# On the cogirth of binary matroids

Cameron Crenshaw  Mathematics Department
Louisiana State University
Baton Rouge, Louisiana, USA<EMAIL_ADDRESS>and  James Oxley  Mathematics
Department
Louisiana State University
Baton Rouge, Louisiana, USA<EMAIL_ADDRESS>
###### Abstract.

The cogirth, $g^{\ast}(M)$, of a matroid $M$ is the size of a smallest
cocircuit of $M$. Finding the cogirth of a graphic matroid can be done in
polynomial time, but Vardy showed in 1997 that it is NP-hard to find the
cogirth of a binary matroid. In this paper, we show that
$g^{\ast}(M)\leq\frac{1}{2}|E(M)|$ when $M$ is binary, unless $M$ simplifies
to a projective geometry. We also show that, when equality holds, $M$
simplifies to a Bose-Burton geometry, that is, a matroid of the form
$PG(r-1,2)-PG(k-1,2)$. These results extend to matroids representable over
arbitrary finite fields.

###### Key words and phrases:

binary matroid, representable matroid, cogirth

##  1\. Introduction

For an arbitrary graph $G$, the well-known fact that the degree sum of $G$ is
twice the number of edges of $G$ implies that

$\frac{|E(G)|}{\delta(G)}\geq\frac{1}{2}|V(G)|,$

where $\delta(G)$ is the minimum degree of $G$. In a matroid $M$ of nonzero
rank, the _cogirth_ , $g^{\ast}(M)$, of $M$ is the size of a smallest
cocircuit of $M$. As $U_{r,n}$ shows, $\frac{|E(M)|}{g^{\ast}(M)}$ can be
arbitrarily close to 1 even for simple matroids, although it is bounded below
by $\frac{1}{2}(r(M)+1)$ when $M$ is graphic.

In this paper, we show that, when $M$ is binary,

$\frac{|E(M)|}{g^{\ast}(M)}\geq 2$

unless $M$ simplifies to a projective geometry. We also characterize the
matroids that achieve equality in this bound. Both of these results are
special cases of results for matroids representable over arbitrary finite
fields.

The terminology used here will follow Oxley [2] with the following addition.
We will often use $P_{r}$ and $A_{r}$ to denote $PG(r-1,q)$ and $AG(r-1,q)$,
respectively, where $q$ should be clear from the context. The next two results
are the main results of the paper.

######  Theorem 1.1.

For $r\geq 1$, let $M$ be a rank-$r$ matroid representable over $GF(q)$ whose
simplification is not $P_{r}$. Then

$\frac{|E(M)|}{g^{\ast}(M)}\geq\frac{q}{q-1}.$

Moreover, equality holds if and only if $M$ is loopless and, for a fixed
embedding of $\operatorname{si}(M)$ in $P_{r}$,

  * (i)

the complement of $\operatorname{si}(M)$ in $P_{r}$ is isomorphic to $P_{k}$
for some $k$ with $1\leq k<r$; and

  * (ii)

if $P$ is a copy of $P_{k+1}$ in $P_{r}$ containing the complement of
$\operatorname{si}(M)$, then the parallel classes of the elements in $E(M)\cap
E(P)$ all have the same size; and

  * (iii)

$|E(N)|\geq(q-1)|E(M)-E(N)|$ for every restriction $N$ of $M$ that simplifies
to $A_{r}$.

This theorem excludes the matroids $M$ for which $\operatorname{si}(M)\cong
PG(r-1,q)$. These excluded matroids are covered by the next result.

######  Proposition 1.2.

For $r\geq 1$, let $M$ be a matroid that simplifies to $PG(r-1,q)$. Then

$\frac{|E(M)|}{g^{\ast}(M)}\geq\frac{q^{r}-1}{q^{r-1}(q-1)}.$

Moreover, equality holds if and only if $M$ is loopless and all its parallel
classes have the same size.

Condition (i) in Theorem 1.1 says that $M$ simplifies to a _Bose-Burton
geometry_ [1], that is, a matroid that is obtained from $PG(r-1,q)$ by
deleting some $PG(k-1,q)$ where $1\leq k<r$. In each of our results, the bound
on $\tfrac{|E(M)|}{g^{\ast}(M)}$ is relatively easy to obtain. The core of
each proof involves characterizing when equality holds in the bound. The
proofs appear in Section 3.

##  2\. Preliminaries

In a matroid $M$ of rank at least one, a loop contributes to $|E(M)|$ but not
to $g^{\ast}(M)$. Since our concern here is on bounding
$\tfrac{|E(M)|}{g^{\ast}(M)}$ below, we shall focus on matroids without loops.
It will be convenient here to deal with the parallel classes in such a matroid
$M$ by assigning, to each element of $\operatorname{si}(M)$, a _weight_ $w(e)$
that is equal to the cardinality of the parallel class of $M$ that contains
$e$. Thus we deal with simple matroids with associated weight functions that
take a positive-integer value on each element. For a set $X$ in such a matroid
$N$, we write $w(X)$ for $\sum_{x\in X}w(x)$ and write $w(N)$ for $w(E(N))$.
The weight function of $N\backslash Y$ is the restriction of the weight
function of $N$ to $E(N)-Y$. When $Y$ is contracted from $N$, we replace each
parallel class $P$ by a single element $e_{P}$ whose weight in the contraction
is $w_{N}(P)$. We will call this weighted simple matroid the _weighted
contraction_ of $Y$ and denote it by $N/Y$, even though the underlying matroid
is actually $\operatorname{si}(N/Y)$. The cogirth of a weighted matroid is the
minimum weight of a cocircuit.

##  3\. The Proofs

We begin with a lemma that serves as the base case for both of the inductive
arguments that prove the inequalities in the main results.

######  Lemma 3.1.

Let $M$ be a simple, rank-$2$ matroid representable over $GF(q)$, and let $w$
be a weight function on $M$. Then

$\frac{w(M)}{g^{\ast}(M)}\geq\frac{|E(M)|}{|E(M)|-1},$

with equality if and only if $w$ is constant.

###### Proof.

Let $w_{1}\leq w_{2}\leq\cdots\leq w_{n}$ be the weights of the elements of
$M$. As $g^{\ast}(M)=w_{1}+w_{2}+\cdots+w_{n-1}$, the desired inequality is
equivalent to

$(n-1)(w_{1}+w_{2}+\cdots+w_{n})\geq n(w_{1}+w_{2}+\cdots+w_{n-1}).$

Subtracting $(n-1)(w_{1}+w_{2}+\cdots+w_{n-1})$ from each side, we obtain

(3.1) $(n-1)w_{n}\geq w_{1}+w_{2}+\cdots+w_{n-1},$

which is true since $w_{n}\geq w_{i}$ for all $i$. Note that equality holds in
(3.1) if and only if $w_{i}=w_{n}$ for all $i$. ∎

The following is the main result of the paper. It is equivalent to Theorem 1.1
and is stated here in terms of weights.

######  Theorem 3.2.

Let $M$ be a simple, rank-$r$ matroid representable over $GF(q)$, and let $w$
be a weight function on $M$. Suppose $M\not\cong P_{r}$. Then

$\frac{w(M)}{g^{\ast}(M)}\geq\frac{q}{q-1}.$

Moreover, equality holds if and only if, for a fixed embedding of $M$ in
$P_{r}$,

  * (i)

the complement of $M$ is isomorphic to $P_{k}$, with $1\leq k<r$; and

  * (ii)

if $P$ is a copy of $P_{k+1}$ containing the complement of $M$ in $P_{r}$,
then $w$ is constant on $P$; and

  * (iii)

$w(N)\geq(q-1)w(E(M)-E(N))$ for every $A_{r}$-restriction $N$ of $M$.

###### Proof.

We begin by proving the displayed inequality by induction on $r$. Lemma 3.1
gives the result when $r=2$, so suppose $r\geq 3$. If there is an $e$ in
$E(M)$ with $M/e\not\cong P_{r-1}$, then, by induction,

$(q-1)w(M)>(q-1)w(M/e)\geq qg^{\ast}(M/e)\geq qg^{\ast}(M).$

Thus we may assume that $M/e\cong P_{r-1}$ for all $e$ in $E(M)$. Take a line
of $P_{r}$ that meets both $E(M)$ and $E(P_{r})-E(M)$. Let $X$ be the set of
elements of $M$ on this line and $e$ be a maximum-weight element of $X$. Let
$Y=X-e$. Note that $|Y|\leq q-1$, so

(3.2) $w(Y)\leq w(e)(q-1).$

Observe that $M\backslash Y/e$ has rank $r-1$ but is not isomorphic to
$P_{r-1}$ so, by the induction assumption,

$(q-1)w(M\backslash Y/e)\geq qg^{\ast}(M\backslash Y/e).$

Now

$w(M\backslash Y/e)=w(M)-w(e)-w(Y),$

and

$g^{\ast}(M\backslash Y/e)\geq g^{\ast}(M)-w(Y).$

Thus

$(q-1)w(M)\geq qg^{\ast}(M)+w(e)(q-1)-w(Y),$

so, by (3.2),

$(q-1)w(M)\geq qg^{\ast}(M)$

as desired.

Next we characterize when equality is achieved in the last bound. Let $M^{c}$
be the complement of the fixed embedding of $M$ in $P_{r}$. When $M^{c}\cong
P_{k}$ for $1\leq k<r$, a hyperplane of $P_{r}$ either contains this $P_{k}$
or meets it in a $P_{k-1}$. Thus a cocircuit of $M$ is isomorphic to either
$A_{r}$ or $A_{r}-A_{k}$. We call these _type-I_ and _type-II cocircuits_ ,
respectively, noting that there are no type-I cocircuits when $k=r-1$.

