domain: info scco spi

text:
Corentin Ascone
email: corentin.ascone@uphf.fr

Frédéric Vanderhaegen
email: frederic.vanderhaegen@uphf.fr

C N R S A M I H 8201 - L

Holistic Digital Twin Framework: Designing Human-Machine Systems with an Overall Situation Awareness

Keywords: Human-Machine System, Context Awareness, Holistic Framework, Digital Twin, High Risk Context, Overall Situation Awareness

Context Awareness refers to the use of context to provide task-relevant information and/or services to a user. The Holistic Framework for Human-Machine System (HMS) is an architecture based on multiple factors, including a surroundings consideration, to design Digital Twins (DTs). This paper discusses some of the research challenges in understanding the context and developing context-aware DT systems. In particular, the authors gave some solutions that can improve Human-Machines Systems (HMS), such as the use of human data, meta-data and peri-data or the definition of Capacity, Availability, Possibility levels. Finally, a critical discussion is brought to caution future designs of systems such as HMS in a high risk context.

INTRODUCTION

A Digital Twin (DT) needs a physical asset to operate, for instance, a machine, a human, or both in the case of a Human-Machine system twin. Data flows between the asset, its digital space and every structure that needs access to the twin. A study conducted in [START_REF] Ascone | Towards a holistic framework for digital twins of human-machine systems[END_REF] highlighted that few people in the scientific community approach a holistic view of DTs. The defined Holistic Framework in [START_REF] Ascone | Towards a holistic framework for digital twins of human-machine systems[END_REF] takes into account the service to the user, the management of data, a human model, and the consideration of the surrounding around the asset(s) in the modeling and decision processes. When humans talk to humans, they can use implicit situational information, context, to increase conversational bandwidth. Unfortunately, this ability to communicate ideas does not transfer well to humans interacting with machines [START_REF] Abowd | Towards a better understanding of context and context-awareness[END_REF]. Today, new IoT devices make it possible to achieve this by recording contextual and surrounding data. By improving the machine's access to contextual information, it produces more useful services and increase the richness of communication in the human-machine interaction. The principle of Context Awareness is the use of context to provide taskrelevant information and/or services to a user. Designing efficient models with a Context Awareness depends a lot on the application domain and use case. Taking into account the context of high-risk systems, where situations are often unpredictable, presents real challenges. [START_REF] Abowd | Towards a better understanding of context and context-awareness[END_REF] highlighted 3 challenges around contextaware systems that the authors have tried to address. First, one of the main challenges in the area of contextaware systems is the development of a taxonomy and a uniform representation of context types. The notion of context is defined in section 2 and the data modeling and standardization are presented as the main relevant solutions to solve this problem. The second challenge is the realization of an infrastructure to promote the design, implementation and evolution of context-aware systems. For that, a use case is explained in section 3 to show the importance of such a solution. Then, it is proposed to define levels of awareness based on the CAP model of [START_REF] Vanderhaegen | Toward a petri net based model to control conflicts of autonomy between cyber-physical & human-systems[END_REF] in order to adjust in real time the control of an automated system thanks to an evaluation of the capacities, availability and possibilities of an agent to act (section 4). The last challenge identified by [START_REF] Abowd | Towards a better understanding of context and context-awareness[END_REF] is a discovery of compelling context-aware applications that assist our everyday interactions with ubiquitous computational services. Authors therefore have the idea of creating an intelligent and dynamic interface for identifying peri-information and use CAP factors assessments in real time (section 5). Finally in section 6, a critical discussion is brought to caution future designs of Human-Machine Systems such as Digital Twins (DTs) in a high risk context.

CONTEXT MANAGEMENT

Definitions

Context can be defined as any information that can be used to characterize the environment or the situation of an entity. An entity can be a person, a place, or a physical or computational object. In [START_REF] Ascone | Towards a holistic framework for digital twins of human-machine systems[END_REF], it has been shown that the elements surrounding a system (the context) are poorly used and considered in its modeling. Similarly to meta-data ("meta" means "that encompasses and transcends" in Greek), this kind of contextual information can be defined as peri-data ("peri " means "that is around " in Greek). As such, the authors have developed definitions for terms that are used in this paper, as adapted from [START_REF] Pfautz | The role of meta-information in c2 decision-support systems[END_REF]:

• Data: output from a human or machine system that can be used or not in the decision-making process (e.g.: "the car speed is 57.2km/h"). • Information: identified input from a human or any system that can be used or not in a decision-making behavior or process (e.g.: "the speedometer indicates a speed higher than 50km/h"). • Meta-data: is characteristics or qualifiers of a data (e.g.: if Data is "the car speed is 57.2km/h" then a Meta-data could be "data recorded on 01/01/2023 at 01:01:01:111ms"). • Meta-information: is characteristics or qualifiers of an Information (e.g.: if Information is "the speedometer indicates a speed higher than 50km/h" then a Metainformation could be "information perceived by a speedometer older than 15 years"). • Peri-data: data from an auxiliary human, machine or environmental system indirectly linked with the main system (e.g.: a peri-data of a human-car driving can be "the tire/road friction coefficient is 0.3").

• Peri-information: information from an auxiliary human, machine or environmental system indirectly linked with the main system (e.g.: a peri-information of a human-car driving can be "it rains on the road").

A solution to bring more reliability into the system could be answered by a peri-consideration and a use of meta and peri data/information. These concepts are already defined in the computer science community as Context-Awareness [START_REF] Motta | Towards a more in-depth understanding of the iot paradigm and its challenges[END_REF], [START_REF] Perera | Context aware computing for the internet of things: A survey[END_REF] and in the humanmachine cooperation area as Situation-Awareness (Millot, 2013). Awareness of the system environment described in this paper encompasses these two types of consideration (technologies and humans) in the so-called Overall Situation Awareness.

Capture and Storage

Under uncertainty, understanding and controlling complex environments is only possible by means of information exchange between devices (C ¸aliş Uslu and Firat, 2021). For sure, meta-and peri-information are not always available and it may be challenging to collect them. To make rapid decisions, a network made up of smart devices is needed.

In a Holistic Digital Twin architecture, smart handheld devices, mobile phones or other enterprises systems are present and get peri-information which are transferred and updated consecutively into the digital space [START_REF] Wei | Is human digital twin possible? Computer Methods and Programs in Biomedicine Update[END_REF].

The model can analyze point-in-time data, historical data, and extract meta-information from these data. To this, additional technologies must be added that allow the capture of machine meta-data (consumption, temperature, speed, or modes of operation) as well as the operator metainformation, such as his/her tasks and work objectives. Finally, human data can be recorded through devices such as an eye tracker, heartbeat recorder, and camera, configured to establish principles of vigilance detection or emotional recognition. Such systems that are equipped with identification and sensing capabilities are the bridge from the physical to the virtual realm. Indeed, as time goes by, new situations and behaviors that could not be predicted are appearing. These technologies can be efficient ways to achieve a complete and well-designed Digital Twin taking into account each aspect of the system situation, the user behaviors and both surroundings.

The models and architecture of the system become more complex with these new inputs and technologies, which is why a unified data management infrastructure is needed. Data modeling represents the very foundation of the standardization process, providing reliable high-quality data that can be trusted. This data modeling work will lead to a standard for all data and information of the whole system. Similarly to a meta-data standard, a requirement is intended to establish a common understanding of the meaning or semantics of the data, to ensure correct and proper use and interpretation of the data by its owners and users ( Řezník et al., 2022). To achieve this common understanding, a number of characteristics or attributes of the data have to be defined (meta-data). Data modeling also enables one to make links between agents and assets.

Take the example of "a car in manual driving mode"; to this car is associated an underlying agent, which is "the driver", as well as a larger system which encompasses the two previous ones, which can be called a "driving environment". The principle of standardization will allow links to be created between each agent so that in a given "driving environment" there are one or more "cars" and in each car there is a given "driver". How the data are stored is just as important as how they are captured. If data get lost, misplaced, and the query system is too slow, the use of this meta-and peri-information can bring more harm than good, especially in a high-risk context. Hence the importance of a data management system.

USE CASE OF A HUMAN-CAR DRIVING SITUATION

Let us start our discussion with a modeling work done in [START_REF] Vanderhaegen | Towards increased systems resilience: New challenges based on dissonance control for human reliability in cyber-physical and human systems[END_REF] studies and experiments. Driving is a relatively complex Human-Machine System (HMS) with various Human-Machine Interactions (HMI). The realization of a Digital Twin of this system requires one to take into account a model of the vehicle as well as a second one representing the driver and the interactions between both. Firstly, a conceptual model of a human-car driving was established, then all the rules that govern the driving of a vehicle by a human were listed to form a driving model, see Figure 1.

After validation by a panel of 43 car drivers, these rules of human driving behavior are set up in the human model of a Human-Car Digital Model. A DM (Digital Model) is different from a DT by the fact that it has no direct link with a real and physical system [START_REF] Ascone | Towards a holistic framework for digital twins of human-machine systems[END_REF]. The models can be identical in both configurations but with DM the data are simulated. Simulation tools such as Digital Models can then provide model designer (as human drivers) with pedagogical feedback to support the dissonance control process of a DT [START_REF] Vanderhaegen | Pedagogical learning supports based on human-systems inclusion applied to rail flow control[END_REF]. For example, related to rules R3 and R7 of the Digital Twin model in Figure 1, a new high risk was observed. Indeed, since a tricolor is green, the car driver model will not stop and continue its route (i.e., application of the rule R3). However, the car driver model can decide also to stop before another car in front of them in order to avoid a collision (i.e., application of the rule R7). The application of rules R3 and R7 relates to normal behavior under normal conditions. Such a dynamic situation was simulated by the MissRail Digital Model platform of the Polytechnic University of Hauts-de-France (UPHF, Valenciennes), Figure 2. In the first step, the car moves next to a tram line and behind another car, respecting the safety distance. In the second step, the Human-Car simulation approaches the tricolor signal that is green. Finally, in Step 3, the digital driver stops the car to avoid a frontal collision with another car. However, the digital user stops the car on rail tracks, and then a potential lateral collision between the car and a train can occur.

Fig. 2. Pictures extract from the Missrail Experiment

It should be noted that even models designed by experts and validated by a large panel of experienced users can lead to risky situations in specific conditions. Rule-based models seem to be theoretically efficient and understandable for users, but they do not allow one to take into account all the possible and conceivable situations that can be encountered in our world. Furthermore, the systems to be controlled should no longer be segregated and an allinclusive design should be established within the design community of Cyber-Physical&Human System. Another difference between DM and DT is the possibility for DT to adapt themselves and act according to the "whole" environment. One of the characteristics of holistic and inclusive design is then the precious Context Awareness ability. In practice, models rarely require or significantly encourage the inclusion of peri-information and meta-data [START_REF] Guarino | Towards tramway safety by managing advanced driver assistance systems depending on grades of automation[END_REF]. Peri-information is not always available in the incoming data stream for these models and may need to be obtained separately, either through specific requests or additional computation. Incorporating the proposed Overall Situation Awareness concept in models represents clearly a significant challenge and will allow one to make Digital Twin more inclusive and holistic.

PREVIOUS CAP CONCEPT AND IMPROVEMENTS

Regardless of the type of cooperation modelling, zero risk cannot exist in various and uncertain environments. A different way of design consist to add decision making modules to counteract the risky situations (Millot, 2013) . Indeed, it is more difficult to take contextual factors into account upstream the use, i.e. during the design of the model. In that case, it would be necessary to model all the rules including all possible peri-information ; which is almost impossible or at least would make the model totally complex and impossible to manage, recondition or modify. The easiest way to integrate peri-information into the system architecture is still to add a separate decision making module downstream the system model which will allow the risk factors of the situation to be calculated. A such complete system that can update its model and synchronise itself is assimilated to autonomous system such as Holistic Digital Twins [START_REF] Ascone | Towards a holistic framework for digital twins of human-machine systems[END_REF].

