# ORTAC+ : A User Friendly Domain Specific Language for Multi-Agent Mission
Planning

Caroline Bonhomme Safran Electronics and Defense
Massy, FranceONERA
Toulouse, France<EMAIL_ADDRESS>Safran Electronics and
Defense
Massy, France Jean-Louis Dufour Safran Electronics and Defense
Massy, France<EMAIL_ADDRESS>
###### Abstract

A tactical military unit is a complex system composed of many agents such as
infantry, robots, or drones. Given a mission, an automated planner can find an
optimal plan. Therefore, the mission itself must be modeled. The problem is
that languages like PDDL are too low-level to be usable by the end-user: an
officer in the field. We present ORTAC+, a language and a planning tool
designed for this end-user. Its main objective is to allow a natural modeling
of the mission, to minimize the risk of bad modeling, and thus obtain reliable
plans. The language offers high-level constructs specifically designed to
describe tactical missions, but at the same time has clear semantics allowing
a translation to PDDL, to take advantage of state-of-the-art planners.

##  1 Introduction

A tactical military ground unit is a complex system composed of many agents,
such as infantry sections, armored units, ground robots, or flying drones. A
typical mission lasts one day and is a mix of displacements on a road network
of 20 km between given start and finish points and actions. We deal only with
the reconnaissance action of a route or an area when an enemy presence is
suspected. For us, the reconnaissance action is equivalent to a speed
reduction of the progression on a route and exploration of an area. The
progression and exploration should be done more safely, thus it is slower. It
could be seen as the first visit to a location. This brings us to a variant of
the classic Multi-Agent Path Finding (MAPF) problem [1]. Of course, if we
consider moves as actions, this is also a multi-agent planning problem [2],
but classification as a MAPF problem is more accurate.

Given a mission, if it is described in a programming language, an automated
planner can find an optimal plan. Therefore the mission itself must be
modeled. The automated planning community has developed the PDDL language in
order to specify high level planning problems [3]. However, the specification
of the mission should be done by an officer in the field who is not an expert
in automated planning. Languages like PDDL are too low-level to be usable by
the defined end-user who is not an expert in planning. This is why we present
ORTAC+. Its main objective is to allow a natural modeling of the mission, so
as to minimize the risk of errors on the modeling of the desired mission, and
thus obtain reliable plans. The language is composed by high-level constructs
specifically designed to facilitate the specification of tactical missions,
with a clear semantic allowing a translation to PDDL, to take advantage of
state-of-the-art planners. Planning is done offline, and the plans are
deterministic, with no contingency. The default assumption is that the enemy,
when suspected, is in fact not present. Otherwise, if there is contact with
the enemy, the execution of the plan is de facto interrupted in the field, and
the officer must replan with modified objectives.

We proceed by presenting related work of the paper in section 2. The domain-
specific language ORTAC+ is presented in section 3, the operational context in
section 4, and the conclusion in section 5.

##  2 Related Work

ORTAC stands for ”Optimal Resource and Technical Action Control”. It is a
multi-agent planning language and tool combining constraint-solving techniques
with advanced search strategy to deconflict single-agent plans [4]. At the
language level, ORTAC is innovative in the constraints of coordination between
units introducing the notion of ”support” explained later [4]. The proper term
for that is ”cross-schedule dependencies” [5], typically in the case of
”coalition formation.”

PDDL is the standard planning language used commonly by the automated planning
community. One of the purposes of PDDL was to benchmark planning algorithms.
Since its creation in 1998 for the International Planning Competition [6], the
language evolved to model more complex planning problems as temporal or
numerical planning [3]. The Competition of Distributed and Multiagent Planners
[7] benchmarks some centralized and decentralized algorithms with an extension
of PDDL for multiagent planning as a reference language. ORTAC and PDDL are
powerful tools for modeling high-level missions and finding a corresponding
plan, nevertheless, they cannot be used by an end-user.

In recent years, an effort on designing domain specific languages (DSL) for
non-expert in robotics raised in the research community. Indeed, some of these
languages lack clear semantics [8]. Additionnaly, users are required to define
the behavior of each robot. Consequently, the autonomy of the system is
hindered. Theses systems lack planning capabilities, making it challenging to
initiate the integration of planning tools. The research community uses PDDL
as the standard task planing language. However, its integration into robotics
remains limited. To remedy this issue, efforts were done to integrate planning
in robotics [9].

##  3 ORTAC+

ORTAC+ is a tool that assists a military officer in planning a mission. It is
a language that aims to be expressive with well-defined syntax and semantics.
To do that, the language should allow the description of a given mission and
be compatible with automated planners. A military mission is limited in space
and time. The first version of ORTAC+ emphasizes the spatial aspects of the
mission. The temporal aspects will be added in future works.

###  3.1 Problem Definition

The missions could be seen as a multi-agent planning problem where each agent
collaborates to reach a global goal. However, we only consider agents
movements. Thus, the model is a variant of a MAPF problem. A classical MAPF
problem is represented with a set of agents with an initial position and a
target position, in an undirected graph. The solution of a classical MAPF
problem is a set of single-agent paths that does not collide [1]. ORTAC+
allows the description of this kind of problem but aims to be more general
with the following variants:

  * •

Targets are no longer assigned to each agent but to teams of agents, leading
to the possibility of having less target positions than agents. Several agents
could be assigned to the same target position but only one of them should
reach it.

  * •

Spatial constraints are added to the problem. The constraint can be forbidden
or enforced vertex for specific agent.

  * •

Agents can be heterogeneous. While classical MAPF problems harnesses agents
with similar characteristics, our problem differs in this aspect.

  * •

The position of an agent is not only on vertices but can be on edges too. The
path of an agent is then a succession of vertices and edges.

###  3.2 Modelisation

Our DSL describes the mission within a three axis representation:

  * •

A geographical representation, corresponding to an undirected graph $G=(V,E)$,
with $V$ the space of vertices which models geographic points and $E$ the
space of edges that links one vertice to another. If there is an edge between
two vertices, it means that there is a path between these two points, a road
for example.

  * •

A resource representation, including the agents involved in the mission and
their characteristics. Each element of the description is an object with
attributes.

  * •

An operational representation where the constraints and the goal will be
defined.

This approach allows a structured and organized representation of critical
information, enabling a comprehensive analysis of a complex mission. While the
geographic and resource representations correspond to static meta-data
available at the start of the mission, the operational representation
corresponds to flexible operational and tactical elements from the mission,
allowing the definition of the goal and mandatory waypoints. Furthermore, some
components of the description are objects with attributes. The type of mission
modeled by ORTAC+ deals with three types of objects: the nodes, the edges, and
the agents. By leveraging the three axes, we enable an in-depth portrayal of
mission-critical information with a more expressive and accessible
representation. It is worth noting that the agents involved in the missions
are usually humans, but the tool aims to be compatible with robots.

During the conception phase, the graph is pre-defined or already given. The
mission specification includes the available resources and the operational
elements within the system with high-level predicates. Each predicate has a
logical meaning and is a constraint for a planning algorithm.

####  3.2.1 Resources

The resources encompass the agents, their initial states, and their
characteristics. The agent$\\_$define(init, characteristics) predicate
instantiates the agent, creating the object agent with its initial position
init and its characteristics. It is still possible to add attributes to the
agent after its instantiation.

####  3.2.2 Operational Predicates

This representation is the modular part of the modeling. Indeed, modeling a
complex mission can lead to the absence of solutions. In this part the user
can change some elements of the problem and relax some constraints to find a
plan. The planning tool allocates a path for each agent to reach the goal, and
the tactical interpretation of the mission remains to the user.

Predicates |  Description |  Logic formula
---|---|---
node$\\_$goal(node, agent) |  The location of agent should be node at the final state |  $agent.loc(t_{final})=node$
node$\\_$visit(node, agent) |  Agent should visit node during execution |  $\exists t\in[\\![t_{initial};t_{final}]\\!],\newline agent.loc(t)=node$
edge$\\_$visit(edge, agent) |  Agent should visit edge during execution |  $\exists t\in[\\![t_{initial};t_{final}]\\!],\newline agent.loc(t)=edge$
node$\\_$avoid(node, agent) |  Agent should never visit node |  $\forall t\in[\\![t_{initial};t_{final}]\\!],\newline agent.loc(t)\neq node$
edge$\\_$avoid(edge, agent) |  Agent should never visit edge |  $\forall t\in[\\![t_{initial};t_{final}]\\!],\newline agent.loc(t)\neq edge$
node$\\_$supported$\\_$from(node1, node2) |  An agent can visit node1 only if another agent supports him in node2 |  $\forall t\in[\\![t_{initial};t_{final}]\\!],\forall agent,\newline agent.loc(t)=node_{1}\Rightarrow\newline \exists agent_{2}\neq agent,\newline agent_{2}.loc(t)=node_{2}$
Table 1: Existing predicates in ORTAC+ DSL. Other predicates can be
constructed

Table 1 presents the high-level predicates introducing the constraints. Each
predicate has a description and a logical meaning. For example, the predicate
node$\\_$goal() can allocate a node goal to an agent. The predicates aim to
give a declarative specification.