######  3.2.1.

Suppose $M$ satisfies (i) and (ii). If $C^{\ast}$ is a type-II cocircuit of
$M$, then

$w(C^{\ast})=\frac{q-1}{q}w(M).$

As $M$ satisfies (i), $M^{c}\cong P_{k}$. Since $C^{\ast}$ is a type-II
cocircuit, there is a restriction $A$ of $P_{r}$ isomorphic to $A_{r}$ such
that $A$ meets $M^{c}$ and $C^{\ast}=E(M)\cap E(A)$. Let $H$ be the hyperplane
of $P_{r}$ that is the complement of $A$.

Now, $P_{r}$ consists of $\frac{q^{r-k}-1}{q-1}$ copies of $P_{k+1}$
containing $M^{c}$, and the pairwise intersection of these copies is $M^{c}$.
Thus $M$ is the disjoint union of $\frac{q^{r-k}-1}{q-1}$ copies of $A_{k+1}$.
By (ii), the elements in each $A_{k+1}$ have the same weight. To complete the
proof of 3.2.1, we show that $C^{\ast}$ contains exactly $\frac{q-1}{q}$ of
the elements of each $A_{k+1}$.

Consider the complementary $A_{k+1}$ to $M^{c}$ in a fixed $P_{k+1}$. Note
that $P_{k+1}$ consists of $q+1$ copies of $P_{k}$, including $M^{c}$, that
contain $H\cap E(M^{c})$, which is isomorphic to $P_{k-1}$. Therefore, this
$A_{k+1}$ is the disjoint union of $q$ copies of $A_{k}$. Now $H$ meets
$P_{k+1}$ at a $P_{k}$ distinct from $M^{c}$. Thus $A$ meets $P_{k+1}$ in a
set that is the union of $q$ disjoint copies of $A_{k}$, one of which is in
$M^{c}$. This implies that $C^{\ast}\cap A_{k+1}$ is the disjoint union of
$q-1$ copies of $A_{k}$, and 3.2.1 follows.

Now assume that $\tfrac{w(M)}{g^{\ast}(M)}=\tfrac{q}{q-1}$. Then equality
holds in (3.2) so $|Y|=q-1$ and $w(y)=w(e)$ for all $y\in Y$. The former
implies that every line of $P_{r}$ that meets both $M$ and $M^{c}$ contains
exactly $q$ points of $M$. This means that every line that contains two points
of $M^{c}$ lies entirely in $M^{c}$. Thus $M^{c}$ is a flat of $P_{r}$,
proving (i).

As $w(y)=w(e)$ for all $y\in Y$, it follows that $w$ is constant on each line
of $P_{r}$ that meets both $M$ and $M^{c}$. Since a $P_{k}$ contained in a
$P_{k+1}$ meets every line of the $P_{k+1}$, (ii) is satisfied. It now follows
from 3.2.1 that $\tfrac{w(M)}{g^{\ast}(M)}=\frac{q}{q-1}$ if and only if $M$
satisfies (i) and (ii), and the type-I cocircuits of $M$ have weight at least
$\tfrac{q-1}{q}w(M)$. It is straightforward to check that this third condition
is equivalent to (iii), so the theorem holds. ∎

The reader may find condition (iii) of Theorem 3.2 unsatisfying, and the next
proposition offers a potential replacement, (iii)′. The example that follows
Proposition 3.3 shows that conditions (i), (ii), and (iii)′ do not guarantee
$\frac{w(M)}{g^{\ast}(M)}=\frac{q}{q-1}$ for a matroid $M$ meeting the
hypotheses of Theorem 3.2. In addition, the example illustrates the potential
difficulty of finding a satisfactory replacement for (iii).

######  Proposition 3.3.

Let $M$ be a simple, rank-$r$ matroid representable over $GF(q)$, and let $w$
be a weight function on $M$. Suppose that $M\not\cong P_{r}$ and that
$\tfrac{w(M)}{g^{\ast}(M)}=\frac{q}{q-1}$. Then

  * (iii)′

$q^{r-1}w(e)\leq w(M)$ for all $e$ in $E(M)$.

###### Proof.

By Theorem 3.2, $M^{c}=P_{k}$. If $k=r-1$, then Theorem 3.2(ii) implies that
(iii)′ holds with equality. Thus we may assume that $k<r-1$. Extend the weight
function of $M$ to $P_{r}$ by assigning each element of $M^{c}$ a weight of
one. Then contract $M^{c}$ from $P_{r}$ to form a weighted matroid
$M^{\prime}\cong P_{r-k}$. Fix an element $e$ in $E(M)$, and let $e^{\prime}$
be the image of $e$ in $M^{\prime}$. Note that $w(M^{\prime})=w(M)$ and
$w(e^{\prime})=q^{k}w(e)$ under this transformation. Moreover, the type-I
cocircuits of $M$ correspond to the cocircuits of $M^{\prime}$, so the weight
of each cocircuit of $M^{\prime}$ equals the weight of the corresponding
type-I cocircuit of $M$.

Let $C^{\ast}$ be a cocircuit of $M^{\prime}$ that avoids $e^{\prime}$ and let
$H$ be the complementary hyperplane to $C^{\ast}$ in $M^{\prime}$. Since
$\frac{w(M)}{g^{\ast}(M)}=\frac{q}{q-1}$, it follows that

$\frac{q}{q-1}w(C^{\ast})\geq w(M^{\prime}),$

and subtracting $w(C^{\ast})$ from each side produces

(3.3) $\frac{1}{q-1}w(C^{\ast})\geq w(H).$

Note that (3.3) holds for an arbitrary cocircuit of $M^{\prime}$ avoiding
$e^{\prime}$, so we have such an inequality for every such cocircuit.
Moreover, $C^{\ast}$ and $H$ partition $E(M^{\prime})$ so, for a fixed $f\in
E(M^{\prime}\backslash e^{\prime})$, its weight contributes to exactly one
side of each inequality. Now, there are $\frac{q^{r-k-1}-1}{q-1}$ total
inequalities as this is the number $t$ of hyperplanes of $M^{\prime}$
containing $e^{\prime}$. Similarly, $w(f)$ contributes to the right-hand side
of exactly $\frac{q^{r-k-2}-1}{q-1}$ of these inequalities as this is the
number $s$ of hyperplanes of $M^{\prime}$ containing both $e^{\prime}$ and
$f$. Hence $w(f)$ contributes to the left-hand side of $t-s$ of these
inequalities. Summing these inequalities gives

$\frac{t-s}{q-1}w(M^{\prime}\backslash e^{\prime})\geq sw(M^{\prime}\backslash
e^{\prime})+tw(e^{\prime}),$

and this simplifies to

$w(M^{\prime})\geq q^{r-k-1}w(e^{\prime}).$

Finally, we substitute $w(M)$ for $w(M^{\prime})$ and $q^{k}w(e)$ for
$w(e^{\prime})$ to obtain

$w(M)\geq q^{r-1}w(e)$

as desired. ∎

######  Example 3.4.

Let $q=2$ and let $M=P_{4}-p$ for some $p\in E(P_{4})$. Then $M\cong
P_{4}-P_{1}$. Take a hyperplane $H$ of $P_{4}$ containing $p$, and note that
$H$ has $|P_{3}|$ elements. Then $H-p$ is a hyperplane of $M$ and the
corresponding cocircuit $C^{\ast}$ is type-I and has $|A_{4}|$ elements.

Assign the weight 2 to each element of $H-p$ and the weight 1 to each element
of $C^{\ast}$. Then

$w(M)=2(|P_{3}|-1)+1\cdot|A_{4}|=2(6)+8=20.$

Observe that conditions (i) and (ii) of Theorem 3.2 hold and, since, for all
$e$ in $E(M)$,

$q^{r-1}w(e)\leq 2^{3}(2)<20=w(M),$

so does (iii)′. However, $w(C^{\ast})=8$, so the equation

(3.4) $\frac{w(M)}{g^{\ast}(M)}=\frac{q}{q-1}$

fails.

Now, in $P_{4}$, take a line in $C^{\ast}\cup p$ and another in $H$ that each
meet in $\\{p\\}$. Swap the weights 1 and 2 on the elements of $M$ on these
lines. Note that (i), (ii), and (iii)′ continue to hold, and $w(M)$ is
unchanged. However, it is straightforward to check that the weights of the
type-I cocircuits of $M$ are at least 10, so (3.4) holds by Theorem 3.2. Thus,
characterizing the matroids for which equality holds in Theorem 3.2 requires
not only restricting the weights themselves, but also controlling their
distribution.

Finally, we prove a proposition equivalent to Proposition 1.2 stated here in
terms of weights.

######  Proposition 3.5.

Let $M$ be a matroid isomorphic to $PG(r-1,q)$ and $w$ be a weight function on
$E(M)$. Then

$\frac{w(M)}{g^{\ast}(M)}\geq\frac{q^{r}-1}{q^{r-1}(q-1)}.$

Moreover, equality holds if and only if $w$ is constant.

