Nestor Bonjorn

Fotis Foukalas
email: fotisf@dtu.dk

Member IEEE Paul Pop

Enhanced 5G V2X Services using Sidelink Device-to-Device Communications

Keywords: 5G V2X services, sidelink device-to-device, resource allocation, scheduling, cooperative solution

In this paper, we present advances of sidelink (SL) device-to-device (D2D) communications as a key-enabling technology for 5G enhanced vehicular-to-everything (eV2X) communications. We provide an overview about the resource allocation and scheduling of different SL D2D modes under incoverage and out-of-coverage application scenarios. Moreover, we present the scheduling for SL D2D V2X communications, which relies on semi-persistent scheduling (SPS) as proposed within the 3GPP specification. Simulations were carried out to evaluate the performance in terms of collision probability assuming different values of the key parameters such as resource reselection interval (RRI) and resource selection window. We finally discuss about the open technical challenges for ultrareliable and low-latency communications as distilled from the 5G V2X use cases introduced in 3GPP Rel.15. A cooperative resource allocation and scheduling solution is also given, where a more detailed version is considered our future work on this topic.

I. INTRODUCTION

5G vehicular communication is considered one of the big challenges towards the next generation communication system. Several 5G vehicle-to-everything (V2X) use cases have been already proposed within 3GPP Rel.15 such as vehicle platooning, remote driving or cooperative collision avoidance [START_REF]Study on enhancement of 3GPP Support for 5G V2X Services[END_REF]. In particular, in [START_REF] Seo | LTE Evolution for Vehicle-to-Everything Services[END_REF], the authors provided an overview of the open challenges towards supporting V2X services in high mobility environments and dense locations of UEs. Aiming to guarantee the V2X service requirements in such vehicular communication environment, new solutions are required to address requirements such as reliability and latency. The authors focused on different design requirements such as the air interface, cost-effective network deployment and the support of different communication types. However, they neither provided particular solutions for any of these challenges nor focused on any technical solutions. They referred, though, to the channel structure of the sidelink (SL) deviceto-device (D2D) communications. They also mentioned the 3GPP specification work that is expected to provide a new air interface design for 5G communications in a broader way.

In [START_REF] Uhlemann | Initial Steps Toward a Cellular Vehicle-to-Everything Standard[END_REF], the author provided more details about the 3GPP specification initiatives to the vehicle-to-vehicle (V2V) communications. In particular, the PC5 interface, where the SL D2D communications relies on, and the recently introduced modes 3 and 4 (mode 3-4) are described in detail. An additional reference is provided to the new resource allocation technique that takes into account the near-far effect. A white paper from 5G automotive association (5GAA) aims to promote the vision of connected mobility such as autonomous driving and intelligent transportation. 5GAA provides an endto-end ecosystem through an industrial alliance paying attention to 5G V2X communications as well [START_REF]5G Automotive Vision[END_REF]. Moreover, the authors in [START_REF] Tseng | LTE-Advanced Enhancement for Vehicular Communication[END_REF] paid more attention to the PC5 interface for V2X communications providing a discussion on open challenges to provide efficient resource allocation, reliable and prioritized message type, power control and communication range enhancements. More recently, in [START_REF] Lien | Enhanced LTE Device-to-Device Proximity Services[END_REF], the authors also mention the need for an extension of the PC5 interface for V2X communications provided by the LTE D2D proximity service (ProSe). PC5 interface is actually the interface that will enable the V2V communications under in-coverage and outof-coverage application scenarios [START_REF]V2V Work Item Completion[END_REF]. In [START_REF] Campolo | Better Platooning Control Toward Autonomous Driving : An LTE Device-to-Device Communications Strategy That Meets Ultralow Latency Requirements[END_REF], they proposed a V2X communications solution to support better vehicle platooning towards the 5G V2X communications. The proposed solution relied on the LTE D2D technology addressing the low-latency requirement of messaging within the platoon. In [START_REF] Sun | Support for Vehicle-to-Everything Services Based on LTE[END_REF], the authors provided an overview of the operation scenarios of LTE V2X solutions based on particular use cases.