Besides, the predicates can be generalised so that they operate on lists of
objects and no longer individual objects. To facilitate modeling, when a
”generalised” predicate is applied to an individual object, this is
automatically replaced by the application on the singleton containing that
object. So the preceding predicate is in fact syntactic sugar for:
node_goal([node],[agent]). The general case node_goal(node_list,agent_list)
has the following meaning:

$\forall n\in node\\_list,\exists a\in agent\\_list,a\ is\ in\ n\ at\
t_{final}$ (1)

The $\forall\ n$ is in fact a general pattern to which all ’list-generalized’
predicates conform, so as to facilitate their understanding: the first list
argument can always be ”and-expanded”, which means:

predicate([o1,o2], list2, …) $\Leftrightarrow$ predicate(o1, list2, …) &
predicate(o2, list2, …)

Because of the multi-agent nature of the model, coordination constraints could
be required. We are interested in the coordination presented in ORTAC [4]. The
main coordination predicate uses quantitative time, but if we restrict
ourselves to causality, it boils down to: support(unit1,node1,unit2,node2)
which means: ”when unit1 goes through node1 (if it goes there), unit2 must be
in node2”. For this to make sense, node1 and node2 must be close enough, and
from a tactical point of view, it is the standard way to progress in dangerous
areas. However, we are no longer specifying the mission, but have begun
designing the plan. We will call ORTAC’s description style ”imperative”, and
ORTAC+ will add a new style that we will call ”declarative”:
node_supported_from(node1,node2) which means: ”when an agent goes through
node1, another agent must be in node2”. This first addition to ORTAC will be
called ”anonymization”, it is not a generic evolution because it is only
suitable for certain predicates like support or goal. And concerning goals,
their ”anonymization” corresponds to an interesting variant of MAPF initially
called ”permutation-invariant” [10] and today called ”anonymous” [11].

####  3.2.3 Ontology

Considering the specificity of our language for its domain, the proposal
entails adding a knowledge base to augment its capability. This knowledge base
is an ontology designed to help the mission specification. The characteristics
of the agents is defined with the declarative predicate presented in the
section 3.2.1.

It is possible to specify a constraint involving several agents. For example,
considering a mission with two types of agents, an agent can be a ”UAV” for
unmanned aerial vehicle or a ”UGV” for unmanned ground vehicle. This
characteristics will be define as an attribute of the agents. Instead of
writing: node$\\_$goal(14, [agent1, agent2]), it is possible to write
node$\\_$goal(14, ”UGV”) if agent1 and agent2 have the attribute UGV, meaning
at the end of the mission a ”UGV” should be at the node 14.

Another example of the combined efficiency of objects and ontologies is:
french infantry uses impressive vehicles called ”VBCI” which need some place
to maneuver. If the officer wants to ban the progression of these vehicles on
narrow roads, he has to stipulate: edge_avoid(”width ¡ 10”, ”VBCI”) in spite
of edge_avoid(list_of_edges_with_width_less_than_10, list_of_units_with_VBCI)

Currently, the ontology is hierarchical. An attribute can be a leaf in
knowledge tree. For instance, if agent1 is a UGV on wheels and agent2 is a UGV
on caterpillars. There are the relations: a UGV with wheels is a UGV, and the
UGV on caterpillars is a UGV. They have the same parent. The ontology allows
to specify node$\\_$goal(14, ”UGV”) to involve agent1 and agent2 in the
constraint.

##  4 Operational Context

ORTAC+ is built to specify military missions in order to assist an officer
planning the mission. In the following part, an example of mission is
described, then its representation in the proposed language.

###  4.1 Description of the Mission

Figure 1: Cartography of the mission in Goma, Congo.

Figure 2: Geographical mission modelisation using graphs

Congolese rebels, supported by elements from Rwanda, are located in the
eastern region of Goma. The rebels raise a significant threat to the peace and
security of the city. The mission focused on two specific objectives: securing
the United Nation (UN) point at the border, between Congo and Rwanda, and
preparing for the potential evacuation of civilians at the airport. With
forces comprising 4 distinct military sections and 4 motorized companies
positioned along the west cost of the city, this mission implies the
recognition of crucial points and axes leading to the destination goals.
Operational constraint. Units follow specific rules of engagements during the
execution of their tasks:

  * •

Simultaneous coexistence of two distinct units at the same spatial coordinates
is not allowed.

  * •

Any unit advancing on a main street or an intersection must be supported by
another unit recently deployed to an adjacent position.

Tactical information. The mission encompasses critical data as a foundation
for modeling and planning:

  * •

The airport: the airport is a crucial point and must be controlled during the
mission, with a designated convoy of one company and one unit equipped with
night vision glasses will be tasked to evacuate people from the airport.

  * •

Securing the UN point: The UN point is identified at the border, requiring
strict security measures, with a priority to maintain control over

  * •

Support team: A support team will be positioned at the southern region of the
operation area

###  4.2 Mission specification in ORTAC+

Figure 3: Specification of the ”Secure the UN in Goma” mission in ORTAC+

An example of the mission specification in ORTAC+ is given in Figure 3. The
mission goal is the securing of the United Nations. The cartography of the
mission is given in Figure 2. Friendly forces are blue rectangles at the west
of Goma. Target positions are triangles at the east near to enemies in red
rectangles. The black triangle correspond to the airport and the red one the
UN. The 3 axis of representation (geography, resources and operations) appear
clearly combined with the construction of a hierarchical ontology. Geography
elements of the mission are supposed to be already given. It is represented in
Figure 2 The Goma mission involves two types of agents: compagnies and
sections. The predicates ”agent$\\_$define()” is used to instantiate the
units. At the initial state, the 4 mechanized compagnies are at the same node
(9). This violates the constraint that only one unit can be at a node at the
same time. That’s why the notion of capacity is introduced for nodes and
edges. By default the capacity of a node or an edge will be 1 but it can be
changed for some exceptions. The method ”attribute()” allows the user to add
some characteristics to the units. Then, the high level predicates described
earlier are used to add the constraints of the mission. The final state is
defined with the predicate ”node$\\_$goal()” with the help of the knowledge
base. A compagny should be at the node 11, corresponding to the airport, at
the final state. Other constraints are defined, for instance, the unit1 should
visit the node 9 but avoid the edge (9,8). ORTAC+ introduces coordination
constraints. If an enemy is suspected on a node or an edge, a unit cannot
visit it without support. Therefore, the predicate
”node$\\_$supported$\\_$from()” model this coordination. In the presented
example, it is specified in order to visit the node 3, there must be a support
at the node 18.

##  5 Conclusion

We have presented ORTAC+, a DSL that hopefully permits a military officer to
specify a mission. Compared to its predecessor, ORTAC, the evolution can be
summarized as follows:

  1. 1.

predicates are anonymized when suitable, in particular for the declaration of
goals or supports and operate on lists of homogeneous objects (agents, nodes,
or edges),

  2. 2.

these lists can be described intentionally, with a propositional sub-language
constraining the attributes of objects,

  3. 3.

these propositions, when they involve the ”or” operator, can be shortened
thanks to the declaration of the adequate ontology.

As a result, ORTAC+ allows more compact and declarative models, which will be
more easily written and checked by end-users. At the same time, we take care
to keep semantics to allow a translation to PDDL, making it possible to rely
on the existing planners. Current and future work explores 3 orthogonal
directions: new coordination constraints, quantitative time and resources, and
compatibility with flying drones.

