# Towards semi-classical analysis for sub-elliptic operators

Véronique Fischer  University of Bath, Department of Mathematical Sciences,
Bath, BA2 7AY, UK<EMAIL_ADDRESS>
###### Abstract.

We discuss the recent developments of semi-classical and micro-local analysis
in the context of nilpotent Lie groups and for sub-elliptic operators. In
particular, we give an overview of pseudo-differential calculi recently
defined on nilpotent Lie groups as well as of the notion of quantum limits in
the Euclidean and nilpotent cases.

2010 MSC. 43A80; 58J45, 35Q40.

Keywords. Analysis on nilpotent Lie groups, evolution of solutions to the
Schrödinger equation, micro-local and semi-classical analysis for sub-elliptic
operators, abstract harmonic analysis, $C^{*}$-algebra theory.

###### Contents

  1. 1  Introduction
    1. 1.1 The questions in the elliptic framework.
    2. 1.2 Sub-elliptic operators.
    3. 1.3 Aim and organisation of the paper
    4. 1.4 Acknowledgement
  2. 2 Quantum limits in Euclidean or elliptic settings
    1. 2.1 Micro-local defect measures
    2. 2.2 The viewpoint of quantum limits
    3. 2.3 Semi-classical measures as quantum limits
    4. 2.4 Applications
  3. 3 Pseudo-differential theory and quantum limits on nilpotent Lie groups
    1. 3.1 Preliminaries on nilpotent Lie groups
    2. 3.2 Pseudo-differential calculi on graded nilpotent Lie groups
      1. 3.2.1 Preliminaries on graded groups
      2. 3.2.2 The symbolic pseudo-differential calculus on $G$
      3. 3.2.3 The classical pseudo-differential calculus on $G$
      4. 3.2.4 The semi-classical pseudo-differential calculus on $G$
    3. 3.3 Operator-valued measures
    4. 3.4 Micro-local defect measures on graded Lie groups
    5. 3.5 Semi-classical measures on graded Lie groups
    6. 3.6 Future works

##  1\.  Introduction

Since the 1960’s, the analysis of elliptic operators has made fundamental
progress with the emergence of pseudo-differential theory and the subsequent
developments of micro-local and semi-classical analysis. In this paper, we
consider some questions that are well understood for elliptic operators and we
discuss analogues in the setting of sub-elliptic operators.

###  1.1. The questions in the elliptic framework.

The questions we are interested in concern the tools that have been developed
in the elliptic framework to describe and understand the limits in space or in
phase-space of families of functions. They are of two natures: micro-local and
semi-classical. Micro-local analysis aims at understanding elliptic operators
in high frequency, while semi-classical analysis investigate the mathematical
evolution of functions and operators depending on a small parameter
$\varepsilon$ (akin to the Planck constant in quantum mechanics) that goes to
zero.

A typical micro-local question is, for instance, to ‘understand the
convergence’ as $j\to\infty$ of an orthonormal basis of eigenfunctions
$\psi_{j}$, $j=0,1,2,\ldots$

$\Delta\psi_{j}=\lambda_{j}\psi_{j},\qquad\qquad\mbox{with}\quad
0=\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq\ldots$

of the Laplace operator $\Delta$ on a compact Riemannian manifold $M$. One way
to answer this question is to describe the accumulation points of the sequence
of probability measures $|\psi_{j}(x)|^{2}dx$, $j=0,1,2,\ldots$ If $M$ is the
$n$-dimensional torus or if the geodesic flow of $M$ is ergodic, then the
volume element $dx$ is an accumulation point of $|\psi_{j}(x)|^{2}dx$,
$j=0,1,2,\ldots$ and one can extract a subsequence of density one
$(j_{k})_{k\in\mathbb{N}}$,

$\mbox{i.e.}\quad\lim_{\Lambda\to\infty}\frac{|\\{j_{k}:\lambda_{j_{k}}\leq\Lambda\\}|}{|\\{j:\lambda_{j}\leq\Lambda\\}|}=1,$

for which the convergence holds, that is, for any continuous function
$a:M\to\mathbb{C}$,

(1) $\lim_{k\to+\infty}\int_{M}a(x)\
|\varphi_{j_{k}}(x)|^{2}dx=\int_{M}a(x)\,dx.$

Under the ergodic hypothesis, this is a famous result due to Shnirelman [48],
Colin de Verdière [15], and Zelditch [55] in 1970’s and 80’s and sometimes
called the Quantum Ergodicity Theorem - see also the semi-classical analogue
in [38].

A typical semi-classical problem is to understand the quantum evolution of the
Schrödinger equation

$i\varepsilon\partial_{t}\psi^{\varepsilon}=-\frac{\varepsilon^{2}}{2}\Delta\psi^{\varepsilon},\qquad\mbox{given
an}\ L^{2}\mbox{-bounded family of initial datum}\
\psi^{\varepsilon}|_{t=0}=\psi^{\varepsilon}_{0};$

in this introduction, let us consider the setting of $\mathbb{R}^{n}$ to fix
ideas. Again, a mathematical formulation consists in describing the
accumulation points of the sequence of measures
$|\psi^{\varepsilon}(t,x)|^{2}dxdt$ as $\varepsilon\to 0$.

###  1.2. Sub-elliptic operators.

In this paper, we discuss the extent to which these types of questions have
been addressed for sub-elliptic operators. The main examples of sub-elliptic
operators are sub-Laplacians $\mathcal{L}$ generalising the Laplace operator.
Concrete examples of sub-elliptic and non-elliptic operators include

$\mathcal{L}_{G}=-\partial_{u}^{2}-(u\partial_{v})^{2}\quad\mbox{on}\
\mathbb{R}_{u}\times\mathbb{R}_{v}=\mathbb{R}^{2},$

often called the Grushin operator (the subscript $G$ stands for Grushin). More
generally, Hörmander sums of squares are sub-elliptic operators; they are
operators $\mathcal{L}=-X_{1}^{2}-\ldots-X_{n_{1}}^{2}-X_{0}$ on a manifold
$M^{n}$ where the vector fields $X_{j}$’s together with their iterated
brackets generate the tangent space $TM$ at every point [40]. A more geometric
source of sub-Laplacians is the analysis on sub-Riemannian manifolds, starting
with CR manifolds such as the unit sphere of the complex plane
$\mathbb{C}^{2}$ or even of $\mathbb{C}^{n}$ for any $n\geq 2$, and more
generally contact manifolds. Well-known contact manifolds of dimension three
include the Lie group $SO(3)$ with two of its three canonical vector fields,
as well as the motion group
$\mathbb{R}^{2}_{x,y}\times\mathbb{S}^{1}_{\theta}$ with the vector fields
$X_{1}=\cos\theta\partial_{x}+\sin\theta\partial_{y}$, and
$X_{2}=\partial_{\theta}$. Sub-Laplacians appear in many parts of sciences, in
physics, biology, finance, etc., see [11].

A particular framework of sub-Riemannian and sub-elliptic settings is given by
Carnot groups; the latter are stratified nilpotent Lie groups $G$ equipped
with a basis $X_{1},\ldots,X_{n_{1}}$ for the first stratum $\mathfrak{g}$ of
the Lie algebra of $G$. Using the natural identification of $\mathfrak{g}$
with the space left-invariant vector fields, the canonical sub-Laplacian is
then $\mathcal{L}=-X_{1}^{2}-\ldots-X_{n_{1}}^{2}$. This is an important class
of examples not only because this provides a wealth of models and settings on
which to test conjectures, but also more fundamentally, as any Hörmander sum
of squares can be lifted - at least theoretically [29, 46, 43] \- to a Carnot
group. For instance, the Grushin operator $\mathscr{L}_{G}$ on
$\mathbb{R}^{2}$ described above can be lifted to the sub-Laplacian
$\mathscr{L}_{\mathbb{H}_{1}}=-X_{1}^{2}-X_{2}^{2}$ on the Heisenberg group
$\mathbb{H}_{1}$; here the product on
$\mathbb{H}_{1}\sim\mathbb{R}^{3}_{x,y,t}$ is given by

$(x,y,t)(x^{\prime},y^{\prime},t^{\prime})=\left(x+x^{\prime},y+y^{\prime},t+t^{\prime}+\frac{1}{2}(xy^{\prime}-x^{\prime}y)\right),$

and $X_{1}=\partial_{x}-\frac{y}{2}\partial_{t}$ and
$X_{2}=\partial_{y}+\frac{x}{2}\partial_{t}$.

The examples given above infer that the analysis of sub-elliptic operators
such as Hörmander sums of squares is more non-commutative than in the elliptic
case. Indeed, the commutator of the vector fields $X_{j}$’s in our examples
above usually produces terms that cannot be neglected in any meaningful
elliptic analysis, whereas in the elliptic case the $X_{j}$’s can be chosen to
yield local coordinates and therefore commute up to lower order terms. This
led to a difficult non-commutative analysis in the late 70’s and 80’s around
the ideas of lifting the nilpotent Lie group setting [29, 46, 43], and
subsequently in 80’s and 90’s using Euclidean micro-local tools as well [23,
47, 44]. At the same time, sub-Riemannian geometry was emerging. Although many
functional features are almost identical to the Riemannian case [49], there
are fundamental differences regarding e.g. geodesics, charts or local
coordinates, tangent spaces etc. see e.g. [10, 37, 42, 1].

The analysis of operators on classes of sub-Riemannian manifolds started with
CR and contact manifolds [29], followed by a calculus on Heisenberg manifolds
[9, 45]. In 2010 [53], an index theorem was proved for sub-elliptic operators
on contact manifolds. The key idea was to adapt Connes’ tangent groupoid [17]
from the Riemannian setting to the sub-Riemannian’s. For contact manifolds,
the Euclidean tangent space is then replaced with the Heisenberg group. Since
then, considerable progress has been achieved in the study of spectral
properties of sub-elliptic operators in these contexts (see e.g. [22]) with
the development of these groupoid techniques on filtered manifolds [53, 14,
54].

Few works on sub-elliptic operators followed the path opened by M. Taylor [52]
at the beginning of the 80’s, that is, to use the representation theory of the
underlying groups to tackle the non-commutativity. To the author’s knowledge,
in the nilpotent case, they are essentially [4, 28, 3] and, surprisingly, have
appeared only in the past decade.

###  1.3. Aim and organisation of the paper

This paper describes the scientific journey of the author and of her
collaborator Clotilde Fermanian-Kammerer towards micro-local and semi-
classical analysis for sub-elliptic operators, especially on nilpotent Lie
groups. The starting point of the investigations was to define and study the
analogues of micro-local defect measures. As explained in Section 2, this has
led to adopt the more general view point and the vocabulary from
$C^{*}$-algebras regarding states even in the Euclidean or elliptic case. The
first results regarding micro-local defect measure and semi-classical measures
on nilpotent Lie groups are presented in Section 3, including applications and
future works.

###  1.4. Acknowledgement

The author is grateful to the Leverhulme Trust for their support via Research
Project Grant 2020-037.

This paper summarises the main ideas discussed by the author to the Bruno Pini
Mathematical Analysis Seminar of the University of Bologna in May 2021. The
author would like to thank the organisers for giving her the opportunity to
present the project and for their warm welcome - even in zoom form.

##  2\. Quantum limits in Euclidean or elliptic settings

In this section, we discuss how micro-local defect measures and semi-classical
measures can be seen as quantum limits, that is, as states of
$C^{*}$-algebras.