When the Human-Machine System is an autonomous system, the autonomous model defined by [START_REF] Vanderhaegen | Toward a petri net based model to control conflicts of autonomy between cyber-physical & human-systems[END_REF] can be used and adapted thanks to the three minimum parameters: system Competence, system Availability and system Possibility/Prescription. The so-called Competence-Availability-Possibility (CAP) model is then proposed to represent the capability of an HMS to act alone and to control conflicts of autonomy in terms of competence, availability and/or possibility. The competence or capacity of an agent to act (human or machine) depends essentially on its designed functions and its use through time (wear, breakdown, breakage...). In terms of availability and the possibility to act, several questions arise in this kind of design. How to measure the availability of an agent (especially when it is about a human)? Then, how do we identify the possibility to act of an agent on a given action? An evaluation of the surroundings (peri-data and peri-information) can be very useful to answer this challenge. Based on the CAP modeling principle, the idea is now to define dynamically levels in terms of capacity, availability and possibility, thanks to an awareness of the internals ("meta") and surroundings ("peri") information of each asset. Therefore, due to the capture of metadata and the peri-data recorded in the asset environment, it is necessary to define the awareness levels of each of the three parameters (C,A,P) to adjust the control of the whole Cyber-Physical&Human System (Figure 3). An architecture presenting such a device will allow to assess agents and to assign tasks in real-time according to these evaluation.

Fig. 3. Overall Situation Awareness Levels Architecture

The old CAP method allowed to assess the collaborative system and to highlight the possible dissonances after an undesired event and in a manual way (by a human). This new architecture will allow a dynamic (automatic and online) evaluation of the 3 parameters in order to set C, A, P levels and define an overall awareness of the situation and its context. For the whole system these three criterion can be calculated dynamically (at each instant t ).

First of all, a list has to be made with all the factors (data, meta-data, peri-data) and events (information, metainformation, peri-information) that respectively compose and affect the system and subsystems, see table 1. For each factor, a degree or level of importance must be defined, it correspond to the gravity level of an event. and then associated with the criteria it affects. Once the list is made, a method of and calculation of risk must be put in place. A risk factor of occurrence of an event is defined by its occurrence and its severity ( RF = O * G). Severity or gravity of an factor corresponds to the impact it has on the system (degree column in the previous list). Whether it is a label, as 'high' or 'low', or a numeric value, as a temperature, the value must be normalized and restricted to a given interval. The same is true for its probability of occurrence, so that the product RF (t) is between 0 and 1 . If the RF value equals 0, it means that there is no influence on the system, in contrast, a value of 1 corresponds to an inability to control the system during that event. A weighted sum of all the factors is then done at each instant t to obtain the time Overall Situation Awareness (OSA) level of the system:

OSA(t) = N c n=1 RF c(t) + N a n=1 RF a(t) + N p n=1 RF p(t)
Once the structure developed and each weights defined, the last part to achieve is the linkage of factors to data and information. Periodic or event-triggered, some other devices are then necessary to achieve a complete solution of HMS assessment.

A COMPLETE CONTEXT AWARENESS SOLUTION

In the previous example, it was the consideration of crossing a railway axis that would have allowed to change the outcome of the scenario. Once captured, this metainformation could have been used to set a larger stopping distance between the two vehicles, thus avoiding this risky situation. Consideration of contextual factors can often appear upstream of the design, during the test phases by simulation, i.e. using the MissRail tool. In each simulation of scenarios, new accidents and new dissonances can be simulated. According to these identified gaps, new rules can be established in the model, such as rule R100: 'If there is no place to pass through a railway axis, then do not pass.' or rule R101: "if there is a railway axis, then Increase the inter-distance by 10 meters.". In more complex models peri-data are fixed and have no flexibility or variability during a scenario. Models are adapted to work in a certain area. More simply, peri-data response times and efficiency can be improved by a real-time update of these data and thanks to a transformation of these previous parameters into real-time variables. There is a real need for dynamic evaluation of CAP levels. 2021) can help to realize such a system. The figure describes the same scenario as in section 3 "use case". The scenario is also split into images reflecting the 3 steps, but on which are identified in "green" the new contextual information that had not previously been detected without the aidsystem (unlike those in "red" that were). In Step 1, the system enable the identification of the tram infrastructure (C1) and a busy car traffic in the background (C2). These peri-information can be used by the intelligent system to notice the user, make him/her more vigilant and so increase its availability. But for the purposes of this article, such a tool can be used to detect objects, signs, etc., to finally affect the 'Occurrence' part of the respective risk factor RF . In Step 2 of Figure 4, 'the traffic light is green' (D1) was an information already perceived by the Digital Twin, but the system had not given importance to the ground and the presence of rail on it (C3). Preventing the car from getting on the tracks too quickly would have increased the vehicle's possibility to cope with the arrival of a tram. Another peripheral information, such as C4 (the roundabout sign), provides more information to the driver and affects his driving choices. For example, the driver should have anticipated and looked to see if the traffic in the roundabout would have allowed him to pass and not end up in that situation (D2: to avoid a frontal collision with another car, the user stops the car). To all this, one could add the identification of peri-information such as 'collision with an imminent tram' and 'no vehicle/object behind', which, when combined, would have allowed one to drive in reverse and avoid the collision.

Authors, think also to use other peripheral devices like camera, to observe and measure the availability of the driver during the driving phase. The concept of eyetracking already makes it proofs in academic and industrial areas and allow to diagnose the driver attention [START_REF] Carr | The role of eye tracking technology in assessing older driver safety[END_REF], [START_REF] Bendjoudi | Audio-visual and heart signals for attention and emotion analysis[END_REF]. We can push the design to it extreme and realize a complex human model of the driver. It is important to mention in this type of model that the orders of magnitude of human parameters are relatively variable in literature (La [START_REF] Delfa | Design of a cooperative eco-driving rail control system: an experimental study[END_REF]. If these parameters are so changeable, it is because they depend on the human being and other external factors. In a holistic system design with Context Awareness and Meta-Information on the human being, there should not be so many parameters but rather variables that are driven by factors such as capabilities or availability. This can be achieved through human data capture devices and on a register of user's meta-data and meta-information.

To complete the whole assessment system, a minimalist interface has been developed to display the results of the proposed C,A,P level technique (Figure 5).

Fig. 5. Context Awareness Levels Interface

The interface is easy to implement on a car dashboard, for instance. Tests and experiments are still to be done in order to confirm on a large public the necessity of a such interface in future Human-Machine Systems.

DISCUSSIONS

In the holistic architecture, taking a maximum of data into account makes it possible to define an integral awareness of the situation of an agent (Overall Situation Awareness).

The capacity factor can be influenced by many items ( [START_REF] Vanderhaegen | Cooperation and learning to increase the autonomy of adas[END_REF]. For example, the capacity of a human to perform a specific task can be assessed by integrating personal data into the models. Physical factors such as height, weight, corpulence (etc.) define operators' capacity to perform certain tasks (i.e., "grabbing an object at a height"). In hospitals, medical records are centralised and allow for more accurate and rapid diagnosis of a patient [START_REF] Wei | Is human digital twin possible? Computer Methods and Programs in Biomedicine Update[END_REF]. Taking into account an operator's disability (physical or psycho-motor) makes it possible to evaluate more effectively a human's ability to perform a given task.

Age is also a good factor to evaluate the reactivity of humans. All this requires the personification of human models, but also raises questions about the ethics of this data. Capacity of a machine is easier to capture because they are linked to design information, maintenance, damage (i.e., "capacity of a machine to pack 500 biscuits per hour ; feasible or not?"). Then, the availability factor can be assessed using different techniques. Availability can be simply answered by the question "Is the machine or human available to do that task?". The availability of a human or a machine to do a specific task can be done thanks to a camera to see if the operator is not work overloaded ("he/she is already doing something else") or thanks to asset meta-and peri-data, for instance, the state of machines' semaphores and machines availability information (maintenance state or process mode). Semaphores can be described as counters used to ensure synchronization between processes for shared resources. Finally, the prescription of an asset to perform a task can also be defined as the time at which an obligation cannot be continued. Therefore, it is a question of the notion of the possibility of doing a specific thing. For example, the possibility of a human to enter in a certain zone, or the permissions and requirements to hold an object (due to the temperature or if its a too much delicate thing). In terms of machines, it is more an authorization of an action (i.e., "Is the car allowed to activate the horn by itself in a certain situation?").

Unfortunately, it seems that simple rule-based systems are no longer sufficient on their own to maintain a requirement such as safety in decision-making of risky systems. There are artificial neural networks and other machine learning techniques that enable Artificial Intelligence by implementing an algorithm that trains a machine on how to learn by itself to adapt upcoming changes without human intervention. In that sense, these new rules can also be created and used by the system itself completely automatically, without the need for designer confirmation. One could even imagine having a very simple model and training it via simulations and by defining the operating, safety, comfort, economic (etc.) objectives alone. With this method, it is possible to carry out a large number of tests to identify design flaws and improve the design before deployment. However, the diversity of scenarios is limited only by the imagination of those who create them. That means, it is highly impossible to model all possible risk scenarios and therefore the model may not know what to do in an unknown situation. Whatever the performance of such techniques, there is always a doubt about the trust that can be placed in them, especially for system control.

In fact, the notion of "without human intervention" instills fear in all communities, at the ethical, political, or safety level [START_REF] Anderson | Why is ai so scary?[END_REF].

CONCLUSION

In the present paper, authors brought solutions that can improve models of Human-Machine Systems, such as the use of human data, meta-data and peri-data. They defined a state of the Context Awareness around the machine and the human that can be used in the decision making or even models. This principle is reminiscent to the Holistic Framework where a decision can be made only if the context of the process and the situation of the operator in it are well known. The Context Awareness impacts on the possibilities and availability of an agent to act. Meta-information is more present to modify the ability of an asset to act on a certain task. Additional systems as Decision Maker are efficient to keep safe systems.

The proposed decision technique can also be augmented by a tool which will enable the identification of human behaviors and feelings such as the works of [START_REF] Jegham | Safe Driving : Driver Action Recognition using SURF Keypoints[END_REF] or [START_REF] Bendjoudi | Audio-visual and heart signals for attention and emotion analysis[END_REF]. The human meta-data recorded and the car meta-information available can then allow the variation of skill and capacity level. In general, it is certain that the vehicle's maintenance booklet will influence the capacity level of the asset, as well as the history of accidents that can modify the car's functions. CAP levels can be a good way to take all factors into account and sum up the context and situation states of a system in operation. With all these techniques combined, it is certain that a significant number of peri-information and human-information will be captured. They can then be used by the decision making system directly or via an interface for the user to be defined. Taking into account the environment and context in visual representations of system environments will improve realism, reliability, and situational awareness on the user side too. These elements are added to the models to achieve the overall holistic architecture for Digital Twins.

Fig. 1 .

 1 Fig. 1. Frame of reference of the human car driving control

Fig. 4 .