It is obvious from the above discussion that V2X communications will play a role towards 5G. It is also evident that the SL D2D communication is considered as a keyenabling technology towards enhanced 5G V2X services. In this paper, we provide an overview of the new SL D2D modes 3 and 4, as well as details about the baseband processing and resource allocation. A link-level simulator was developed relying on [START_REF] Zarrinkoub | Understanding LTE with MATLAB: from mathematical modeling to simulation and prototyping[END_REF] and simulation results are provided highlighting the different trade-offs in terms of BLER for different modulations, signal-to-noise ratios (SNRs) and payload sizes. Next, the scheduling for SL D2D V2X communications is analysed. A semi-persistent scheduling (SPS) was proposed within the 3GPP specification and, thus, we have developed a system-level simulator to evaluate the performance considering a number of vehicular UEs. it is observed from the simulation that the collision probability is affected by different design factors of the standardized SPS. Finally, a cooperative solution that improves the reliability of the system is devised, offering reliable 5G V2X services. Our future work on this topic will further develop the proposed cooperative resource allocation ISBN 978-3-903176-05-8 c 2018 IFIP and scheduling solution for future enhanced 5G V2X services.

The rest of this paper is organized as follows. Sec.II provides an overview of SL D2D communications with special focus on resource allocation for modes 3 and 4. Sec.III describes the semi-persistent-scheduling proposed for SL D2D modes 3 and 4 within the 3GPP standardization body. Sec.IV provides a short list of design requirements for ultra-reliable and low-latency 5G eV2X communications and a cooperative solution to address reliability requirements. Finally, Sec.V concludes this work.

II. SIDELINK DEVICE-TO-DEVICE LTE V2X

COMMUNICATIONS

In Mode 1, the UEs are assisted by the eNB and they use dedicated radio resources for data transmission. In Mode 2, the UEs randomly select the radio resources from a resource pool that was previously sent by the eNB. Both modes share the same resource allocation structure, in which the transmission of data is scheduled within the so-called physical SL control channel (PSCCH) period. Within this period, a set of subframes are determined for the PSCCH transmission (light blue region in Fig. 1a) and a different set of subframes are determined for the physical SL shared channel (PSSCH) (yellow region in Fig. 1a). The corresponding PSCCH for a given PSSCH is always sent before the PSSCH data. The PSCCH contains the sidelink control information (SCI), also called scheduling assignment (SA), which is used by the receiver to know the occupation of the PSSCH radio resources. In both modes, the SCI is configured in format 0 and it is transmitted identically in two different subframes. This retransmission is always necessary due to the lack of feedback channel in SL communication. The receiver blindly detects the SCI by trying out all possible PSCCH resources. The PSSCH transport block is transmitted four times in four consecutive subframes within the subframe pool, allowing the receiver UE to implement open loop hybrid automatic repeat request (HARQ) by combining the four redundancy versions of the PSSCH transport block.

Likewise, the difference between Mode 3 and Mode 4 is also that in Mode 3 the UEs resource allocation is assisted by the eNB while in Mode 4 the UEs choose the radio resources autonomously. However, mode 3-4 share a completely different structure than mode 1-2 described above in order to allocate the PSCCH and PSSCH. First, there is no PSCCH period in order to distribute the transmission of both physical channels into different temporal periods. In contrast, the PSCCH and PSSCH channels are separated in the frequency domain. The resource grid is divided into sub-bands or sub-channels in which the first resource blocks (lowest frequencies) of these sub-channels form the PSCCH pool (light blue region in Fig. 1b) and, the other resource blocks, the PSSCH pool (yellow region in Fig. 1b). Although PSCCH and PSSCH can be transmitted on non-adjacent resource blocks, in this work, we rely on the adjacent configuration (as illustrated in Fig. 1b). Two identical SCIs (in format 1) to combat channel distortion and their corresponding PSSCH transport block are sent in the same subframe. In mode 3-4, a transport block can be sent either once or twice. In case of two transmitting attempts, another subframe is used with the same structure: two SCIs and their corresponding PSSCH transport block. In this case, the receiver also implements HARQ.