###  2.1. Micro-local defect measures

The notion of micro-local defect measure, also called H-measure, emerged
around 1990 independently in the works of P. Gérard [33] and L. Tartar [50].
Their motivations came from PDEs, in relation to the div-curl lemma and more
generally to describe phenomena of compensated compactness. The following
result gives the existence of micro-local defect measures:

######  Theorem 2.1 ([33]).

Let $\Omega$ be an open subset of $\mathbb{R}^{n}$. Let
$(f_{j})_{j\in\mathbb{N}}$ be a bounded sequence in $L^{2}(\Omega,loc)$
converging weakly to 0. Then there exist a subsequence
$(j_{k})_{k\in\mathbb{N}}$ and a positive Radon measure $\gamma$ on
$\Omega\times\mathbb{S}^{n-1}$ such that the convergence

$(Af_{j},f_{j})_{L^{2}}\longrightarrow_{j=j_{k},k\to\infty}\int_{\Omega\times\mathbb{S}^{n-1}}a_{0}(x,\xi)d\gamma(x,\xi)$

holds for all classical pseudo-differential operator $A$, $a_{0}$ denoting its
principal symbol.

Here, $\mathbb{S}^{n-1}$ denotes the unit sphere in $\mathbb{R}^{n}$. The
classical pseudo-differential calculus refers to all the Hörmander pseudo-
differential operators of non-positive order, with symbols admitting a
homogeneous expansion and with integral kernel compactly supported in
$\Omega\times\Omega$.

The measure $\gamma$ in Theorem 2.1 is called a micro-local defect measure for
$(f_{j})$, or the (pure) micro-local defect measure for $(f_{j_{k}})$.
Examples of micro-local defect measures include

  * •

an $L^{2}$-concentration in space $f_{j}(x)=j^{n/2}\chi(j(x-x_{0}))$ about a
point $x_{0}$ (here, $\chi\in C_{c}^{\infty}(\mathbb{R}^{n})$ is some given
function), whose micro-local defect measure is
$\gamma(x,\xi)=\delta_{x_{0}}(x)\otimes d_{\chi}(\xi)d\sigma(\xi)$, where
$\sigma$ is the uniform measure on $\mathbb{S}^{n-1}$ (i.e. the standard
surface measure on the unit sphere) and
$d_{\chi}(\xi):=\int_{r=0}^{\infty}|\widehat{\chi}(r\xi)|^{2}r^{n-1}dr$ ,

  * •

an $L^{2}$-concentration in oscillations $f_{j}(x)=\psi(x)e^{2i\pi
j\xi_{0}\cdot x}$ about a frequency $\xi_{0}\in\mathbb{S}^{n-1}$ (here, $\psi$
is some given smooth function with compact support in $\mathbb{R}^{n}$), whose
micro-local defect measure is
$\gamma(x,\xi)=|\psi(x)|^{2}dx\otimes\delta_{\xi_{0}}(\xi)$.

Theorem 2.1 extends readily to manifolds by replacing
$\Omega\times\mathbb{S}^{n-1}$ with the spherical co-tangent bundle.

The introduction of this paper mentions the Quantum Ergodicity Theorem, see
(1). This is in fact the reduced version ‘in position’. A modern presentation
of the full Quantum Ergodicity Theorem can be expressed as saying that the
Liouville measure $dx\otimes d\sigma(\xi)$ is a micro-local defect measure of
the sequence $(\psi_{j})_{j\in\mathbb{N}_{0}}$, for which the subsequence
$(j_{k})$ is of density one.

###  2.2. The viewpoint of quantum limits

The author’s definition of quantum limits is a notion along the line of the
following:

######  Definition 2.1.

The _quantum limit_ of a sequence $(f_{j})$ of unit vectors in a Hilbert space
$\mathcal{H}$ is any accumulation point of the functional
$A\mapsto(Af_{j},f_{j})_{\mathcal{H}}$ on a sub-$C^{*}$-algebra of
$\mathscr{L}(\mathcal{H})$.

One may still keep the vocabulary ‘quantum limits’ in slightly more general
contexts. For instance, one often encounters a subalgebra of
$\mathscr{L}(\mathcal{H})$ that may need to be completed into a
$C^{*}$-algebra, possibly after quotienting by (a subspace of) the kernel of
the mapping $A\mapsto\limsup_{j\to\infty}|(Af_{j},f_{j})_{\mathcal{H}}|$. We
may also consider a bounded family $(f_{j})$ in $\mathcal{H}$ rather than unit
vectors, leaving the normalisation for the proofs of further properties.

The applications we have in mind involve pseudo-differential calculi as
subalgebras of $\mathscr{L}(\mathcal{H})$ where the Hilbert space
$\mathcal{H}$ is some $L^{2}$-space. A quantum limit in this context will
often turn out to be a state (or a positive functional if the
$\|f_{j}\|_{\mathcal{H}}$’s are only bounded) on a space of symbols, hence a
positive Radon measure in the commutative case. Indeed, from functional
analysis, we know that a bounded linear functional on the space of continuous
functions on a (say) compact space is given by a Radon measure, and if the
functional is also positive, the measure will be positive as well.

Let us now explain how the viewpoint of quantum limits and states gives
another proof of Theorem 2.1 by first obtaining the following result:

######  Lemma 2.1.

Let $\Omega$ be an open bounded subset of $\mathbb{R}^{n}$. Let
$(f_{j})_{j\in\mathbb{N}}$ be a bounded sequence in $L^{2}(\bar{\Omega})$
converging weakly to 0 as $j\to\infty$. Then there exists a subsequence
$(j_{k})_{k\in\mathbb{N}}$ and a positive Radon measure on
$\bar{\Omega}\times\mathbb{S}^{n-1}$ such that

$(Af_{j},f_{j})_{L^{2}(\bar{\Omega})}\longrightarrow_{j=j_{k},k\to\infty}\int_{\bar{\Omega}\times\mathbb{S}^{n-1}}a_{0}(x,\xi)d\gamma(x,\xi)$

holds for all classical pseudo-differential operator $A$ whose principal
symbol $a_{0}$ is $x$-supported in $\bar{\Omega}$.

###### Sketch of the proof of Lemma 2.1.

If $\limsup_{j\to\infty}\|f_{j}\|_{L^{2}(\bar{\Omega})}=0$, then $\gamma=0$.
Hence, we may assume that
$\limsup_{j\to\infty}\|f_{j}\|_{L^{2}(\bar{\Omega})}=1$. We consider the
sequence of functionals $\ell_{j}:A\mapsto(Af_{j},f_{j})_{L^{2}}$ on the
algebra $\mathcal{A}_{0}$ of classical pseudo-differential operators $A$ whose
symbols are $x$-supported in $\bar{\Omega}$.

The weak convergence of $(f_{j})$ to zero means that
$\lim_{j\to\infty}\ell_{j}(A)=0$ for every operator $A$ in

$\mathcal{K}=\\{\mbox{compact operator in}\
\mathcal{A}_{0}\\}\sim\\{\mbox{operators in}\ \mathcal{A}_{0}\ \mbox{of order
}<0\\},$

by Rellich’s theorem.

The properties of the pseudo-differential calculus imply that a limit of
$(\ell_{j})_{j\in\mathbb{N}}$ is a state on the closure of the quotient
$\overline{\mathcal{A}_{0}/\mathcal{K}}$; we recognise the abelian
$C^{*}$-algebra generated by the principal symbols $x$-supported in
$\bar{\Omega}$, that is, the space of continuous functions on the compact
space $\bar{\Omega}\times\mathbb{S}^{n-1}$. Such a state is given by a
positive Radon measure on $\bar{\Omega}\times\mathbb{S}^{n-1}$. ∎

Let us now give the new proof of Theorem 2.1 announced above. Adopting the
setting of the statement, we find a sequence of open sets $\Omega_{k}$,
$k=1,2,\ldots$ such that $\bar{\Omega}_{k}$ is a compact subset of
$\Omega_{k+1}$ and $\cup_{k\in\mathbb{N}}\Omega_{k}=\Omega$. Applying Lemma
2.1 to each $\Omega_{k}$ together with a diagonal extraction yield Theorem
2.1.

The author and her collaborator Clotilde Fermanian Kammerer are forever
indebted to Professor Vladimir Georgescu for his enlightening explanations on
the proof of the existence of micro-local defect measures given above.
Vladimir Georgescu’s comments describe also the states of other
$C^{*}$-algebras of operators bounded on $L^{2}$ from profound works by O.
Cordes and his collaborators on Gelfand theory for pseudo-differential calculi
[18, 19, 20, 21, 51]. They also provide a framework which generalises the two
original proofs of the existence of H-measure / micro-local defect measures:

  * •

the one by L. Tartar [50] which uses operators of multiplication in position
and Fourier multipliers in frequencies, and

  * •

the one by P. Gérard’s [33] relying on properties of the classical pseudo-
differential calculus, especially the Gårding inequality.

###  2.3. Semi-classical measures as quantum limits

The semi-classical calculus used here is ‘basic’ in the sense that it is
restricted to the setting of $\mathbb{R}^{n}$ and to operators ${\rm
Op}_{\varepsilon}(a)$ with $a\in
C_{c}^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})$ for instance. Here ${\rm
Op}_{\varepsilon}(a)={\rm Op}(a_{\varepsilon})$ is ‘the’ pseudo-differential
operator with symbol $a_{\varepsilon}(x,\xi)=a(x,\varepsilon\xi)$ via a chosen
$t$-quantisation on $\mathbb{R}^{n}$ \- for instance the Weyl quantisation
($t=1/2$) or the Kohn-Nirenberg quantisation ($t=0$, also known as PDE
quantisation and often written as ${\rm Op}(a)=a(x,D)$). More sophisticated
semi-classical calculi can be defined, for instance allowing the symbols $a$
to depend on $\varepsilon$ and in the context of manifolds, see e.g. [57].

Semi-classical measures were introduced in the 90’s in works such as [34, 35,
36, 41]. In this section, we show how the viewpoint of quantum limits gives a
simple proof of their existence as in the case of micro-local defect measures
(see Section 2.2). With the Weyl quantisation, the existence of semi-classical
measures can be proved using graduate-level functional analysis and the
resulting measures are called Wigner measures. But our proof below is
independent of the chosen quantisation.

######  Theorem 2.2.

Let $(f_{\varepsilon})_{\varepsilon>0}$ be a bounded family in
$L^{2}(\mathbb{R}^{n})$. Then there exists a sequence $\varepsilon_{k}$,
$k\in\mathbb{N}$ with $\varepsilon_{k}\to 0$ as $k\to\infty$, and a positive
Radon measure $\gamma$ on $\mathbb{R}^{n}\times\mathbb{R}^{n}$ such that

$\forall a\in C_{c}^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})\qquad({\rm
Op}_{\varepsilon}(a)f_{\varepsilon},f_{\varepsilon})_{L^{2}}\longrightarrow_{\varepsilon=\varepsilon_{k},k\to\infty}\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}a(x,\xi)d\gamma(x,\xi).$

###### Sketch of the proof of Theorem 2.2.

We may assume $\limsup_{\varepsilon\to 0}\|f_{\varepsilon}\|_{L^{2}}=1$. We
set $\ell_{\varepsilon}(a):=({\rm
Op}_{\varepsilon}(a)f_{\varepsilon},f_{\varepsilon})_{L^{2}}$. For each $a\in
C_{c}^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})$, $\ell_{\varepsilon}(a)$
is bounded so its limits exist as $\varepsilon\to 0$. A diagonal extraction
and the separability of $C_{c}^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})$
yield the existence of $\ell=\lim_{k\to\infty}\ell_{\varepsilon_{k}}$ on
$C_{c}^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})$. From the properties of
the semi-classical calculus, one checks that $\ell$ extends to a state of the
commutative $C^{*}$-algebra
$\overline{C_{c}^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})}$, hence a
positive Radon measure on $\mathbb{R}^{n}\times\mathbb{R}^{n}$. ∎

The semi-classical analogues of the examples of micro-local defect measures
are:

  * •

an $L^{2}$-concentration in space
$f_{\varepsilon}(x)=\varepsilon^{-n/2}\chi(\frac{x-x_{0}}{\varepsilon})$ about
a point $x_{0}$ (again, $\chi\in C_{c}^{\infty}(\mathbb{R}^{n})$ is some given
function), whose semi-classical measure is
$\gamma(x,\xi)=\delta_{x_{0}}(x)\otimes|\widehat{\chi}(\xi)|^{2}d\xi$,

  * •

an $L^{2}$-concentration in oscillations
$f_{\varepsilon}(x)=\psi(x)e^{2i\pi\xi_{0}\cdot x/\varepsilon}$ about a
frequency $\xi_{0}\in\mathbb{R}^{n}$ (again, $\psi\in
C_{c}^{\infty}(\mathbb{R}^{n})$ is some given function), whose semi-classical
measure is $\gamma=|\psi(x)|^{2}dx\otimes\delta_{\xi_{0}}(\xi)$.