 4 Fig. 4. Pictures extract from the Missrail Experiment with the Context-Identification tool The example in the following figure shows a system for identifying peri-information in real time through an image recognition process based on artificial intelligence techniques such as Deep Learning (Figure 4). Work such as Maaloul (2018), Lebrun et al. (2013), or Habib et al. (2021) can help to realize such a system. The figure describes the same scenario as in section 3 "use case". The scenario is also split into images reflecting the 3 steps, but on which are identified in "green" the new contextual information that had not previously been detected without the aidsystem (unlike those in "red" that were). In Step 1, the system enable the identification of the tram infrastructure (C1) and a busy car traffic in the background (C2). These peri-information can be used by the intelligent system to notice the user, make him/her more vigilant and so

Table 1 .

 1 Example of Factors Definition List

	Label	Info	Criteria	Degree
	F1	info1	C	low or 0.04
	F2	info2	A	normal or 0.15
	F3	info2	P	high or 0.36
	...	...	...	...


Felix Ali Mehmeti
email: felix.ali-mehmeti@uphf.fr

Kaïs Ammari
email: kais.ammari@fsm.rnu.tn

Serge Nicaise
email: serge.nicaise@uphf.fr

Serge Nicaise Dispersive Effects

Ali Felix

Ka Ïs Mehmeti

AND Serge Ammari

Nicaise

DISPERSIVE EFFECTS FOR THE SCHR ÖDINGER EQUATION ON FINITE METRIC GRAPHS WITH INFINITE ENDS

Keywords: Mathematics Subject Classification. 34B45, 47A60, 34L25, 35B20, 35B40 Dispersive estimate, Schrödinger operator, connected graph. 1

whether they are published or not. The documents may come

Introduction

A characteristic feature of the Schrödinger equation is the loss of the localization of wave packets during evolution, the dispersion. This effect can be measured by L ∞ -time decay, which implies a spreading out of the solutions, due to the time invariance of the L 2 -norm. The well known fact that the free Schrödinger group in R n considered as an operator family from L 1 to L ∞ decays exactly as c • t -n/2 follows easily from the explicit knowledge of the kernel of this group [24, p. 60].

In this paper we derive analogous L ∞ -time decay estimates for the Schrödinger equation on some connected graphs.

Let us now introduce some notations which will be used throughout the rest of the paper.

R is a connected graph made of edges e k , k ∈ I = I B ∪ I U , where I B , I U are finite sets of N (the set of positive integers) and I U is non empty. R B is the compact core of R, i.e., the subgraph of R made of the edges e k , k ∈ I B . R U = R \ R B corresponds to the non-compact part of R that is made of a finite number of infinite star graphs, edges of R U are called outer edges.

For all k ∈ I B , e k has a finite length ℓ k , hence e k is identified with (0, ℓ k ).

For all k ∈ I U , e k has a infinite length, hence e k is identified with (0, +∞).

V is the set of vertices of R.

For all v ∈ V , I v = set of edges having v as vertex.

V ext is the set of exterior vertices of R (#I v = 1), while V int is the set of interior vertices of R (#I v ≥ 2).

Finally let V connecting be the set of connecting vertices of R, namely the subset of v ∈ V int such that I v ∩ I U ̸ = ∅. Without loss of generality, we can assume that for any edge e ∈ V connecting , #I v ≥ 3.

By convention, the vertices of V int remain interior vertices of R B , hence the set of exterior vertices of R B is V ext . An example of a such a graph is presented in Figure 1, there vertices are represented by a dot, and V ext = {v 1 , v 2 }, V int = {v 3 , v 4 , v 5 , v 6 }, V connecting = {v 4 , v 5 , v 6 }. Consequently, R B is made of edges joining v 1 v 6 , v 6 v 5 , v 5 v 4 , v 5 v 3 , v 3 v 4 and v 3 v 2 (hence #I B = 6), and R B is made of four half-lines (one emanetating from v 4 , two emanetating from v 5 and one emanetating from v 6 ).

• • • • • • v 1 v 2 v 3 v 4 v 5 v 6
Introduce H = L 2 (R) = ⊕ k∈I L 2 (e k ) the Hilbert space direct sum of the L 2 (e k ), k ∈ I with inner product

((u k ), (v k )) H = k∈I (u k , v k ) L 2 (e k ) .
The present paper is organized as follows. The kernel of the resolvent needed for the proof of Theorem 3.1, is given in section 2. The calculation of the resolvent is equivalent to the resolution of a linear system which describes the impact of the transmission conditions on the coefficients of the scattering matrix. In the general setting of this section we assume that determinant of the matrix of this system factorizes in a factor whose zeros are the eigenvalues of the Hamilton operator and a nonzero factor together with hypotheses on the structure of the Cramer solution formula. These hyptheses imply the existence of a decomposition of the kernel of the resolvent into a meromorphic term in the whole complex plane with poles on the positive real axis and a term which is continuous on the real line but discontinuous when crossing it.

The poles are shown to be the eigenvalues of the operator, the continuous term creates the absolutely continuous spectrum. The absence of further terms proves the absence of a singular continuous spectrum. Note that in [START_REF] Cacciapuoti | Topology-induced bifurcations for the nonlinear Schrödinger equation on the tadpole graph[END_REF][START_REF] Noja | Bifurcations and stability of standing waves in the nonlinear Schrödinger equation on the tadpole graph[END_REF] the eigenvalues and eigenvectors are given without this detailed analysis of the rest of the spectrum; there the authors further consider the bifurcation of solutions of the nonlinear Schrödinger equation from the eigenvectors of the linear operator.

In section 3 we give the proof of the main result of the paper (Theorem 3.1). To this end we replace the formula of the resolvent into Stone's formula and prepare all terms for the application of the Lemma of van der Corput. For this section we assume that the second factor of the matrix mentioned above is bounded away from zero. This hypthesis has an impact on the geometry of the metric graph as can be seen in section 5. It implies that the inverse of the second factor is a quasiperiodic function and can thus be developed in a trigonometric series thanks to a theorem of harmonical analysis. This leads to a decomposition of the solution formula in terms which can be treated by the van der Corput Lemma without amplitude, implying a time decay-rate at least as t -1/2 . Section 4 is devoted to the analysis of the evolution of the local probability density in terms of the probability flow. The Schrödinger group is a group of isometries. This is linked to the interprétation of |u| 2 as the probability density of the position of the particle. Due to this isometry property, probability behaves as an incompressible fluid, which leads to the possibility to define a probability flow. Using this idea, we prove that the rate of change of the probability of the position of the particle to be in the sub-graph R B made of the bounded edges is equal to i times the flow from R B to its complement R U . This type of equation is in general called continuity equation (for the basic concept see [START_REF] Likharev | Essential Graduate Physics -Quantum Mechanics[END_REF], section 1.4). It allows us to prove the boundedness of the rate of change of the probability in R B . For the proof we use the fact that the values of the flow at the contact vertices between R B and R U can be controlled by the H 2 -norm of the solution on R U , using the Sobolev embedding H 2 (e k ) ⊂ C 1 (e k ). Using similar arguments we prove that the rate of change of the probability in R B decays at least as t -1

In the last section we consider several special metric graphs with semi-infinite ends. These examples show that our hypotheses to obtain the time decay-rate at least as t -1/2 include the possibility of circles (made of one or several edges). In the study of the Y-graph and the tadpole with two heads we show that our hypotheses for time-decay require that the two bounded edges have commensurable lengths. For the tadpole with two heads we find nice fomulas in the resolvent and the solution on the unbounded edge, which are valid regardless the ratio of the lenghts of the two bounded edges. Note that the solution formula includes all frequencies up to infinity, a result which is inaccessible in the general context of sections 2 and 3.

Kernel of the resolvent

Given z ∈ C + := {z ∈ C : ℑz > 0} and f ∈ L 2 (R), we are looking for u ∈ D(H) solution of (2.1) ∆ R u + z 2 u = f in R.
Hence as in [9, § 3.2] we look for u in the form

u k (x) = c k e -izx + d k e izx (2.2) + 1 2iz ℓ k 0 e iz|x-y| f k (y) dy, ∀x ∈ (0, ℓ k ), k ∈ I B , u k (x) = d k e izx (2.3) + 1 2iz ∞ 0 e iz|x-y| f k (y) dy, ∀x ∈ (0, +∞), k ∈ I U ,
where f k is the restriction of f to the edge e k ; c k , d k are constants (that depend on z and f ) fixed in order to satisfy the transmission conditions. Indeed from these expansions, we clearly see that for any k ∈ I, one has:

u ′′ k + z 2 u k = f k in e k . Now we see that the transmission conditions are equivalent to (2.4) D(z)((c k , d k ) k∈I B , (d k ) k∈I U ) ⊤ = b(z, f ),
where D(z) is a square matrix of size N = 2#I B + #I U whose entries are linear combinations of 1 and of e ±izℓ k (with k ∈ I B ) with coefficients independent of z and the right-hand side b(z, f ) is a column of size N whose entries are linear combinations of

1 z ∞ 0 e izy f k (y) dy, ∀k ∈ I U ,
and of 1 z

ℓ k 0 e izy f k (y) dy, 1 z ℓ k 0 e iz(ℓ k -y) f k (y) dy, ∀k ∈ I B ,
still with coefficients independent of z. Hence a solution to (2.4) exists if and only if the determinant of D(z) is different from zero. According to Theorem 3.2 of [START_REF] Exner | Equivalence of resolvent and scattering resonances on quantum graphs[END_REF], the zeroes of D(z) correspond to the scattering resonances of the graph (see [START_REF] Berkolaiko | Introduction to quantum graphs[END_REF][START_REF] De Verdìère | Topological Resonances on Quantum Graphs[END_REF]), which are situated in ℑz ≤ 0. Nevertheless, let us analyze this determinant, for any z ∈ C + ∪ R * , with R * = R \ {0}. For that purpose, we introduce a subset S d of σ(-∆ Dir R B ) defined as the set of λ ∈ σ(-∆ Dir R B ) such that there exists an eigenvector φ ∈ D(∆ Dir R B ) such that (2.5)

φ(v) = 0, ∀v ∈ V connecting .
As V connecting is not empty, we deduce that 0 cannot be in S d .

For future uses, if S d is non empty, we denote by S d := {λ 2 m } m∈N , with λ m > 0, where λ 2 m is repeated according to its multiplicity (corresponding to the number of associated eigenvectors φ satisfying (2.5)). For all m, we then denote by φ (m) the eigenvector of -∆ Dir R B associated with λ 2 m (and satisfying (2.5)). Proof. As usual, det D(z) = 0 if and only if the homogeneous system associated with (2.4) has a non-trivial solution, hence if and only if a non-trivial solution u of (2.1) in the form (2.2)-(2.3) with f = 0 exists. In this case, as u satisfies u ′′ k + z 2 u k = 0 in e k , ∀k ∈ I, by multiplying this identity by ūk , integration by troncating the infinite edges up to a positive real number R, we have

k∈I B e k (u ′′ k + z 2 u k )ū k dx + k∈I U R 0 (u ′′ k + z 2 u k )ū k dx = 0.
Hence by integration by parts and using the transmission conditions, we find that

k∈I B e k (|u ′ k | 2 -z 2 |u k | 2 ) dx + k∈I U R 0 (|u ′ k | 2 -z 2 |u k | 2 ) dx - k∈I U u ′ k (R)ū k (R) = 0.
But due to (2.3) with f = 0 we have u ′ k = izu k , ∀k ∈ I U , and therefore the previous identity becomes

k∈I B e k (|u ′ k | 2 -z 2 |u k | 2 ) dx + k∈I U R 0 (|z| 2 -z 2 )|u k | 2 dx -iz k∈I U |u k (R)| 2 = 0.
Writting z = a + ib with b ≥ 0, we get equivalently

k∈I B e k (|u ′ k | 2 + (b 2 -a 2 -2iab)|u k | 2 ) dx (2.6) +2(b 2 -iab) k∈I U |d k | 2 R 0 e -2bx dx -i(a + ib) k∈I U |d k | 2 e -2bR = 0.
If a = 0, then taking the real part of this identity, we get

k∈I B e k (|u ′ k | 2 + b 2 |u k | 2 ) dx +2b 2 k∈I U |d k | 2 R 0 e -2bx dx + b k∈I U |d k | 2 e -2bR = 0.
And since b > 0, we deduce that u is zero, which is impossible.