## References

  * [1] Roni Stern, Nathan Sturtevant, Ariel Felner, Sven Koenig, Hang Ma, Thayne Walker, Jiaoyang Li, Dor Atzmon, Liron Cohen, TK Kumar, et al.  Multi-agent pathfinding: Definitions, variants, and benchmarks.  In Proceedings of the International Symposium on Combinatorial Search, volume 10, pages 151–158, 2019.  doi:10.48550/arXiv.1906.08291.
  * [2] Alejandro Torreño, Eva Onaindia, Antonín Komenda, and Michal Štolba.  Cooperative multi-agent planning: A survey.  ACM Comput. Surv., 50(6), 2017.  doi:10.1145/3128584.
  * [3] Patrik Haslum, Nir Lipovetzky, Daniele Magazzeni, Christian Muise, Ronald Brachman, Francesca Rossi, and Peter Stone.  An introduction to the planning domain definition language, volume 13.  Springer, 2019.  doi:10.1007/978-3-031-01584-7.
  * [4] Christophe Guettier, Willy Lamal, Israel Mayk, and Jacques Yelloz.  Design and experiment of a collaborative planning service for netcentric international brigade command.  In Proceedings of the AAAI Conference on Artificial Intelligence, volume 29, pages 3967–3974, 2015.  doi:10.1609/aaai.v29i2.19055.
  * [5] G Ayorkor Korsah, Anthony Stentz, and M Bernardine Dias.  A comprehensive taxonomy for multi-robot task allocation.  The International Journal of Robotics Research, 32(12):1495–1512, 2013.  doi:10.1177/0278364913496484.
  * [6] Derek Long, Henry Kautz, Bart Selman, Blai Bonet, Hector Geffner, Jana Koehler, Michael Brenner, Joerg Hoffmann, Frank Rittinger, Corin R. Anderson, Daniel S. Weld, David E. Smith, Maria Fox, and Derek Long.  The aips-98 planning competition.  AI Magazine, 21(2):13, 2000.  doi:10.1609/aimag.v21i2.1505.
  * [7] Michal Stolba, Antonin Komenda, and Daniel Kovacs.  Competition of distributed and multiagent planners (codmap).  In Proceedings of the AAAI Conference on Artificial Intelligence, volume 30, 2016.  doi:10.1609/aaai.v30i1.9870.
  * [8] Swaib Dragule, Thorsten Berger, Claudio Menghi, and Patrizio Pelliccione.  A survey on the design space of end-user-oriented languages for specifying robotic missions.  Software and Systems Modeling, 20(4):1123–1158, February 2021.  doi:10.1007/s10270-020-00854-x.
  * [9] Erez Karpas and Daniele Magazzeni.  Automated planning for robotics.  Annual Review of Control, Robotics, and Autonomous Systems, 3:417–439, 2020.  doi:10.1146/annurev-control-082619-100135.
  * [10] S. Kloder and S. Hutchinson.  Path planning for permutation-invariant multirobot formations.  IEEE Transactions on Robotics, 22(4):650–665, August 2006.  doi:10.1109/tro.2006.878952.
  * [11] Hang Ma and Sven Koenig.  Optimal target assignment and path finding for teams of agents.  arXiv preprint arXiv:1612.05693, 2016.  doi:10.48550/arXiv.1612.05693.
  * [12] Jingjin Yu and Steven M. LaValle.  Multi-agent path planning and network flow.  In Springer Tracts in Advanced Robotics, pages 157–173. Springer Berlin Heidelberg, 2013.  doi:10.1007/978-3-642-36279-8_10.
  * [13] Bernhard Nebel.  On the computational complexity of multi-agent pathfinding on directed graphs.  Proceedings of the International Conference on Automated Planning and Scheduling, 30:212–216, June 2020.  doi:10.1609/icaps.v30i1.6663.
  * [14] Alexander Kott, Ray Budd, Larry Ground, Lakshmi Rebbapragada, and John Langston.  Building a tool for battle planning: Challenges, tradeoffs, and experimental findings.  Applied Intelligence, 23(3):165–189, December 2005.  doi:10.1007/s10489-005-4606-z.
  * [15] Hang Ma and Sven Koenig.  Optimal target assignment and path finding for teams of agents, 2016.  doi:10.48550/ARXIV.1612.05693.


$\displaystyle\sum_{k=1}^{\ell+2}\Big{\langle}\Upsilon_{0},K_{k}\Upsilon_{\ell+2-k}\Big{\rangle}=-E_{\ell}+\Big{\langle}\Upsilon_{0},\widetilde{K}_{\ell}\Upsilon_{0}\Big{\rangle}+\sum_{k=1}^{\ell+1}\Big{\langle}\Upsilon_{0},K_{k}\Upsilon_{\ell+2-k}\Big{\rangle}=0$
(4.22)

by Lemma 4.3. This completes the proof of the lemma.

∎

###  4.2 Proof of Lemma 4.4

To prove Lemma 4.4, we derive the following implication: If there are
constants $C_{2}>0$, $\ell\in\mathbb{N}$ such that $|E_{k}|\leq
C_{2}^{k}\sqrt{(k+2)!}$ for all $k\in\\{1,\ldots,\ell-1\\}$, then

$\displaystyle\sum_{k=1}^{\ell+1}\|P\mathbb{P}K_{k}\Upsilon_{l+2-k}\|\leq
C_{2}^{\ell}\sqrt{(\ell+2)!}.$ (4.23)

Note that for notational convenience, we omit the tensor product in operators
of the form $P\otimes\mathbb{P}$ throughout this section. With Lemma 4.3, we
can use this implication to arrive at $|E_{\ell}|\leq
C_{2}^{\ell}\sqrt{(\ell+2)!}$ for all $\ell\geq 1$ by iteration. Having
established the bounds on $E_{\ell}$, the proof of the second statement of the
lemma follows in close analogy to the derivation of (4.23).

The proof of (4.23) requires some preparation. We start by introducing the
auxiliary operators

$\displaystyle V_{\ell,\beta}:=\begin{cases}-E_{\ell-2}&\text{ for
}\beta=\ell\geq 1,\\\ K_{\ell}+E_{\ell-2}&\text{ for }\beta=0\text{ and
}\ell\geq 1,\\\ 0&\text{ otherwise, }\end{cases}$ (4.24)

where we set $E_{-1}=0$. With these at hand, we define

$\displaystyle\mathbb{T}_{\ell,\beta}:=\sum_{p=1}^{\ell}\sum_{\begin{subarray}{c}\ell_{1},\ldots,\ell_{p}\geq
1\\\
\ell_{1}+\ldots+\ell_{p}=\ell\end{subarray}}\sum_{\begin{subarray}{c}\beta_{1},\dots,\beta_{p}\\\
0\leq\beta_{i}\leq\ell_{i}\\\
\beta_{1}+\ldots+\beta_{p}=\beta\end{subarray}}V_{\ell_{1},\beta_{1}}\\!\left(RV_{\ell_{2},\beta_{2}}\right)\dots\left(RV_{\ell_{p},\beta_{p}}\right)\\!P.$
(4.25)

Inserting $K_{\ell}=V_{\ell,0}+V_{\ell,\ell}$ in (4.1), it is straightforward
to see that

$\displaystyle\mathbb{K}_{\ell}=P\sum_{\beta=0}^{\ell+2}\mathbb{T}_{\ell+2,\beta}\quad\text{and}\quad\widetilde{\mathbb{K}}_{\ell}=P\sum_{\beta=0}^{\ell+1}\mathbb{T}_{\ell+2,\beta},$
(4.26)

where the second identity follows from the first by
$\mathbb{T}_{\ell+2,\ell+2}=V_{\ell+2,\ell+2}P=-E_{\ell}P$ and
$\widetilde{\mathbb{K}}_{\ell}=\mathbb{K}_{\ell}-E_{\ell}$. While this way of
writing $\mathbb{K}_{\ell}$ and $\widetilde{\mathbb{K}}_{\ell}$ involves many
summands that are zero by definition, it turns out useful to keep track of the
combinatorial factors.

Furthermore, let us define the spaces
$\mathcal{H}_{k}:=\mathbb{U}^{*}\\!\left(\mathcal{F}_{\leq k}\right)$, where
$\mathbb{U}$ is the unitary defined in Eq. (1.34) and $\mathcal{F}_{\leq
k}\subseteq\mathcal{F}$ is the subspace of states involving at most $k$
particles. Before we prove Lemma 4.4, we need some auxiliary results.