###### Proof.

We prove the inequality by induction on $r$. It is trivial when $r=1$ and is
true for $r=2$ by Lemma 3.1, so suppose $r\geq 3$. Let $C^{\ast}$ be a
cocircuit of $M$ of weight $g^{\ast}(M)$ and let $H$ be the complementary
hyperplane to $C^{\ast}$ in $M$. Choose $Z$ as a maximum-weight hyperplane of
$H$. Then, letting $Y$ be the complement of $Z$ in $H$, we have

(3.5)
$\frac{w(M)}{g^{\ast}(M)}=\frac{w(C^{\ast})+w(Y)+w(Z)}{w(C^{\ast})}=1+\frac{w(Y)+w(Z)}{w(C^{\ast})}.$

Observe that the weighted contraction of $Z$ from $M$ is isomorphic to $P_{2}$
so, by the inequality for $r=2$, we get that

$\frac{w(M/Z)}{g^{\ast}(M/Z)}\geq\frac{q+1}{q}.$

Now, since $C^{\ast}$ is also a minimum-weight cocircuit of $M/Z$, we rewrite
this inequality as

(3.6) $\frac{w(Y)+w(C^{\ast})}{w(C^{\ast})}\geq\frac{q+1}{q}.$

It follows that $qw(Y)\geq w(C^{\ast})$. Substituting into (3.5), we obtain

(3.7) $\frac{w(M)}{g^{\ast}(M)}\geq 1+\frac{1}{q}\cdot\frac{w(Y)+w(Z)}{w(Y)}.$

Finally, the hyperplane $H$ is isomorphic to $P_{r-1}$, and our choice of $Z$
makes $Y$ a minimum-weight cocircuit of $H$. Thus, by induction,

(3.8) $\frac{w(Y)+w(Z)}{w(Y)}\geq\frac{q^{r-1}-1}{q^{r-2}(q-1)}.$

Substituting (3.8) into (3.7) gives the desired inequality.

One easily checks that, when $w$ is constant,

(3.9) $\frac{w(M)}{g^{\ast}(M)}=\frac{q^{r}-1}{q^{r-1}(q-1)}.$

We now use induction on $r$ to prove that the elements of $M$ have the same
weight when (3.9) holds. When $r=1$, this is trivial, and Lemma 3.1 handles
the rank-2 case.

Suppose $r\geq 3$. Since (3.9) holds, equality holds in (3.6) and (3.8). It
follows from the latter using the induction assumption that the weight
function $w$ on $E(M)$ is constant on the hyperplane $H$ of $M$. From the
former, we deduce that, in $M/Z$, every point has equal weight. It follows
that every hyperplane $H^{\prime}$ of $M$ containing $Z$ has the same weight.
Hence the cocircuit $E(M)-H^{\prime}$ has the same weight as $C^{\ast}$.
Replacing $H$ by $H^{\prime}$, we deduce that $w$ is constant on the elements
of $H^{\prime}$. Letting $H^{\prime}$ range over all of the hyperplanes of $M$
containing $Z$, we deduce that $w$ is constant on $E(M)$. ∎

## References

  * [1] R.C. Bose, R.C. Burton, A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and MacDonald codes, J. Combinatorial Theory 1 (1966), 96–104.
  * [2] J. Oxley, Matroid Theory, Second edition, Oxford University Press, New York, 2011.
  * [3] A. Vardy, The intractability of computing the minimum distance of a code, IEEE Trans. Inform. Theory 43 (1997), 1757–1766.


# Comparative Analysis of CHATGPT and the evolution of language models

1,2Oluwatosin Ogundare, 1Gustavo Quiros Araya

(1Siemens Technology
2California State University, San Bernardino
)

###### Abstract

Interest in Large Language Models (LLMs) has increased drastically since the
emergence of ChatGPT and the outstanding positive societal response to the
ease with which it performs tasks in Natural Language Processing (NLP). The
triumph of ChatGPT, however, is how it seamlessly bridges the divide between
language generation and knowledge models. In some cases, it provides anecdotal
evidence of a framework for replicating human intuition over a knowledge
domain. This paper highlights the prevailing ideas in NLP, including machine
translation, machine summarization, question-answering, and language
generation, and compares the performance of ChatGPT with the major algorithms
in each of these categories using the Spontaneous Quality (SQ) score. A
strategy for validating the arguments and results of ChatGPT is presented
summarily as an example of safe, large-scale adoption of LLMs.

##  1 Introduction to Classical Theories in Computational Linguistics

A modern history of language models must admit that computational reasoning
over any grammar or a set of symbols necessitates structure. Ferdinand de
saussure’s doctrine on structuralism lays out the key concepts of a scientific
model of language with the view that it encapsulates a finite set of symbols
that is independent of the intention or the psychological process that
generates them [11]. Noam Chonksky improved on this idea by positing that all
humans share an innate cognitive grammar structure and that this universal
grammar structure generalizes principles on which language derives it’s form.
This idea forms the basis of generative grammar. Consequently, language can be
modelled as a state machine that has infinite realization from a generation
and deciphering standpoint [4]. A main result of generative grammar in terms
of the theory of computational linguistics is the introduction of a formal
framework on which logical operations can be performed on linguistics objects
known as the minimalist framework [3]. P.W. Culicover and L.McNaly advocates
for more sophistication in the tools used for analysis in generative syntax
and potentially other ideas that have emerged from Chomsky’s generative
grammar, an indication that there is a limitation in what can be acheived by
models that rely on the minimalist framework for formal linguistic reasoning
[6]. For example, classical linguistic ideas like anaphora, an expression
whose meaning depends on the antecedent or the context established by the
antecedent, remains difficult within the framework of the minimalist program.
Other theories that investigate the semantic meaning of language like
functional linguistics improved the state of the art by introducing its own
formalism that encompasses taxonomy and the notation that analyzes the
structure of language which tend to perform better than the minimalist program
when ‘meaning’ or ‘intent’ is the principal focus of a language task [8].
Another major early contribution to computational language analysis is Part-
of-Speech (POS) tagging. At its inception, many of the approaches were rule
based but famously in 1989, Kenneth Church automated POS tagging using a
Hidden Markov Model (HMM) [5]. Part-of-Speech tagging remains a major Natural
Language Processing (NLP) task in both syntax and semantic analysis of text. A
direct consequence of automatic POS tagging is the idea of classifying the
nouns based on the domain of reference or generally based on language specific
roles. This type of NLP task is referred to as Named Entity Recognition (NER).
Named Entity Recognition identifies the subjects and objects of sentences
based on the position of certain parts of speech as shown by Bikel, Miller &
Schwartz in 1998 [2]. Other techniques for NLP tasks quickly developed with
the success of stochastic approaches to language modelling. For example,
Ngrams was used widely in the early days for text categorization and still
remains popular in modern NLP models[12]. More recently stochastic approaches
have very concrete applications in evaluating document similarity, performing
semantic role labelling and other mainstream NLP tasks [10].

##  2 ChatGPT and Prevailing Ideas in NLP

The landscape of prevailing computational theories in Natural Language
Processing (NLP) shifts in response to what constitute the popular NLP tasks.
Currently, the major tasks are Machine Translation, Machine Summarization,
Question Answering and Language Generation. There are minor tasks like text
classification or control program generation that can be solved efficiently
using simple statistical learning or machine learning models when the domain
is well understood[9]. Although the quality of the performance of mainstream
algorithms over the set of identified tasks have progressively improved
through the decades, especially since the advent of deep learning, none has
been able to perform quite as well over the all of the major tasks like
ChatGPT. To demonstrate this claim, we select the major mainstream algorithm
in every category and compare the performance of the chosen algorithm against
ChatGPT using what we refer to as the Spontaneous Quality (SQ) score which is
a measure of perceived quality, i.e., how well the output reflects the
intention at first glance without in-depth linguistic analysis. The
Spontaneous Quality score computes a value for a reference language model by
accessing its performance over 3 criteria.

  1. 1.

Accuracy

  2. 2.

Clarity

  3. 3.

Native Speaker Likeness

For scale in-variance, the accuracy score, the clarity score and Native
Speaker Likeness score are on the interval (0, 1) and weighted in the SQ score
by {$\alpha_{i}$}.

Mathematically,

. SQ Score = $\sum_{i}\alpha_{i}P_{i}$

. Where $\alpha_{i}=$ Coefficient of Importance
. $P_{i}=$ Performance criteria

The reference test selects a paradigmatic average case in a mainstream
language and another archetype of the average case in an obscure language. The
main idea is to construct a shallow normalized performance metric, that can be
repeated and averaged over a series of random pairs of mainstream and obscure
languages. The reference test score is computed as the average of the series
of geometric mean of the Spontaneous Quality (SQ) score of the algorithm in
the mainstream case and the Spontaneous Quality (SQ) score of the algorithm in
the obscure case.
Mathematically,

Reference Test (RT) Score = $\frac{1}{k}\sum_{i=1}^{k}\sqrt{(SQ\hskip
2.84526ptScore_{main}^{i}\times SQ\hskip 2.84526ptScore_{obscure}^{i})}$

###  2.1 Machine Translation

REFERENCE TEST
Using coefficients of importance 0.5, 0.25, 0.25 over the performance criteria
accuracy, clarity, native speaker likeness

  * •

Mainstream case $::English\rightarrow Spanish$

  * •

Obscure case $::English\rightarrow Yoruba$

Test Phrase: ”The man is a man that is a unique man”
Spanish:

  * •

Google Translate:: ”El hombre es un hombre que es un hombre único.”

  * •

Chat GPT:: ”El hombre es un hombre que es un hombre único.”

Both translations are identical with the same SQ score.
SQ Score = $0.5\times 1+0.25\times 1+0.25\times 1=1$
Yoruba:

  * •

Google Translate:: ”Okunrin naa je okunrin ti o je eniyan alailegbe”

  * •

Chat GPT:: ”Okunrin ni okunrin ti o ni okunrin kan pupo”

SQ Score for Google Translate = $0.5\times 0.9+0.25\times 0.8+0.25\times
0.5=0.725$
SQ Score for Chat GPT = $0.5\times 0.3+0.25\times 0.2+0.25\times 0.7=0.375$
For a series of 1 language pair, the RT score is simply the geometric mean as
shown below:
The Reference Test (RT) Score for Google Translate = $\sqrt{1*0.725}$ = 0.8514
The Reference Test (RT) Score for Chat GPT = $\sqrt{1*0.375}$ = 0.6123

SUMMARY OF GOOGLE TRANSLATE VS CHATGPT OVER SELECT LANGUAGES

Model |  Language |  SQ Score |  Language |  SQ Score |  Instance RT Score
---|---|---|---|---|---
Google Translate |  Russian |  0.95 |  Tajik |  0.9 |  0.925
Chat GPT |  Russian |  1.0 |  Tajik |  0.85 |  0.922
Google Translate |  French |  1.0 |  Hungarian |  0.9 |  0.949
Chat GPT |  French |  1.0 |  Hungarian |  1.0 |  1.0
Google Translate |  Hindi |  1.0 |  Telugu |  0.7 |  0.837
Chat GPT |  Hindi |  0.9 |  Telugu |  0.7 |  0.794
Google Translate |  Arabic |  0.75 |  Hausa |  0.7 |  0.725
Chat GPT |  Arabic |  0.75 |  Hausa |  0.1 |  0.274
Table 1: Performance of Google Translate VS ChatGPT in Machine Translation.