If a ̸ = 0, by taking the imaginary part of the identity (2.6), we find

d k = 0, ∀k ∈ I U .
This means that u is identically equal to zero on the outer edges of the graph, and hence z 2 = a 2 is an eigenvalue of the Laplace operator on the graph R B with Dirichlet boundary conditions at all exterior vertices v ∈ V ext with eigenvector u. In other words, z 2 belongs to the set of eigenvalues σ(-∆ Dir R B ). Since additionally, u = 0 on the outer edges of the graph, u is zero at the vertices v ∈ I connecting , and therefore z 2 belongs to S d . □ Lemma 2.1 suggests that the point spectrum corresponds to S d , this is indeed the case.

2.2.

Corollary. With S d introduced before, it holds

σ pp (H) = S d .
Proof. From the previous Lemma, any element z 2 of S d belongs to σ pp (H). On the contrary let λ > 0 be an eigenvalue of H of eigenvector u, then for any k ∈ I, u k ∈ L 2 (e k ) is in the form

u k (x) = c k e -izx + d k e izx , ∀x ∈ e k ,
with z = √ λ and c k , d k are complex number fixed to satisfy the transmission conditions. From the L 2 property of u k , k ∈ I U , we deduce that (2.9)

c k = d k = 0, ∀k ∈ I U .
Indeed for any R > 0, direct calculations yield

R 0 |c k e -izx + d k e izx | 2 dx = R(|c k | 2 + |d k | 2 ) + 2ℜ c k dk 1 2iz (e 2izR -1) .
Hence by Cauchy-Schwarz's and Young's inequalities, we get

R 0 |c k e -izx + d k e izx | 2 dx ≥ R - 1 z |c k | 2 + |d k | 2 .
Letting R goes to infinity, we find that (2.9) holds, since the left-hand side of this estimate is uniformly bounded in R. This implies that u is identically equal to zero on the infinite edges and the arguments of the proof of Lemma 2.1 imply that λ belongs to S d .

In the case λ = 0, u satisfying

u ′′ k = 0 on e k , ∀k ∈ I U , we get k∈I e k u ′′ k ūk dx = 0,
and by integration by parts and using the transmission conditions, we arrive at

k∈I e k |u ′ k | 2 dx = 0,
hence u is constant on each edge. By the L 2 property of u, we then deduce that

u k = 0 on e k , ∀k ∈ I U ,
and by the continuity property, we conclude that u is zero and therefore 0 is not an eigenvalue of H. □

This result is in accordance with Corollary 2.1 of [START_REF] De Verdìère | Topological Resonances on Quantum Graphs[END_REF].

The previous results also allow to deduce the kernel of the resolvent.

2.3. Theorem. Let f ∈ H. Then, for x ∈ R and z ∈ C such that ℑz > 0, recalling that R(z 2 , H) = (z 2 I -H) -1 , we have (2.10) [R(z 2 , H)f ](x) = R K(x, x ′ , z 2 )f (x ′ ) dx ′ ,
where, by setting d(z) = det D(z), the kernel K is defined as follows:

(2.11) K(x, y, z 2 ) = 1 2iz e iz|x-y| + 1 d(z) σ=±, τ =± c k,σ,τ (z)e iz(σx+τ y) , ∀x, y ∈ e k , k ∈ I B , (2.12) K(x, y, z 2 ) = 1 2izd(z) σ=± c k,k ′ ,σ (z)e iz(σx+y) , ∀x ∈ e k , y ∈ e k ′ , k ∈ I B , k ′ ∈ I U , (2.13) K(x, y, z 2 ) = 1 2iz e iz|x-y| + c k (z) d(z) e iz(x+y) , ∀x, y ∈ e k , k ∈ I U , (2.14) K(x, y, z 2 ) = 1 2izd(z) τ =± c k,k ′ ,τ (z)e iz(x+τ y) , ∀x ∈ e k y ∈ e k ′ , k ∈ I U , k ′ ∈ I B ,
where the functions c k,σ,τ , c k,k ′ ,σ , c k and c k,k ′ ,τ are linear combinations of 1 and of e ±izℓj (with j ∈ I B ) with coefficients independent of z but that may depend on k, k ′ , σ and τ (hence holomorphic functions).

Proof. By the previous Lemma, d(z) is different from zero for any z ∈ C such that ℑz > 0, therefore by solving the linear system (2.4), inserting the obtained values of c k and

d k in (2.2)- (2.
3) and using the form of the coefficients of the matrix D(z) and of the right-hand side b(z, f ), we arrive at (2.10) with kernels in the announced forms. □

As usual, to obtain the resolution of the identity of H, we want to use the limiting absorption principle that consists to pass to the limit in K(x, y, z 2 ) as ℑz goes to zero. But in view of the presence of the factor d(z) in the denominator of the kernels, this limit is a priori not allowed as d may have zeroes on the real line. The solution is to split up the kernel K(x, y, z 2 ) into a meromorphic part K p in the whole complex plane with poles at the points ±λ m and a part K c which is continuous on the real line but discontinuous when crossing it. To be able to do so, we make the following assumption, that are satisfied for the operator H on a large class of graphs R.

2.4.

Assumption. The determinant d(z) can be factorized in the form

(2.15) d(z) = d d (z)d c (z), ∀z ∈ C + ∪ R,
with two holomorphic functions d d , d c on C that are linear combinations of 1 and of e ±izℓ k (with k ∈ I B ) with coefficients independent of z and such that

d d (z) = 0 ⇔ z 2 ∈ S d ,
and therefore

d c (z) ̸ = 0, ∀z ∈ R.
There exist two matrix valued functions

M c : C + ∩ R * → C N ×N : z → M c (z), M p : C + ∩ R * → C N ×N : z → M p (z),
whose entries are linear combinations of 1 and of e ±izℓ k (with k ∈ I B ) with coefficients independent of z (hence holomorphic on C) such that the adjugate of D(z) admits the splitting

(2.16) adj(D(z)) = M p (z) + d d (z)M c (z), ∀z ∈ C + ∩ R * .
Further for all m ∈ N, there exists a matrix M m , such that

(2.17) D(z) -1 = 1 z -λ m M m + R m (z),
for all z in a neighborhood of λ m , with R m holomorphic at λ m .

As D(z) -1 = d(z) -1 adj(D(z)), (2.16) directly implies that D(z) -1 can be written as

(2.18) D(z) -1 = 1 d(z) M p (z) + 1 d c (z) M c (z), ∀z ∈ C + .
Note further that the assumption (2.17) implies in particular that

I = 1 z -λ m M m D(z) + R m (z)D(z),
for all z in a neighborhood of λ m and consequently passing to the limit in z → λ m , we find that

(2.19) M m D(λ m ) = 0.
With this assumption, we can then split up the ratios

c k,σ,τ (z) d(z) , c k,k ′ ,σ (z) d(z) , c k (z) d(z) and c k,k ′ ,τ (z) d(z)
into a continuous part and a meromorphic one:

c k,σ,τ (z) d(z) = c p k,σ,τ (z) d(z) + c c k,σ,τ (z) d c (z) , c k,k ′ ,σ (z) d(z) = c p k,k ′ ,σ (z) d(z) + c c k,k ′ ,σ (z) d c (z) , c k (z) d(z) = c p k (z) d(z) + c c k (z) d c (z) , c k,k ′ ,τ (z) d(z) = c p k,k ′ ,τ (z) d(z) + c c k,k ′ ,τ (z) d c (z) .
These splittings lead to the natural splitting of the kernel K into

(2.20) K(x, y, z 2 ) = K c (x, y, z 2 ) + K p (x, y, z 2 ),
where

(2.21) K c (x, y, z 2 ) = 1 2iz e iz[x-y] + σ=±, τ =± c c k,σ,τ (z) 2izd c (z) e iz(σx+τ y) , ∀x, y ∈ e k , k ∈ I, (2.22) K p (x, y, z 2 ) = σ=±, τ =± c p k,σ,τ (z) 2izd(z) e iz(σx+τ y) , ∀x, y ∈ e k , k ∈ I B , (2.23) K p (x, y, z 2 ) = σ=± c p k,k ′ ,σ (z) 2izd(z) e iz(σx+y) , ∀x ∈ e k , y ∈ e k ′ , k ∈ I B , k ′ ∈ I U , (2.24) K p (x, y, z 2 ) = c p k (z) 2izd(z) e iz(x+y) , ∀x, y ∈ e k , k ∈ I U , (2.25) K p (x, y, z 2 ) = τ =± c p k,k ′ ,τ (z) 2izd(z) e iz(x+τ y) , ∀x ∈ e k y ∈ e k ′ , k ∈ I U , k ′ ∈ I B .
Before going on, we want to show that some coefficients c p 's are zeroes. For that purposes, we first give a relationship between an orthogonality property of f with the range of the matrix D(λ m ).

2.5. Lemma. Let f be in L 2 (R) with a compact support. The system (2.4) with z = λ m , namely

(2.26) D(λ m )X m = b(λ m , f ),
admits a solution if and only if f is orthogonal to all eigenvectors φ of -∆ Dir R B associated with the eigenvalue λ 2 m .

Proof. As D(λ m ) is a square matrix, the system (2.26) has a solution if and

only if b(λ m , f ) is orthogonal to K m vectors Y ℓ , ℓ = 1, • • • K m
, where K m is the dimension of ker D(-λ m ), that here coincides with the multiplicity of λ 2 m (as eigenvalue of -∆ Dir R B with the additional nullity at the interior nodes belonging to the infinite edges). As f has a compact support, using the form of b(λ m , f ), this is equivalent to

(2.27) (f, ψ m ) R = 0, ∀m = 1, • • • , K m ,
for some ψ m ∈ L 2 loc (R). To show that the ψ m 's correspond to the eigenvectors of -∆ Dir R B associated with the eigenvalue λ 2 m , it suffices to notice that for any u ∈ D(H) with a compact support, the system (compare with (2.1)) ∆ R u + λ 2 m u = f in R has a solution with f defined as the left-hand side of this identity. Using the procedure of the beginning of this section, the system (2.26) has a solution and therefore ψ m satisfies (∆ R u + λ 2 m u, ψ m ) R = 0, for any u ∈ D(H) with a compact support. By taking different functions u, we deduce that ψ m satisfies the differential equation ψ ′′ m + λ m ψ m = 0, on all edges as well as the continuity and Kirchoff law at the interior nodes and Dirichlet bc at the exterior nodes. Hence our previous arguments show that indeed ψ m corresponds to the eigenvectors of -∆ Dir R B associated with the eigenvalue λ 2 m . □ 2.6. Corollary. Let f ∈ L 2 (R) with a compact support be orthogonal to all eigenvectors φ of -∆ Dir R B associated with the eigenvalue λ 2 m . Then the solution

X(z) = ((c k (z), d k (z)) k∈I B , (d k (z)) k∈I U ) ⊤ of (2.4) remains bounded in a neighborhood of λ m , namely ∥X(z)∥ 2 ≤ C∥f ∥ L 1 (R) ,
with a positive constant C independent of f but depending on λ m .