###  2.4. Applications

Let us give an application of quantum limits to semi-classical analysis
already mentioned in the introduction in the form of the following result
taken from [26, Appendix A]. This is an elementary version of properties that
hold in more general settings and for more general Hamiltonians, including
integrable systems (see [2, 13]).

######  Proposition 2.1.

Let $(\psi^{\varepsilon}_{0})_{\varepsilon>0}$ be a bounded family in
$L^{2}(\mathbb{R}^{n})$ and the associated solutions to the Schrödinger
equation,

$i\varepsilon^{\tau}\partial_{t}\psi^{\varepsilon}=-\frac{\varepsilon^{2}}{2}\Delta\psi^{\varepsilon},\qquad\
\psi^{\varepsilon}|_{t=0}=\psi^{\varepsilon}_{0}.$

where $\Delta=-\sum_{1\leq j\leq n}\partial_{x_{j}}^{2}$ is the standard
Laplace operator on $\mathbb{R}^{n}$. We assume that the oscillations of the
initial data are exactly of size $1/\varepsilon$ in the sense that we have:

$\exists
s,C_{s}>0,\qquad\forall\varepsilon>0\qquad\varepsilon^{s}\|\Delta^{s/2}\psi^{\varepsilon}_{0}\|_{L^{2}(\mathbb{R}^{n})}+\varepsilon^{-s}\|\Delta^{-{s/2}}\psi^{\varepsilon}_{0}\|_{L^{2}(\mathbb{R}^{n})}\leq
C_{s}.$

Any limit point of the measures $\left|\psi^{\varepsilon}(t,x)\right|^{2}dxdt$
as $\varepsilon\to 0$ is of the form $\varrho_{t}(x)dt$ where $\varrho_{t}$ is
a measure on $\mathbb{R}^{n}$ satisfying:

  1. (1)

$\partial_{t}\varrho_{t}=0$ for $\tau\in(0,1)$,

  2. (2)

$\varrho_{t}(x)=\int_{\mathbb{R}^{n}}\gamma_{0}(x-t\xi,d\xi)$ for $\tau=1$,

  3. (3)

$\varrho_{t}=0$ for $\tau>1$.

###### Proof.

Using for instance the notion of quantum limits, we obtain time-dependent
semi-classical measures in the sense of the existence of a subsequence
$(\varepsilon_{k})$ and of a continuous map $t\mapsto\gamma_{t}$ from
$\mathbb{R}$ to the space of positive Radon measures such that

$\int_{\mathbb{R}}\theta(t)\,({\rm
Op}_{\varepsilon}(a)f_{\varepsilon},f_{\varepsilon})_{L^{2}}\longrightarrow_{\varepsilon=\varepsilon_{k},k\to\infty}\iint_{\mathbb{R}\times\mathbb{R}^{2n}}\theta(t)\,a(x,\xi)\,d\gamma_{t}(x,\xi)dt,$

for any $\theta\in C_{c}^{\infty}(\mathbb{R})$ and $a\in
C_{c}^{\infty}(\mathbb{R}^{2n})$. Now, up to a further extraction of a
subsequence, we obtain using the Schrödinger equation:

  1. (1)

for $\tau\in(0,1)$, $\gamma_{t}(x,\xi)=\gamma_{0}(x,\xi)$ for all times
$t\in\mathbb{R}$,

  2. (2)

for $\tau=1$,
$\partial_{t}\gamma_{t}(x,\xi)=\xi\cdot\nabla_{x}\gamma_{t}(x,\xi)$ in the
sense of distributions,

  3. (3)

for $\tau>1$, $\gamma_{t}=0$ for all times $t\in\mathbb{R}$.

Taking the $x$-marginals of the measures $\gamma_{t}$ gives the measures
described in the statement. ∎

The usual Schrödinger equation corresponds to $\tau=1$, as in the introduction
of this paper. In this case, the description of the semi-classical measure
above provides the link between the quantum world and the classical one:
$\gamma_{t}$ is the composition of $\gamma_{0}$ with the Hamiltonian flow from
classical mechanics.

##  3\. Pseudo-differential theory and quantum limits on nilpotent Lie groups

In this section, we will present the works [24, 25, 26] of Clotilde Fermanian-
Kammerer and the author about quantum limits on nilpotent Lie groups. We will
only describe briefly the setting and the notation, referring the interested
reader to the literature for all the technical details. We will end with a
word on future developments.

###  3.1. Preliminaries on nilpotent Lie groups

Let us consider a nilpotent Lie group $G$; we will always assume that
nilpotent Lie groups are connected and simply connected. If we fix a basis
$X_{1},\ldots,X_{n}$ of its Lie algebra $\mathfrak{g}$, via the exponential
mapping $\exp_{G}:\mathfrak{g}\to G$, we identify the points
$(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$ with the points
$x=\exp_{G}(x_{1}X_{1}+\cdots+x_{n}X_{n})$ in $G$. This also leads to a
corresponding Lebesgue measure on $\mathfrak{g}$ and the Haar measure $dx$ on
the group $G$, hence $L^{p}(G)\cong L^{p}(\mathbb{R}^{n})$ and we allow
ourselves to denote by $C_{c}^{\infty}(G),\,\mathcal{S}(G)$ etc, the spaces of
continuous functions, of smooth and compactly supported functions or of
Schwartz functions on $G$ identified with $\mathbb{R}^{n}$, and similarly for
distributions.

The group convolution of two functions $f_{1}$ and $f_{2}$, for instance
square integrable, is defined via

$(f_{1}*f_{2})(x):=\int_{G}f_{1}(y)f_{2}(y^{-1}x)dy.$

The convolution is not commutative: in general, $f_{1}*f_{2}\not=f_{2}*f_{1}$.

A vector of $\mathfrak{g}$ defines a left-invariant vector field on $G$ and,
more generally, the universal enveloping Lie algebra
$\mathfrak{U}(\mathfrak{g})$ of $\mathfrak{g}$ is isomorphic to the space of
the left-invariant differential operators; we keep the same notation for the
vectors and the corresponding operators.

Let $\pi$ be a representation of $G$. Unless otherwise stated, we always
assume that such a representation $\pi$ is strongly continuous and unitary,
and acts on a separable Hilbert space denoted by $\mathcal{H}_{\pi}$.
Furthermore, we keep the same notation for the corresponding infinitesimal
representation which acts on $\mathfrak{U}(\mathfrak{g})$ and on the space
$\mathcal{H}_{\pi}^{\infty}$ of smooth vectors. It is characterised by its
action on $\mathfrak{g}$

$\pi(X)=\partial_{t=0}\pi(e^{tX}),\quad X\in\mathfrak{g}.$

We define the _group Fourier transform_ of a function $f\in L^{1}(G)$ at $\pi$
by

$\pi(f)\equiv\widehat{f}(\pi)\equiv\mathcal{F}_{G}(f)(\pi)=\int_{G}f(x)\pi(x)^{*}dx.$

We denote by ${\widehat{G}}$ the unitary dual of $G$, that is, the unitary
irreducible representations of $G$ modulo equivalence and identify a unitary
irreducible representation with its class in ${\widehat{G}}$. The set
${\widehat{G}}$ is naturally equipped with a structure of standard Borel
space. The Plancherel measure is the unique positive Borel measure $\mu$ on
${\widehat{G}}$ such that for any $f\in C_{c}(G)$, we have:

(2)
$\int_{G}|f(x)|^{2}dx=\int_{{\widehat{G}}}\|\mathcal{F}_{G}(f)(\pi)\|_{HS(\mathcal{H}_{\pi})}^{2}d\mu(\pi).$

Here $\|\cdot\|_{HS(\mathcal{H}_{\pi})}$ denotes the Hilbert-Schmidt norm on
$\mathcal{H}_{\pi}$. This implies that the group Fourier transform extends
unitarily from $L^{1}(G)\cap L^{2}(G)$ to $L^{2}(G)$ onto
$L^{2}({\widehat{G}}):=\int_{{\widehat{G}}}\mathcal{H}_{\pi}\otimes\mathcal{H}_{\pi}^{*}d\mu(\pi)$
which we identify with the space of $\mu$-square integrable fields on
${\widehat{G}}$.

A _symbol_ is a measurable field of operators
$\sigma(x,\pi):\mathcal{H}_{\pi}^{\infty}\to\mathcal{H}_{\pi}^{\infty}$,
parametrised by $x\in G$ and $\pi\in{\widehat{G}}$. We formally associate to
$\sigma$ the operator ${\rm Op}(\sigma)$ as follows

${\rm Op}(\sigma)f(x):=\int_{G}{\rm
Tr}\left(\pi(x)\sigma(x,\pi)\widehat{f}(\pi)\right)d\mu(\pi),$

where $f\in\mathcal{S}(G)$ and $x\in G$. If $G$ is the abelian group
$\mathbb{R}^{n}$, this corresponds to the Kohn-Nirenberg quantisation.

Regarding symbols, when no confusion is possible, we will allow ourselves some
notational shortcuts, for instance writing $\sigma(x,\pi)$ when considering
the field of operators
$\\{\sigma(x,\pi):\mathcal{H}_{\pi}^{\infty}\to\mathcal{H}_{\pi}^{\infty},(x,\pi)\in
G\times{\widehat{G}}\\}$ with the usual identifications for
$\pi\in{\widehat{G}}$ and $\mu$-measurability.

This quantisation has already been observed in [52, 4, 28] for instance. It
can be viewed as an analogue of the Kohn-Nirenberg quantisation since the
inverse formula can be written as

$f(x):=\int_{G}{\rm Tr}\left(\pi(x)\widehat{f}(\pi)\right)d\mu(\pi),\quad
f\in\mathcal{S}(G),\ x\in G.$

This also shows that the operator associated with the symbol $\text{\rm
I}=\\{\text{\rm I}_{\mathcal{H}_{\pi}},(x,\pi)\in G\times{\widehat{G}}\\}$ is
the identity operator ${\rm Op}(\text{\rm I})=\text{\rm I}$.

Note that (formally or whenever it makes sense), if we denote the (right
convolution) kernel of ${\rm Op}(\sigma)$ by $\kappa_{x}$, that is,

${\rm Op}(\sigma)\phi(x)=\phi*\kappa_{x},\quad x\in G,\
\phi\in\mathcal{S}(G),$

then it is given by

$\pi(\kappa_{x})=\sigma(x,\pi).$

Moreover the integral kernel of ${\rm Op}(\sigma)$ is

$K(x,y)=\kappa_{x}(y^{-1}x),\quad\mbox{where}\quad{\rm
Op}(\sigma)\phi(x)=\int_{G}K(x,y)\phi(y)dy.$

We shall abuse the vocabulary and call $\kappa_{x}$ the kernel of $\sigma$,
and $K$ its integral kernel.