######  Lemma 4.5.

Assume that $|E_{k}|\leq C_{2}^{k}\sqrt{(k+2)!}$ for some $C_{2}>0$ and all
$k\in\\{1,\ldots,\ell-1\\}$. Then there exists a constant $D>0$, such that

$\displaystyle\|(YV_{\ell,0}Z)|_{\mathcal{H}_{r}}\|\leq
D\sqrt{(r+1)\dots(r+\ell)},$ (4.27)
$\displaystyle\|\left(V_{\ell,0}Z\right)|_{\mathcal{H}_{r}}\|\leq\left(1+\sqrt{\Lambda}\right)D\sqrt{(r+1)\dots(r+\ell)},$
(4.28)
$\displaystyle\|(\mathbb{R}PV_{\ell_{1},0}RV_{\ell_{2},0}R^{\frac{1}{2}})|_{\mathcal{H}_{r}}\|\leq
D\sqrt{(r+1)\dots(r+\ell_{1}+\ell_{2}-2)}$ (4.29)

with $Y,Z\in\\{(-R)^{\frac{1}{2}},P\\}$.

###### Proof.

The bounds (4.27) and (4.28) follow immediately from Lemma 3.3 and Lemma 2.5.
In order to verify Eq. (4.29), note that $\mathbb{R}^{2}\leq
C\mathbb{U}^{\dagger}(\mathcal{N}+1)^{-2}\mathbb{U}$, and therefore

$\displaystyle\|(\mathbb{R}PV_{\ell_{1},0}RV_{\ell_{2},0}R^{\frac{1}{2}})|_{\mathcal{H}_{r}}\|\leq\sqrt{C}\|(\mathbb{U}^{\dagger}(\mathcal{N}+1)^{-1}\mathbb{U}PV_{\ell_{1},0}RV_{\ell_{2},0}R^{\frac{1}{2}})|_{\mathcal{H}_{r}}\|.$
(4.30)

To process this expression further, let us decompose
$V_{\ell,0}|_{\mathcal{H}_{r}}=\sum_{i=-2}^{2}W_{\ell}^{i}$, where
$W_{\ell}^{i}$ is defined by $W_{\ell}^{-i}=\mathds{1}(\ell\leq
4)\,\pi_{\mathcal{H}_{r+i}}V_{\ell,0}|_{\mathcal{H}_{r}}$ and
$\pi_{\mathcal{H}_{r}}$ is the orthogonal projection onto $\mathcal{H}_{r}$.
Making use of Lemma 3.3, we know that
$\|(YW_{\ell}^{i}Z)|_{\mathcal{H}_{r}}\|\leq D\sqrt{(r+1)\dots(r+\ell)}$ and
consequently

$\displaystyle\|(PW_{\ell_{1}}^{i_{1}}RW_{\ell_{2}}^{i_{2}}R^{\frac{1}{2}})|_{\mathcal{H}_{r}}\|\leq
D^{2}\sqrt{(r+1)\dots(r+\ell_{1}+\ell_{2})}.$ (4.31)

Further note that
$Z_{\ell_{1},\ell_{2}}^{i_{1},i_{2}}:=PW_{\ell_{1}}^{i_{1}}RW_{\ell_{2}}^{i_{2}}R^{\frac{1}{2}}$
satisfies
$Z_{\ell_{1},\ell_{2}}^{i_{1},i_{2}}\mathcal{H}_{=r}\subset\mathcal{H}_{=r+i_{1}+i_{2}}$,
where $\mathcal{H}_{=r}:=\mathcal{H}_{r}\cap\mathcal{H}_{r-1}^{\perp}$, and
consequently $Z_{\ell_{1},\ell_{2}}^{i_{1},i_{2}}\mathcal{H}_{=r}\perp
Z_{\ell_{1},\ell_{2}}^{i_{1},i_{2}}\mathcal{H}_{=s}$ for $r\neq s$.
Decomposing an arbitrary state $\Psi\in\mathcal{H}_{r}$ into
$\Psi=\sum_{s=0}^{r}\Psi_{s}$ such that $\Psi_{s}\in\mathcal{H}_{=s}$, we
obtain

$\displaystyle\|\mathbb{U}^{\dagger}(\mathcal{N}+1)^{-1}\mathbb{U}PW_{\ell_{1}}^{i_{1}}RW_{\ell_{2}}^{i_{2}}R^{\frac{1}{2}}\Psi\|^{2}=\sum_{s=s_{0}}^{r}\|\mathbb{U}^{\dagger}(\mathcal{N}+1)^{-1}\mathbb{U}PW_{\ell_{1}}^{i_{1}}RW_{\ell_{2}}^{i_{2}}R^{\frac{1}{2}}\Psi_{s}\|^{2}$
$\displaystyle\ \
=\sum_{s=s_{0}}^{r}\frac{1}{(s+i_{1}+i_{2}+1)^{2}}\|PW_{\ell_{1}}^{i_{1}}RW_{\ell_{2}}^{i_{2}}R^{\frac{1}{2}}\Psi_{s}\|^{2}\leq\sum_{s=s_{0}}^{r}D^{2}\frac{(s+1)\dots(s+\ell_{1}+\ell_{2})}{(s+i_{1}+i_{2}+1)^{2}}\|\Psi_{s}\|^{2}$
(4.32)

with $s_{0}:=\max\\{0,-i_{1}-i_{2}\\}$, where we have used Eq. (4.31). Since
we have the inequality
$\frac{(s+1)\dots(s+\ell_{1}+\ell_{2})}{(s+i_{1}+i_{2}+1)^{2}}\leq
7(s+1)\dots(s+\ell_{1}+\ell_{2}-2)\leq 7(r+1)\dots(r+\ell_{1}+\ell_{2}-2)$ we
obtain
$\|(\mathbb{U}^{\dagger}(\mathcal{N}+1)^{-1}\mathbb{U}PW_{\ell_{1}}^{i_{1}}RW_{\ell_{2}}^{i_{2}}R^{\frac{1}{2}})|_{\mathcal{H}_{r}}\|\leq\sqrt{7}D\sqrt{(r+1)\dots(r+\ell_{1}+\ell_{2}-2)}$,
which concludes the proof. Here we used that $W_{\ell}^{-i}=0$ if $\ell\geq
5$. ∎

######  Lemma 4.6.

Assume that $|E_{k}|\leq C_{2}^{k}\sqrt{(k+2)!}$ for some $C_{2}>0$ and all
$k\in\\{1,\ldots,\ell-1\\}$. Then there exist constants $C_{0},C_{1}>0$, such
that

$\displaystyle\left\|(X\mathbb{T}_{\ell,\beta})|_{\mathcal{H}_{s}}\right\|\leq
C_{0}\,C_{1}^{\ell-\beta}C_{2}^{\max\\{\beta-2,0\\}}\sqrt{(s+1)\dots(s+\ell-\beta)}\sqrt{\beta!}$

for all $\beta\leq\ell$ and $X\in\\{(-R)^{\frac{1}{2}},P\\}$, where we use the
convention that $(s+1)\dots(s+\ell-\beta)=1$ for $\beta=\ell$. Furthermore,
$\left\|(\mathbb{R}P\mathbb{T}_{\ell+2,0})|_{\mathcal{H}_{s}}\right\|\leq
C_{1}^{\ell}\sqrt{(s+1)\dots(s+\ell)}$.