Proof. Recalling that X m is a solution of (2.26), we notice that

D(z) -1 b(λ m , f ) = D(z) -1 D(λ m )X m .
Then using the assumption (2.17) as well as (2.19) we get

D(z) -1 b(λ m , f ) = R m (z)D(λ m )X m , which means that D(z) -1 b(λ m , f ) is holomorphic at λ m .
Coming back to X(z), we may write

X(z) = D(z) -1 b(z, f ) = D(z) -1 b(λ m , f ) + D(z) -1 (b(z, f ) -b(λ m , f )),
and again using the assumption (2.17), we arrive at

X(z) = D(z) -1 b(λ m , f ) + R m (z)(b(z, f ) -b(λ m , f )) + 1 z -λ m M m (b(z, f ) -b(λ m , f )). Since b(z, f ) is meromorphic at λ m , the ratio b(z, f ) -b(λ m , f ) z -λ m
admits a limit as z goes to λ m , hence X(z) as well. □ 2.7. Corollary. We have c p k = 0, ∀k ∈ I U . as well as

c p k0,k,σ = 0, ∀k 0 ∈ I B , k ∈ I U , σ = ±.
Proof. Fix k ∈ I U and take an arbitrary function φ ∈ D(0, ∞) and define f on R as equal to φ on e k and zero elsewhere. As f clearly satifies the assumption of the previous corollary, we have

e k c p k (z) d(z) e iz(x+y) f (y) dy ≤ C∥φ∥ L 1 (0,∞) ,
for all z in a neighborhood of λ m , with a positive constant C independent of φ but depending on λ m . Hence for a fixed y 0 ∈ e k , by taking a sequence of φ n such that

φ n → δ y0 in D ′ (0, ∞),
and such that

∞ 0 φ n (y) dy = 1, we find that c p k (z) d(z) e iz(x+y0) ≤ C,
for all z in a neighborhood of λ m . This obviously implies that c p k (z) is zero near λ m . The same argument can be used for the c p k0,k,σ 's. □ 2.8. Corollary. We have

c p k,k ′ ,τ = 0, ∀k ∈ I U , k ′ ∈ I B , τ = ±.
Proof. Direct consequence of the symmetry

K(x, x ′ , z 2 ) = K(x ′ , x, z 2 ),
and of the previous Corollary. □

With these results, we are able to give the expression of the resolution of the identity E of H.

2.9. Theorem. Let the assumption 2.4 be satisfied. Take f, g ∈ H with a compact support and let 0 < a < b < +∞. Then for any holomorphic function h on the complex plane, we have

(h(H)E(a, b)f, g) H = - 1 π R (a,b) h(λ) R f (x ′ )ℑK c (x, x ′ , λ) dx ′ dλ ḡ(x)dx + m∈N:a<λ 2 m <b h(λ 2 m )(f, ψ m ) H (ψ m , g) H , (2.28)
where for all λ > 0, K c (x, x ′ , λ) was defined before.

Proof. First by Stone's formula, we have (see for instance Lemma 3.13 of [START_REF] Mehmeti | Spectral Theory and L ∞ -time Decay Estimates for Klein-Gordon Equations on Two Half Axes with Transmission: the Tunnel Effect[END_REF] or Proposition 4.5 of [START_REF] Ali Mehmeti | Multiple tunnel effect for dispersive waves on a star-shaped network: an explicit formula for the spectral representation[END_REF])

(h(H)E(a, b)f, g) H = lim δ→0 lim ε→0 1 2iπ b-δ a+δ [h(λ)R(λ -iε, H) -R(λ + iε, H)] dλf, g H .
First using (2.20), we can write

(h(H)E(a, b)f, g) H = I c + I p ,
where

I c = lim δ→0 lim ε→0 1 2iπ b-δ a+δ h(λ) [R c (λ -iε, H) -R c (λ + iε, H)] dλf, g H , I p = lim δ→0 lim ε→0 1 2iπ b-δ a+δ h(λ) [R p (λ -iε, H) -R p (λ + iε, H)] dλf, g H .
where R p (resp. R c ) is the operator corresponding to the kernel K p (resp. K c ).

As R c (λ -iε, H) = R c (λ + iε, H), we can write

I c = - 1 π lim δ→0 lim ε→0 b-δ a+δ h(λ)ℑR c (λ + iε, H)dλf, g H = - 1 π lim δ→0 lim ε→0 b-δ a+δ h(λ)(ℑR c (λ + iε, H)f, g) H dλ = - 1 π lim δ→0 lim ε→0 b-δ a+δ h(λ) R R ℑK c (x, x ′ , λ + iε)f (x ′ )dx ′ g(x)dxdλ.
At this stage, we take advantage of Theorem 2.3 and (2.20). First by (2.11) to (2.14) and by (2.20), we see that

K c (x, y, λ + iε) -→ K c (x, y, λ), as ε -→ 0.
Furthermore we see that

|K c (x, y, λ + iε)| ≤ C |λ + iε| , ∀λ ∈ (a, b), x, y ∈ R,
for x, y ∈ R for some positive constant C independent of x, y. This allows to pass to the limit in ε -→ 0 by using the convergence dominated theorem to obtain that

lim δ→0 lim ε→0 b-δ a+δ h(λ) R R ℑK c (x, x ′ , λ + iε)f (x ′ )dx ′ g(x)dxdλ = b a h(λ) R R ℑK c (x, x ′ , λ)f (x ′ )dx ′ g(x)dxdλ.
Hence it remains to manage the term I p . As before we can write

I p = 1 2iπ lim δ→0 lim ε→0 b-δ a+δ R B R B h(λ)(K p (x, x ′ , λ -iε) -K p (x, x ′ , λ + iε))f (x ′ )dx ′ g(x)dxdλ.
First by Lemma 2.11 below, we can show that

lim ε→0 b-δ a+δ R B R B h(λ)(K p (x, x ′ , λ -iε) -K p (x, x ′ , λ + iε))f (x ′ )dx ′ g(x)dxdλ = lim ε→0 b-δ a+δ R B R B (h(λ -iε)K p (x, x ′ , λ -iε) -h(λ + iε)K p (x, x ′ , λ + iε))f (x ′ )dx ′ g(x)dxdλ.
Now, for a fixed δ > 0, we can always assume that a + δ and b -δ are always different from λ 2k and therefore

lim ε→0 ε -ε R B R B h(c δ + iy)K p (x, x ′ , c δ + iy)f (x ′ )dx ′ g(x)dxdy = 0,
for c δ = a + δ or b -δ. Consequently if we define the contour C ε,δ by the lines λ -iε, λ + iε with λ ∈ (a + δ, b -δ) and a + δ + iy, b -δ + iy, with y ∈ (-ε, ε), we deduce that

I p = -lim δ→0 lim ε→0 1 2iπ C ε,δ R B R B h(λ)K p (x, x ′ , λ)f (x ′ )dx ′ g(x)dxdλ.
Now recalling the expression (2.22), we perform the change of variable µ = √ λ, that leads to

- 1 2iπ C ε,δ R B R B h(λ)K p (x, x ′ , λ)f (x ′ )dx ′ g(x)dxdλ = 1 2π D ε,δ h(µ 2 ) k∈I B ,τ =±, σ=± c p k,σ,τ (µ) d(µ) e k e k e iµ(σx+τ x ′ ) f (x ′ )dx ′ g(x)dxdµ,
where

D ε,δ = { √ λ : λ ∈ C ε,δ
} is another contour in the complex plane. At this stage we can apply residue theorem and deduce, after simple calculations of the residues, that

- 1 2iπ C ε,δ R B R B h(λ)K p (x, x ′ , λ)f (x ′ )dx ′ g(x)dxdλ = m∈N:a+δ<λ 2 m <b-δ h(λ 2 m ) k∈I B ,τ =±, σ=± c p k,σ,τ (µ) d(µ) e k f (x ′ ) e iλmτ x ′ dx ′ e k g(x) e iλmσx dx = m∈N:a+δ<λ 2 m <b-δ h(λ 2 m )(f, ψ m ) H (ψ m , g) H .
This proves the result by passing to the limit in δ → 0. □ As direct consequences of the previous Theorem, we get the next results.

2.10. Corollary. If the assumption 2.4 holds, then the operator H has no singular spectrum and

P ac f = f -P pp f, ∀f ∈ H,
where

P pp f = m∈N (f, ψ m ) H ψ m .
2.11. Lemma. Under the assumption of the previous theorem, one has

lim ε→0 b-δ a+δ R B R B h(λ)K p (x, x ′ , λ ± iε)f (x ′ )dx ′ g(x)dxdλ = lim ε→0 b-δ a+δ R B R B h(λ ± iε)K p (x, x ′ , λ ± iε)f (x ′ )dx ′ g(x)dxdλ.
Proof. The proof is based on the use of the Lebesgue's convergence theorem. Let us prove it in the case λ + iε.

Writing b-δ a+δ h(λ)K p (x, x ′ , λ + iε)dλ = b-δ a+δ (h(λ) -h(λ + iε))K p (x, x ′ , λ + iε)dλ + b-δ a+δ h(λ + iε)K p (x, x ′ , λ + iε)dλ,
we only need to show that

lim ε→0 b-δ a+δ R B R B (h(λ) -h(λ + iε))K p (x, x ′ , λ + iε)f (x ′ )dx ′ g(x)dxdλ = 0.
Since for any λ ̸ = λ 2 m , one has (h(λ) -h(λ + iε))K p (x, x ′ , λ + iε) → 0, as ε → 0, to apply Lebesgue's convergence theorem, it suffices, for instance, to show that (h(λ) -h(λ + iε))K p (x, x ′ , λ + iε) is uniformly bounded (in ε). But in view of the definition (2.22) of K p , we only need to estimate the ratio

q(λ, ε) := h(λ) -h(λ + iε) d( √ λ + iε) .
Now using a Taylor expansion, we can say that for z small enough, say |z| < η, we have

d( λ 2 m + z) = d ′ (λ m ) 2λ m z + o(z).
Hence applying this property to λ -λ 2 m + iε for one m ∈ N, we find that

d( √ λ + iε) ∼ d ′ (λ m ) 2λ m (λ -λ 2 m + iε), for (λ -λ 2 m ) 2 + ε 2 < η 2 . Hence for |λ -λ 2 m | < η 2 and ε < η 2 , we deduce that d √ λ + iε ∼ 1 2λ m |λ -λ 2 m + iε| ≥ ε 2λ m .
Since h is holomorphic, we clearly have

|h(λ) -h(λ + iε)| ≤ Cε, ∀λ ∈ [a, b],
for some C > 0 (independent of λ and ε). Therefore for any λ such that |λ -λ 2 m | < η 2 for some m ∈ N, and for any ε < η 2 , one has

|q(λ, ε)| ≤ C ′ ,
for some C ′ > 0 (independent of λ and ε).

It then remains to treat the case where |λ -λ 2 m | ≥ η 2 for all m ∈ N. But in that case, as

√ λ + iε -λ m = λ + iε -λ 2 m √ λ + iε -λ m ,
for ε small enough, we get

| √ λ + iε -λ m | ≥ C"η,
for some C" > 0 (independent of λ and ε). This implies that

d λ + iε) ≥ δ,
for some δ > 0 (independent of λ and ε) and again implies that q(λ, ε) is uniformly bounded in that case as well.