The RT score for Google Translate over 4 language pairs = 0.969
The RT Score for ChatGPT over 4 language pairs = 0.748

###  2.2 Machine Summarization

REFERENCE TEST
The reference text is the last section of the well known poem titled ”Piano”
by D.H. Lawrence

”So now it is vain for the singer to burst into clamour With the great black
piano appassionato. The glamour Of childish days is upon me, my manhood is
cast Down in the flood of remembrance, I weep like a child for the past”

  * •

QuillBot.com Summarizer:: ”The singer’s manhood is cast aside and he weeps for
the past.”

  * •

Chat GPT:: ”As I recall the memories of my childhood, I am overcome with
nostalgia and weep uncontrollably, feeling as though my manhood has been
washed away by the flood of reminiscence.”

SQ Score for QuillBot.com Summarizer = $0.5\times 0.6+0.25\times
0.8+0.25\times 0.8=0.7$
SQ Score for Chat GPT = $0.5\times 0.9+0.25\times 1+0.25\times 1=0.95$

The reference text for the obscure case is taken from D.O. Fagunwa’s ”Ireke
Onibudo”, P.11

”Sugbon ibiti a de naa dara tobe ti mo feran ma le pe e ni ile rara, oniruurun
ododo oloorunrun didun ni mo ri nibi sugbon ona ti won se si aarin awon ododo
naa goolu ni won fi te ile ibe” [1] [7]

  * •

QuillBot.com Summarizer:: ”Ibiti a de naa dara, oniruuru ododo oloorunrun
didun ni mo ri nibe.”

  * •

Chat GPT:: ”Mo ni ki nfi ife oorun iwo kan bi aye mi, mo ni ki nse ilu mi,
sugbon awon igbagbo ni won ni, niwaju gbogbo eto ni lo di okan, lati fi ogun
isoro gbogbo ranmo ni ile.”

SQ Score for QuillBot.com Summarizer = $0.5\times 0.65+0.25\times
0.9+0.25\times 0.8=0.75$
SQ Score for Chat GPT = $0.5\times 0.3+0.25\times 0.2+0.25\times 0.4=0.3$

The RT Score for QuillBot.com Summarizer = $\sqrt{0.7*0.75}$ = 0.724, The RT
Score for Chat GPT = $\sqrt{0.95*0.3}$ = 0.534

SUMMARY OF QuillBot.com SUMMARIZER VS CHATGPT OVER SELECT LANGUAGES

Model |  Language |  SQ Score |  Language |  SQ Score |  Instance RT Score
---|---|---|---|---|---
QuillBot.com |  Russian |  0.9 |  Tajik |  0.35 |  0.561
Chat GPT |  Russian |  0.75 |  Tajik |  0.65 |  0.698
QuillBot.com |  French |  0.75 |  Hungarian |  0.8 |  0.774
Chat GPT |  French |  0.8 |  Hungarian |  0.9 |  0.849
QuillBot.com |  Hindi |  0.2 |  Telugu |  0.025 |  0.071
Chat GPT |  Hindi |  0.9 |  Telugu |  0.9 |  0.9
QuillBot.com |  Arabic |  0.65 |  Hausa |  0.8 |  0.721
Chat GPT |  Arabic |  0.9 |  Hausa |  0.025 |  0.15
Table 2: Performance of QuillBot.com Summarizer VS ChatGPT in Machine
Translation.

The RT score for QuillBot.com Summarizer and ChatGPT over 4 language pairs
respectively is 0.531, 0.649

###  2.3 Language Generation

One of the triumphs of ChatGPT is how beautifully it produces words that are
not unusual within the vocabulary of a target language and for mainstream
languages as shown prior, it produces sentences that follow the rules of
grammar in those languages. Even in some obscure cases when it generates
sentences whose meaning are divergent from the intent, it follows the rules of
grammar and it strives to use words that are not unusual in the local
vernacular. In contrast to natural language, ChatGPT performs consistently
well when generating programs across a variety of programming languages and
paradigms. In this section, we will select a reference test that generates a
program in a mainstream programming language and see if ChatGPT’s performance
degrade when it generates PLC programs for industrial automation.

REFERENCE TEST
Generate a control program for a fast acting valve in a chemical plant in
python as well as Siemens S7 ladder logic.

Chat GPT Python:

⬇

# Fast-Acting Valve Control Program

import time

# Set the initial state of the valve to closed

valve_open = False

# Define the function to control the valve

def control_valve(open_time):

global valve_open

valve_open = True

# Send command to open the valve

print(”Opening valve…”)

# Wait for the open_time duration

time.sleep(open_time)

valve_open = False

# Send command to close the valve

print(”Closing valve…”)

Chat GPT S7 Ladder Logic:

⬇

Program: Fast-Acting Valve Control Program

Step 1: Inputs

I0.0: Start button

I0.1: Stop button

Step 2: Outputs

Q0.0: Valve

Step 3: Program

LD R0, I0.0 ; Load input I0.0 into register R0

LD R1, I0.1 ; Load input I0.1 into register R1

LD Q0.0, 0 ; Set the valve to closed

TND 1.0s ; Debouncing timer for the start button

LD R2, I0.0 ; Load input I0.0 into register R2

AND R3, R2, NOT R0 ; Detect rising edge of start button press

LD R4, I0.1 ; Load input I0.1 into register R4

AND R5, R4, NOT R1 ; Detect rising edge of stop button press

XOR R6, R3, R5 ; Check if either start or stop button has been pressed

ANB R7, R6, Q0.0 ; Check if valve is currently closed

ORM Q0.0, R7 ; Open valve if it is closed

TND 1.0s ; Debouncing timer for the stop button

LD R8, I0.1 ; Load input I0.1 into register R8

AND R9, R8, NOT R4 ; Detect rising edge of stop button press

ANB Q0.0, NOT R9 ; Close valve if stop button is pressed

END

We compute the SQ score for ChatGPT for the generated algorithms factoring the
lack of detail in the problem statement as follows:

SQ Score for ChatGPT Python = $0.5\times 1+0.25\times 0.9+0.25\times 0.9=0.95$
SQ Score for Chat GPT = $0.5\times 1+0.25\times 1+0.25\times 1=1$
The Reference Test (RT) Score over the 2 cases = 0.975

SUMMARY OF CHATGPT PERFORMANCE IN GENERATING SOURCE OVER SELECT PROGRAMMING
LANGUAGES

Model |  Language |  SQ Score |  Language |  SQ Score |  Instance RT Score
---|---|---|---|---|---
Chat GPT |  C |  0.95 |  Fortran77 |  0.95 |  0.95
Chat GPT |  CSharp |  0.95 |  Rust |  0.95 |  0.95
Chat GPT |  Javascript |  0.95 |  Ruby |  0.95 |  0.95
Chat GPT |  Haskell |  1.0 |  FSharp |  1.0 |  1.0
Chat GPT |  Scala |  1.0 |  Lisp |  1.0 |  1.0
Chat GPT |  Java |  0.95 |  Golang |  0.95 |  0.95
Table 3: Performance of ChatGPT in Language Generation.

The RT score for ChatGPT over 6 language pairs respectively is 0.967

##  3 Information differentiation Strategy in Knowledge Models

We introduce the notion of the mediator within the machine generation
workflow, as shown in fugure 1, as a crucial step in ensuring the quality and
ethical standards of the generated content. The mediator acts as a gatekeeper
to check the output of the LLM chatbot against various guidelines,
regulations, policy, design recommendations, and prevailing sanctions. These
guidelines may include legal and ethical frameworks such as data protection
laws, hate speech regulations, or product-specific guidelines. By verifying
the generated content against these guidelines, the mediator ensures that the
content is compliant with the relevant regulations.

In addition to checking against guidelines, the mediator also checks the
output against the constraints imposed by other engineering or product
artifacts. These constraints may include technical limitations or product-
specific requirements. For example, a chatbot designed for a specific platform
may need to follow certain design standards or incorporate specific features.
By ensuring that the output is consistent with these constraints, the mediator
ensures that the generated content is technically feasible and aligns with the
product’s goals.

If the generated content fails to pass any of these mandatory checks, the
mediator raises a review flag, and the output is automatically rejected. This
automatic rejection ensures that any content that does not meet the
established standards is caught early in the process, minimizing the potential
negative impact. If the output passes all the checks, the mediator approves
the LLM output for further downstream processing, including integration with
other systems, deployment, or publication.

Figure 1: A process mediator working in tandem with an LLM.

##  4 Conclusion & Further Work

In conclusion, language models have come a long way since the early days of
computational linguistics. From the introduction of generative grammar to the
current state-of-the-art Large Language Models (LLMs) like ChatGPT, the field
has made a lot of improvements in understanding and modeling language as well
as bridging the gap between language models and knowledge models. With an
increasing amount of available data for systematic training and the
advancements in the computational theory of machine learning, we can expect
language models to continue to improve and push the boundaries of what is
possible in natural language processing especially applications in industrial
automation. Future research on Large Language Models (LLMs) should investigate
ethical concerns regarding the expanded functionality of LLMs as well attempt
an explanation of the performance improvements in language generation.
Additionally, a computational formalism of the notion of spontaneous human-
like intuition that is sometimes evident in how ChatGPT performs on machine
summarization would aid widespread adoption.