Fig. 1 illustrates different scenarios within the different communication modes and it enables to highlight the main differences between mode 1-2 and mode 3-4. In the example depicted in Fig. 1, three packets with different sizes arrive from higher layers at different instants. The arrows within the grid show the PSSCH occupations that an SCI determines. One of the main drawbacks of mode 3-4 in comparison with mode 1-2 is the spectral efficiency since the sub-channels are not always full of data while being used. It is observed in Fig. 1b that, in the second transport block transmission (7-bytes packet), the data (orange part) occupies only a short portion of the entire sub-channel. The other resource blocks of the sub-channel are not used in this transmission and cannot be used by other UEs with the same grid configuration without colliding, even though there was enough space for both UEs to transmit in the same sub-channel. This is because of the sub-channel granularity in the resource allocation in mode 3-4, in which the transmission of data always starts at the lowest frequencies of the sub-channel. This constraints the scheduling flexibility since different UEs can be allocated in fewer different positions than in mode 1-2, where the scheduling granularity is resource block. Therefore, there is not only a lower spectral efficiency in mode 3-4 than in mode 1-2, but there is also a higher collision probability when several nearby UEs are transmitting at the same time and the sub-channels are partially empty. The main advantage of mode 3-4 is the latency since a packet arrived from upper layers is immediately scheduled in any of the available following subframes within the subframe pool, as illustrated in Fig. 1b. Instead, in mode 1-2 the packet transmission might have to wait for the next PSCCH period to start, as illustrated for the second packet in Fig. 1a. Moreover, mode 3-4 introduce additional demodulation reference signal (DMRS) symbols in order to handle high-Doppler effects in high-mobility scenarios. For these reasons, Modes 3 and 4 were proposed in Rel.14 as the communication modes for V2X applications for in-coverage and out-of-coverage scenarios, respectively. Fig. 2 depicts the average block error rate (BLER) over SNR in dB for mode 3-4. The simulation is carried out for 1000 packet transmissions using different message payload sizes such as 200, 400 and 600 bytes and different types of modulation such as QPSK and 16-QAM. The results are achievable as long as there is no interference from other UEs transmissions. We can see that the higher the SNR is, the lower the BLER is, as expected. Moreover, using different modulation types (e.g., QPSK or 16-QAM) has a big impact on the performance in terms of BLER. As expected, a higher order modulation, i.e. 16-QAM, leads to a higher BLER due to a higher number of bits transmission per frequency channel. It is interesting though, that the message payload size does not contribute significantly to the performance results.

III. SIDELINK D2D SCHEDULING FOR V2X COMMUNICATIONS

A. Scheduling for V2X communications

Due to the periodic nature and predictable size of packets in V2X transmissions, a sensing-based semi-persistent scheduling (SPS) is proposed within 3GPP in order to optimize the use of the resource grid and minimize the transmission collisions between different UEs. The standardized SPS procedure is illustrated in Fig. 3 and described below.

In the standardized SPS approach, the UE transmits in a certain resource every resource reservation interval (RRI). After each transmission, the UE's MAC entity decreases the value of a counter called SL RESOURCE RESELECTION COUNTER by one. When this counter reaches zero, the UE either keeps its current resources with probability p RK (probResourceKeep) or triggers resource reselection with probability (1 -p RK ). In both cases, the UE randomly selects a new integer value for the counter within a specified range [C1, C2] that depends on the RRI value [START_REF]3GPP, Technical Specification Group Radio Access Network; Evolved Universal Terrestrial Radio Access (E-UTRA); Medium Access Control (MAC) protocol specification[END_REF] (see Table I). In case that resource reselection is triggered, the UE generates a set of resource candidates in the resource selection window, which corresponds to the range of subframes [n+T 1, n+T 2], where n is the current subframe, T 1 depends on the process delay of the UE (T 1 ≤ 4) and T 2 shall fulfill the latency requirements (20 ≤ T 2 ≤ 100) [START_REF]3GPP, Technical Specification Group Radio Access Network; Evolved Universal Terrestrial Radio Access (E-UTRA)[END_REF]. The resource candidates are generated based on what is previously sensed in the sensing window (1000ms) and the UE randomly chooses one of these resource candidates. The UE continues transmitting in the selected frequency resource until resource reselection is triggered back again.