□

3. The dispersive estimates 3.1. Theorem. In addition to the assumption 2.4, suppose that there exists a positive constant α such that

(3.29) |d c (µ)| ≥ α, ∀µ ∈ R.
Then there a positive constant C such that for all t ̸ = 0,

(3.30) e -itH P ac L 1 (R)→L ∞ (R) ≤ C |t| -1/2 ,
where in a natural way

L 1 (R) = ⊕ k∈I L 1 (e k ), and
L ∞ (R) = ⊕ k∈I L ∞ (e k ).
Proof. For any 0 < a < b < ∞, by Theorem 2.9 for any x, y ∈ R, we have found the following expression for the kernel of the operator e -itH I (a,b) P ac :

+∞ 0 e -itλ I (a,b) (λ)E ac (dλ)(x, y) = - 1 π b a e -itλ I (a,b) (λ)ℑK c (x, y, λ)dλ,
and by the change of variables λ = µ 2 , we get

+∞ 0 e -itλ I (a,b) E ac (dλ)(x, y) = - 2 π √ b √ a e -itµ 2 I (a,b) (µ 2 )ℑK c (x, y, µ 2 )µ dµ.
Now recalling the definition of K c , we have to distinguish between the following cases:

(

1) If x, y ∈ R U , then 2iµK c (x, y, µ 2 ) = e iµ|x-y| + c c k (µ) d c (µ) e iµ(x+y) , k ∈ I U .
Hence in that case, we have to estimate

√ b
√ a e -itµ 2 e iµ|x-y| dµ , and

√ b √ a e -itµ 2 c c k (µ) d c (µ) e iµ(x+y) dµ .
The first term is directly estimated by van der Corput Lemma [26, Proposition 2, p. 332]:

√ b √ a e -itµ 2 e iµ|x-y| dµ ≤ 4 √ 2 √ t , ∀ t > 0.
For the second term, as c c k (µ) is a linear combination of 1 and e ±iµℓ k , k ∈ I B , with coefficients independent of µ, we are reduced to estimate This property guarantees that

√ b √ a e -itµ 2 1 d c (µ) e iµ(x+y+ℓ) dµ ≤ λ∈D |d λ | √ b
√ a e -itµ 2 e iµ(x+y+ℓ+λ) dµ .

Again this right-hand side is estimated by van der Corput Lemma to conclude that

√ b √ a e -itµ 2 1 d c (µ) e iµ(x+y+ℓ) dµ ≤ 4 √ 2 √ t λ∈D |d λ |.
All together we have proved that

(3.32) +∞ 0 e -itλ I (a,b) (λ)E ac (dλ)(x, y) ≤ C √ t , ∀ x, y ∈ R U , t > 0,
where C is a positive constant independent of x, y, t, a, b.

(2) If x ∈ R B and y ∈ R U , then 2iµK c (x, y, µ 2 ) = σ=± c c k,k ′ ,σ (µ) d c (µ) e iµ(σx+y) , k ∈ I B , k ′ ∈ I U .
The same arguments as before imply that

(3.33) +∞ 0 e -itλ I (a,b) (λ)E ac (dλ)(x, y) ≤ C √ t , ∀x ∈ R B , y ∈ R U , t > 0,
where C > 0 is independent of x, y, t, a, b.

(3) If x ∈ R U and y ∈ R B , then 2iµK c (x, y, µ 2 ) = τ =± c c k,k ′ ,τ (µ) d c (µ) e iµ(x+τ y) , k ∈ I U , k ′ ∈ I B .
We deduce as before that

(3.34) +∞ 0 e -itλ I (a,b) (λ)E ac (dλ)(x, y) ≤ C √ t , ∀x ∈ R U , y ∈ R B , t > 0,
where

C > 0 is independent of x, y, t, a, b. (4) If x, y ∈ R B we have 2iµK c (x, y, µ 2 ) = e iµ|x-y| + σ=±,τ =± c c k,σ,τ (µ) d c (µ) e iµ(σx+τ y) , k ∈ I B .
As before we deduce that

(3.35) +∞ 0 e -itλ I (a,b) (λ)E ac (dλ)(x, y) ≤ C √ t , ∀x, y ∈ R B , t > 0,
where C > 0 is independent of x, y, t, a, b.

The estimates (3.32) to (3.35) imply that for all

f ∈ L 2 (R) ∩ L 1 (R) (3.36) ∥e -itH I (a,b) (H)P ac f ∥ ∞ ≤ 2C √ t ∥f ∥ 1 , ∀t > 0,
where C > 0 is independent of t, a, b.

As e -itH I (a,b) (H)P ac f converges to e -itH P ac f in L 2 (R) as a → 0 and b → ∞, by extracting a subsequence, we have that e -itH I (a,b) (H)P ac f → e -itH P ac f a.e., and therefore (3.36) implies that

(3.37) ∥e -itH P ac f ∥ ∞ ≤ C √ t ∥f ∥ 1 , ∀t > 0,
where C > 0 is independent of t. By the density of L 2 (R) ∩ L 1 (R) into L 1 (R), we conclude that this estimate remains valid in the whole L 1 (R). □

Without the assumption (3.29), we can prove the same decay but for data in a fixed frequency band, namely we have the next result.

3.2.

Theorem. Let the assumption 2.4 be satisfied. Then for all 0 ≤ a < b < ∞ there a positive constant C (that may depend on b) such that for all t ̸ = 0,

(3.38) e -itH I (a,b) (H)P ac L 1 (R)→L ∞ (R) ≤ C |t| -1/2 .
Proof. The proof is the same as before except for terms like (3.31). Indeed in that case, we simply notice that d c (µ) is non zero on [0, b] and is smooth (in the variable µ), consequently by a variant of van der Corput Lemma [26, Corollary, p. 334], we have

√ b √ a e -itµ 2 1 d c (µ) e iµ(x+y+ℓ) dµ ≤ 4 √ 2 √ t 1 |d c ( √ b)| + √ b √ a |d ′ c (µ)| |d c (µ)| 2 dµ .

This directly implies that

√ b √ a e -itµ 2 1 d c (µ) e iµ(x+y+ℓ) dµ ≤ C √ t ,
with a positive constant C that may depend on b. □

Probability densities and probability flows

Let us start with the following definition.

Definition: If u(t) = e -itH u 0 , t ∈ R, with u 0 ∈ D(H), then for all k ∈ I, we define its probability density ρ k and probability flow j k by

ρ k = |u k | 2 , j k = u k u k,x -u k u k,x .
We first recall the continuity equations on each edge of the graph. 4.1. Lemma. Let u 0 ∈ D(H) and u(t) = e -itH u 0 , t ∈ R. Then we have

i d dt e k ρ k = i d dt ∥u k ∥ 2 L 2 (e k ) = j k (0) -j(l k ), ∀k ∈ I B , (4.39) i d dt e k ρ k = i d dt ∥u k ∥ 2 L 2 (e k ) = j k (0), ∀k ∈ I U . (4.40)
Proof. These identities as based on simple integration by parts. Namely as u k satisfies iu k,t = -u k,xx on e k , we have

i d dt ∥u k ∥ 2 L 2 (e k ) = e k (iu k,t u k -u k iu k,t ) = - e k (u k,xx u k -u k u k,xx ).
By an integration by parts, we directly find (4.39) and (4.40). □ 4.2. Theorem. Let u 0 ∈ D(H) and u(t) = e -itH u 0 , t ∈ R. Then we have

i d dt k∈I B e k ρ k = i d dt k∈I B ∥u k ∥ 2 L 2 (e k ) = - k∈I U j k (0). (4.41) Proof. As H is a selfajoint operator, ∥e -itH ∥ L(H) = 1, consequently k∈I ∥u k (t)∥ 2 L 2 (e k ) = k∈I ∥u 0,k ∥ 2 L 2 (e k ) , ∀t ∈ R.
This implies that

i d dt k∈I B ∥u k ∥ 2 L 2 (e k ) = -i d dt k∈I U ∥u k ∥ 2 L 2 (e k ) , (4.42)
and we conclude using (4.40). □

The previous theorem means that the time derivative of the square of the L 2 -norm (of the solution of the Schrödinger equation) on the bounded edges of the graph equals i k∈I U j k (0). In other words, the rate of change of the total probability on the bounded part of the graph is equal to i times the probability flow at the connecting vertices.

Let us now show that the hypothesis on u 0 implies that the probability flow remains bounded uniformly in time.

4.3.

Corollary. Let u 0 ∈ D(H) and u(t) = e -itH u 0 , t ∈ R. Then there exists a positive constant C such that

d dt k∈I B e k ρ k (t) = d dt k∈I B ∥u k (t)∥ 2 L 2 (e k ) ≤ C∥u 0 ∥ 2 D(H) , ∀t ∈ R. (4.43)
Proof. As u 0 belongs to D(H), we have Hu(t) = e -itH (Hu 0 ), and therefore

∥Hu(t)∥ L 2 (R) = ∥Hu 0 ∥ L 2 (R) , ∀t ∈ R.
This means that u k belongs to H 2 (e k ), for all k ∈ I and that there exists a positive constant C 1 such that for all t ∈ R, one has

k∈I ∥u k (t)∥ 2 H 2 (e k ) ≤ C 1 k∈I (∥u k (t)∥ 2 L 2 (e k ) + ∥Hu k (t)∥ 2 L 2 (e k ) ) = C 1 (∥u 0 ∥ 2 L 2 (R) + ∥Hu 0 ∥ 2 L 2 (R) ) = C 1 ∥u 0 ∥ 2 D(H) .
Now using the embedding of H 2 (e k ) into C 1 (ē k ), we deduce that there exists a positive constant

C 2 such that |j k (t, 0)| ≤ C 2 ∥u 0 ∥ 2 D(H) , ∀k ∈ I, t ∈ R.
Using this estimate in (4.41), we obtain (4.43), reminding that the number of unbounded edges is finite. □

We end up this section by showing that if the L 1 -L ∞ decay is in |t| -1/2 , then the sum of the probability flow at the endpoint of the infinite edges tends to zero as |t| -1 for smoother initial data.

4.4. Corollary. Let the assumptions of Theorem 3.1 be satisfied and assume that u 0 ∈ D(H) ∩ L 1 (R), that u 0 = P ac u 0 , as well as Hu 0 ∈ L 1 (R) and let u(t) = e -itH u 0 , t ∈ R. Then there exists a positive constant C such that

d dt k∈I B ∥u k (t, •)∥ 2 L 2 (e k ) ≤ C|t| -1 ∥u 0 ∥ L 1 (R) (∥u 0 ∥ L 1 (R) + ∥Hu 0 ∥ L 1 (R) ), ∀t ∈ R \ {0}. (4.44)
Consequently, for all t 2 > t 1 > 0, one has

k∈I B ∥u k (t 2 , •)∥ 2 L 2 (e k ) - k∈I B ∥u k (t 1 , •)∥ 2 L 2 (e k ) (4.45)
≤ C ln t 2 t 1 (∥u 0 ∥ L 1 (R) (∥u 0 ∥ L 1 (R) + ∥Hu 0 ∥ L 1 (R) ).
Proof. By our assumption on u 0 , Theorem 3.1 yields

(4.46) ∥u(t, •)∥ L ∞ (R) ≤ C 1 |t| -1/2 ∥u 0 ∥ L 1 (R) and ∥Hu(t, •)∥ L ∞ (R) ≤ C 1 |t| -1/2 ∥Hu 0 ∥ L 1 (R) ,
for some positive constant C 1 (independent of t). Now fix k ∈ I U , then the restriction of u k on (0, 1) satisfies

∥u k (t, •)∥ H 2 (0,1) ≤ C 2 (∥u k (t, •)∥ L 2 (0,1) + ∥Hu k (t, •)∥ L 2 (0,1) ),
for some positive constant C 2 (independent of t). Hence

∥u k (t, •)∥ H 2 (0,1) ≤ C 2 (∥u k (t, •)∥ L ∞ (0,1) +∥Hu k (t, •)∥ L ∞ (0,1) ) ≤ C 2 1 |t| -1/2 (∥u 0 ∥ L 1 (R) +∥Hu 0 ∥ L 1 (R) )
. By the Sobolev embedding theorem we deduce that

|u k,x (t, 0)| ≤ C 3 ∥u k (t, •)∥ H 2 (0,1) ≤ C 3 C 2 1 |t| -1/2 (∥u 0 ∥ L 1 (R) + ∥Hu 0 ∥ L 1 (R) ), for some positive constant C 3 (independent of t).
This estimate and (4.46) directly lead to

|j k (t, 0)| ≤ C 4 |t| -1 ∥u 0 ∥ L 1 (R) (∥u 0 ∥ L 1 (R) + ∥Hu 0 ∥ L 1 (R) ), ∀t > 0, k ∈ I U ,
for some positive constant C 4 (independent of t). Using this estimate in (4.41), we obtain (4.44).