## References

  * [1] Gabriel Olujide Ayoola.  A Critical Introduction and Annotated Translation of DO Fagunwa’s Ireke-Onibudo from Yoruba into English.  PhD thesis, University of Georgia, 2019.
  * [2] Daniel M Bikel, Scott Miller, Richard Schwartz, and Ralph Weischedel.  Nymble: a high-performance learning name-finder.  arXiv preprint cmp-lg/9803003, 1998.
  * [3] Noam Chomsky.  The minimalist program.  MIT press, 2014.
  * [4] Noam Chomsky and David W Lightfoot.  Syntactic structures.  Walter de Gruyter, 2002.
  * [5] Kenneth Ward Church.  A stochastic parts program and noun phrase parser for unrestricted text.  In International Conference on Acoustics, Speech, and Signal Processing,, pages 695–698. IEEE, 1989.
  * [6] Peter W Culicover and Louise McNally.  On exploring the limits of syntax.  In The limits of syntax, pages 1–17. Brill, 1998.
  * [7] Daniel O Fagunwa.  Ireke Onibudo.  Nelson, 1949.
  * [8] Michael Alexander Kirkwood Halliday and Christian MIM Matthiessen.  Halliday’s introduction to functional grammar.  Routledge, 2013.
  * [9] Oluwatosin Ogundare, Gustavo Quiros Araya, and Yassine Qamsane.  No code ai: Automatic generation of function block diagrams from documentation and associated heuristic for context-aware ml algorithm training.  In 2022 7th International Conference on Mechanical Engineering and Robotics Research (ICMERR), 2022.
  * [10] Oluwatosin Ogundare and Nathanial Wiggins.  Identifying sub-documents in a composite scanned document using naive bayes, levenshtein distance and domain driven knowledge base.  In 2018 5th International Conference on Soft Computing and Machine Intelligence (ISCMI), pages 84–87, 2018.
  * [11] Gary P Radford and Marie L Radford.  Structuralism, post-structuralism, and the library: de saussure and foucault.  Journal of documentation, 2005.
  * [12] Ching Y Suen.  N-gram statistics for natural language understanding and text processing.  IEEE transactions on pattern analysis and machine intelligence, (2):164–172, 1979.


# Controllability of the Schrödinger equation on unbounded domains without
geometric control condition

Matthias Täufer  FernUniversität in Hagen, Fakultät für Mathematik und
Informatik

###### Abstract.

We prove controllability of the Schrödinger equation in $\mathbb{R}^{d}$ in
any time $T>0$ with internal control supported on nonempty, periodic, open
sets. This demonstrates in particular that controllability of the Schrödinger
equation in full space holds for a strictly larger class of control supports
than for the wave equation and suggests that the control theory of Schrödinger
equation in full space might be closer to the diffusive nature of the heat
equation than to the ballistic nature of the wave equation. Our results are
based on a combination of Floquet-Bloch theory with Ingham-type estimates on
lacunary Fourier series.

###### Key words and phrases:

Controllability, Schrödinger equation, Observability Estimate, Floquet-Bloch
theory

###### 2010 Mathematics Subject Classification:

93Bxx, 35Q93

##  1\. Controllability of the Schrödinger equation in $\mathbb{R}^{d}$

Consider the Schrödinger equation in $\mathbb{R}^{d}$ with internal control

(1) $\begin{cases}i\frac{\mathrm{d}}{\mathrm{d}t}u+\Delta
u=\mathbf{1}_{S}f&\quad\text{in $\mathbb{R}^{d}\times[0,T]$},\\\
u(0)=u_{0}&\quad\in L^{2}(\mathbb{R}^{d})\\\ \end{cases}$

where $S\subset\mathbb{R}^{d}$ is the _control set_ , $\mathbf{1}_{S}$ its
indicator function, and $f\in L^{2}(S\times(0,T))$ is called the _control_.
For any choice of $f$ and $u_{0}$, the solution of (1) is given by the Duhamel
formula

$u(t)=\mathrm{e}^{it\Delta}u_{0}+\int_{0}^{t}\mathrm{e}^{i(t-s)\Delta}\mathbf{1}_{S}f(s)\mathrm{d}s.$

######  Definition 1.

Equation (1) is called _controllable in time $T>0$_ if for all $u_{0},u_{T}\in
L^{2}(\mathbb{R}^{d})$ there exists $f\in L^{2}(S\times(0,T))$ such that the
solution of (1) satisfies $u(T)=u_{T}$.

Our main result is:

######  Theorem 2.

Let $S\subset\mathbb{R}^{d}$ be nonempty, periodic and open. Then (1) is
controllable in any time $T>0$.

######  Remark 3.

Since the free Schrödinger group $(\mathrm{e}^{it\Delta})_{t\in\mathbb{R}}$ is
unitary, controllability in time $T$ is equivalent to _null-controllability in
time $T$_, that means that for every $u_{0}\in L^{2}(\mathbb{R}^{d})$ there is
a control $f$ such that $u_{T}=0$. Indeed, a null-control for the initial
state

$\tilde{u}_{0}:=u_{0}-\mathrm{e}^{-iT\Delta}u_{T}$

will send the initial state $u_{0}$ to $u_{T}$ in time $T$. Therefore, from
now on, we will focus on null-controllability and assume that the target state
$u_{T}=0$.

Let us comment on the implications of Theorem 2: In recent years, control
problems on unbounded domains have gained attention – on the one hand
motivated by applications to kinetic theory [DSV22], on the other hand in
order to further the geometric understanding of problems on bounded domains by
reconciling phenomena on bounded and unbounded domains and using unbounded
domains as a proxy for large domains or a family of growing domains where the
control set has a particular structure [TT17a, Täu18, NTTV18, NTTV20, WWZZ19].
In parts this has coincided with progress in the understanding of Anderson
localization for random Schrödinger operators where conditions on the support
of random potentials and the form of randomness causing localization have been
succesively relaxed [TV15, NTTV15, TV16, TT17b, TV21] Thus, also the task of
identifying all sets $S\subset\mathbb{R}^{d}$ that ensure controllability of
controlled systems has recently gained attention – for the controlled
Schrödinger equation as well as for its two cousins – the _controlled wave
equation_

(2) $\begin{cases}\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}u-\Delta
u=\mathbf{1}_{S}f&\quad\text{in $\mathbb{R}^{d}\times[0,T]$},\\\
\left((u(0),\frac{\partial}{\partial t}u(0)\right)=(u_{0},u_{1}),&\\\
\end{cases}$

and the _controlled heat equation_

(3) $\begin{cases}\frac{\mathrm{d}}{\mathrm{d}t}u-\Delta
u=\mathbf{1}_{S}f&\quad\text{in $\mathbb{R}^{d}\times[0,T]$},\\\
u(0)=u_{0}.&\\\ \end{cases}$

For the latter two, all sets $S$ leading to null-controllability in time $T$
have recently been identified:

######  Definition 4.

A set $S\subset\mathbb{R}^{d}$ satisfies the _Geometric Control Condition_
(GCC) if there are $\delta,L>0$ such that every straight line of length $L$
has intersection at least $\delta$ with $S$ (with respect to the one-
dimensional Hausdorff measure on the line).
A set $S\subset\mathbb{R}^{d}$ is called _thick_ if there are $\gamma,\rho>0$
such that $\operatorname{Vol}(S\cap B_{\rho}(x))\geq\gamma$ for all
$x\in\mathbb{R}^{d}$.

Nonempty, open, periodic.Set satisfying the GCC.Generic thick set. Figure 1.
Different types of control sets discussed in this article. Note that the
nonempty, open, periodic arrangement of balls on the left does not satisfy the
GCC since there are arbitrarily long straight lines in its complement.

######  Remark 5.

The GCC is a strictly stronger condition that thickness. One example of thick
sets not satisfying the GCC are periodic arrangements of balls with small
radii as permitted in Theorem 2.

In the literature, one often finds the GCC formulated in terms of geodesics
for a Riemannian metric, i.e. tailored to controllability of differential
operators of the form $\operatorname{div}A\nabla$ instead of the pure
Laplacian, and there are versions for domains or manifolds with non-empty
boundary. Since we are interested in the pure Laplacian in full space, we
refrain from formulating these more general variants. Also note that the GCC
reflects the ballistic nature of the wave equation whereas thickness reflects
the diffusive nature of the heat equation.

In 2016, Burq and Joly proved that the GCC is the necessary and sufficient
criterion on the set $S\subset\mathbb{R}^{d}$ for controllability of the wave
equation.

######  Proposition 6 ([BJ16, Section 6.2]).

The controlled wave equation in $\mathbb{R}^{d}$ is controllable in some time
$T$ if and only if $S\subset\mathbb{R}^{d}$ satisfies the GCC.

Note that for the wave equation there will be a – in general positive –
infimal time $T$ (depending on the parameters $L,\delta$ in the definition of
the GCC) in which it is controllable – a consequence of finite speed of
propagation.

As for the heat equation in full space, if null-controllability holds in some
time then it holds for all times and, in 2018 and 2019 two articles by Egidi
and Veselic [EV18] as well as by Wang, Wang, Zhang and Zhang [WWZZ19]
independently proved that the necessary and sufficient criterion is thickness.

######  Proposition 7 ([EV18, WWZZ19]).

The controlled heat equation in $\mathbb{R}^{d}$ (3) is null-controllable in
any time $T>0$ if and only if $S\subset\mathbb{R}^{d}$ is thick.

