9.53$ | $9.87$
Table 4. Table of values corresponding to the case of focusing nonlinearity
$\nu=-10$ with inverted oscillator potential
$V(x)=-\frac{1}{4}\omega^{2}x^{2}$. The exact expectation value is $\langle
x\rangle^{3}=9.8685$ from (37). Figure 3. Full line is the graph of the
function $\rho_{2}^{\infty}$, broken line is the graph of the function
$\rho_{1}^{\infty}$; the two graphs differ because of a small translation of
the spatial coordinate.

##  5\. Conclusions

Theorem 1 states that the result reported in this paper has at least as much
theoretical validity as the method based on the standard spectral splitting
approximation.

In fact, numerical experiments suggest that this new method has a
significantly higher speed of convergence than the standard method and
therefore it seems more suitable for performing sophisticated numerical
experiments. This higher speed of convergence is evident in Table 4 and in
Figure 3, but in reality it could already be observed in other cases as well,
although the effect is much less evident, and it is due to the fact that in
our proposed spectral splitting method the linear term is exactly treated and
not approximated as in the usual approximation. Indeed, if one compares the
absolute errors $\Delta_{1}^{n,m}$ and $\Delta_{2}^{n,m}$ that respectively
occur in the usual spectral splitting approximation and in the spectral
splitting approximation proposed in this paper then it turns out (see Table 5)
that the second approximation proposed here is more efficient in all the
situations and, in particular, in the case of focusing nonlinearity and
inverted oscillator potential. The fact that the method proposed in this paper
is faster and more accurate than the usual spectral splitting approximation
also emerges when comparing the expected values $\langle x\rangle_{j}^{t}$,
$j=1,2$, of the position observable obtained through the approximate solutions
with the exact value $\langle x\rangle^{t}$ obtained through Ehrenfest’s
Theorem.

$V(x)$ | $\nu$ | $n$ | $m$ | ${\Delta_{2}^{n,m}}/{\Delta_{1}^{n,m}}$
---|---|---|---|---
$\frac{1}{4}x^{2}$ | $+10$ | $270$ | $9000$ | $0.55$
$\frac{1}{4}x^{2}$ | $-10$ | $270$ | $9000$ | $0.94$
$-\frac{1}{4}x^{2}$ | $+10$ | $135$ | $45000$ | $0.66$
$-\frac{1}{4}x^{2}$ | $-10$ | $135$ | $45000$ | $0.046$
Table 5. Comparison of absolute errors for different nonlinearities and
harmonic/inverted oscillator potentials.

Not only that, this advantage could become decisive when numerical experiments
are performed when the spatial dimension is greater than $1$ and it would be
interesting to perform a series of experiments to clarify this issue.

On the other hand, the price to pay is due to the fact that the evolution
operator associated with the linear Schrödinger operator is not always
explicitly known; however, one could at least partially overcome this defect
by using numerical solvers of the Schrödinger equation that are sufficiently
efficient and fast.

##  Appendix A Mehler’s formula

Here we recall the expression for the evolution operator associated to the
linear Schrödinger operator $H$ with quadratic potential; this expression is
named Mehler’s formula.

Since the potential is quadratic then the linear operator
$H=-\frac{d^{2}}{dx^{2}}+\alpha x^{2}$, $\alpha\in{\mathbb{R}}$, admits a
self-adjoint extension on the domain ${\mathcal{D}}$ and the evolution
operator $e^{-iHt}$ is well defined.

Let $H_{0}=-\frac{\partial^{2}}{\partial x^{2}}$ be the free Schrödinger
operator; then the associated evolution operator has the form

$\displaystyle\left[e^{-itH_{0}}\psi_{0}\right](x)=\int_{{\mathbb{R}}}K_{0}(x,y;t)\psi_{0}(y)dy$
(39)

where [15]

$\displaystyle K_{0}(x,y;t)=\frac{1}{\sqrt{4\pi it}}e^{i(x-y)^{2}/4t}\,.$ (40)

Let $H_{HO}=-\frac{\partial^{2}}{\partial x^{2}}+\frac{1}{4}\omega^{2}x^{2}$,
$\omega>0$, be the Harmonic Oscillator Schrödinger operator; then the
evolution operator has the form

$\displaystyle\left[e^{-itH_{HO}}\psi_{0}\right](x)=\int_{{\mathbb{R}}}K_{HO}(x,y;t)\psi_{0}(y)dy$
(41)

where [11]

$\displaystyle K_{HO}(x,y;t)=\sqrt{\frac{\omega}{4\pi i\sin(\omega
t)}}\exp\left\\{i\frac{\omega}{4\sin(\omega t)}\left[(x^{2}+y^{2})\cos(\omega
t)-2xy\right]\right\\}\,.$ (42)

Let $H_{IO}=-\frac{\partial^{2}}{\partial x^{2}}-\frac{1}{4}\omega^{2}x^{2}$,
$\omega>0$, be the Inverted Oscillator Schrödinger operator; then the
evolution operator has the form