For the second assertion, for t 2 > t 1 > 0, as

k∈I B ∥u k (t 2 , x)∥ 2 L 2 (e k ) - k∈I B ∥u k (t 1 , x)∥ 2 L 2 (e k ) = t2 t1 d dt k∈I B ∥u k (t, x)∥ 2 L 2 (e k ) dt ≤ t2 t1 d dt k∈I B ∥u k (t, x)∥ 2 L 2 (e k ) dt,
we arrive at (4.45) using (4.44) □

Note that (4.42) and (4.44) directly imply that

d dt k∈I U ∥u k (t, •)∥ 2 L 2 (e k ) ≤ C|t| -1 ∥u 0 ∥ L 1 (R) (∥u 0 ∥ L 1 (R) + ∥Hu 0 ∥ L 1 (R) ), ∀t ∈ R \ {0}.
Note also that the estimate (4.45) means that the energy on the bounded part of the graph can oscillate between 0 and the total energy at time t = 0 but the variation is at most of logarithm type.

Some phenomena on special metric graphs

According to Theorem 3.1, the L 1 -L ∞ decay of e itH P ac in t -1 2 holds as far as the assumptions (2.15), (2.16), (2.17), and (3.29) are satisfied. Such assumptions are satisfied when the graph R B is a tree with V ext = ∅ as checked in [START_REF] Banica | Dispersion for the Schrödinger Equation on Networks[END_REF]. The simplest graph that does not enter in this class is the tadpole that we treated in [START_REF] Ali Mehmeti | Dispersive effects for the Schrödinger equation on the tadpole graph[END_REF].

In this section, we analyze different examples of more complex graphs that do not enter into the above class and for which all these assumptions hold (or not). 5.1. R is a tree with only one Dirichlet vertex. If R is a tree with only one Dirichlet vertex (see figure 2), then S d is empty, therefore the assumption (2.16) holds. where D R (z) is the matrix associated with R introduced in section 2, while DR (z) is the same matrix than D R (z), where in the starting system we replace on one infinite edge the term d k e izx by dk e -izx .

Note that the proof of the implication P(n) ⇒ P(n + 1) is given in [START_REF] Banica | Dispersion for the Schrödinger Equation on Networks[END_REF], hence we only need to check that P(1) holds. If R has one interior vertex of valency ℓ ≥ 3 (hence one finite edge that we can suppose to be 1 by a scaling argument) and ℓ -1 infinite edges, by simple calculations, we notice that detD R (z) = X -1 {(2 -ℓ)X 2 + ℓ}, with X = e iz . Therefore as |X| = 1, we deduce that

|detD R (z)| = |(2 -ℓ)X 2 + ℓ| ≥ ℓ -(ℓ -2) = 2.
Similarly simple calculations yield

det DR (z) = X -1 {(4 -ℓ)X 2 + ℓ -3}. Consequently we have det DR (z) det D R (z) = |(4 -ℓ)X 2 + ℓ -3| |(2 -ℓ)X 2 + ℓ| .
Now using the splitting e iz = cos z + i sin z and some trigonometric formulas, we see that

|(4 -ℓ)X 2 + ℓ -3| 2 |(2 -ℓ)X 2 + ℓ| 2 = (ℓ -3) 2 + 2(ℓ -3)(4 -ℓ) cos z + (4 -ℓ) 2 ℓ 2 + 2ℓ(2 -ℓ) cos z + (2 -ℓ) 2 ,
hence we are reduced to estimate the ratio

r(c) = (ℓ -3) 2 + 2(ℓ -3)(4 -ℓ)c + (4 -ℓ) 2 ℓ 2 + 2ℓ(2 -ℓ)c + (2 -ℓ) 2 , ∀c ∈ [-1, 1].
Since this function r is continuous in [-1, 1], it suffices to show that

(5.47) r(c) < 1, ∀c ∈ [-1, 1],
to deduce that P(1) holds. But this estimate is equivalent to

(ℓ -3) 2 + 2(ℓ -3)(4 -ℓ)c + (4 -ℓ) 2 < ℓ 2 + 2ℓ(2 -ℓ)c + (2 -ℓ) 2 , ∀c ∈ [-1, 1].
And by developping the left-hand side and right-hand side, we get equivalenty c < 10ℓ -21 10ℓ -24 .

We can conclude that (5.47) holds since 10ℓ-21 10ℓ-24 > 1 for all ℓ ≥ 3. □ 5.2. Bounded subgraph with commensurable lengths. In this case by a scaling argument, we can always assume that the lengths of the edges of the bounded graph R B are positive integers and consequently det D(z) is a polynomial P of the variable X = e iz . Hence (2.15) and (2.16) always hold, while (2.17) hold if all eigenvalues of S d are simple. Furthermore the assumption (3.29) also holds in such a case, because the zeroes of P are either in the unit sphere giving rise to the point spectrum or are outside this unit sphere leading to so-called resonances but that remains bounded away from the unit sphere since only a finite number of such eigenvalues exist.

Let us now consider some examples where the bounded graph has a cycle and/or with some incommensurable lengths.

5.3.

A triangle with infinite half-lines at the corners. We assume that R B is a triangle with three edges of length 1 and assume that an infinite edge is attached at each vertex, see Fig. 3. In that case, simple calculations show that 4. Let us notice that the case N = 1 corresponds to the tadpole [START_REF] Ali Mehmeti | Dispersive effects for the Schrödinger equation on the tadpole graph[END_REF]. By a scaling argument, we can assume that the length of the circle is 1. For the same reason than in the case of the tadpole, In such a case, it is not difficult to see that

S d = 2πN
S d = {(2kπ)
S d = {4k 2 π 2 : k ∈ N} ∪ 4k 2 π 2 ℓ 2 : k ∈ N ,
if ℓ is an irrational number or if ℓ = p q is a rational number with p and q two positive relatively prime numbers such that p or q is even, while

S d = {4k 2 π 2 : k ∈ N} ∪ 4k 2 π 2 ℓ 2 : k ∈ N ∪ {((2m -1)qπ) 2 : m ∈ N} if ℓ = p
q is a rational number with p and q two positive relatively prime odd numbers. For the first sequence, for all k ∈ N, the eigenvector u (1,k) associated with 4k 2 π 2 is in the form u (1,k) = (sin(2kπ•), 0, 0), while for the second sequence, for all k ∈ N, the eigenvector u (2,k)

associated with 4k 2 π 2 ℓ 2 is in the form u (2,k) = 0, sin( 2kπ ℓ •), 0 .
In case of a rational number ℓ = p q with p and q two positive relatively prime odd numbers, the eigenvector associated with ((2m -1)qπ) 2 is in the form u (3,m) = (sin((2m -1)qπ•), -sin((2m -1)qπ•), 0), hence, this eigenvalue is simple.

Consequently if ℓ is a rational number, then the eigenvalues 2kπq are double and the other ones are simple, while if ℓ is irrational, all eigenvalues are simple.

Let us now check that the assumption (2.15) holds. Indeed direct calculations yield

D(z) =        1 1 0 0 -1 0 0 1 1 -1 1 -X 1 -X -1 0 0 0 0 0 1 -X ℓ 1 -X -ℓ 0 X -1 1 -X -1 X ℓ -1 1 -X -ℓ 1       
, where X = e -iz and for shortness X ℓ = e -iℓz . Hence we easily check that det D

(z) = (1 -X)(X ℓ -1)X -(1+ℓ) (-3 + X + X ℓ + 5X 1+ℓ ).
If ℓ is an irrational number or if ℓ = p q is a rational number with p and q two positive relatively prime numbers such that p or q is even, then (2.15) holds with d p (z) = (1 -e -iz )(e -izℓ -1), and d c (z) = e iz(1+ℓ) (-3 + e -iz + e -izℓ + 5e -iz(1+ℓ) ), since in this case the sole roots z of det D(z) are the zeroes of e -iz -1 and of e -izℓ -1. In the case of a rational number ℓ = p q with p and q two positive relatively prime odd numbers, we set Y = e -i z q , and notice that Y q = e -iz and Y p = e -izℓ , which allows to transform (-3 + e -iz + e -izℓ + 5e -iz(1+ℓ) ) into the polynomial

Q(Y ) = -3 + Y q + Y p + 5Y p+q .
Since Q(-1) = 0, we deduce that Q is divisible by Y + 1, hence there exists a polynomial R of degree p + q -1 such that

Q(Y ) = (Y + 1)R(Y ).
Furthermore, as Q ′ (-1) = -4(p + q) is different from zero, R(-1) is different from zero. Coming back to the original variable z, we have found that -3 + e -iz + e -izℓ + 5e -iz(1+ℓ) = (1 + e -i z q )R(e -i z q ). This proves that (2.15) holds with

d d (z) = (1 -e -iz
)(e -izℓ -1)(1 + e -i z q ), and d c (z) = e iz(1+ℓ) R(e -i z q ). Direct calculations show that

adj(D(z)) = X -(1+ℓ) (X -1)D 1 (z) + (1 -X ℓ )D 2 (z) + (1 -X)(X ℓ -1)D 3 (z) , with D 1 (z) =        0 0 0 0 0 0 0 0 0 0 0 0 0 q ℓ,2 (X) 0 0 0 0 s ℓ (X) 0 0 0 0 0 0        , D 2 (z) =        0 0 q ℓ,1 (X) 0 0 0 0 r ℓ (X) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        , and
D 3 (z) =        3X ℓ -1 2(1 -X ℓ ) 0 X ℓ -1 X ℓ + 1 X(3X ℓ -1) 2X(1 -X ℓ ) 0 X(X ℓ -1) X(1 + X ℓ ) 2(1 -X) 3X -1 X -1 0 X + 1 2X ℓ (1 -X) X ℓ (3X -1) X ℓ (X -1) 0 X ℓ (X + 1) 2(X ℓ + 1)(1 -X) 2(X -1)(1 -X ℓ ) (X ℓ 1)(X -1) (X + 1)(X ℓ -1) (X + 1)(X ℓ + 1)        , where q ℓ,1 (X) = -4X 1+ℓ + X ℓ + 1, q ℓ,2 (X) = -X + 4X 1+ℓ -1, r ℓ (X) = X 2 -2X + X 2+ℓ + 2X 1+ℓ , s ℓ (X) = -X 1+2ℓ -X 2ℓ -2X 1+ℓ + 2X ℓ .
Hence, if ℓ is an irrational number or if ℓ = p q is a rational number with p and q two positive relatively prime numbers such that p or q is even, the assumptions (2.16) and (2.17) hold (with

M c (z) = X -(1+ℓ) D 3 (z) and M p (z) = X -(1+ℓ) ((X -1)D 1 (z) + (1 -X ℓ )D 2 (z)), replacing X (resp. X ℓ
) by e -iz (resp. e -iℓz )).