B. Simulation results and discussion

A system-level simulator was developed in order to evaluate the performance of the presented resource scheduling approach. This simulator allows us to discuss about the different parameters setting as well as to propose and test potential enhancements into the system. The simulated SPS algorithm that runs on each UE is given in Algorithm 1. The steps in lines 2 to 4 correspond to the initialization of the algorithm, in which random resources are assigned to each UE. The variable subfr is defined as the time consumed throughout the simulation. Next, the if-statement in line 7 is regarding the event "if the UE transmits in the current subframe", since txSubfr denotes the subframe where the UE transmits. The simulation parameters values are summarized in Table II. The simulation scenario, as well as some assumptions and simplifications, are described below.

The simulation setup consists of a group of 10 vUEs that communicate to each other using SL D2D modes 3-4. At the Algorithm 1 Pseudo code of the standardized SPS approach end while 25: end procedure beginning of the simulation, the resources used by each UE are randomly initialized, as well as the current value of their counters. In order to simulate a dynamic scenario, we add a new UE into the system with random resources every 10 seconds. At the same time, one of the existent 10 vehicles leave the group so that we keep having 10 vehicles in the platooning scenario. Both the considered number of vUEs and the "join/leave" event frequency are reasonable parameters for a typical vehicle platoon example.

We assume that all vehicles transmit periodically with the same rate due to the periodic nature of V2X traffic. In other words, the different vehicles use the same RRI value and send packets of equal size. For the sake of simplicity, we also assume that all subframes may be used for SL operation. Regarding the generation of resource candidates when performing resource reselection, we consider that the resources used by any other vehicle are occupied while the others are available. To this end, we do not use any power threshold in order to consider a resource as available or not.

The simulation runs for 1.000.000 subframes, which represent 1000sec in the system. The average collision rate from the beginning of the simulation is calculated every 1sec. In this In Fig. 4, we depict simulation results of the average collision probability with the 5 possible p RK values. We can see that the lowest average collision probability is obtained with p RK = 0.4, while the highest average collision probability is obtained with p RK = 0.8. With p RK = 0.4 reselection is performed when the counter expires with a probability of 60%, while the resources are retained with a probability of 40%. For the following simulations, p RK = 0.4 is used.

The rest of variable parameters within the standardized SPS approach are RRI and T 2. These two parameters depend on the specific application requirements, mainly in the maximum end-to-end latency. While RRI can take the values 20, 50 and 100 [START_REF]3GPP, Technical Specification Group Radio Access Network; Evolved Universal Terrestrial Radio Access (E-UTRA); Medium Access Control (MAC) protocol specification[END_REF] (also other higher values that are not considered in our use case), T 2 can take any value between 20 and 100 [START_REF]3GPP, Technical Specification Group Radio Access Network; Evolved Universal Terrestrial Radio Access (E-UTRA)[END_REF]. In order to observe the system behaviour for different combinations of RRI and T 2, we perform nine different simulations using all the combinations within the numbers 20, 50 and 100. We depict all the corresponding results in Fig. 5. As expected, a higher RRI number gives a lower collision probability due to a lower traffic density. We also observe that the lowest average collision probability is obtained with RRI = 100ms and T 2 = 20ms. In fact, given a specific RRI the lowest collision probability always seems to be given by the minimum T 2 value. Nevertheless, the combination of RRI = 100ms and T 2 = 20ms can only be used if the maximum end-to-end delay requirement is 100ms or higher. For example, if the latency requirement is 50ms, the best combination is given by RRI = 50ms and T 2 = 20ms. Finally, if the latency requirement is 20 ms, only the combination of RRI = 20ms and T 2 = 20ms can be used from all different considered solutions.