###### Proof of Lemma 4.6.

Using the fact that
$V_{\ell,\beta}\mathcal{H}_{r}\subset\mathcal{H}_{r+\ell-\beta}$ and that
there exists a constant $D>0$ such that
$\|(YV_{\ell,\beta}Z)|_{\mathcal{H}_{r}}\|\leq
D^{\ell-\beta}C_{2}^{\max\\{\beta-2,0\\}}\sqrt{(r+1)\dots(r+\ell-\beta)}\sqrt{\beta!}$
for $Y,Z\in\\{(-R)^{\frac{1}{2}},P\\}$, see Lemma 4.5, we can bound
$\left\|(X\mathbb{T}_{\ell,\beta})|_{\mathcal{H}_{s}}\right\|$ from above by

$\displaystyle
D^{\ell-\beta}C_{2}^{\max\\{\beta-2,0\\}}\sqrt{(s\\!+\\!1)\dots(s\\!+\\!\ell\\!-\\!\beta)}\sum_{p=1}^{\beta}\sum_{\begin{subarray}{c}\ell_{1},\dots,\ell_{p}\in\mathbb{N}\\\
\ell_{1}+\ldots+\ell_{p}=\ell\end{subarray}}\sum_{\begin{subarray}{c}\beta_{1},\dots,\beta_{p}:0\leq\beta_{i}\leq\ell_{i}\\\
\beta_{1}+\ldots+\beta_{p}=\beta\end{subarray}}\sqrt{\beta_{1}!\dots\beta_{p}!}$
$\displaystyle\
=D^{\ell-\beta}C_{2}^{\max\\{\beta-2,0\\}}\sqrt{(s\\!+\\!1)\dots(s\\!+\\!\ell\\!-\\!\beta)}\sum_{p=1}^{\beta}\sum_{\begin{subarray}{c}\beta_{1},\dots,\beta_{p}\geq
0\\\
\beta_{1}+\ldots+\beta_{p}=\beta\end{subarray}}\sqrt{\beta_{1}!\dots\beta_{p}!}\left(\sum_{\begin{subarray}{c}\ell_{1},\dots,\ell_{p}\\\
\ell_{i}\geq\max\\{\beta_{i},1\\}\\\
\ell_{1}+\ldots+\ell_{p}=\ell\end{subarray}}1\right).$

For given $\beta_{1},\dots,\beta_{p}$ satisfying
$\beta_{1}+\dots+\beta_{p}=\beta$, let us define the set
$I=I(\beta_{1},\dots,\beta_{p}):=\\{j:\beta_{j}\geq 1\\}$, the number
$p^{\prime}=p^{\prime}(\beta_{1},\dots,\beta_{p}):=|I|$ and the indicator
function $\chi=\chi(\beta_{1},\dots,\beta_{p}):=\mathds{1}_{p^{\prime}\geq
p-\ell+\beta}$. Note that the existence of a sequence
$\ell_{1},\dots,\ell_{p}$ satisfying $\ell_{1}+\dots+\ell_{p}=\ell$ and
$\ell_{j}\geq\max\\{\beta_{i},1\\}$ immediately implies

$\displaystyle p-p^{\prime}=|\\{1,\dots,p\\}\setminus
I|\leq\sum_{i\in\\{1,\dots,p\\}\setminus
I}\ell_{i}=\sum_{i\in\\{1,\dots,p\\}\setminus
I}(\ell_{i}-\beta_{i})\leq\sum_{i=1}^{p}(\ell_{i}-\beta_{i})=\ell-\beta,$

and therefore $\chi(\beta_{1},\dots,\beta_{p})=1$. As a consequence

$\displaystyle\sum_{\begin{subarray}{c}\ell_{1},\dots,\ell_{p}:\ell_{i}\geq\max\\{\beta_{i},1\\}\\\
\ell_{1}+\ldots+\ell_{p}=\ell\end{subarray}}1$
$\displaystyle\leq\sum_{\begin{subarray}{c}\ell_{1},\dots,\ell_{p}:\ell_{i}\geq\beta_{i}\\\
\ell_{1}+\ldots+\ell_{p}=\ell\end{subarray}}\chi=\sum_{\begin{subarray}{c}\delta_{1},\dots,\delta_{p}:\delta_{i}\geq
0\\\
\delta_{1}+\ldots+\delta_{p}=\ell-\beta\end{subarray}}\chi=\sum_{q=1}^{p}\binom{p}{q}\sum_{\begin{subarray}{c}\delta_{1},\dots,\delta_{p}\geq
1\\\ \delta_{1}+\ldots+\delta_{p}=\ell-\beta\end{subarray}}\chi$
$\displaystyle\leq\sum_{q=1}^{p}\binom{p}{q}2^{\ell-\beta}\chi\leq
2^{p}2^{\ell-\beta}\chi.$ (4.33)

With this at hand, we can bound

$\displaystyle\sum_{p=1}^{\beta}\sum_{\begin{subarray}{c}\beta_{1},\dots,\beta_{p}\geq
0\\\
\beta_{1}+\ldots+\beta_{p}=\beta\end{subarray}}\sqrt{\beta_{1}!\dots\beta_{p}!}\left(\sum_{\begin{subarray}{c}\ell_{1},\dots,\ell_{p}:\ell_{i}\geq\max\\{\beta_{i},1\\}\\\
\ell_{1}+\ldots+\ell_{p}=\ell\end{subarray}}1\right)$ $\displaystyle\leq
2^{\ell-\beta}\\!\sum_{p=1}^{\beta}2^{p}\sum_{\begin{subarray}{c}\beta_{1},\dots,\beta_{p}\geq
0\\\
\beta_{1}+\ldots+\beta_{p}=\beta\end{subarray}}\sqrt{\beta_{1}!\dots\beta_{p}!}\,\chi=2^{\ell-\beta}\sum_{p=1}^{\beta}\sum_{p^{\prime}=p-\ell+\beta}^{p}\\!\\!\\!\\!2^{p}\binom{p}{p^{\prime}}\sum_{\begin{subarray}{c}\beta_{1},\dots,\beta_{p^{\prime}}\geq
1\\\
\beta_{1}+\ldots+\beta_{p^{\prime}}=\beta\end{subarray}}\sqrt{\beta_{1}!\dots\beta_{p^{\prime}}!}\,.$

In order to estimate this further, we are going to verify

$\displaystyle\sum_{\begin{subarray}{c}\beta_{1},\dots,\beta_{q}\geq 1\\\
\beta_{1}+\ldots+\beta_{q}=\beta\end{subarray}}\sqrt{\beta_{1}!\dots\beta_{q}!}\leq\frac{12^{q}}{\sqrt{q!}}\sqrt{\beta!}$
(4.34)

by induction. For $q=1$ the statement is trivial. Regarding the step $q\mapsto
q+1$, note that

$\displaystyle\sum_{\begin{subarray}{c}\beta_{1},\dots,\beta_{q+1}\geq 1\\\
\beta_{1}+\ldots+\beta_{q+1}=\beta\end{subarray}}\sqrt{\beta_{1}!\dots\beta_{q+1}!}=\sum_{t=1}^{\beta-q}\sqrt{t!}\sum_{\begin{subarray}{c}\beta_{1},\dots,\beta_{q}\geq
1\\\
\beta_{1}+\ldots+\beta_{q}=\beta-t\end{subarray}}\sqrt{\beta_{1}!\dots\beta_{q}!}\leq\frac{12^{q}}{\sqrt{q!}}\sum_{t=1}^{\beta-1}\sqrt{t!}\sqrt{(\beta-t)!}$
$\displaystyle\ \ \ \
=\frac{12^{q}}{\sqrt{q!}}\sqrt{\beta!}\sum_{t=1}^{\beta-1}\binom{\beta}{t}^{-\frac{1}{2}}=\frac{12^{q}}{\sqrt{q!}}\sqrt{\beta!}\left(\frac{2}{\sqrt{\beta}}+\frac{4}{\sqrt{\beta(\beta-1)}}+\sum_{t=3}^{\beta-3}\binom{\beta}{t}^{-\frac{1}{2}}\right)$
$\displaystyle\ \ \ \
\leq\frac{12^{q}}{\sqrt{q!}}\sqrt{\beta!}\left(\frac{2}{\sqrt{\beta}}+\frac{4}{\sqrt{\beta(\beta-1)}}+(\beta-5)\binom{\beta}{3}^{-\frac{1}{2}}\right)\leq\frac{12^{q+1}}{\sqrt{q!}\sqrt{\beta}}\sqrt{\beta!},$
(4.35)

where we have used
$\binom{\beta}{t}^{-\frac{1}{2}}\leq\binom{\beta}{3}^{-\frac{1}{2}}$ for
$3\leq t\leq\beta-3$. Since
$\sum_{\begin{subarray}{c}\beta_{1},\dots,\beta_{q+1}\geq 1\\\
\beta_{1}+\ldots+\beta_{q+1}=\beta\end{subarray}}\sqrt{\beta_{1}!\dots\beta_{q+1}!}=0$
for $\beta<q+1$, we can assume $\beta\geq q+1$, which concludes the induction.
Combining what we have so far allows us to bound
$\left\|(X\mathbb{T}_{\ell,\beta})|_{\mathcal{H}_{s}}\right\|$ from above by

$\displaystyle(2D)^{\ell-\beta}C_{2}^{\max\\{\beta-2,0\\}}\sqrt{(s\\!+\\!1)\dots(s\\!+\\!\ell\\!-\\!\beta)}\sqrt{\beta!}\sum_{p=1}^{\beta}\sum_{p^{\prime}=\max\\{p-\ell+\beta,0\\}}^{p}\\!\\!\\!\\!2^{p}\binom{p}{p^{\prime}}\frac{12^{p^{\prime}}}{\sqrt{p^{\prime}!}}.$