$\displaystyle\left[e^{-itH_{IO}}\psi_{0}\right](x)=\int_{{\mathbb{R}}}K_{HO}(x,y;t)\psi_{0}(y)dy$
(43)

where [2, 4]

$\displaystyle K_{IO}(x,y;t)=\sqrt{\frac{\omega}{4\pi i\sinh(\omega
t)}}\exp\left\\{i\frac{\omega}{4\sinh(\omega
t)}\left[(x^{2}+y^{2})\cosh(\omega t)-2xy\right]\right\\}\,.$

######  Remark 7.

It is well known that

$\displaystyle\|e^{-iH\delta}\psi_{0}\|_{L^{2}}=\|\psi_{0}\|_{L^{2}}$

for any self-adjoint operator $H$. Furthermore, in the case of self-adjoint
operator $H$ with quadratic potential then from (40), (42) and (43) it follows
that

$\displaystyle\|e^{-iHt}\psi_{0}\|_{L^{\infty}}\leq
Ct^{-1/2}\|\psi_{0}\|_{L^{1}}$

for any $\alpha=\pm\frac{1}{4}\omega^{2}\in{\mathbb{R}}$ and for any $t\leq
t^{\star}$, where $t^{\star}<\frac{\pi}{\omega}$ is fixed, and for some
$C=C(t^{\star},\omega)$.

## References

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correlation plots make use of the acf function of the stats package in R under
default settings [33]. More lags are considered when estimating $\phi_{s,n}$.
Fig. 2: Model Diagnostics for $\text{CO}^{2}$ Residuals

Altogether, there is sufficient evidence that the previous year’s monthly or
yearly average $\text{log}(\text{CO}^{2})$ levels predict a non-negligible
increase in the average global temperature of the following month or year.
Since this approach places no specific constraint on the underlying structure
of statistical dependencies, e.g., it does not adopt preconceptions about
strong or weak mixing or some form of $m$-dependence, the results of this
analysis are arguably stronger. They arrive with fewer theoretical caveats.
Nonetheless, important limitations lurk. Model diagnostics still make use of
the residuals, which are a biased and constrained representation of the true
error. This situation, however, applies to all regression diagnostic
procedures that make use of the residuals and does not rule out their use for
the estimation of restricted portions of auto-correlation. Ultimately, these
constraints complicate, but do not eliminate, the utility of these objects’
employment as a diagnostic mechanism.

##  7 Concluding remarks

This manuscript accomplished three main objectives. First, it established a
small set of related identities for the variance of a vector of random sums.
These identities require reasoning about the mean number of outcome variables
correlated within a sample and their average correlation only. Since the true
dependency structure of any collection of random variables is safely posited
to be unknown and empirically unidentifiable in whole, removing the strict
need to specify $n\choose 2$ parameters is useful. It was shown that a
researcher can elect to reason about these two intuitive summary constants
instead, or that she can employ them in conjunction with popular covariance
modeling methods to capture at least some of the variability that is missed by
an invalid specification. Although these constants are unknown, so are the
$n\choose 2$ covariance parameters that statisticians specify on a daily
basis. Furthermore, the cogent defense of a conservative choice for these
values is a much less demanding task than the alternative in a majority of
circumstances. Pertinently, these identities were also used to affirm the
consistency of additive estimators—including cluster-robust variance
estimators w.r.t. their identified portion of the overall variance—under the
very general condition that the average number of correlated variables in a
sample is asymptotically sub-linear as a function of $n$. For cluster variance
estimators, this was shown to be the case even when no valid partition of the
sample exists. A second accomplished objective was to extend these results to
estimating equations and hence to the estimators of statistical approaches
such as the generalized linear model.

The third and most important contribution of this paper was to prove a
sharpened version of Hoeffding’s inequality for a class of commonly
encountered random variables. Notably, it was proven that this inequality can
apply even when every single outcome variable in a sample is statistically
dependent, insofar as the magnitude of their average correlation is at least
moderately controlled. This result is certainly valuable for many fields where
the assumption of weak or local dependence is especially untenable, such as in
climate science, social network analysis, finance, and really any ecological
or sociological domain. That said, more work is due. Like all statistical
models, the valid application of this inequality relies on a set of
assumptions that can only be feasibly verified in practice. Although
imperfect, however, the diagnostic processes available are equivalent to those
used to check the assumptions of common regression models. In this sense, the
approach established here at least possesses nomological validity.

## CRediT Statement

Shane Sparkes: Conceptualization, Methodology, Validation, Formal analysis,
Data Curation, Writing - Original Draft, Writing - Review and Editing,
Visualization. Lu Zhang: Writing - Review and Editing, Supervision, Project
administration.

## Acknowledgments

We would like to thank the referees for their thoughtful commentary,
encouragement, and insights. Moreover, we would also like to acknowledge Dr.
Thomas Valente and Dr. Juan Pablo Lewinger for their commentary on a first
scalar form of the variance identity.

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