IV. ULTRA-RELIABLE AND LOW-LATENCY DESIGN REQUIREMENTS AND SOLUTION FOR 5G V2X SERVICES

A. Ultra-Reliable and Low-Latency Design Requirements

The requirements to enhance 3GPP support for 5G V2X service are described in [START_REF]Study on enhancement of 3GPP Support for 5G V2X Services[END_REF], where design requirements for 25 different 5G V2X use cases are presented. A list of the most important use cases as specified in 3GPP Rel.15 is presented below:

• eV2X support for vehicle platooning: information exchange such as join and leave, announcement warning, etc. • eV2X support for remote driving: remote driving differs from autonomous driving from the fact that the vehicle is controlled remotely. • Automated cooperative driving for short-distance grouping: automated cooperative driving is considered a combination of vehicle platooning with high-demanding communication among the vehicles. • Collective perception of the environment: vehicles can exchange real time information, collected by vehicle sensors, among each other.

• Cooperative collision avoidance: vehicles should be able to evaluate the probability of an accident by coordinating manoeuvres using cooperative awareness messages and data from sensors. We would like to summarize the design requirements of the 5G V2X use cases mentioned above in terms of the payload size, payload message reliability and latency per use case. To this end, Table III summarizes the different use cases with the corresponding KPIs. We can observe in Table III that both latency and reliability requirements are very stringent for 5G communications. In fact, latencies of 10ms as in the automated cooperative driving use case are unachievable within the described system since both RRI and T2 values are considered to be 20ms at least within 3GPP specifications in Rel.14. On the other hand, in order to guarantee message reliabilities of 99.99% in a real traffic scenario, new enhancements will need to be made for both resource allocation and scheduling.

B. Cooperative Resource Allocation and Scheduling

In the standardized SPS approach, collisions among different vUEs occur while their resource selection windows overlap. In this section, we present a solution based on the standardized SPS approach that deals with this problem more efficiently in terms of reliability. The proposal consists of transmitting the counter values in each packet transmission so that the vUEs are aware of the current counters of their surrounding vUEs. Then, the vUEs will trigger counter reselection if any of the received counters in the last RRI coincide with their own current counter. In this way, we aim to avoid the fact that two UEs arrive to counter 0 and, thus, perform resource reselection close in time, leading to a free-of-collision resource selection window.

We present below the counter reselection procedure. The counters considered for counter reselection (set A, hereafter) consists of the counters lower than the current counter. The selection of lower counters aims to not delay the resource reselection triggering. In this way, if the UE transmission involved in the counter reselection is colliding with another transmission, we do not prolong the collision time. Then, considering the received counter values during the last RRI as C RX = [c 0 , c 1 , ..., c N -1 ], where c i is the i-th received counter value and N is the number of received counters, the non-available counters (set B, hereafter) consists of C RX ∪ (C RX -1), where

(C RX -1) = [c 0 -1, c 1 -1, ..., c N -1 -1].
(C RX -1) are also considered as non-available counters because, in case one of them is chosen, it will coincide with some surrounding UE counter when this one has gone through another RRI. This is due to the fact that the collision event will be appeared that we are trying to avoid. Finally, when performing counter reselection, the UEs will randomly choose, with equal probability, one of the counters in the set end while 30: end procedure Another difference comparing to the standardized SPS approach is based on the counter value that is chosen when the current counter expires (i.e., reaches value 0). In the standardized approach, a counter range is specified so that the UEs randomly choose, with equal probability, one of the counters in the counter range defined as [C1, C2]. In our proposal, we fix the chosen counter to 50, i.e. C1 = C2 = 50. This is done because, since the counter values among close UEs are separated by our counter reselection approach, they will remain so by choosing a fixed counter value when the counter expires. Instead, if they randomly chose a new counter each time it expires, the counters from different UEs could 'collide' again. Finally, the parameter p RK is removed from the system (or set to 0) since it no longer improves the performance of the algorithm. The operational procedure of the overall proposal running in each UE is given in Algorithm 2.

C = A \ B.
Fig. 6 depicts the results obtained for the proposed approach using the parameter values from Table II (apart from C1 and C2), as well as the results obtained for the simulated standardized approach (in this case, using p RK = 0.4). It is observed that the proposed approach clearly achieves lower collision probability (0.14% after 1000 seconds in the simulated scenario) than the standardized approach (0.64%). In fact, all the collisions in the system come from the new vUEs that eventually enter into the system. This is demonstrated in Fig. 7, where a static scenario, i.e., without vUEs leaving and joining, is considered. In this case, we can see that the average collision probability decreases throughout all the simulation period. This is because after some collisions at the beginning of the simulation, which are caused by a random initialization of the UEs resources, there are no more collisions.