Making use of the estimate

$\displaystyle\sum_{p=1}^{\beta}\sum_{p^{\prime}=\max\\{p-\ell+\beta,0\\}}^{p}\\!\\!\\!\\!2^{p}\binom{p}{p^{\prime}}\frac{12^{p^{\prime}}}{\sqrt{p^{\prime}!}}$
$\displaystyle=\sum_{p^{\prime}=0}^{\beta}\frac{12^{p^{\prime}}}{\sqrt{p^{\prime}!}}\sum_{p=\max\\{1,p^{\prime}\\}}^{\min\\{p^{\prime}+\ell-\beta,\beta\\}}\\!\\!\\!\\!2^{p}\binom{p}{p^{\prime}}\leq\sum_{p^{\prime}=0}^{\beta}\frac{12^{p^{\prime}}}{\sqrt{p^{\prime}!}}\sum_{p=1}^{p^{\prime}+\ell-\beta}\\!\\!\\!\\!4^{p}$
$\displaystyle\leq\sum_{p^{\prime}=0}^{\beta}\frac{12^{p^{\prime}}}{\sqrt{p^{\prime}!}}\frac{4}{3}4^{p^{\prime}+\ell-\beta}\leq
C_{0}4^{\ell-\beta}$

with
$C_{0}:=\frac{4}{3}\sum_{p^{\prime}=0}^{\beta}\frac{(48)^{p^{\prime}}}{\sqrt{p^{\prime}!}}<\infty$
concludes the proof of the first statement of the lemma with $C_{1}:=8D$.

Regarding the proof of the second statement of the lemma, let us write

$\displaystyle\mathbb{R}P\mathbb{T}_{\ell+2,0}=\sum_{\ell_{1},\ell_{2}=1}^{4}\mathbb{R}PV_{\ell_{1},0}RV_{\ell_{2},0}R^{\frac{1}{2}}R^{\frac{1}{2}}\mathbb{T}_{\ell+2-\ell_{1}-\ell_{2},0}.$
(4.36)

Using
$\|(\mathbb{R}PV_{\ell_{1},0}RV_{\ell_{2},0}R^{\frac{1}{2}})|_{\mathcal{H}_{r}}\|\leq
D\sqrt{(r+1)\dots(r+\ell_{1}+\ell_{2}-2)}$ for a suitable constant $D>0$, see
Lemma 4.5, yields

$\displaystyle\left\|(\mathbb{R}P\mathbb{T}_{\ell+2,0})|_{\mathcal{H}_{s}}\right\|\leq
DC_{0}C_{1}^{\ell+2-\ell_{1}-\ell_{2}}\sqrt{(s+1)\dots(s+\ell)}.$ (4.37)

∎

We are now prepared to prove Lemma 4.4.

###### Proof of Lemma 4.4.

As explained at the beginning of this section, the main step is to derive
(4.23). To this end, let us write
$P\mathbb{P}K_{k}\Upsilon_{\ell+2}=P\mathbb{P}Z_{1}+P\otimes\mathbb{P}Z_{2}$
with $Z_{1}$ defined as

$\displaystyle\sum_{p=1}^{\ell}\\!\sum_{\beta=0}^{\ell+2p}\\!\sum_{\begin{subarray}{c}\ell_{1},\dots,\ell_{p}\geq
1\\\
\ell_{1}+\ldots+\ell_{p}=\ell\end{subarray}}\sum_{\begin{subarray}{c}\beta_{1},\dots,\beta_{p}\\\
0\leq\beta_{i}\leq\ell_{i}+2\\\
\beta_{1}+\ldots+\beta_{p}=\beta\end{subarray}}\\!\\!\\!V_{k,0}(R\mathbb{T}_{\ell_{1}+2,\beta_{2}})(\mathbb{R}P\mathbb{T}_{\ell_{2}+2,\beta_{1}})\dots\\!(\mathbb{R}\\!P\mathbb{T}_{\ell_{p}+2,\beta_{p}})\Upsilon_{0}\\!\\!$
(4.38)

and $Z_{2}$ defined as

$\displaystyle\sum_{p=1}^{\ell+2}\\!\sum_{\beta=0}^{\ell+2+2p}\\!\sum_{\begin{subarray}{c}\ell_{1},\dots,\ell_{p}\geq
1\\\
\ell_{1}+\ldots+\ell_{p}=\ell+2\end{subarray}}\sum_{\begin{subarray}{c}\beta_{1},\dots,\beta_{p}\\\
0\leq\beta_{i}\leq\ell_{i}+2\\\
\beta_{1}+\ldots+\beta_{p}=\beta\end{subarray}}\\!\\!V_{k,0}(\mathbb{R}P\mathbb{T}_{\ell_{1}+2,\beta_{1}})(\mathbb{R}P\mathbb{T}_{\ell_{2}+2,\beta_{1}})\dots(\mathbb{R}P\mathbb{T}_{\ell_{p}+2,\beta_{p}})\Upsilon_{0},$
(4.39)

where we have used that $PK_{k}R=PV_{k,0}R$ and
$\mathbb{P}K_{k}\mathbb{R}=\mathbb{P}V_{k,0}\mathbb{R}$, see (4.24). According
to Lemma 4.6 we have
$\|(R^{\frac{1}{2}}\mathbb{T}_{\ell_{1}+2,0})|_{\mathcal{H}_{3s}}\|\leq(\sqrt{3}C_{1})^{\ell_{1}+2}\sqrt{(s+1)\dots(s+\ell_{1}+2)}$
as well as for $\beta_{i}\geq 3$

$\displaystyle\|(X\mathbb{T}_{\ell_{i}+2,\beta_{i}})|_{\mathcal{H}_{3s}}\|\leq\frac{C_{0}}{\tau}(\sqrt{3}C_{1})^{\ell_{i}+2-\beta_{i}}C_{2}^{\beta_{i}-2}\sqrt{(s+1)\dots(s+\ell_{i}+2-\beta_{i})}$

where $X\in\\{\mathbb{R}\\!P,(-R)^{\frac{1}{2}}\\}$ and
$\tau:=\min\\{\|\mathbb{R}\|^{-1},1\\}$. For $\beta_{i}=0$ we use the estimate

$\displaystyle\left\|(\mathbb{R}P\mathbb{T}_{\ell+2,0})|_{\mathcal{H}_{3s}}\right\|\leq(\sqrt{3}C_{1})^{\ell}\sqrt{(s+1)\dots(s+\ell)}$

by Lemma 4.6. Further, $\|\left(PV_{k,0}Y\right)|_{\mathcal{H}_{3s}}\|\leq
C_{1}^{k}\sqrt{(s+1)\dots(s+k)}$ for $Y\in\\{(-R)^{\frac{1}{2}},P\\}$, see
Lemma 4.5. Note at this point that we only need to consider values of $q$
bounded by $p-1$, since $X\mathbb{T}_{\ell_{p}+2,\beta_{p}}\Upsilon_{0}=0$ for
$X\in\\{\mathbb{R}P,(-R)^{\frac{1}{2}}\\}$ allowing us to restrict the sum in
Eq. (4.38) and Eq. (4.39) to $\beta_{p}=0$. Furthermore, the index $\beta$ is
in both Equations bounded by $\beta_{*}:=\ell+k-(p-q)+2q$, since
$\beta=\sum_{j=1}^{p}\beta_{j}\leq\sum_{j:\beta_{j}\neq
0}(\ell_{j}+2)=\sum_{j:\beta_{j}\neq 0}\ell_{j}+2q$ and in the case of Eq.
(4.38) we have $\sum_{j:\beta_{j}\neq
0}\ell_{j}\leq\ell-(p-q)\leq\ell+k-(p-q)$, while we have
$\sum_{j:\beta_{j}\neq 0}\ell_{j}\leq\ell+2-(p-q)\leq\ell+k-(p-q)$ in the case
of Eq. (4.39). Note that we have used the fact that $PV_{k,0}P=0$ for $k=1$
and therefore we can assume $k\geq 2$ in Eq. (4.39). Making use of
$\mathbb{T}_{\ell+2,0}\mathcal{H}_{3r}\subset\mathcal{H}_{3(r+\ell)}$ ,
$\mathbb{T}_{\ell+2,\beta}\mathcal{H}_{3r}\subset\mathcal{H}_{3(r+\ell+2-\beta)}$
and $\Upsilon_{0}\in\mathcal{H}_{s}$, we obtain for a suitable constant
$\widetilde{C}_{0}$ and both $Z_{1}$ and $Z_{2}$ the estimate