###  3.2. Pseudo-differential calculi on graded nilpotent Lie groups

####  3.2.1. Preliminaries on graded groups

Graded groups are connected and simply connected Lie group whose Lie algebra
$\mathfrak{g}$ admits an $\mathbb{N}$-gradation
$\mathfrak{g}=\oplus_{\ell=1}^{\infty}\mathfrak{g}_{\ell}$ where the
$\mathfrak{g}_{\ell}$, $\ell=1,2,\ldots$, are vector subspaces of
$\mathfrak{g}$, almost all equal to $\\{0\\}$, and satisfying
$[\mathfrak{g}_{\ell},\mathfrak{g}_{\ell^{\prime}}]\subset\mathfrak{g}_{\ell+\ell^{\prime}}$
for any $\ell,\ell^{\prime}\in\mathbb{N}$. These groups are nilpotent.
Examples of such groups are the Heisenberg group and, more generally, all
stratified groups (which by definition correspond to the case
$\mathfrak{g}_{1}$ generating the full Lie algebra $\mathfrak{g}$); with a
choice of basis or of scalar product on $\mathfrak{g}_{1}$, the latter are
called Carnot groups.

Graded groups are homogeneous in the sense of Folland-Stein [30] when equipped
with the dilations given by the linear mappings
$D_{r}:\mathfrak{g}\to\mathfrak{g}$, $D_{r}X=r^{\ell}X$ for every
$X\in\mathfrak{g}_{\ell}$, $\ell\in\mathbb{N}$. We may re-write the set of
integers $\ell\in\mathbb{N}$ such that $\mathfrak{g}_{\ell}\not=\\{0\\}$ into
the increasing sequence of positive integers
$\upsilon_{1},\ldots,\upsilon_{n}$ counted with multiplicity, the multiplicity
of $\mathfrak{g}_{\ell}$ being its dimension. In this way, the integers
$\upsilon_{1},\ldots,\upsilon_{n}$ become the weights of the dilations and we
have $D_{r}X_{j}=r^{\upsilon_{j}}X_{j}$, $j=1,\ldots,n$, on a basis
$X_{1},\ldots,X_{n}$ of $\mathfrak{g}$ adapted to the gradation.

We denote the corresponding dilations on the group via

$rx=\exp(D_{r}X),\quad\mbox{for}\ x=\exp(X)\in G.$

This leads to homogeneous notions for functions, distributions and operators.
For instance, the homogeneous dimension of $G$ is the homogeneity of the Haar
measure, that is, $Q:=\sum_{\ell\in\mathbb{N}}\ell\dim\mathfrak{g}_{\ell}$;
and the differential operator $X^{\alpha}$ is homogeneous of degree
$[\alpha]:=\sum_{j}\upsilon_{j}\alpha_{j}$.

####  3.2.2. The symbolic pseudo-differential calculus on $G$

In the monograph [28], the (Fréchet) space $S^{m}(G)$ of symbols of degree
$m\in\mathbb{R}$ on $G$ is defined and the properties of the corresponding
space of operators $\Psi^{m}(G)={\rm Op}(S^{m}(G))$ are studied. Naturally,
when $G$ is the abelian group $\mathbb{R}$, the classes of symbols and of
operators are the ones due to Hörmander.

In the monograph, it is proved that
$\Psi^{*}(G):=\cup_{m\in\mathbb{R}}\Psi^{m}(G)$ is a symbolic pseudo-
differential calculus in the following sense:

  * •

$\Psi^{*}(G)$ is an algebra of operators, with an asymptotic formula for ${\rm
Op}(\sigma_{1}){\rm Op}(\sigma_{2})={\rm Op}(\sigma)$.

  * •

$\Psi^{m}(G)$ is adjoint-stable, i.e. ${\rm Op}(\sigma)^{*}={\rm
Op}(\tau)\in\Psi^{m}(G)$ when $\sigma\in S^{m}(G)$, with an asymptotic formula
for $\tau$.

  * •

$\Psi^{*}(G)$ contains the left-invariant differential calculus as
$X^{\alpha}\in\Psi^{[\alpha]}(G)$.

  * •

$\Psi^{*}(G)$ contains the spectral calculus of the positive Rockland
operators. Note that in the context of graded groups, the positive Rockland
operators are the analogues of the elliptic operators

  * •

$\Psi^{*}(G)$ acts continuously on the Sobolev spaces adapted to the graded
groups with $\Psi^{m}(G)\ni T:L^{p}_{s}(G)\hookrightarrow L^{p}_{s-m}(G)$.

####  3.2.3. The classical pseudo-differential calculus on $G$

Part of the paper [25] is devoted to defining the notions of homogeneous
symbols and of classes $\dot{S}^{m}(G)$ of homogeneous symbols of degree $m$.
Indeed, the dilations on the group $G$ induce an action of $\mathbb{R}^{+}$ on
the dual ${\widehat{G}}$ via

(3) $r\cdot\pi(x)=\pi(rx),\qquad\pi\in{\widehat{G}},\ r>0,\ x\in G.$

The homogeneous symbols are then measurable fields of operators on
$G\times\Sigma_{1}$ where

$\Sigma_{1}:=({\widehat{G}}/\mathbb{R}^{+})\setminus\\{1_{{\widehat{G}}}\\}.$

is the analogue of the sphere on the Fourier side in the Euclidean case.

This then allows us to consider symbols admitting a homogeneous expansion. The
space of operators in $\Psi^{m}(G)$ which admits a homogeneous expansion and
whose integral kernel is compactly supported is denoted by $\Psi^{m}_{cl}(G)$.
It is proved that $\Psi_{cl}^{*}(G):=\cup_{m\in\mathbb{R}}\Psi^{m}_{cl}(G)$ is
also a symbolic pseudo-differential calculus in the same sense as in Section
3.2.2. Furthermore, there is a natural notion of principal symbol associated
to a symbol; the principal symbol is homogeneous by construction.

Again, when $G$ is the abelian group $\mathbb{R}^{n}$, this calculus is the
well-known classical pseudo-differential calculus, and the notion of principal
symbol is the usual one.

We set $\Psi_{cl}^{\leq 0}(G):=\cup_{m\leq 0}\Psi^{m}_{cl}(G)$. Depending on
the context, the classical pseudo-differential calculus on $G$ may refer to
the space of operators of any order in $\Psi_{cl}^{*}(G)$ or to the space of
operators of non-positive orders $\Psi_{cl}^{\leq 0}(G)$.

####  3.2.4. The semi-classical pseudo-differential calculus on $G$

The semi-classical pseudodifferential calculus was presented in the context of
groups of Heisenberg type in [26], but in fact extends readily to any graded
group $G$.

We consider the class of symbols ${\mathcal{A}}_{0}$ of fields of operators
defined on $G\times{\widehat{G}}$

$\sigma(x,\pi)\in{\mathcal{L}}(\mathcal{H}_{\pi}),\;\;(x,\pi)\in
G\times{\widehat{G}},$

that are of the form

$\sigma(x,\pi)=\mathcal{F}_{G}\kappa_{x}(\pi),$

where $\kappa_{x}(y)$ is smooth and compactly supported in $x$ while being
Schwartz in $y$; more technically, the map $x\mapsto\kappa_{x}$ is in
$C_{c}^{\infty}(G:\mathcal{S}(G))$. The group Fourier transform yields a
bijection from $C_{c}^{\infty}(G:\mathcal{S}(G))$ onto $\mathcal{A}_{0}$, and
we equip $\mathcal{A}_{0}$ with the Fréchet topology so that this mapping is
an isomorphism of topological vector spaces.

Let $\varepsilon\in(0,1]$ be a small parameter. For every symbol
$\sigma\in\mathcal{A}_{0}$, we consider the dilated symbol obtained using the
action of $\mathbb{R}^{+}$ on ${\widehat{G}}$, see (3),

$\sigma^{(\varepsilon)}:=\\{\sigma(x,\varepsilon\cdot\pi):(x,\pi)\in
G\times{\widehat{G}}\\},$

and then the associated operator

${\rm Op}^{\varepsilon}(\sigma):={\rm Op}(\sigma^{(\varepsilon)}).$

As in the case of $\mathbb{R}^{n}$ (see Section 2.3), this yields a (basic)
semi-classical calculus in the following sense:

  * •

Each operator ${\rm Op}^{\varepsilon}(\sigma)$, $\sigma\in\mathcal{A}_{0}$, is
bounded on $L^{2}(G)$ with

$\|{\rm
Op}^{\varepsilon}(\sigma)\|_{\mathscr{L}(L^{2}(G))}\leq\|\sigma\|_{\mathcal{A}_{0}}:=\int_{G}\sup_{x\in
G}|\kappa_{x}(y)|dy,$

where $\kappa_{x}$ is the kernel of $\sigma$; $\|\cdot\|_{\mathcal{A}_{0}}$
defines a continuous semi-norm on $\mathcal{A}_{0}$.

  * •

The singularities of the operators concentrate around the diagonal of the
integral kernels as $\varepsilon\to 0$:

$\forall N\in\mathbb{N}\quad\exists C_{N}>0\quad\forall\varepsilon\in(0,1],\
\sigma\in\mathcal{A}_{0}\quad\|\sigma-\mathcal{F}_{G}\left(\kappa_{x}\chi(\varepsilon\,\cdot)\right)\|_{\mathcal{A}_{0}}\leq
C{\varepsilon}^{N}$

where $\chi\in C_{c}^{\infty}(G)$ is a fixed function identically equal to 1
on a neighbourhood of 0.

  * •

There is a calculus in the sense of expansions in powers of $\varepsilon$ in
$\mathscr{L}(L^{2}(G))$ for products ${\rm Op}^{(\varepsilon)}(\sigma_{1}){\rm
Op}^{(\varepsilon)}(\sigma_{2})$ and for adjoints ${\rm
Op}^{(\varepsilon)}(\sigma)^{*}$; here
$\sigma_{1},\sigma_{2},\sigma\in\mathcal{A}_{0}$.

###  3.3. Operator-valued measures

In Section 2, we explained why quantum limits in Euclidean or elliptic
settings are often described with positive Radon measures on the spaces of
symbols as these spaces are then commutative $C^{*}$-algebras. In the context
of nilpotent Lie groups, the symbols are operator-valued, and we will see
below that our examples of quantum limits will then be described in terms of
operator-valued measures as introduced in [24, 25]. Let us recall the precise
definition of this notion:

######  Definition 3.1.

Let $Z$ be a complete separable metric space, and let
$\xi\mapsto\mathcal{H}_{\xi}$ be a measurable field of complex Hilbert spaces
of $Z$.

  * •

The set $\widetilde{\mathcal{M}}_{ov}(Z,(\mathcal{H}_{\xi})_{\xi\in Z})$ is
the set of pairs $(\gamma,\Gamma)$ where $\gamma$ is a positive Radon measure
on $Z$ and $\Gamma=\\{\Gamma(\xi)\in{\mathcal{L}}(\mathcal{H}_{\xi}):\xi\in
Z\\}$ is a measurable field of trace-class operators such that

$\|\Gamma d\gamma\|_{\mathcal{M}}:=\int_{Z}{\rm
Tr}_{\mathcal{H}_{\xi}}|\Gamma(\xi)|d\gamma(\xi)<\infty.$

Here ${\rm Tr}_{\mathcal{H}_{\xi}}|\Gamma(\xi)|$ denotes the standard trace of
the trace-class operator $|\Gamma(\xi)|$ on the separable Hilbert space
$\mathcal{H}_{\xi}$.