V. SUMMARY AND FUTURE WORK

In this work, we present the advances within 3GPP standardization body regarding enhanced vehicular-to-everything (eV2X) communications services, which is actually a step towards 5G vehicular communications. First, the SL D2D communications technology for V2X services is presented known as modes 3 and 4 giving details about the baseband processing and the resource allocation. Simulation results of the link-level simulator are obtained in terms of achievable BLER for different payload sizes, SNRs and modulation types. In the sequel, a system-level simulator is presented assuming the semi-persistent scheduling (SPS) approach proposed within 3GPP. Performance evaluation in terms of collision probability is carried out showing the trade-offs using different parameters. Taking into account the 5G eV2X service requirements, we propose a cooperative ultra-reliable resource allocation and scheduling approach that will be presented in more detail for different 5G V2X use cases in our future work.

Fig. 1 .

 1 Fig. 1. Resource allocation structure: a) Mode 1-2, b) Mode 3-4.

Fig. 2 .

 2 Fig. 2. Achievable BLER vs SNR for different payload sizes and modulations.

Fig. 3 .

 3 Fig. 3. Semi-persistend scheduling (SPS) according to 3GPP Rel.14.

1 :

 1 procedure SPS(RRI,T1,T2,C1,C2,N subCH ,L subCH ) 2: txSubch ← a random set of L subCH contiguous subchannels within N subCH 3: txSubfr ← a random number between 1 and RRI 4: counter ← a random number between 1 number between 0 and 1) ≤ p RK then 15: txSubfr ← txSubfr + RRI 16: else 17: txSubfr, txSubch ← Resource reselection within the interval [subf r + T 1, subf r + T 2]

Fig. 4 .

 4 Fig. 4. Average collision probability pc with different p RK values.

Fig. 5 .

 5 Fig. 5. Average collision probability pc with different combinations of RRI and T 2.

Algorithm 2 7 :

 27 Pseudo code of the proposed cooperative SPS approach 1: procedure PROPOSAL(RRI,T1,T2,C2,N subCH ,L subCH ) 2: txSubch ← a random set of L subCH contiguous subchannels within N subCH 3: txSubfr ← a random number between 1 and RRI 4: counter ← a random number between 1 if subfr = txSubfr then 8: if counter = 0 then 9: rxCounters ← received counters in the last RRI 10: if counter in rxCounters then 11: A ← {1, ..., counter -1} 12: B ← rxCounters∪(rxCounters-1) counter in subchannels txSubch 23: txSubfr, txSubch ← Resource reselection within the interval [subf r + T 1, subf r + T 2] based on sensing 24:

Fig. 6 .

 6 Fig. 6. Average collision probability in dynamic scenario: standardized vs proposed approaches.

Fig. 7 .

 7 Fig. 7.Average collision probability in static scenario: standardized vs proposed approaches.

TABLE I SL

 I RESOURCE RESELECTION COUNTER RANGE DEPENDING ONRRI VALUE[START_REF]3GPP, Technical Specification Group Radio Access Network; Evolved Universal Terrestrial Radio Access (E-UTRA); Medium Access Control (MAC) protocol specification[END_REF]

	RRI	[C1, C2]
	≥ 100ms	[5, 15]
	50ms	[10, 30]
	20ms	[25, 75]

ACKNOWLEDGEMENT

The research leading to these results has been performed in the SafeCOP project, that received funding from the ECSEL Joint Undertaking under grant agreement no. 692529, and from National funding.