Now, the Schrödinger equation shares properties both with the wave equation
(time-reversability) and with the heat equation (infinite speed of
propagation). It is occasionally described as a chimera of the two or as a
_wave equation with infinite speed of propagation_. This raises the questions
which criterion is necessary and sufficient for controllability of the
Schrödinger equation. So far, this has only been answered in dimension one,
where the concept of thickness and the GCC happen to coincide. Indeed, the
references [MPS20, HWW22] independently proved that they are the necessary and
sufficient criterion for controllability of the one-dimensional Schrödinger
equation.

In higher dimensions, there are only preliminary results for the controlled
Schrödinger equation so far. It is known that, if the control set is a
complement of a ball, the Schrödinger equation is null-controllable [RZ09],
with explicit estimates on the cost [WWZ19]. Also, the following theorem
stating that the GCC is sufficient for controllability in any dimension sees
to be folklore:

######  Theorem 8.

If the control set satisfies the GCC, then the Schrödinger equation is null
controllable in any time $T>0$.

###### Proof.

If $S$ satisfies the GCC then the _wave equation_ (2) is controllable in some
time $T^{\ast}>0$ by Proposition 6. But a result by Miller immediately turns
this into controllability of the Schrödinger equation in any time $T>0$.
Indeed this is the statement of, [Mil05, Theorem 3.1], with $A=-\Delta$.
111Strictly speaking, the application of [Mil05, Theorem 3.1] leads to
controllability of the Schrödinger equation with a different sign convention
$i\frac{\mathrm{d}}{\mathrm{d}t}u-\Delta u=\mathbf{1}_{S}f$, but this merely
amounts to a complex conjugation. ∎

Likewise, thickness is a known necessary condition:

######  Theorem 9 (Theorem 1.6 in [MPS20] and Theorem 1.1 in [HWW22]).

Let the controlled Schrödinger equation be controllable in some time $T>0$.
Then $S$ must be thick.

For the sake of self-containedness we provide a proof in Section 3. It differs
from the argument in [MPS20, HWW22] which rely on works in [Mil05] in
combination with Kovrijkine’s version of the Logvinenko-Sereda theorem [LS74,
Kov00]. Instead, we use an elementary construction of explicit counterexamples
to an observability inequality for non-thick sets.

GCCopen,nonempty,periodicthickheat equation controllableSchrödinger equation
controllablewave equation controllable Figure 2. Known necessary and
sufficient conditions on the control sets $S\subset\mathbb{R}^{d}$ for the
heat, Schrödinger and wave equation.

Now, in order to return to the interpretation of Theorem 2, we can summarize
our findings in the following

######  Corollary 10.

The set of all $S\subset\mathbb{R}^{d}$ such that the controlled Schrödinger
equation in $\mathbb{R}^{d}$ with control in $S$ is controllable in any time
$T>0$ is contained in the corresponding set for the heat equation (thick
sets), but _strictly larger_ than the corresponding set for the wave equation
(sets satisfying the GCC).

One consequence is that when designing control systems for the Schrödinger
equation, one has a larger choice for the control support
$S\subset\mathbb{R}^{d}$ than for the wave equation.

It would be an interesting question to investigate whether the GCC still
yields any benefit for controllability of the Schrödinger equation – e.g. on
the level of the control cost. Likewise, the question of identifying the class
of _all_ sets $S\subset\mathbb{R}^{d}$ which lead to controllability of the
Schrödinger equationin full space remains open.

We also expect our results not necessarily be restricted to full space.
Indeed, by reflection techniques, see for instance [ES22] for analogous
arguments in the case of the heat equation, Theorem 2 will also holds for
half-spaces, sectors or certain infinite cones. Also, infinite strips or slabs
can be treated by a modification of the proof of Theorem 2 in Section 2.1,
taking the Floquet transform only in those directions in which the domain is
unbounded.

##  2\. Proofs

Without loss of generality, we may assume that the periodic set $S$ is a union
of balls of radius $R>0$, centered at the lattice points of
$2\pi\mathbb{Z}^{d}$

$S=\bigcup_{k\in 2\pi\mathbb{Z}^{d}}B_{R}(k).$

By the Hilbert Uniqueness method [Lio88], (null)-controllability in time $T$
is equivalent to the observability inequality

(4) $\lVert u\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\leq
C\int_{0}^{T}\lVert\mathbf{1}_{S}\mathrm{e}^{it\Delta}u\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\mathrm{d}t\quad\text{for
all}\quad u\in L^{2}(\mathbb{R}^{d}).$

###  2.1. Floquet-Bloch decomposition

We use Floquet-Bloch theory to reduce the observability estimate (4) to a
family of observability estimates on compact domains with varying boundary
conditions. Let
$\mathbb{T}^{d}:=\mathbb{R}^{d}/(2\pi\mathbb{Z}^{d})\cong(-\pi,\pi]^{d}$
denote the $d$-dimensional torus.

######  Definition 11.

Denote by $\mathcal{F}\colon L^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{T}^{d}\times\mathbb{T}^{d})$ the _Floquet transform_

$\displaystyle(\mathcal{F}u)(y,\theta)$
$\displaystyle={(2\pi)^{-d/2}}\sum_{k\in
2\pi\mathbb{Z}^{d}}\mathrm{e}^{i\theta k}u(y+k).$

This is an isometric isomorphism that commutes with periodic functions in the
sense that if $u\in L^{2}(\mathbb{R}^{d})$ and $f\in
L^{\infty}(\mathbb{R}^{d})$ is $2\pi\mathbb{Z}^{d}$-periodic, then

$\mathcal{F}(f\cdot
u)(y,\theta)=f\mid_{\mathbb{T}^{d}}(y)\cdot(\mathcal{F}u)(y,\theta).$

One also has for $u\in H^{2}(\mathbb{R}^{d})$

$\mathcal{F}(\Delta u)(y,\theta)=\Delta_{\theta}(\mathcal{F}u)(y,\theta)$

for almost all $\theta\in\mathbb{T}^{d}$ where $\Delta_{\theta}$ is the _self-
adjoint Laplacian on $\mathbb{T}^{d}$ (in the $y$-coordinate) with
$\theta$-pseudoperiodic boundary conditions_, that means

$\mathcal{D}(\Delta_{\theta})=\left\\{\phi\in
L^{2}(\mathbb{T}^{d})\colon\begin{aligned} &\text{there exists\
}\tilde{\phi}\in H^{2}_{\mathrm{loc}}(\mathbb{R}^{d})\ \text{with}\
\tilde{\phi}\mid_{\mathbb{T}^{d}}=\phi,\ \text{and}\\\ &\tilde{\phi}(x+2\pi
k)=\exp(ik\cdot\theta)\tilde{\phi}(x)\ \text{for all $k\in\mathbb{Z}^{d}$,
$x\in\mathbb{R}^{d}$}\end{aligned}\right\\}.$

Indeed, these statements can be seen by straightforward calculations on
sufficiently smooth functions and the general case follows by density. We also
refer to [Sjö91, Kuc93] for further reading on Floquet theory for partial
differential operators. In particular, one also has for all $u\in
L^{2}(\mathbb{R}^{d})$

$\mathcal{F}(\mathrm{e}^{it\Delta}u)(y,\theta)=\mathrm{e}^{it\Delta_{\theta}}(\mathcal{F}u)(y,\theta).$

The spectral decomposition of the operator $\Delta_{\theta}$ is well-known:

######  Lemma 12.

The spectral decomposition of $\Delta_{\theta}$ is given by the eigenbasis

$(\mathrm{e}^{i\gamma y})_{\gamma\in\Gamma_{\theta}},\text{where}\
\Gamma_{\theta}:=\frac{\theta}{2\pi}+\mathbb{Z}^{d}$

with corresponding eigenvalues

$(\lvert\gamma\rvert^{2})_{\gamma\in\Gamma_{\theta}}.$

###### Proof.

Since $(\mathrm{e}^{iky})_{k\in\mathbb{Z}^{d}}$ is an orthogonal basis of
$L^{2}(\mathbb{T}^{d})$, so is $(\mathrm{e}^{i\gamma
y})_{\gamma\in\Gamma_{\theta}}$. The eigenvalue property and the
$\theta$-pseudoperiodicity are verified by explicit computation. ∎

By isometry of the the Floquet transform, the l.h.s. of (4) becomes

$\displaystyle\int_{0}^{T}\int_{S}\lvert(\mathrm{e}^{it\Delta}u)(x)\rvert^{2}\mathrm{d}x\
\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\mathrm{d}t\left\langle\mathrm{e}^{it\Delta}u,\mathbf{1}_{S}\mathrm{e}^{it\Delta}u\right\rangle_{\mathbb{R}^{d}}$
$\displaystyle=\int_{0}^{T}\mathrm{d}t\left\langle\mathcal{F}(\mathrm{e}^{it\Delta}u),\mathcal{F}(\mathbf{1}_{S}\mathrm{e}^{it\Delta}u)\right\rangle_{\mathbb{T}^{d}\times\mathbb{T}^{d}}$
$\displaystyle=\int_{\mathbb{T}^{d}}\mathrm{d}\theta\int_{0}^{T}\mathrm{d}t\left\langle\mathrm{e}^{it\Delta_{\theta}}(\mathcal{F}u)(\cdot,\theta),\mathbf{1}_{B_{R}}\mathrm{e}^{it\Delta_{\theta}}(\mathcal{F}u)(\cdot,\theta)\right\rangle_{\mathbb{T}^{d}}$
$\displaystyle=\int_{\mathbb{T}^{d}}\mathrm{d}\theta\int_{0}^{T}\mathrm{d}t\int_{B_{R}}\mathrm{d}y\lvert\mathrm{e}^{it\Delta_{\theta}}(\mathcal{F}u)(y,\theta)\rvert^{2}$
$\displaystyle=\int_{\mathbb{T}^{d}}\mathrm{d}\theta\int_{0}^{T}\mathrm{d}t\int_{B_{R}}\mathrm{d}y\big{\lvert}\sum_{\gamma\in\Gamma_{\theta}}\mathrm{e}^{i(t\lvert\gamma\rvert^{2}+y\gamma)}\alpha_{\theta,\tilde{\gamma}}\big{\rvert}^{2}$

where the
$\alpha_{\theta}=(\alpha_{\theta,\tilde{\gamma}})_{\gamma\in\Gamma_{\theta}}\in\ell^{2}(\tilde{\Gamma}_{\theta})$
are the Fourier coefficients of $\mathbb{T}^{d}\ni
y\mapsto\mathcal{F}(u)(y,\theta)\in L^{2}(\mathbb{T}^{d})$ with respect to the
eigenbasis from Lemma 12 for fixed $\theta\in\mathbb{T}^{d}$. Writing