$\displaystyle\|Z_{i}\|\leq\sum_{p=1}^{\ell+2}\sum_{q=0}^{p-1}\sum_{\beta=0}^{\beta_{*}}\binom{p}{q}\tilde{C}_{0}^{q}(\sqrt{3}C_{1})^{\ell+k+2p-\beta}C_{2}^{\beta-2q}$
$\displaystyle\quad\times\sqrt{(s\\!+\\!1)\dots(s\\!+\\!\ell\\!+\\!k\\!+\\!2(q\\!+\\!1)\\!-\\!\beta)}\,\sum_{\begin{subarray}{c}\beta_{1},\dots,\beta_{q}\geq
3\\\
\beta_{1}+\ldots+\beta_{q}=\beta\end{subarray}}\sqrt{\beta_{1}!\dots\beta_{q}!}\sum_{\begin{subarray}{c}\ell_{1},\dots,\ell_{p}\geq\max\\{1,\beta_{i}-2\\}\\\
\ell_{1}+\ldots+\ell_{p}=\ell+2\end{subarray}}1$ $\displaystyle\leq
C\\!\sum_{p=1}^{\ell+2}\sum_{q=0}^{p-1}\sum_{\beta=0}^{\beta_{*}}\binom{p}{q}\tilde{C}_{0}^{q}(\sqrt{3(s+1)}C_{1})^{\ell+k+2p-\beta}C_{2}^{\beta-2q}$
$\displaystyle\ \
\times\sqrt{(\ell+k+2(q+1)-\beta)!}\,2^{p}2^{\ell+2p-\beta}\sum_{\begin{subarray}{c}\beta_{1},\dots,\beta_{q}\geq
3\\\
\beta_{1}+\ldots\beta_{q}=\beta\end{subarray}}\,\sqrt{\beta_{1}!\dots\beta_{q}!},$

see also Eq. (4.33) regarding the last estimate, where we have further used
$(s+1)\dots(s+\ell+k+2(q+1)-\beta)\leq(s+1)^{\ell+k+2(q+1)-\beta}(\ell+k+2(q+1)-\beta)!$.
We next show by induction with respect to $q$ that the remaining sum over
$\beta_{1},\ldots,\beta_{q}$ is bounded by
$\frac{C_{*}^{q}}{\sqrt{q!}}\sqrt{(\beta+2-2q)!}$ for a suitable constant
$C_{*}$. The case $q=1$ is clear. Regarding the induction step $q\mapsto q+1$
we estimate, similarly as in the proof of Lemma 4.6,

$\displaystyle\sum_{\begin{subarray}{c}\beta_{1},\ldots,\beta_{q+1}\geq 3\\\
\beta_{1}+\ldots+\beta_{q+1}=\beta\end{subarray}}\sqrt{\beta_{1}!\dots\beta_{q+1}!}\leq\frac{C_{*}^{q}}{\sqrt{q!}}\sum_{t=3q}^{\beta-3}\sqrt{(t+2-2q)!}\sqrt{(\beta-t)!}$
$\displaystyle\ \
\leq\frac{C_{*}^{q}}{\sqrt{q!}}\sqrt{(\beta\\!-\\!2q\\!-\\!1)!}\left(2\sqrt{6}+2\sqrt{\frac{24}{\beta\\!-\\!2q\\!-\\!1}}+(\beta\\!-\\!3q\\!-\\!2)\sqrt{\frac{120}{(\beta\\!-\\!2q\\!-\\!2)(\beta\\!-\\!2q\\!-\\!1)}}\right)$
$\displaystyle\ \
\leq\frac{C_{*}^{q+1}}{\sqrt{q!}}\sqrt{(\beta-2q-1)!}\leq\frac{C_{*}^{q+1}}{\sqrt{(q+1)!}}\sqrt{(\beta-2q)!},$

where we have used that $t+2-2q$ and $\beta-t$ are bounded from above by
$\beta-2q-1$. Since $\beta+2-2q$ as well as $\ell+k+2(q+1)-\beta$ are bounded
from above by $\ell+k+2$ we further obtain
$\sqrt{(\ell+k+2(q+1)-\beta)!}\sqrt{(\beta+2-2q)!}\leq\binom{\ell+k+4}{\ell+k+2}^{-\frac{1}{2}}\sqrt{(\ell+k+4)!}=\sqrt{2}\sqrt{(\ell+k+2)!}$,
and therefore we can bound

$\displaystyle\|P\mathbb{P}K_{k}\Upsilon_{\ell+2}\|\\!\leq\\!D\\!\sum_{p=1}^{\ell+2}\sum_{q=0}^{p-1}\sum_{\beta=0}^{\beta_{*}}\\!2^{p}\binom{p}{q}\\!\\!\left(\\!2\sqrt{3(s\\!+\\!1)}C_{1}\\!\right)^{\ell+k+2p-\beta}\\!\\!\\!\\!C_{2}^{\beta\\!-\\!2q}\\!\frac{(\tilde{C}_{0}C_{*})^{q}}{\sqrt{q!}}\sqrt{(\ell\\!+\\!k\\!+\\!2)!}$
$\displaystyle\ \ \ \ \leq
D\sqrt{(\ell+k+2)!}\sum_{q=0}^{\ell+1}\frac{(\tilde{C}_{0}C_{*})^{q}}{\sqrt{q!}}\sum_{p=q+1}^{\ell+2}\sum_{\beta=0}^{\beta_{*}}\left(8\sqrt{3(s+1)}C_{1}\right)^{\ell+k+2p-\beta}C_{2}^{\beta-2q}$

for a suitable constant $D$, where we have used that
$p\leq\frac{1}{2}(\ell+k+2p-\beta)$ and therefore $2^{p}\binom{p}{q}\leq
4^{p}\leq 2^{\ell+k+2p-\beta}$. Using the shorthand
$C_{3}:=4\sqrt{3(s+1)}C_{1}$, we obtain for $C_{2}>C_{3}$

$\displaystyle\sum_{\beta=0}^{\beta_{*}}$
$\displaystyle\\!\\!\left(4\sqrt{3(s\\!+\\!1)}C_{1}\right)^{\ell+k+2p-\beta}\\!\\!C_{2}^{\beta-2q}\\!=\\!K^{\ell+k+2p}C_{2}^{-2q}\frac{\left(\frac{C_{2}}{K}\right)^{\beta_{*}}-\frac{K}{C_{2}}}{1-\frac{K}{C_{2}}}\leq\\!\frac{C_{2}^{\ell+k}}{1-\frac{K}{C_{2}}}\left(\frac{K^{3}}{C_{2}}\right)^{p-q}.$

Furthermore, we have
$\sum_{p=q+1}^{\ell+2}\left(\frac{C_{3}^{3}}{C_{2}}\right)^{p-q}C_{2}^{\ell+k}\leq\frac{1}{1-\frac{C_{3}^{3}}{C_{2}}}C_{3}^{3}C_{2}^{\ell+k-1}$
for $C_{2}>C_{3}^{3}$, and consequently we obtain for a suitable constant
$\widetilde{D}$

$\displaystyle\|P\mathbb{P}K_{k}\Upsilon_{\ell+2}\|\leq\frac{DK^{3}\sqrt{(\ell+k+2)!}C_{2}^{\ell+k-1}}{\left(1-\frac{K}{C_{2}}\right)\left(1-\frac{K^{3}}{C_{2}}\right)}\sum_{q=0}^{\ell+1}\frac{(\tilde{C}_{0}C_{*})^{q}}{\sqrt{q!}}\leq\widetilde{D}\sqrt{(\ell+k+2)!}C_{2}^{\ell+k-1}.$

Consequently we obtain for $C_{2}\geq 3\widetilde{D}$

$\displaystyle\sum_{k=1}^{\ell+1}\big{\|}P\mathbb{P}K_{k}\Upsilon_{\ell-k+2}^{(n)}\big{\|}=\\!\\!\\!\\!\\!\sum_{k=1}^{\max\\{3,\ell+1\\}}\\!\\!\\!\big{\|}P\otimes\mathbb{P}K_{k}\Upsilon_{\ell-k+2}\big{\|}\\!\leq\\!3\widetilde{D}\sqrt{(\ell+2)!}C_{2}^{\ell-1}\\!\leq\\!C_{2}^{\ell}\sqrt{(\ell+2)!}.$

This completes the derivation of (4.23). As a consequence of Lemma 4.3, we
thus obtain the first part of the lemma.