Louis Jeanjean
email: louis.jeanjean@univ-fcomte.fr

Stefan Le Coz
email: slecoz@univ-fcomte.fr

INSTABILITY FOR STANDING WAVES OF NONLINEAR KLEIN-GORDON EQUATIONS VIA MOUNTAIN-PASS ARGUMENTS

Keywords: Mathematics Subject Classification. 35Q53, (35B35, 35A15, 35Q51)

We introduce mountain-pass type arguments in the context of orbital instability for Klein-Gordon equations. Our aim is to illustrate on two examples how these arguments can be useful to simplify proofs and derive new results of orbital stability/instability. For a power-type nonlinearity, we prove that the ground states of the associated stationary equation are minimizers of the functional action on a wide variety of constraints. For a general nonlinearity, we extend to the dimension N = 2 the classical instability result for stationary solutions of nonlinear Klein-Gordon equations proved in 1985 by Shatah in dimension N 3.

Introduction

The aim of the present paper is to show how recent methods and results concerning the variational characterizations of the ground states for elliptic equations of the form [START_REF] Adams | Sobolev spaces[END_REF] -∆ϕ = g(ϕ), ϕ ∈ H 1 (R N ; C) can be used to study the orbital stability/instability of the standing waves of various nonlinear equations such as Schrödinger equations, Klein-Gordon equations, generalized Boussinesq equations, etc. Our work is motivated by recent developments (see for instance [START_REF] Chen | Strong instability of standing waves for a nonlocal Schrödinger equation[END_REF][START_REF] Liu | Strong instability of solitary-wave solutions to a Kadomtsev-Petviashvili equation in three dimensions[END_REF][START_REF] Liu | Strong instability of solitary waves for nonlinear Klein-Gordon equations and generalized Boussinesq equations[END_REF][START_REF] Liu | Instability of standing waves of the Schrödinger equation with inhomogeneous nonlinearity[END_REF][START_REF] Ohta | Strong instability of standing waves for nonlinear Klein-Gordon equations[END_REF][START_REF] Ohta | Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system[END_REF]) of the techniques introduced by Berestycki and Cazenave [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] to prove the instability of standing waves for nonlinear evolution equations. We present our approach on two examples involving nonlinear Klein-Gordon equations of the form

(2)

u tt -∆u + ρu = f (u)
where ρ > 0, u : R × R N → C and f : [0, +∞) → R is extended to C by setting f (z) = f (|z|)z/|z| for z ∈ C \ {0}.

A standing wave of ( 2) is a solution of the form e iωt ϕ ω (x) for ω ∈ R and ϕ ω ∈ H 1 (R N ; C). Thus ϕ ω satisfies [START_REF] Berestycki | Équations de champs scalaires euclidiens non linéaires dans le plan[END_REF] -∆ϕ ω + (ρω 2 )ϕ ωf (ϕ ω ) = 0.

Clearly, (3) is of the form [START_REF] Adams | Sobolev spaces[END_REF]. From now on we write H 1 (R N ) for H 1 (R N ; C). The least energy level m is defined by ( 4)

m := inf{S(v) v ∈ H 1 (R N ) \ {0}, v is a solution of (1)}
where S : H 1 (R N ) → R is the natural functional (often called action) corresponding to ( 1) 1) is said to be a ground state, or least energy solution, if

S(v) := 1 2 ∇v 2 2 - R N G(v)dx, with G(s) := |s| 0 g(t)dt. A solution ϕ ∈ H 1 (R N ) of (
S(ϕ) = m.
The study of the existence for solutions of (1) goes back to the work of Strauss [START_REF] Strauss | Existence of solitary waves in higher dimensions[END_REF] (see also [START_REF] Coleman | Action minima among solutions to a class of Euclidean scalar field equations[END_REF]). The most general result in that direction is due to Berestycki and Lions [START_REF] Berestycki | Nonlinear scalar field equations I[END_REF] for N = 1 and N 3 and Berestycki, Gallouet and Kavian [START_REF] Berestycki | Équations de champs scalaires euclidiens non linéaires dans le plan[END_REF] for N = 2.