$t\lvert\gamma\rvert^{2}+y\gamma=\left\langle\begin{pmatrix}y\\\ t\\\
\end{pmatrix},\begin{pmatrix}\gamma\\\ \lvert\gamma\rvert^{2}\\\
\end{pmatrix}\right\rangle=:z\cdot\tilde{\gamma},$

for $\tilde{\gamma}:=(\gamma,\lvert\gamma\rvert^{2})\in\mathbb{R}^{d+1}$,
using unitarity of the Schrödinger group to shift the integration in $t$ from
$[0,T]$ to $[-\frac{T}{2},\frac{T}{2}]$ and using that, possibly after
shrinking R, $[-\frac{T}{2},\frac{T}{2}]\times B_{R}$ certainly contains the
$(d+1)$-dimensional ball of radius $R$ around the origin (which we again
denote by $B_{R}$), we can lower bound this by

$\int_{B_{R}}\big{\lvert}\sum_{\tilde{\gamma}\in\tilde{\Gamma}_{\theta}}\exp\left(z\cdot\tilde{\gamma}\right)\alpha_{\theta,\tilde{\gamma}}\big{\rvert}^{2}\mathrm{d}z.$

We have thus reduced the proof of the observability estimate (4) to the
following statement: There exists $C>0$ such that for all
$\theta\in\mathbb{T}^{d}$ one has

(5)
$\int_{B_{R}}\big{\lvert}\sum_{\tilde{\gamma}\in\tilde{\Gamma}_{\theta}}\exp\left(z\cdot\tilde{\gamma}\right)\alpha_{\theta,\tilde{\gamma}}\big{\rvert}^{2}\mathrm{d}z\geq
C\lVert\alpha_{\theta}\rVert_{\ell^{2}(\tilde{\Gamma}_{\theta})}^{2}\quad\text{for
all $\alpha_{\theta}\in\ell^{2}(\tilde{\Gamma}_{\theta})$},$

where

$\tilde{\Gamma}_{\theta}:=\left\\{\begin{pmatrix}\gamma\\\
\lvert\gamma\rvert^{2}\end{pmatrix}\colon\gamma\in\frac{\theta}{2\pi}+\mathbb{Z}^{d}\right\\}\subset\mathbb{R}^{d+1}.$

###  2.2. Lower bounds on exponential sums

We proceed to prove (5). For $\theta=0$, this can be found in the
$2$-dimensional case in [Jaf90] as an ingredient in a proof of controllability
of the Schrödinger equation on rectangles, and for higher dimensions in
[KL05]. While the proofs therein immediately generalize to _fixed_ $\theta$,
we need to ensure that the resulting constants are $\theta$-independent – a
statement not explicitly provided in [KL05]. We therefore follow the central
parts of the proofs given therein.

For a discrete set $X\subset\mathbb{R}^{d+1}$, let its _gap_ be given by

$\operatorname{gap}(X):=\inf\\{\lvert x-y\rvert\colon x,y\in X,x\neq y\\}.$

The following lemma can be inferred in this form from Theorem 8.1 in [KL05],
see also [Jaf90, Proposition 1]. In one dimension, the lemma is due to Ingham
[Ing36], while the higher dimensional case is essentially due to Kahane
[Kah62], see also [BKL99].

######  Lemma 13.

There is $c>0$, depending only on the dimension such that for all $\delta>0$
there is $\tilde{C}>0$, depending only on the dimension and on $\delta$, such
that for all $R\geq\frac{c}{\delta}$ and all $\Gamma\subset\mathbb{R}^{d+1}$
with $\operatorname{gap}(\Gamma)\geq\delta$ we have

$\int_{B_{R}}\lvert\sum_{\gamma\in\Gamma}\alpha_{\gamma}\exp(i\gamma
x)\rvert^{2}\mathrm{d}x\geq\tilde{C}\sum_{\gamma\in\Gamma}\lvert\alpha_{\gamma}\rvert^{2}\quad\text{for
all $\alpha\in\ell^{2}(\Gamma)$}.$

######  Remark 14.

There is an explicit expression for the infimum of all possible $c$, see the
remark after [KL05, Theorem 8.1]. More precisely, in dimension $d\geq 2$, the
infimal $c$ is the first positive root of $J_{\frac{d-1}{2}}$, the Bessel
function of first kind with parameter $\frac{d-1}{2}$. Choosing a strictly
larger $c$, one obtains a uniform $\tilde{C}$ for all $R\geq\frac{c}{\delta}$.

By demanding $R\geq\frac{c}{\delta}$, Lemma 13 imposes a reciprocal relation
between the radius $R$ of the ball and the gap $\delta$. Since the gap of
$\Gamma_{\theta}$ is of order one, this would exclude small $R$. The following
lemma remedies this by allowing to decompose a discrete $\Gamma$ into finitely
many $\Gamma_{j}$ with larger gaps.

######  Lemma 15.

Let $R>0$, $N\in\mathbb{N}$, and $c>0$ be the constant from Lemma 13. Let
$\Gamma=\Gamma_{1}\cup\dots\cup\Gamma_{N}$ be such that
$\operatorname{gap}(\Gamma)\geq 1$, and
$\operatorname{gap}(\Gamma_{i})\geq\delta_{i}$. Then, there is $C>0$,
depending only on the dimension, on $N$, and on the $\delta_{i}$ such that for
all

(6) $R\geq 2\left(\frac{c}{\delta_{1}}+\dots+\frac{c}{\delta_{N}}\right)$

we have

$\int_{B_{R}}\lvert\sum_{\gamma\in\Gamma}\alpha_{\gamma}\exp(i\gamma
x)\rvert^{2}\mathrm{d}x\geq
C\sum_{\gamma\in\Gamma}\lvert\alpha_{\gamma}\rvert^{2}\quad\text{for all
$\alpha\in\ell^{2}(\tilde{\Gamma})$}.$

Here, we use the convention that for a one-element set
$\operatorname{gap}(\Gamma)=\infty$ and $\frac{1}{\infty}=0$. A qualitative
variant of Lemma 15 is due to [Kah62], whereas we refer here again to [KL05,
Proposition 8.4] from which the quantitative version stated above can be
inferred in combination with Lemma 13. While the statement a priori holds for
all

$R>\left(\frac{c}{\delta_{1}}+\dots+\frac{c}{\delta_{N}}\right),$

the resulting constant $C$ will depend on a $\epsilon>0$ with

$R=\left(\frac{c}{\delta_{1}}+\dots+\frac{c}{\delta_{N}}\right)+\epsilon,$

chosen in the proof in [KL05]. The prefactor $2$ in (6) thus allows for a
uniform choice, depending only on the dimension, on $N$, and on the
$\delta_{i}$.

Now, the next lemma provides an appropriate decomposition of the set
$\Gamma_{\theta}$ into appropriate subsets. For $p=2$, the
$\theta$-independent version is in [Jaf90] in dimension $d=2$ and in [KL05] in
the higher-dimensional case. A $\theta$-independent version for $p>1$ can also
be obtained as a corollary of [Kah62, Théorème III.4.1]. However, for the
required $\theta$-independence of the constant $C$ in Lemma 15, we provide a
proof here.

######  Lemma 16.

Let $R>0$, and $p>1$. There is $N\in\mathbb{N}$ such that for all
$\theta\in\mathbb{T}^{d}$, we can decompose

$\tilde{\Gamma}_{\theta}=\left\\{\begin{pmatrix}\gamma\\\
\lvert\gamma\rvert^{2}\end{pmatrix}\colon\gamma\in\frac{\theta}{2\pi}+\mathbb{Z}^{d}\right\\}=\bigcup_{j=1}^{N}\tilde{\Gamma}_{\theta,j}$

where for $\delta_{j}:=\operatorname{gap}(\tilde{\Gamma}_{\theta,j})$ one has

$2\left(\frac{c}{\delta_{1}}+\dots+\frac{c}{\delta_{N}}\right)\leq R$

###### Proof.

We split $\tilde{\Gamma}_{\theta}$ into three components.

_1\. Points with $(x_{1},\dots,x_{d})$ bounded away from the axes._ Defining
for $\alpha\geq 1$

$A_{\alpha}:=\left\\{x\in\mathbb{R}^{d+1}\colon\lvert x_{j}\rvert\geq\alpha\
\forall j\in\\{1,\dots d\\}\right\\}$

one has

$\operatorname{gap}(\tilde{\Gamma}_{\theta}\cap A_{\alpha})\geq\alpha.$

Indeed, let $\gamma\in\Gamma_{\theta}$ with
$(\gamma,\lvert\gamma\rvert^{p})\in A_{\alpha}$ and $e_{k}=(0,\dots,0,\pm
1,0,\dots,0)\in\mathbb{Z}^{d}$ be the $k$-th unit vector. Then,

$\displaystyle\left\lvert\begin{pmatrix}\gamma\\\
\lvert\gamma\rvert^{2}\end{pmatrix}-\begin{pmatrix}\gamma+e_{k}\\\
\lvert\gamma+e_{k}\rvert^{2}\end{pmatrix}\right\rvert^{2}$
$\displaystyle\geq\left(\lvert\gamma\rvert^{2}-\lvert\gamma+e_{k}\rvert^{2}\right)^{2}.=\left(\lvert\gamma\rvert^{2}-\lvert\gamma\pm
e_{k}\rvert^{2}\right)^{2}$
$\displaystyle\geq\left(\gamma_{k}^{2}-(\gamma_{k}\pm
1)^{2}\right)^{2}\geq(\pm 2\gamma_{k}+1)^{2}\geq(\alpha-1)^{2}\geq\alpha^{2}.$

Thus, choosing

$\alpha=\max\left\\{\frac{6c}{R},1\right\\}$

and setting $\tilde{\Gamma}_{\theta,1}=A_{\alpha}\cap\tilde{\Gamma}_{\theta}$
ensures $\frac{2c}{\delta_{1}}\leq\frac{R}{3}$.