  * •

Two pairs $(\gamma,\Gamma)$ and $(\gamma^{\prime},\Gamma^{\prime})$ in
$\widetilde{\mathcal{M}}_{ov}(Z,(\mathcal{H}_{\xi})_{\xi\in Z})$ are
equivalent when there exists a measurable function
$f:Z\to\mathbb{C}\setminus\\{0\\}$ such that

$d\gamma^{\prime}(\xi)=f(\xi)d\gamma(\xi)\;\;{\rm
and}\;\;\Gamma^{\prime}(\xi)=\frac{1}{f(\xi)}\Gamma(\xi)$

for $\gamma$-almost every $\xi\in Z$. The equivalence class of
$(\gamma,\Gamma)$ is denoted by $\Gamma d\gamma$, and the resulting quotient
set is denoted by ${\mathcal{M}}_{ov}(Z,(\mathcal{H}_{\xi})_{\xi\in Z})$.

  * •

A pair $(\gamma,\Gamma)$ in
$\widetilde{\mathcal{M}}_{ov}(Z,(\mathcal{H}_{\xi})_{\xi\in Z})$ is positive
when $\Gamma(\xi)\geq 0$ for $\gamma$-almost all $\xi\in Z$. In this case, we
may write
$(\gamma,\Gamma)\in\widetilde{\mathcal{M}}_{ov}^{+}(Z,(\mathcal{H}_{\xi})_{\xi\in
Z})$, and $\Gamma d\gamma\geq 0$ for $\Gamma
d\gamma\in{\mathcal{M}}_{ov}^{+}(Z,(\mathcal{H}_{\xi})_{\xi\in Z})$.

By convention and if not otherwise specified, a representative of the class
$\Gamma d\gamma$ is chosen such that ${\rm Tr}_{\mathcal{H}_{\xi}}\Gamma=1$.
In particular, if $\mathcal{H}_{\xi}$ is $1$-dimensional, $\Gamma=1$ and
$\Gamma d\gamma$ reduces to the measure $d\gamma$. One checks readily that
$\mathcal{M}_{ov}(Z,(\mathcal{H}_{\xi})_{\xi\in Z})$ equipped with the norm
$\|\cdot\|_{{\mathcal{M}}}$ is a Banach space.

When the field of Hilbert spaces is clear from the setting, we may write

$\mathcal{M}_{ov}(Z)=\mathcal{M}(Z,(\mathcal{H}_{\xi})_{\xi\in
Z}),\quad\mbox{and}\quad\mathcal{M}_{ov}^{+}(Z)=\mathcal{M}^{+}(Z,(\mathcal{H}_{\xi})_{\xi\in
Z}),$

for short. For instance, if $\xi\mapsto\mathcal{H}_{\xi}$ is given by
$\mathcal{H}_{\xi}=\mathbb{C}$ for all $\xi$, then $\mathcal{M}(Z)$ coincides
with the space of finite Radon measures on $Z$. Another example is when $Z$ is
of the form $Z=Z_{1}\times\widehat{G}$ where $Z_{1}$ is a complete separable
metric space, and $\mathcal{H}_{(z_{1},\pi)}=\mathcal{H}_{\pi}$, where the
Hilbert space $\mathcal{H}_{\pi}$ is associated with the representation
$\pi\in\widehat{G}$.

###  3.4. Micro-local defect measures on graded Lie groups

In [25], the following analogue to Theorem 2.1 is proved in the setting of
graded groups. It uses the classical pseudo-differential calculus and the
sphere $\Sigma_{1}$ of the dual as mentioned in Section 3.2.3 and the notion
of operator-valued measure (see Definition 3.1).

######  Theorem 3.1.

Let $\Omega$ be an open subset of $G$. Let $(f_{j})_{j\in\mathbb{N}}$ be a
bounded sequence in $L^{2}(\Omega,loc)$ converging weakly to 0. Then there
exists a subsequence $(j_{k})_{k\in\mathbb{N}}$ and an operator-valued measure
$\Gamma d\gamma\in\mathcal{M}_{ov}^{+}(G\times\Sigma_{1})$ such that

$(Af_{j},f_{j})_{L^{2}}\longrightarrow_{j=j_{k},k\to\infty}\int_{\Omega\times\Sigma_{1}}{\rm
Tr}\left(\sigma_{0}(x,\dot{\pi})\
\Gamma(x,\dot{\pi})\right)d\gamma(x,\dot{\pi})\,,$

holds for all classical pseudo-differential operator $A\in\Psi^{\leq
0}_{cl}(G)$, $\sigma_{0}$ denoting its principal symbol.

The proof of Theorem 3.1 given in [25] follows the same ideas as the ones
presented in Section 2.2 with the adaptations that come from dealing with a
more non-commutative $C^{*}$-algebra of symbols.

Examples of micro-local defect measures developed in [25] include

  * •

an $L^{2}$-concentration in space,

  * •

an $L^{2}$-concentration in oscillations using matrix coefficients of
representations.

An application to compensated compactness is also deduced. It would be
interesting to relate this to the works by B. Franchi and his collaborators
[7, 8, 31, 32] on compensated compactness on the Heisenberg group.

###  3.5. Semi-classical measures on graded Lie groups

In [24], the semi-classical analysis developed on $G$ yields the same property
of existence of (group) semi-classical measures:

######  Theorem 3.2.

Let $(f_{\varepsilon})_{\varepsilon>0}$ be a bounded family in $L^{2}(G)$.
Then there exists a sequence $\varepsilon_{k}$, $k\in\mathbb{N}$ with
$\varepsilon_{k}\to 0$ as $k\to\infty$, and an operator-valued measure $\Gamma
d\gamma\in\mathcal{M}_{ov}^{+}(G\times{\widehat{G}})$ satisfying

$\forall\sigma\in\mathcal{A}_{0}\qquad({\rm
Op}_{\varepsilon}(\sigma)f_{\varepsilon},f_{\varepsilon})_{L^{2}}\longrightarrow_{\varepsilon=\varepsilon_{k},k\to\infty}\int_{\Omega\times{\widehat{G}}}{\rm
Tr}\left(\sigma(x,\pi)\ \Gamma(x,\pi)\right)d\gamma(x,\pi).$

The (group) semi-classical analogues of the (group) micro-local defect
measures for an $L^{2}$-concentration in space and an $L^{2}$-concentration in
oscillations is also given in [24] in the context of the groups of Heisenberg
type; naturally, the former holds on any graded group.

In [26], we prove an analogue of the application given in Section 2.4 but for
the sub-Laplacian on any group of Heisenberg type. We obtain a description of
the $t$-dependent group semi-classical measures corresponding to the solutions
to the Schrödinger equations, and therefore of their weak limits after taking
the $x$-marginals. However, there is not one threshold $\tau=1$ as in the
Euclidean case, but two, namely $\tau=1$ and $\tau=2$. More precisely, the
semi-classical measures and the weak limits can be written into two parts:

  * •

one with a Euclidean behaviour and threshold $\tau=1$, and

  * •

one with threshold $\tau=2$.

With our methods, this comes from the splitting of the unitary dual
${\widehat{G}}$ into the following two subsets:

  * •

the subsets of infinite dimensional representations (for instance realised as
the Scrödinger representations), and

  * •

the subset of finite dimensional representations, in fact of dimension one and
given by the (abelian or Euclidean) characters of the first stratum.

This splitting is also present in other works that do not involve
representation theory; see for instance [12] about the Grushin-Schrödinger
equation and [56, 16] about sublaplacians on contact manifolds. In fact, this
phenomenon of slower dispersion than in Euclidean settings has already been
observed for other sub-Riemannian PDEs, see e.g. [6, 39, 5].

###  3.6. Future works

The tools developed so far in [24, 25, 26] can be adapted to (graded)
nilmanifolds along the lines of [27]. Nilmanifolds are quotients of nilpotent
Lie groups by a discrete subgroup. When the subgroup is also co-compact, this
results in a compact manifold which is locally given by the group. This
provides an excellent setting for the applications to PDEs of the theory
developed in [24, 25, 26] .

The extension to sub-Riemannian manifolds will certainly be more difficult.
However, given the recent progress in groupoids on filtered manifolds [53, 14,
54], the author feels confident that the semi-classical and micro-local
analysis already developed on graded groups will be transferable to the
setting of equiregular sub-Riemannian manifolds in the near future.

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# Hawking Temperature for 4D-Einstein-Gauss-Bonnet Black Holes from
uncertainty principle

Sara Azizi<EMAIL_ADDRESS>Department of Physics, K. N. Toosi
University of Technology, P.O. Box 15875-4416, Tehran, Iran  Sareh Eslamzadeh
<EMAIL_ADDRESS>Department of Theoretical Physics, Faculty of Basic
Sciences, University of Mazandaran, Babolsar, IRAN  Javad T. Firouzjaee
<EMAIL_ADDRESS>Department of Physics, K. N. Toosi University of
Technology, P.O. Box 15875-4416, Tehran, Iran  School of Physics, Institute
for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran
Kourosh Nozari<EMAIL_ADDRESS>Department of Theoretical Physics, Faculty
of Basic Sciences, University of Mazandaran, P. O. Box 47416-95447, Babolsar,
IRAN

###### Abstract

Inspired by string theory, Heisenberg’s uncertainty principle can be
generalized to include the photon-electron gravitational interaction, which
leads to the Generalized Uncertainty Principle (GUP). Although GUP considers
gravitational uncertainty at the minimum fundamental length scale in physics,
it does not consider the effects of spacetime curvature on quantum mechanical
uncertainty relations. The Extended Uncertainty Principle (EUP) is a
generalization of Heisenberg’s Uncertainty Principle that, unlike the GUP,
applies to large length scales. GEUP is also a linear combination of EUP and
GUP that creates minimal uncertainty on large length scales. The Einstein-
Gauss-Bonnet theory (EGB) can be considered as one of the most promising
candidates for modified gravity. In this paper, by using GUP, EUP, and GEUP,
we intend to obtain the Hawking temperature of a four-dimensional EGB black
hole in the asymptotically flat and (Anti)-de Sitter spacetime. We show that
coupling constant, cosmological constant, mass, and radius significantly
affect Hawking temperature and decrease or increase Hawking temperature
depending on the chosen horizons.

Keywords: Generalized Uncertainty Principle, Extended Uncertainty Principle,
Einstein-Gauss-Bonnet theory, (Anti)-de Sitter spacetime, Hawking temperature,
Cosmological constant.