On the contrary if ℓ = p q is a rational number with p and q two positive relatively prime odd numbers, we need to extract a factor (1 + e -i z q ) from D 3 (z). This is easily obtained again by using the change of unknown Y = e -i z q , and yields

Y q = e -iz = X and Y p = e -izℓ = X ℓ .
Consequently only the factor X + 1 and X ℓ + 1 contributes to this phenomenon since

X + 1 = Y q + 1 = (Y + 1)r q (Y ), X ℓ + 1 = Y p + 1 = (Y + 1)r p (Y ),
where

r p (Y ) = p-1 m=0 (-1) m Y p-1-m . This means that D 3 (z) = D 4 (z) + (1 + Y )D 5 (z),
Proof. The proof of the identity (5.50) consists in using the approach of section 2 in particular by solving the 5 × 5 system (2.4) with the specify right-hand side.

To show the property (5.51) we may notice that

γ(z) = X 1+ℓ -3 + X -1 + X -ℓ + 5X 1+ℓ -3 + X + X ℓ + 5X 1+ℓ .
Since X -1 = e iz , we remark that X -1 = X, therefore

-3 + X -1 + X -ℓ + 5X 1+ℓ = -3 + X + X ℓ + 5X 1+ℓ ,
in other word the numerator of the previous fraction is the conjuguate of its denominator. This directly leads to |γ(z)| = |X 1+ℓ | = 1.

□

The property (5.51) allows us to conjecture that the L 1 -L ∞ decay in |t| -1/2 for the third component of the solution of Schrödinger equation with an initial datum that is zero on the heads remains valid. Since we need to estimate the quantity (see the arguments of section 3)

√ b
√ a e itµ 2 γ(µ))e iµ(y-x) dµ , the variant of van der Corput Lemma [START_REF] Stein | Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals[END_REF]Corollary,p. 334] is useless in this case since the derivative of γ is not integrable due to the property (5.49), therefore another tool should be used to prove this conjecture.

Note that the previous theorem also allows to give an explicit expression of the solution of the Schrödinger equation on the graph on the infinite edge once the intial data on the heads are zero and the one on the infinite edge is compactly supported. Namely we have 5.3. Theorem. Assume that e 3 is the infinite edge of the graph. Assume that u 0 = (0, 0, u 0,3 ) ∈ L 2 (R) with u 0,3 ∈ D(0, ∞), then the solution u = (u 1 , u 2 , u 3 ) of Schrödinger equation on the graph G satisfies (5.52) u 3 (t, x) = F -1 e -it• 2 (1 + γ)F(ũ 0,3 ) (x), ∀x > 0, t ∈ R, where ũ0,3 means the extension of u 0,3 by zero in (-∞, 0) and

F(ũ 0,3 )(z) = 1 √ 2π R e -izx ũ0,3 (x) dx, ∀z ∈ R.
denotes the usual Fourier transform of ũ0,3 .

Proof. For 0 < a < b, we have -sin(µ(x -y))ℑγ(µ) dy dµ.

By using the expression of the cosinus and the sinus in term of the imaginary powers, we see that e -itµ 2 (1 + ℜγ(µ))e iµx F(ũ 0,3 )(µ) dµ

+ i ( √ a, √ b)∪(- √ b,- √ a)
e -itµ 2 ℑγ(µ)e iµx F(ũ 0,3 )(µ) dµ

= 1 √ 2π ( √ a, √ b)∪(- √ b,- √ a)
e -itµ 2 (1 + γ(µ))e iµx F(ũ 0,3 )(µ) dµ.

Since F(ũ 0,3 ) is rapidly decreasing, hence integrable in R and due to the property (5.51), we can pass to the limit as a goes to 0 and b goes to infinity and find (5.52). □

But multiplying (5.53) by qnπ ℓ , we get

|q n π -z n | < π ℓq n ,
which means that we can write z n = q n π + r n , with

|r n | < π ℓq n ,
that tends to zero as n goes to infinity. This implies that d(z n ) = 2(-1) pn (1 -e 2irn ), that tends to zero as n goes to infinity.

Note that a similar phenomenon that the one described in Theorem 5.2 appears if ℓ is an irrational number.

Conclusion

For graphs for which some of our assumptions (2.15), (2.16), (2.17), and (3.29) do not hold, the question of the L 1 -L ∞ decay of e itH P ac remains open. We believe that (2.15), (2.16), and (2.17) are always satisfied, while we have seen that this is not always the case for (3.29). We actually conjecture that when (3.29) does not hold, the decay in t -1 2 is no more valid but deteriorates into t -s , with 0 < s < 1 2 .

Figure 1 .

 1 Figure 1. A graph with #I B = 6 and #I U = 4

2. 1 .

 1 Lemma. It holds det D(z) = 0 with z ∈ C + ∪ R * if and only if z 2 belongs to S d .

|d k | 2 R 0 e|d k | 2

 202 -2by dx + e -2bR k∈I U |d k | 2 = 0. We now distinguish the case b > 0 from the case b = 0. 1. If b > 0, then by direct calculation of the second term of the previous identity we get equivalently 2b k∈I B e k |u k | 2 dx + k∈I U |d k | 2 = 0. Consequently d k = 0 for all k ∈ I U and u k = 0 on e k , ∀k ∈ I B , which directly implies that c k = d k = 0 for all k ∈ I B . Hence det D(z) cannot be equal to zero. 2. If b = 0, the identity (2.7) directly implies that k∈I U

e -itµ 2 1

 1 d c (µ) e iµ(x+y+ℓ) dµ where ℓ can take the value 0 or ±ℓ k , k ∈ I B . Now we exploit the assumption (3.29) and the fact that d c is a quasi-periodic function to deduce by Corollary 1 of [17, p. 164] that (see also [9, Lemma 3.4]) 1 d c (µ) = λ∈D d λ e iµλ , where D is a discrete set of real numbers and d λ ∈ C for all λ ∈ D such that λ∈D |d λ | < ∞.

Figure 2 .

 2 Figure 2. A tree with only one Dirichlet vertex For this class of examples, the assumption (3.29) always holds. 5.1. Theorem. If R is a tree with only one Dirichlet vertex, then the assumption (3.29) always holds.

Figure 3 .

 3 Figure 3. A triangle graph

•

Figure 4 .•Figure 5 .

 45 Figure 4. A spider graph

(0

  I (a,b) u)(t) = (e -itH I (a,b) )u 0 ,and therefore by Theorem 2.9, we get(I (a,b) u) 3 (t, x) = -ℑK 33 (x, y, µ 2 )u 0,3 (y) dyµ dµ.Hence using the expression of the kernel (5.50), we get(I (a,b) u) 3 (t, x) = 1 π √ b √ a e -itµ 2 ∞ 0 u 0,3(y) cos(µ(x -y))(1 + ℜγ(µ))

2 ( 1 + 2 ( 1 + 2 2 ( 1 + 2 ( 1 + 2

 2121221212 3 (y) cos(µ(x -y))dy = √ 2π 2 e iµx F(ũ 0,3 )(µ) + e -iµx F(ũ 0,3 )(-µ) ,∞ 0 u 0,3 (y) cos(µ(x -y))dy = √ 2π 2i e iµx F(ũ 0,3 )(µ) -e -iµx F(ũ 0,3 )(-µ) .Inserting these two expressions in the previous one, we obtain(I (a,b) u) 3 (t, x) ℜγ(µ)) e iµx F(ũ 0,3 )(µ) + e -iµx F(ũ 0,3 )(-µ) +iℑγ(µ) e iµx F(ũ 0,3 )(µ) -e -iµx F(ũ 0,3 )(-µ) dµ ℜγ(µ))e iµx F(ũ 0,3 )(µ) dµ + √ b √ a e -itµ 2 (1 + ℜγ(µ))e -iµx F(ũ 0,3 )(-µ) dµ ℑγ(µ)e -iµx F(ũ 0,3 )(-µ) dµ .Performing the change of variable z = -µ in the second and fourth integrals, we find(I (a,b) u) 3 (t, x) ℜγ(µ))e iµx F(ũ 0,3 )(µ) dµ + ℜγ(-z)e izx F(ũ 0,3 )(z) dz + i √ b√ a e -itµ 2 ℑγ(µ)e iµx F(ũ 0,3 )(µ) dµ -ℑγ(-z)e izx F(ũ 0,3 )(z) dz .By noticing that γ(-z) = γ(z), we then have ℜγ(-z) = ℜγ(z), ℑγ(-z) = -ℑγ(z), and consequently we find (I (a,b) u) 3 (t, x)

  each eigenvalue being simple. Consequently by subsection 5.2, all the assumptions (2.15), (2.16), (2.17), and (3.29) hold.

* ,

  2 } k∈N ,

	each eigenvalue being simple. Consequently by subsection 5.2, all the assumptions (2.15), (2.16),
	(2.17), and (3.29) hold.

where

And again the assumptions (2.16) and (2.17

Consequently in the case of a rational number ℓ all the assumptions (2.15), (2.16), (2.17), and (3.29) are satisfied.

On the contrary, if ℓ is an irrational number, even if d(z) = det D(z) is different from zero for all real numbers z (and consequently Assumption 2.4 holds), let us check that (3.29) does not hold. Indeed by the Theorem from [START_REF] Hartman | Sur une condition supplémentaire dans les approximations diophantiques[END_REF] (see also [14, p. 18]) with a = b = 1 and s = 2, there exist two sequences of positive and odd integers p n and q n (that both tend to infinity) such that

Then we take the sequence z n = q n π.

With this choice we see that

But z n ℓ = q n πℓ and multiplying (5.48) by q n π, we deduce that

with |r n | < 8π qn , that hence tends to zero 0 as n goes to infinity. Consequently, one has (5.49)

Before passing to another example, let us study an interesting particular case, the case when the initial data are zero on the two heads. We first state a result for the resolvent.

Theorem.

Assume that e 3 is the infinite edge of the graph. Assume that f

where the kernel K 33 is defined by

with (recalling that X = e -iz )

Note that passing to the limit in (5.52) as t goes to zero, we find

and since

we deduce that F -1 (γF(ũ 0,3 )) (x) = 0, ∀x > 0, for any u 0,3 ∈ D(0, ∞). This is a surprising property that is not easy to check using the form of γ but is true due to our previous Theorem.

A Y graph.

Here we assume that R B is made of two intervals, one of lenght 1 and the other one of lenght ℓ, and that an infinite edge is attached at one of their extremities (the sole vertex of the graph), see Fig. 6.

In this case, one can show that det D(z) = e i(1+ℓ)z (1 + e -2iz + e -i2ℓz -3e -i2(1+ℓ)z ), and S d = ∅ if ℓ is an irrational number, otherwise

each eigenvalue being simple, if ℓ = p q is a rational number with p and q two positive relatively prime numbers.

Hence if ℓ is a rational number, by subsection 5.2 and the simplicity of the eigenvalues, all the assumptions (2.15), (2.16), (2.17), and (3.29) hold.

On the contrary, if ℓ is an irrational number, even if d(z) = det D(z) is different from zero for all real numbers z (and consequently Assumption 2.4 holds), let us check that (3.29) does not hold. Indeed in this case, by Liouville's theorem, there always exist two sequences of positive integers p n and q n (that both tend to infinity as n goes to infinity) such that (5.53) ℓ -p n q n < 1 q 2 n , ∀n ∈ N.

Let us now define the sequence z n = p n π ℓ .

With this choice we see that d(z n ) = 2(-1) pn (1 -e 2izn ).