_2\. Neighbourhoods of the axes in the first $d$ coordinates_. Next, for
$\beta>\alpha$ let

$B_{\alpha,\beta}:=A_{\alpha}^{c}\cap\\{x\in\mathbb{R}^{d+1}\colon\exists
j\in\\{1,\dots,d\\}\colon\lvert x_{j}\rvert\geq\beta\\}.$

The set $B_{\alpha,\beta}$ is a union of $2d$ many slabs, consisting of
rectangular neighbourhoods around positive the $x_{1}$ to $x_{d}$ axes in
$\mathbb{R}^{d}$, and extending infinitely in the $x_{d+1}$ direction.

$\mathbb{R}^{d}$$\alpha$$\beta$$A_{\alpha}$$B_{\alpha,\beta}$Rest Figure 3.
Projections onto $\mathbb{R}^{d}$ of the decomposition of
$\tilde{\Gamma}_{\theta}$.

Let $B_{\alpha,\beta,k}^{\pm}\subset B_{\alpha,\beta}$ denote the union of the
two components which are arranged along the positive and negative part of the
$x_{k}$-axis. Then,

$\tilde{\Gamma}_{\theta}\cap
B_{\alpha,\beta,k}^{\pm}=\left\\{\begin{pmatrix}\gamma\\\
\lvert\gamma\rvert^{2}\end{pmatrix}\colon\begin{aligned}
&\gamma\in\mathbb{Z}^{d}+\theta,\ \\\
&\lvert\gamma_{k}+\frac{\theta_{k}}{2\pi}\rvert\geq\beta,\\\
&\lvert\gamma_{i}+\frac{\theta_{i}}{2\pi}\rvert\leq\alpha\text{ for all }i\neq
k\end{aligned}\right\\}$

and we see that for sufficiently large $\beta$, the distance to the next
neighbour in $x_{k}$ direction grows quadratically. Since
$B_{\alpha,\beta,k}^{\pm}$ has a finite profile in $(x_{1},\dots
x_{k-1},x_{k+1},\dots,x_{d})$ direction, we can split
$B_{\alpha,\beta,k}^{\pm}\cap\tilde{\Gamma}_{\theta}$ into

$N_{\alpha}:=\\#\\{[-\alpha-1,\alpha+1]^{d-1}\cap\mathbb{Z}^{d-1}+(\theta_{1},\dots,\theta_{k-1},\theta_{k+1},\dots,\theta_{d})\\}$

many components, each with fixed $(x_{1},\dots,x_{k-1},x_{k+1},\dots,x_{d})$
coordinates, and with gap at least

$\lvert\beta+1\rvert^{2}-\lvert\beta\rvert^{2}=2\beta+\beta^{2}\geq 3\beta.$

Uniting all $d$ many components $\\{B_{\alpha,\beta,k}^{\pm}\\}_{k=1}^{d}$ and
choosing

$\beta\geq\frac{2dN_{\alpha}c}{R}$

we ensure

$2\sum_{j=2}^{dN_{\alpha}+1}\frac{c}{\delta_{j}}\leq\frac{R}{3}.$

_3\. The bounded rest_. We have covered all but at most
$\left(2\beta+1\right)^{d}$ many points of $\tilde{\Gamma}_{\theta}$. Let each
of these points form another $\tilde{\Gamma}_{\theta,j}$ which has gap equal
to $\infty$. We have found for each $\theta\in\mathbb{T}^{d}$ a decomposition
$\tilde{\Gamma}_{\theta}=\cup_{j=1}^{N}\tilde{\Gamma}_{\theta,j}$ where $N$ is
uniformly bounded in $\theta$ and (6) holds. ∎

###### Proof of Theorem 2.

Combining Lemma 16 with Lemma 15 yields the $\theta$-independent observability
estimate (5). Due to the reductions made at the beginning of this section this
proves the observability inequality (4), and consequently Theorem 2. ∎

##  3\. Thickness is necessary for Schrödinger

We give an elementary proof of Theorem 9 (previously demonstrated using a
different method in [MPS20]) stating that in the case of controllability of
the Schrödinger equation the control set $S$ must be thick.

###### Proof.

By the Hilbert uniqueness method [Lio88], (null-)controllability in time $T$
is equivalent to the observability inequality

$\lVert u\rVert^{2}\leq
C\int_{0}^{T}\lVert\mathbf{1}_{S}\mathrm{e}^{it\Delta}u\rVert^{2}\mathrm{d}t\quad\text{for
all}\quad u\in L^{2}(\mathbb{R}^{d}).$

Thus, it suffices to find for any $\epsilon>0$ a function $u\in
L^{2}(\mathbb{R}^{d})$ with

$\int_{0}^{T}\lVert\mathbf{1}_{S}\mathrm{e}^{i\Delta
t}u\rVert^{2}\mathrm{d}t\leq\epsilon\lVert u\rVert^{2}.$

Clearly, we have for any $u\in L^{2}(\mathbb{R}^{d})$

$\int_{0}^{T}\lVert\mathbf{1}_{S}\mathrm{e}^{i\Delta
t}u\rVert^{2}\mathrm{d}t\leq
2\int_{0}^{T}\lVert\mathbf{1}_{S}\left[\mathrm{e}^{i\Delta
t}-\mathrm{e}^{\Delta
t}\right]u\rVert^{2}\mathrm{d}t+2\int_{0}^{T}\lVert\mathbf{1}_{S}\mathrm{e}^{\Delta
t}u\rVert^{2}\mathrm{d}t=(A)+(B).$

For $E>0$ to be chosen below let $u=u_{1}+u_{2}$ with $u_{1}$ having Fourier
support in $B_{E}(0)$ and $u_{2}$ Fourier support in $B_{E}^{c}(0)$. Since
$0\leq\mathbf{1}_{S}\leq\operatorname{Id}$ and since the operator norm of
$\mathbf{1}_{S}\left[\mathrm{e}^{i\Delta t}-\mathrm{e}^{\Delta t}\right]$ is
at most $2$, we can estimate

$(A)\leq 2\int_{0}^{T}\lVert\left[\mathrm{e}^{i\Delta t}-\mathrm{e}^{\Delta
t}\right]u_{1}\rVert^{2}\mathrm{d}t+4T\lVert u_{2}\rVert^{2}=(A_{1})+(A_{2})$

By Plancherel, we further estimate

$(A_{1})=2\int_{0}^{T}\lVert\left[\mathrm{e}^{-i\xi^{2}t}-\mathrm{e}^{-\xi^{2}t}\right]\widehat{u_{1}}\rVert^{2}\mathrm{d}t\leq
2T\max_{\lvert\xi\rvert\leq
E,t\in[0,T]}\lvert\mathrm{e}^{-i\xi^{2}t}-\mathrm{e}^{-\xi^{2}t}\rvert\cdot\lVert
u\rVert^{2}\leq cT^{2}E^{2}\lVert u\rVert^{2}.$

Taking $E$ sufficiently small (depending only on $T$) this can be made smaller
than $\frac{\epsilon}{3}\lVert u\rVert^{2}$, independently of $u$.

We now choose $u$ as a (square root of a) normalized Gaussian with standard
deviation $\nu$, and center $x_{0}$ that is

$u(x):=\frac{1}{\left(2\pi\nu\right)^{d/4}}\exp\left(-\frac{\lVert
x-x_{0}\rVert^{2}}{4\nu}\right).$

where $\nu>0$, $x_{0}\in\mathbb{R}^{d}$ will be chosen below. We have

$\widehat{u}(\xi)=\left(\frac{2\nu}{\pi}\right)^{d/4}\exp\left(-\xi^{2}\nu\right),\quad\text{and}\quad\widehat{u}_{2}(\xi)=\chi_{\lvert\xi\rvert\geq
E}\left(\frac{2\nu}{\pi}\right)^{d/4}\exp\left(-\xi^{2}\nu\right).$

The function $\hat{u}$ is the (square root of a) Gaussian with standard
deviation proportional to $\nu^{-1}$ and for $\nu\gg 1$, its mass will
concentrate on $B_{E}(0)$. Hence, choosing $\nu$ sufficiently large (depending
on $E$ which is already fixed), we obtain
$(A_{2})=4T\lVert\widehat{u}_{2}\rVert^{2}\leq\frac{\epsilon}{3}\lVert
u\rVert^{2}$.

It remains to estimate $(B)$. This is the time evolution part of the heat
equation and can be made small by putting $x_{0}$ into a center of a
sufficiently large (depending on $\epsilon$ and $\nu$) ball on which $\omega$
has little mass, as has been done for the heat equation, see for instance
[EV18] for details. Such a clearing with center $x_{0}$ exists by assumption
since $S$ is assumed not to be thick. ∎

## Acknowledgements

The author thanks Kotaro Inami and Soichiro Suzuki for pointing out an error
in an earlier version, Walton Green for discussions, and the anonymus referees
for helpful comments.

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