###### Contents

  1. I Introduction
  2. II The GUP and Hawking temperature in an asymptotically flat 4D-EGB black hole
  3. III Hawking temperature through EUP and GEUP in an asymptotically Anti-de Sitter 4D-EGB black hole
  4. IV Hawking temperature through EUP and GEUP in an asymptotically de Sitter 4D-EGB black hole
  5. V Conclusion

##  I Introduction

General relativity (GR) and quantum field theory (QFT) are leading theories
that describe gravitational interactions and phenomena at the atomic and
subatomic scales, respectively. Both of these theories are in good agreement
with the experiments. In GR, Einstein predicted the existence of black holes,
which was eventually proven by observations made on the gravitational wave
channel by LIGO / VIRGO 1 ; 2 ; 3 . In addition, the existence of black holes
was once again confirmed by the Event Horizon Telescope, which recorded the
first image of a black hole shadow 4 ; 5 . On the other hand, QFT also
accurately predicts constants of many microscopic structures using
experimental measurements 6 ; 7 ; 8 . Although GR and QFT are in good
agreement with experiments, the current perspective of gravity on the Planck
scale does not provide a unified description of theoretical physics. Using
Dyson’s power-counting method, it can be shown that GR is not a theory that
can be re-normalized 9 . Therefore, alternative theories of gravity are
introduced that can be re-normalized. Correction of quadratic-order curvature
in GR may give rise to renormalizable gravity theories 10 . It is expected
that the problem of singularity formation in GR will be solved with the help
of such theories of gravity 11 . In GR, the Einstein-Hilbert action is
modified to include the so-called Gauss-Bonnet term, which the resulting
theory is called the Einstein-Gauss-Bonnet gravity (EGB) 12 . The EGB theory
is an alternative theory of gravity that could be one of the most promising
candidates for alternative theories of gravity. The action of EGB gravity in a
D-dimensional spacetime with a positive cosmological constant is as follows

$S_{EGB}=S_{EH}+S_{GB}=\frac{1}{16\pi G_{D}}\int
d^{D}x\sqrt{-g}[R-2\Lambda+\alpha\mathcal{L}_{GB}],$ (1)

where $G_{D}$ is the D-dimensional Newton’s constant, $R$ is the Ricci scalar,
$\Lambda$ is the cosmological constant, and $\alpha$ is the Gauss-Bonnet
coupling constant. The Gauss-Bonnet term has been introduced as follows

$\mathcal{L}_{GB}=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}.$

In the past, it was believed that EGB gravity was nontrivial only in the 4 + 1
dimensions and higher dimensions, and would therefore only be used in high-
dimensional models. In fact, EGB was reduced to a topological level term in
the lower dimensions 14 . Although this expression was quadratic in the
Riemann tensor and the Ricci tensor, terms with more than two partial
derivatives of the metric would be omitted, and the Euler-Lagrange equations
in the metric would be converted to second-order quasi-linear partial
differential equations. As a result, like $f(R)$ gravity, there would be no
other dynamical degrees of freedom 15 .

In a recent paper, 16 , D. Glavan and C. Lin argued that if the Gauss-Bonnet
coupling constant scale in EGB theory can be changed to $D>4$, then if the
computations were made when $D\rightarrow 4$, the contribution of the Gauss-
Bonnet expression in the 4-dimensions can be obtained nontrivial. The general
metric equation and symmetric spherical solution of the EGB black hole in
4-dimensional spacetime are as follows

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2}_{2},$ (2)

$f(r)=1+\frac{r^{2}}{2\alpha}\left(1-\sqrt{1+\dfrac{8\alpha
M}{r^{3}}+\dfrac{4\alpha\Lambda}{3}}\right),$ (3)

where $d\Omega^{2}_{2}$ is the line element of the two-dimensional unit sphere
$S^{2}$. The 4D-EGB gravity has been studied in various areas such as
obtaining observational constraints on regular 4-dimensional EGB gravity 17 ,
investigating the power spectra of the Hawking radiation, and greybody factor
of a massless scalar 18 , etc. 19 ; 20 ; 21 ; 22 ; 23 ; 24 ; 25 ; 26 ; 27 ; 28
. In Refs. 22 ; 23 , the authors obtained the energy emission rate and
greybody factor in various fields, including Dirac, electromagnetic and
gravitational field, using the asymptotically flat 4D-EGB theory, and
calculated the lifetime of the black hole for different values of coupling
constants. In addition, in the Refs. 25 ; 26 ; 27 ; 28 , the evaporation
process of the AdS EGB black holes in cases with D dimensions and different
properties of the charged black holes in AdS 4D-EGB spacetime at various
coupling constants have also been investigated.

In addition to the success of this theory, it is worth mentioning some of its
criticisms. Some authors believe that the 4D-EGB solution developed by Glavan
and Lin needs to be modified 29 ; 30 ; 31 ; 32 ; 33 ; 34 . In Ref. 29 , the
authors obtained a set of field equations that can be written in closed form
in four dimensions by inserting a counter term in the action. The method
presented in this reference leads to a theory of gravity that includes an
additional scalar gravitational degree of freedom. In the Ref. 30 , Kobayashi
uses the Kaluza-Klein reduction to promote the warp factor of the internal
space to a scalar degree of freedom and shows that it is possible to describe
the regularized Lovelock gravity through a certain subclass of Horndeski
theory. In addition, Aoki, Gorji, and Mukayama recently presented a consistent
theory of 4D-EGB gravity with two degrees of freedom based on reasonable
assumptions. According to Hamiltonian formalism, they showed that it is not
possible to obtain a consistent gravitational theory with two degrees of
freedom when we incline the $(d+1)$-dimensional EGB gravity to $d\rightarrow
3$. They pointed out that either an extra degree of freedom has to be
introduced into the theory or temporal diffeomorphism has to be violated to
obtain a consistent $(3+1)$-dimensional theory 31 .

One of the main features of a black hole is its temperature which is
characterized through the Hawking process. The origin of this radiation is the
creation of quantum particles in the background vacuum due to the Uncertainty
Principle firouzjaee2 . The Heisenberg Uncertainty Principle (HUP) forms the
cornerstone of quantum physics. Heisenberg showed that a microscopic
particle’s exact position and momentum, the size of an electron, could never
be measured precisely simultaneously 36 . In fact, this principle constitutes
a fundamental limitation for the measurement of classical trajectories at the
atomic or sub-atomic scales. However, in HUP, there is no absolute maximum or
minimum uncertainty for momentum and position. HUP does not consider the
effects of quantum gravity on photon interactions and spacetime curvature 37 .
Calculations performed in recent research in the fields of non-commutative
geometries 38 , string theory 39 , black hole physics 40 , Hawking radiation
spectrum Firouzjaee:2015wps , and quantum gravity 42  show the existence of a
minimum length 43 . As a result of these arguments, a generalization of the
Heisenberg uncertainty principle was introduced, called the generalized
uncertainty principle (GUP), which predicts the minimum length value on the
Planck length scale 44 ; 45 . The absolute minimum in the position uncertainty
restricts the black hole evaporation and, unlike the HUP, prevents Hawking
temperature divergence. Increasing Hawking temperature by GUP with absolute
minimum length ($r_{+}=2al_{p}$) leads to a faster decay of the Schwarzschild
black hole in any dimension 46 ; 47 .

Similar to HUP, GUP does not have a precise limit for maximum position
uncertainty, so it cannot be used on large length scales such as de Sitter or
Anti-de Sitter space. In HUP or GUP, the Hawking temperature of Schwarzschild
black holes cannot be reproduced in the (Anti)-de Sitter space 37 . The
extended uncertainty principle (EUP) covers large scales and predicts the
minimum amount of measurable momentum 48 . In the (Anti)- de Sitter
background, the EUP contains a correction that corresponds to the (Anti)- de
Sitter radius. This expression can also be derived from (Anti)- de Sitter
spacetime geometry. Using EUP, one can accurately obtain Hawking temperatures
in (Anti)- de Sitter black holes, unlike HUP and GUP 49 . In Ref. 37 , the EUP
is generalized to include position uncertainties and calculate Hawking
temperatures correctly. The generalized extended uncertainty principle (GEUP)
can be obtained from the linear combination of GUP and EUP to simultaneously
achieve high energy and large-scale changes 49 .

In this paper, we obtain the Hawking temperature of the 4D-EGB Schwarzschild
black hole using the GUP, EUP, and GEUP and investigate how the coupling
constant affects the Hawking temperature. In addition, we study Hawking
temperature behavior based on the mass, cosmological constant, and radii.

This article is organized as follows: In Section II, we obtain the Hawking
temperature through the GUP for 4D-EGB black holes in asymptotically flat
space. Sections III and IV are devoted to obtaining the Hawking temperature in
Anti-de Sitter and de Sitter space for the 4D-EGB black hole using EUP and
GEUP, respectively, and investigating the mass, the cosmological constant, and
coupling constant effects on Hawking temperature. Finally, Section V
summarizes the works done in this article.

##  II The GUP and Hawking temperature in an asymptotically flat 4D-EGB black
hole

In this section, we recall the generalized uncertainty principle and then
obtain the Hawking temperature for the asymptotic flat 4D-EGB black hole. We
consider the following form of GUP

$\Delta x_{i}\Delta
p_{j}\geq\hbar\delta_{ij}\left[1+a^{2}l^{2}_{P}\frac{(\Delta
p_{j})^{2}}{\hbar^{2}}\right],$ (4)

where $x_{i}$ and $p_{j}$ are spatial and momentum coordinates, respectively,
$l_{p}$ is Planck length, and $a$ is a dimensionless real constant of order
one. GUP has a variety of representations, and Eq. (4) is just one of its
representations, so the precise value of $a$ depends on the chosen model 50 .
The second term on the right-hand side creates an absolute minimum on position
uncertainty

$\Delta x_{i}\geq 2al_{P},$ (5)

which non-trivially gives a minimal measurable length. The uncertainty in the
momentum is obtained from Eq. (4) as

$\frac{\hbar\Delta
x_{i}}{2a^{2}l^{2}_{P}}\left[1-\sqrt{1-\frac{4a^{2}l^{2}_{P}}{(\Delta
x_{i})^{2}}}\right]\leq\Delta p_{i}\leq\frac{\hbar\Delta
x_{i}}{2a^{2}l^{2}_{P}}\left[1+\sqrt{1-\frac{4a^{2}l^{2}_{P}}{(\Delta
x_{i})^{2}}}\right].$ (6)

The left-hand inequality produces small corrections to the Heisenberg’s
uncertainty principle for $\Delta x_{i}\gg al_{P}$. For the corrections in the
GUP formula to be viable, the inequality on the right-hand side cannot be
arbitrarily large. We now derive the Hawking temperature from the generalized
uncertainty principle and examine the general properties of black holes in
flat spacetime. In flat spacetime, the symmetric spherical solution of the
4D-EGB black hole is as follows

$f(r)=1+\frac{r^{2}}{2\alpha}\left[1\pm\sqrt{1+\frac{8\alpha
M}{r^{3}}}\right],$ (7)

where the cosmological constant in flat space vanishes in Eq. (3), and $M$ is
a positive gravitational mass. From the metric $f(r)$, two horizons $r_{\pm}$
can be obtained

$r_{\pm}=M\left[1\pm\sqrt{1-\frac{\alpha}{M^{2}}}\right],$ (8)

where $r_{+}$ is the black hole horizon, and $r_{-}$ is the white hole
horizon. According to Eq. (8), there is a critical mass that depends on the
coupling constant $\alpha$. When $M>M_{cri}=\sqrt{\alpha}$, the black hole has
two horizons: the white hole horizon and the black hole horizon. When
$M=M_{cri}$, the black hole horizon, and the white hole horizon will merge,
and if $M<M_{cri}$, there will be no horizon 13 . Then, based on the Ref. 37 ,
we model a 4-dimensional black hole like a black box with radius $r$, the
uncertainty in the position of a particle propagated by the radius of the
horizon $r_{\pm}$ is equal to

$\Delta x_{i}\approx r_{\pm},$ (9)

where

$r_{\pm}=M\pm\sqrt{M^{2}-\alpha}.$ (10)

We ignore the mass of the emitted particle, so the energy uncertainty of the
particle is equal to $E$, which can be considered as the characteristic
temperature of Hawking radiation. If we assume $E$ is equal to left inequality
in relation (6), the Hawking temperature can be obtained as follows

$T_{GUP}=\left(\frac{d-3}{4\pi}\right)\frac{\hbar
r_{+}}{2a^{2}l^{2}_{P}}\left[1-\sqrt{1-\frac{4a^{2}l^{2}_{P}}{r^{2}_{+}}}\right],$
(11)

where $``(d-3)/4\pi"$ is the calibration factor and we set $\hbar=l_{P}=1$ and
$d=4$ here 51  to find

$T_{GUP}=\frac{r_{+}}{2a^{2}4\pi}\left[1-\sqrt{1-\frac{4a^{2}}{r^{2}_{+}}}\right],$
(12)

We treat the Hawking temperature behavior versus the mass and coupling
constant changes by replacing the black hole horizon radius (10) in Eq. (12)
and plotting the $T-M$ and $T-\alpha$ diagrams. The first plot on the left
panel of Fig. (1) shows the Hawking temperature and $T_{EGB-GUP}$ with
different coupling constants. Hawking temperature increases and diverges
without GUP as the mass decreases. It is illustrated that $T_{EGB-GUP}$
increases with decreasing mass, and $T_{EGB-GUP}$ is limited at rest-mass
remnant and black hole evaporation ends 52 ; 53 ; 54 ; 55 .