The second statement of the lemma follows in close analogy, by using the
established bounds for $|E_{\ell}|$. The only major difference is to use the
bound
$\|\left(V_{k,0}Y\right)|_{\mathcal{H}_{3s}}\|\leq(1+\sqrt{\Lambda}\,)C_{1}^{k}\sqrt{(s+1)\dots(s+k)}$
instead of the previously used estimate
$\|\left(PV_{k,0}Y\right)|_{\mathcal{H}_{3s}}\|\leq
C_{1}^{k}\sqrt{(s+1)\dots(s+k)}$, cf. Lemma 4.5. This completes the proof of
Lemma 4.4. ∎

Financial support. M.B. gratefully acknowledges funding from the ERC Advanced
Grant ERC-AdG CLaQS, grant agreement n. 834782.

Mathematical Physics, Analysis and Geometry

## References

  * [1] G. Auberson. Borel summability for a nonpolynomial potential. _Comm. Math. Phys._ 84, 531–546. (1982)
  * [2] V. Betz and S. Polzer. A functional central limit theorem for polaron path measures _Comm. Pure. Appl. Math._ 75, 11, 2345–2392. (2022)
  * [3] V. Betz and S. Polzer. Effective mass of the polaron: a lower bound. _Comm. Math Phys._ 399, 173–188. (2023)
  * [4] N.N. Bogoliubov. On a new form of the adiabatic theory of disturbances in the problem of interaction of particles with a quantum field. (Russian) _Ukr. Mat. Zh 2,3-24_. (1950)
  * [5] L. Boßmann, S. Petrat, P. Pickl and A. Soffer. Beyond Bogoliubov dynamics. _Pure Appl. Anal._ 3(4), 677–726. (2021)
  * [6] L. Boßmann, S. Petrat and R. Seiringer. Asymptotic expansion of low-energy excitations for weakly interacting bosons. _Forum of Mathematics, Sigma_ 9, E28. (2021)
  * [7] M. Brooks and R. Seiringer. The Fröhlich polaron at strong coupling – part I: The quantum correction to the classical energy. Preprint. _arXiv:2207.03156_. (2022)
  * [8] M. Brooks and R. Seiringer. The Fröhlich polaron at strong coupling – part II: Energy-momentum relation and effective mass. Preprint. _arXiv:2211.03353_. (2022)
  * [9] M. Donsker and S.R.S. Varadhan. Asymptotics for the polaron. _Comm. Pure Appl. Math._ 36, 505–528. (1983)
  * [10] W. Dybalski and H. Spohn. Effective mass of the polaron – revisited. Ann. Henri Poincaré 21, 1573–1594. (2020)
  * [11] D. Feliciangeli and R. Seiringer. Uniqueness and non-degeneracy of minimizers of the Pekar functional on a ball. _SIAM J. Math. Anal._ 52, 1. (2020)
  * [12] D. Feliciangeli and R. Seiringer. The strongly coupled polaron on the torus: Quantum corrections to the Pekar asymptotics. _Arch. Rat. Mech. Anal._ 242(3), 1835–1906. (2021)
  * [13] R. Frank and B. Schlein. Dynamics of a strongly coupled polaron. _Lett. Math. Phys._ 104 (8), 911–929. (2014)
  * [14] R. Frank and R. Seiringer. Quantum corrections to the Pekar asymptotics of a strongly coupled polaron. _Comm. Pure Appl. Math._ 74(3), 544–588. (2021)
  * [15] H. Fröhlich. Theory of electrical breakdown in ionic crystals. _Proc. R. Soc. Lond. A_ 160, 230–241. (1937)
  * [16] S. Graffi, V. Grecchi and B. Simon. Borel summability: Application to the anharmonic oscillator. _Phys. Lett. B_ 32(7), 631–634. (1970)
  * [17] P. Grech and R. Seiringer. The excitation spectrum for weakly interacting bosons in a trap. _Commun. Math. Phys._ 322(2):559–591. (2013)
  * [18] M. Griesemer and A. Wünsch. On the domain of the Nelson Hamiltonian. _J. Math. Phys_. 59(4), 042111. (2018)
  * [19] E.P. Gross. Strong coupling polaron theory and translational invariance. _Ann. Phys. 99_ 1–29. (1976)
  * [20] T. Kato. Perturbation theory for linear operators. Springer. (1995)
  * [21] J. Lampart, D. Mitrouskas and K. Myśliwy. On the global minimium of the energy-momentum relation of the polaron. _Math. Phys. Anal. Geom._ 26, 17. (2023)
  * [22] E. Lenzmann. Uniqueness of ground states for pseudorelativistic Hartree equations. _Anal. PDE_ 2, 1–27. (2009)
  * [23] M. Lewin, P. Nam, S. Serfaty and J. Solovej. Bogoliubov spectrum of interacting Bose gases. _Comm. Pure Appl. Math._ 68, 413–471. (2015)
  * [24] E.H. Lieb. Existence and uniqueness of the minimizing solution of Choquard’s non-linear equation. _Studies in Appl. Math._ 57, 93–105. (1977)
  * [25] E. Lieb and J. Solovej. Ground state energy of the one-component charged Bose gas. _Comm. Math. Phys._ 217, 127–163. (2001)
  * [26] E.H. Lieb and L.E. Thomas. Exact ground state energy of the strong-coupling polaron. Comm. Math. Phys. 183(3), 511–519. (1997)
  * [27] E.H. Lieb and K. Yamazaki. Ground-state energy and effective mass of the polaron. _Phys. Rev. 111, 728_. (1958)
  * [28] D. Mitrouskas, K. Myśliwy and R. Seiringer. Optimal parabolic upper bound for the energy-momentum relation of a strongly coupled polaron. _Forum of Mathematics, Sigma_ , 11, E49. (2023)
  * [29] D. Mitrouskas and R. Seiringer. Ubiquity of bound states for the strongly coupled polaron. Preprint. _arXiv:2211.03606_. (2022)
  * [30] S.J. Miyake. The ground state of the optical polaron in the strong-coupling case. _J. Phys. Soc. Jpn._ 41, 747–752. (1976)
  * [31] J.S. Møller, The polaron revisited. _Rev. Math. Phys._ 18, 485–517. (2006)
  * [32] C. Mukherjee, S. R. S. Varadhan. Identification of the Polaron Measure I: Fixed Coupling Regime and the Central Limit Theorem for Large Times _Comm. Pure. Appl. Anal._ 73, 2, 350–383. (2020)
  * [33] S. Polzer. Renewal approach for the energy-momentum relation of the Fröhlich polaron. _Lett. Math. Phys._ 113, 90. (2023)
  * [34] R. Seiringer. The excitation spectrum for weakly interacting bosons. _Commun. Math. Phys._ 306(2):565–578. (2011)
  * [35] R. Seiringer. The polaron at strong coupling. _Math. Rev. Phys._ 33, 1. (2021)
  * [36] M. Selke. Almost quartic lower bound for the Fröhlich polaron’s effective mass via Gaussian domination. Preprint. _arXiv:2212.14023_. (2022)
  * [37] B. Simon. Borel summability of the ground-state energy in spatially cutoff $(\phi^{4})_{2}$. _Phys. Rev. Lett._ 25, 1583. (1970)
  * [38] B. Simon. Determination of eigenvalues by divergent perturbation series. _Adv. Math._ 7, 240–253. (1971)
  * [39] B. Simon. Large orders and summability of eigenvalue perturbation theory: A mathematical overview. _Int. J. Quantum Chem._ 21(1), 3–25. (1982)
  * [40] A. Sokal. An improvement of Watson’s theorem on Borel summability. _J. Math. Phys._ 21, 261. (1980)
  * [41] H. Spohn. The polaron at large total momentum. _J. Phys. A: Math. Gen._ 21, 1199. (1988)
  * [42] S.W. Tjablikow. Adiabatische Form der Störungstheorie im Problem der Wechselwirkung eines Teilchens mit einem gequantelten Feld. _Abhandl. Sowj. Phys. 4, 54–68_. (1954)