Figure 1: Hawking temperature of black hole horizon versus mass, coupling
constant, and radius for asymptotic flat 4D-EGB black hole. Left diagram:
Hawking temperature has been plotted for different masses from left to right
with $\alpha=0.1,0.3,0.5,0.7$. Right diagram: Hawking temperature versus
coupling constant has been plotted with $M=1$. Bottom diagram: Hawking
temperature versus radius has been plotted with $\alpha=0.1,0.3,0.5,0.7$.

In Ref. 13 , the $T_{EGB}-M$ is plotted and this plot shows that EGB corrects
the temperature behavior of the black hole at a small radius and stops
temperature divergence. Although the EGB predicts zero temperature for the end
of the black hole evaporation and has a mass remnant, its temperature is zero.
However, comparing this plot with the $T-M$ in Fig. (1), it can be concluded
that when the GUP enters the EGB, it stops the evaporation of the black hole
where the temperature has not yet reached zero. In fact, a black hole with a
finite mass within the Planck mass range achieves a limited temperature 56 .

As can be seen from the $T-\alpha$ plot, the Hawking temperature increases
with increasing $\alpha$. In EGB-GUP, if we consider a more significant
$\alpha$, we will obtain a larger temperature for the remnant mass anticipated
at the end of the black hole evaporation.

When we plot the $T-r$ with different $\alpha$, one can see that the Hawking
temperature change with the radius of the black hole is as expected, and when
$r$ decreases, the Hawking temperature increases. Having a minimum length due
to GUP will limit the Hawking temperature and prevents it from diverging at
small lengths 37 . The effect of the coupling constant on temperature is
negligible; however, increasing the positive coupling constants causes a
slight decrease in Hawking temperature.

##  III Hawking temperature through EUP and GEUP in an asymptotically Anti-de
Sitter 4D-EGB black hole

As mentioned earlier, GUP cannot be used in large-length spaces such as
(Anti)-de Sitter spaces. Therefore, we obtain the Hawking temperature from the
uncertainty relation in such spaces by extending Heisenberg’s uncertainty
principle to large-length scales 37 ; 57 . The extended uncertainty relation
is

$\Delta x_{i}\Delta p_{j}\geq\hbar\delta_{ij}\left[1+\beta^{2}\frac{(\Delta
x_{i})^{2}}{l^{2}}\right],$ (13)

where $l$ is the characteristic large-length scale and $\beta$ is a
dimensionless real constant of order one 49 . The EUP in AdS spacetime
requires an absolute minimum of momentum uncertainty

$\Delta p_{i}\geq\frac{\hbar}{\Delta x_{i}}+\frac{\hbar\beta^{2}\Delta
x_{i}}{l}\geq\frac{2\hbar\beta}{l}.$ (14)

Furthermore, GEUP is a combination of GUP and EUP that includes both a minimum
length scale and a minimum momentum scale

$\Delta x_{i}\Delta
p_{j}\geq\hbar\delta_{ij}\left[1+a^{2}l^{2}_{P}\frac{(\Delta
p_{j})^{2}}{\hbar^{2}}+\beta^{2}\frac{(\Delta x_{i})^{2}}{l^{2}}\right].$ (15)

In GEUP, there is also an absolute minimum for the position uncertainty due to
the Planck length

$(\Delta
x_{i})^{2}\geq\frac{4a^{2}l^{2}_{P}}{1-4a^{2}l^{2}_{P}\beta^{2}/l^{2}},$ (16)

and the EUP here also creates an absolute minimum for the momentum uncertainty

$(\Delta
p_{i})^{2}\geq\frac{4\hbar^{2}\beta^{2}/l^{2}}{1-4a^{2}l^{2}_{P}\beta^{2}/l^{2}}.$
(17)

Studies show that the existence of minimum length always raises the
temperature and accelerates evaporation of a black hole, both in flat space
and in (Anti)-de Sitter space 46 ; 47 ; 58 . In this section, we first obtain
the Hawking temperature of the 4D-EGB black hole for the Anti-de Sitter
spacetime. In the following, we will investigate the effect of $\alpha$,
$\Lambda$, and $M$ on the Hawking temperature of the Anti-de Sitter 4D-EGB
black hole, and we will analyze the effect of minimum length in GEUP on the
black hole, as well. Considering the solution (2) in the AdS space, the metric
function is as follows

$f(r)=1+\frac{r^{2}}{2\alpha}\left(1-\sqrt{1+\dfrac{8M\alpha}{r^{3}}-\dfrac{4\alpha\Lambda}{3}}\right).$
(18)

Here the cosmological constant $\Lambda=-(d-1)(d-2)/2{l^{2}_{{}_{AdS}}}$ 37 .
We adopted the negative sign into the metric $f(r)$, so the $\Lambda$ would be
non-zero and positive. In Ref. 13 , $f(r)$ versus $r$ is illustrated.
According to this diagram, as in the previous section, there is a critical
mass. If $M\textgreater M_{cri}$, the black hole will have two horizons: a
black hole horizon and a cosmological horizon, respectively. The AdS 4D-EGB
black hole horizons can be found by obtaining the roots of the metric Eq. (18)

$r=\pm\frac{1}{2}\Big{[}\sqrt{\eta}\mp\sqrt{\frac{\eta-12M}{\eta\Lambda}}\Big{]},$
(19)

where

$\eta=\frac{2}{\Lambda}+\frac{3\times
2^{1/3}(1+4\alpha\Lambda)}{\Lambda\xi^{1/3}}+\frac{\xi^{1/3}}{3\times
2^{1/3}\Lambda},$ (20)

and

$\xi=972\Lambda M^{2}-648\alpha\Lambda+54+\sqrt{(972\Lambda
M^{2}-648\alpha\Lambda+54)^{2}-4(9+36\alpha\Lambda)^{3}}.$ (21)

Of the four roots of $r$, only two can be identified as the black hole horizon
and the cosmological horizon. Using the same method as in the previous
section, one can obtain the general $T_{EGB-EUP}$ and $T_{EGB-GEUP}$ equations
with $T\approx\Delta p_{i}$ (with $c=1$)

$T_{EUP(AdS)}=\left(\frac{d-3}{4\pi}\right)\hbar\left[\frac{1}{r_{+}}+\left(\frac{d-1}{d-3}\right)\frac{r_{+}}{l^{2}_{{}_{AdS}}}\right],$
(22)

$T_{GEUP(AdS)}=\left(\frac{d-3}{4\pi}\right)\frac{\hbar
r_{+}}{2a^{2}l^{2}_{P}}\left[1-\sqrt{1-\frac{4a^{2}l^{2}_{P}}{r^{2}_{+}}\left[1+\left(\frac{d-1}{d-3}\right)\frac{r^{2}_{+}}{l^{2}_{AdS}}\right]}\right],$
(23)

where $``(d-3)/4\pi"$ is the calibration factor, $\beta=\sqrt{(d-1)/(d-3)}$
and $l_{AdS}$ is the length scale of the AdS space. Here, we place $d=4$,
$l_{p}=0.1$, and $a=0.01$ in both cases. Now, we put the radius of the
cosmological horizon in Eq. (22) and Eq. (23),

$r_{CH}=\frac{1}{2}\Big{[}\sqrt{\eta}+\sqrt{\frac{\eta-12M}{\eta\Lambda}}\Big{]},$
(24)

and plot $T-M$, $T-\alpha$, $T-\Lambda$, and $T-r$ for different values of
$M$, $\alpha$, and $\Lambda$ (Fig. (2)).

Figure 2: Hawking temperature of cosmological horizon versus mass, coupling
constant, cosmological constant and radius for AdS 4D-EGB black hole. From the
top left to right, Hawking temperature has been plotted versus masses and
coupling constant with $\alpha=0.5$ and $M=1$, respectively. From bottom left
to right, Hawking temperature versus cosmological constant and radius has been
illustrated with $\alpha=0.3$. Here in all cases $l_{AdS}=2$ and
$\Lambda=0.7$.

Regarding the $T-\alpha$ plot, one can see that the Hawking temperature has
been severely suppressed and decreases as the $\alpha$ coupling constant
increases. Considering the $T-M$ diagram, the temperature decreases with
decreasing mass; however, the Hawking temperature rises again as the mass
decreases. The behavior of the $T-M$ can be compared with the $T-r$ plot of
Ref. 37 . As expected, EUP and GEUP change Hawking temperature behavior at
larger masses, in which case they predict higher Hawking temperatures. The T-M
diagram illustrates that temperature behaves like temperature behavior in a
flat space in small masses. $T-\Lambda$ shows $T_{EGB-EUP}$ and $T_{EGB-GEUP}$
rise with the increasing cosmological constant. In this case, $T_{EGB-GEUP}$
indicates higher Hawking temperatures at larger cosmological constants than
$T_{EGB-EUP}$.

Then, we place the second root, which represents the radius of the black hole
horizon, in Eq. (22) and Eq. (23)

$r_{BH}=\frac{1}{2}\Big{[}\sqrt{\eta}-\sqrt{\frac{\eta-12M}{\eta\Lambda}}\Big{]}.$
(25)

$T-M$ plot shows that the Hawking temperature has decreased significantly with
decreasing mass. The effect of $\alpha$, as the cosmological horizon, is
dominant here, and increasing the coupling constant will reduce the Hawking
temperature.

Figure 3: Hawking temperature of black hole horizon versus mass, coupling
constant, cosmological constant and radius for AdS 4D-EGB black hole. From the
top left to right, Hawking temperature has been plotted versus different
masses with $\alpha=0.5$, and coupling constant with $M=1$, respectively. From
bottom left to right, Hawking temperature versus cosmological constant and
radius has been illustrated with $\alpha=0.3$. Here in all cases $l_{AdS}=2$,
$\Lambda=0.7$.

According to the $T-\Lambda$ diagram, it can be seen that on the black hole
horizon, when the cosmological constant increases, the Hawking temperature
remarkably rises. In Ref. 24 , the behavior of the $T-\Lambda$ for the “de
Sitter” spacetime has been plotted. This behavior of $T_{EGB-EUP}$ and
$T_{EGB-GEUP}$ versus $\Lambda$ in the “Anti-de Sitter” spacetime is the
opposite of the $T_{EUP}$ behavior predicted in Ref. 24  for “de Sitter”
spacetime. Furthermore, we plotted $T-r$ for both the black hole horizon and
the cosmological horizon in Fig. (2) and Fig. (3). Here, $T_{EGB-EUP}$ and
$T_{EGB-GEUP}$ with increasing radius show a decreasing behavior which is in
accordance with the $T-r$ diagram shown in Ref. 37 .

##  IV Hawking temperature through EUP and GEUP in an asymptotically de Sitter
4D-EGB black hole

In this section, we also want to study Hawking temperature through EUP and
GEUP in de Sitter spacetime. For this purpose, it is enough to put
$l^{2}=-l^{2}$ in Eq. (22) and Eq. (23)

$\Delta x_{i}\Delta p_{j}\geq\hbar\delta_{ij}\left[1-\beta^{2}\frac{(\Delta
x_{i})^{2}}{l^{2}}\right],$ (26)

$\Delta x_{i}\Delta
p_{j}\geq\hbar\delta_{ij}\left[1+a^{2}l^{2}_{P}\frac{(\Delta
p_{j})^{2}}{\hbar^{2}}-\beta^{2}\frac{(\Delta x_{i})^{2}}{l^{2}}\right],$ (27)

In de Sitter spacetime, the EUP admits an absolute maximum to $\Delta
x_{i}^{2}$, so that $\Delta p_{i}^{2}$ is always positive

$\Delta x_{i}\leq\frac{l}{\beta},$ (28)

this upper boundary represents the Nariai boundary, where the black hole
horizon and the cosmological horizon meet

$r_{i}\leq\sqrt{\frac{d-3}{d-1}}l_{ds}.$ (29)

At long distances in dS space, the Nariai solution is the limit of the largest
black hole, which consists of two horizons: the Schwarzschild black hole
horizon and a cosmological dS horizon. At the Nariai bound, the black hole
becomes larger and larger until the event horizon of the black hole has an
area equal to the cosmological de Sitter horizon. At this point, spacetime
evolves regularly, and the singularity of the black hole tends to infinity. A
spacetime symmetry in this space will connect the black hole and cosmological
horizons 59 ; 60 . Therefore, according to Eq. (28), there is an upper
boundary for the values of the horizons, which if the value of the obtained
radius is greater than this limit, the horizon of the black hole will not be
formed. In fact, it implies that position uncertainty never transcends the
cosmological horizon. In addition, GEUP also creates only an absolute minimum
on $\Delta x_{i}^{2}$ in the dS space 37

$(\Delta
x_{i})^{2}\geq\frac{4a^{2}l^{2}_{P}}{1+4a^{2}l^{2}_{P}\beta^{2}/l^{2}}.$ (30)

Furthermore, for $\Delta p_{i}$ not to be negative, a condition similar to Eq.
(27) must exist. Similar to the previous sections, we apply the desired
assumptions and obtain the Hawking temperature of a dS 4D-EGB black hole, with
$\Lambda=+(d-1)(d-2)/2{l^{2}_{dS}}>0$,

$T_{EUP(dS)}=\left(\frac{d-3}{4\pi}\right)\hbar\left[\frac{1}{r_{\pm}}-\left(\frac{d-1}{d-3}\right)\frac{r_{\pm}}{l^{2}_{dS}}\right],$
(31)

$T_{GEUP(dS)}=\left(\frac{d-3}{4\pi}\right)\frac{\hbar
r_{\pm}}{2a^{2}l^{2}_{P}}\left[1-\sqrt{1-\frac{4a^{2}l^{2}_{P}}{r^{2}_{\pm}}\left[1-\left(\frac{d-1}{d-3}\right)\frac{r^{2}_{\pm}}{l^{2}_{dS}}\right]}\right].$
(32)

The 4D-EGB black hole in dS has three horizons as follows

$r=\pm\frac{1}{2}\Big{[}\sqrt{\eta}\mp\sqrt{\frac{\eta-12M}{\eta\Lambda}}\Big{]},$
(33)

where

$\eta=\frac{2}{\Lambda}+\frac{3\times
2^{1/3}(1-4\alpha\Lambda)}{\Lambda\xi^{1/3}}+\frac{\xi^{1/3}}{3\times
2^{1/3}\Lambda},$ (34)

and

$\xi=972\Lambda M^{2}-648\alpha\Lambda-54+\sqrt{(972\Lambda
M^{2}-648\alpha\Lambda-54)^{2}-4(9-36\alpha\Lambda)^{3}}.$ (35)

where $r_{-}=r_{BH}$ is the black hole horizon, $r_{+}=r_{CH}$ is the
cosmological horizon, and the $r_{0}$ is the inner horizon 13 . As in the
previous section, we illustrate the $T-\alpha$, $T-\Lambda$, $T-M$, and $T-r$
plots for the dS 4D-EGB black hole, and then we will examine how Hawking
temperature changes with these quantities.

First, we place the values of $\beta$ and $l_{dS}$ in Eq. (27) and obtain the
maximum allowable value. By placing different values $\alpha$, $\Lambda$ and
$M$, we have obtained the radius of the cosmological horizon. In this case,
the cosmological horizon radius resulting from the 4D-EGB metric is always
more extended than the upper bound that the EUP places on $r$. In other words,
due to the 4D-EGB metric feature, the size of the cosmological horizon will
always be larger than the size of dS space.

In the case of the black hole horizon, we first calculate the black hole
Hawking temperature through the HUP, regardless of the Planck length and large
scale length. To accomplish this, it is enough to put different values of
$\alpha$ in $r_{BH}$ and calculate $T_{HUP}$.

According to Table. 1, we see that the dS radius of the black hole horizon
decreases by increasing coupling constants. According to the $T_{HUP}$
equation, the Hawking temperature will rise as the radius decreases. As a
result, increasing the coupling constant raises the Hawking temperature.

$\alpha$ | $r$ | $T_{HUP}$
---|---|---
0.1 | 2.44 | 0.40
0.2 | 2.34 | 0.42
0.3 | 2.24 | 0.44
0.4 | 2.13 | 0.46
0.5 | 2.03 | 0.49
Table 1: Hawking temperature for the black hole horizon in the dS 4D-EGB black
hole for different $\alpha$ and $r_{BH}$. Here we chose $M=1$ and
$\Lambda=0.1$.

Figure 4: Hawking temperature of black hole horizon versus mass, coupling
constant, cosmological constant and radius for dS 4D-EGB black hole. From the
top left to right, Hawking temperature has been plotted versus different
masses and coupling constant with $\alpha=0.1$ and $M=1$, respectively. From
bottom left to right, Hawking temperature versus cosmological constant and
radius has been illustrated with $\alpha=0.1$, receptively. Here in all cases
$\hbar=l_{p}=1$, $l_{dS}=5.5$, $\Lambda=0.1$, and $a=0.2$.

Regarding the Fig. (4), in contrast to the Anti-de Sitter spacetime, the
$T-\alpha$ diagram for the black hole horizon shows that as the $\alpha$
increases positively, $T_{EGB-EUP}$ and $T_{EGB-GEUP}$ also rise. The behavior
of $T_{EGB-EUP}$ and $T_{EGB-GEUP}$ are similar to the changes of $T_{HUP}$
versus $\alpha$. However, when we plot the $T-r$ diagram for the radius of the
black hole horizon, the temperature shows an increasing behavior as the radius
decreases, and the temperature is limited at one point and stops increasing.
This behavior of the $T-r$ diagram is utterly consistent with the results
obtained in Ref. 37 .

The $T-M$ plot predicts higher Hawking temperature than flat and AdS spaces.
$T_{EGB-GEUP}$ has reached the final temperature of evaporation sooner.
Indeed, having a minimum length in GEUP makes the black hole evaporation
faster than when there is no minimum length and ends at a certain point 37 .
If, as in Ref. 13 , we considered only the EGB effect, and we would expect the
black hole to continue evaporation until the temperature drops to zero.
Nevertheless, when we incorporate EUP and GEUP into our equations, the
expected behavior in Ref. 13  will change. In this case, when the Hawking
temperature reaches a limited value, evaporation no longer continues. The
black hole is expected to continue evaporating at EGB-EUP, but here $T_{EGB-
EUP}$ has also stopped at a certain point. This is due to the presence of EGB.
The effect of EGB is more noticeable on the black hole horizon and has a more
significant effect on the final temperature of the dS 4D-EGB black hole. EGB
modifies the structure of the black hole and specifies a minimum mass for it.
If the mass is less than this minimum mass, the black hole will no longer be
considered in the mass range we are studying, i.e., this value will be less
than the mass that three dS black hole horizons should have; this is why
$T_{EGB-EUP}$ has been discarded in this case. However, $T_{EGB-EUP}$ and
$T_{EGB-GEUP}$ behave similarly in substantial masses. Moreover, the
$T-\Lambda$ diagram behaves consistently with Ref. 24 . Increasing the
cosmological constant at different $\alpha$ will cause $T_{EGB-EUP}$ and
$T_{EGB-GEUP}$ to drop.

##  V Conclusion

In previous studies, the EGB solution was assumed to be trivial in four
dimensions, but recently the existence of 4-dimensional Einstein-Gauss-Bonnet
black holes has been investigated by Glavan and Lin. EGB black holes have
different properties in 4-dimension. Hawking radiation and temperature can
also be expected to have unique properties compared to higher dimensions in
this case. In Ref. 37 , it is shown that, in general, the minimum length
generated by GUP can increase the Hawking temperature, regardless of whether
the space is asymptotically flat or (Anti)- de Sitter. In fact, the presence
of the expression $"a^{2}l_{P}/r_{\pm}^{3}"$ in the temperature equation can
be considered a universal trend that causes the black hole to evaporate
faster. We investigated the temperature behavior of the 4D-EGB black hole by
using GUP, EUP, and GEUP. By plotting the temperature in terms of coupling
constants, cosmological constants, mass, and radius, we estimated the
temperature behavior in terms of their variations. The 4D-EGB black hole has
two horizons in flat space: the black hole and the white hole horizon. On the
black hole horizon, temperature exhibits an increasing behavior relative to
the increment in the coupling constant. In addition, the T-M plot illustrates
that when the GUP enters the EGB, it prevents the evaporation of the black
hole where the temperature has not yet reached zero.

In the AdS space, the coupling constant on both the black hole and the
cosmological horizons significantly affects the Hawking temperature and
decreases acutely with increasing positive $\alpha$. Hawking temperature on
the black hole horizon shows increasing behavior with an increasing
cosmological constant, contrary to temperature behavior in the dS space. On
the cosmological horizon, one can see that an increase in $\Lambda$ increases
the Hawking temperature. When plotting the Hawking temperature obtained from
the EUP and GEUP for the black hole horizon versus mass, we see that the
temperature drops sharply to zero when the mass decreases. On the cosmological
horizon, temperature shows the same behavior, except that when the mass
becomes very small, the temperature rises again. $T_{EGB-GEUP}$ predicts a
higher temperature than the $T_{EGB-GEUP}$, and at a particular point, due to
its minimum length, it stops evaporation and prevents Hawking temperature
divergence.

There are expected to be three horizons in the 4D-EGB black hole dS space: the
black hole horizon, the cosmological horizon, and the inner horizon. In the dS
space, EUP and GEUP predict an absolute maximum for position uncertainty. Due
to the Nariai limit, the size of the horizons will be limited and should not
exceed a specific limit. However, the size of our expected 4D-EGB black hole
cosmological horizon will always be more significant than this bound, so the
cosmological horizon will not be formed under this condition.

Ultimately, when we plot $T-r$ for the flat, (A)dS and dS spaces, one can see
that Hawking’s temperature behavior versus radius is entirely in line with the
predictions. GUP and GEUP affect temperature behavior and prevent it from
diverging in a small radius. In fact, the smaller the radius, the greater the
effect of GUP and GEUP on the temperature of the black hole, causing the
temperature to rise sharply and evaporation to accelerate. Finally, the
minimum length stops the black hole from evaporating at a certain point,
preventing Hawking temperature divergence.

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