The assumptions of [START_REF] Berestycki | Équations de champs scalaires euclidiens non linéaires dans le plan[END_REF][START_REF] Berestycki | Nonlinear scalar field equations I[END_REF] when N 2 are : (g0) g is continuous and odd, (g1) if N 3, -∞ < lim inf It is known that the assumptions (g0)-(g3) are almost optimal to insure the existence of a solution of (1) (see [START_REF] Berestycki | Nonlinear scalar field equations I[END_REF]Section 2.2]). In [START_REF] Berestycki | Équations de champs scalaires euclidiens non linéaires dans le plan[END_REF][START_REF] Berestycki | Nonlinear scalar field equations I[END_REF] it is proved that for N 2 and under (g0)-(g3) there exists a positive radial least energy solution ϕ of (1) when the infimum in (4) is taken over the solutions belonging to H 1 (R N , R). Moreover it is easily deduce from the proofs in [START_REF] Berestycki | Équations de champs scalaires euclidiens non linéaires dans le plan[END_REF][START_REF] Berestycki | Nonlinear scalar field equations I[END_REF] that this ϕ is still a least energy solution of (1) when the infimum is, as in (4), taken over the set of all complex valued solutions. See [START_REF] Cingolani | Multi-peak solutions for magnetic NLS equations without non-degeneracy condition[END_REF] for a proof of this statement along with a description of the ground states as being of the form U = e iθ Ũ where θ ∈ R and Ũ is a real positive ground state solution of (1).

In dimension N = 1, the assumptions in [START_REF] Berestycki | Nonlinear scalar field equations I[END_REF] are (h0) g is locally Lipschitz continuous and g(0) = 0, (h1) there exists η 0 > 0 such that G(s) < 0 for all s ∈ (0, η 0 ), G(η 0 ) = 0, g(η 0 ) > 0 and it is proved in [START_REF] Berestycki | Nonlinear scalar field equations I[END_REF] that under (h0) the condition (h1) is necessary and sufficient to guarantee the existence of a unique (up to translation) real positive solution of [START_REF] Adams | Sobolev spaces[END_REF]. Here also, it can be shown (see [START_REF] Cingolani | Multi-peak solutions for magnetic NLS equations without non-degeneracy condition[END_REF]) that the least energy levels coincide for complex and real valued solutions of (1).

Since the pioneer works [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF][START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF], it is known that the stability/instability of the standing waves is closely linked to additional variational characterizations that the associated ground states enjoy. Recently, in [START_REF] Jeanjean | A remark on least energy solutions in R N[END_REF] for N 2 and in [START_REF] Jeanjean | A note on a mountain pass characterization of least energy solutions[END_REF] for N = 1, Jeanjean and Tanaka showed that, under the conditions (g0)-(g3) for N 2 and basically (h0)-(h1) for N = 1, the functional S admits a mountain pass geometry. Precisely they show that setting S(γ(t)) > 0.

Furthermore, they proved that c = m, namely that the mountain pass value gives the least energy level. In fact, the results of [START_REF] Jeanjean | A remark on least energy solutions in R N[END_REF][START_REF] Jeanjean | A note on a mountain pass characterization of least energy solutions[END_REF] are proved within the space H 1 (R N , R) but it is straightforward to show, see Lemma 14, that this equality also holds in H 1 (R N ).

In this paper, we will show, by studying two specific problems, how the ideas and methods developed in [START_REF] Jeanjean | A remark on least energy solutions in R N[END_REF][START_REF] Jeanjean | A note on a mountain pass characterization of least energy solutions[END_REF] can be implemented in the context of instability by blow-up for nonlinear Klein-Gordon equations.

First, working with a nonlinearity of power type (f (s) = |s| p-1 s) we find a set of constraints on which the ground states are minimizers of S. In particular, this gives an alternative, much simpler proof of results in [START_REF] Liu | Strong instability of solitary waves for nonlinear Klein-Gordon equations and generalized Boussinesq equations[END_REF][START_REF] Ohta | Strong instability of standing waves for nonlinear Klein-Gordon equations[END_REF][START_REF] Ohta | Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system[END_REF] concerning the derivation of an additional variational characterization of the ground states. Precisely, we prove Theorem 1. Let α, β ∈ R be such that [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] β < 0, α(p -1) -2β 0 and 2αβ(N -2) > 0 or β 0, α(p -1) -2β 0 and 2α -βN > 0.

Let ω ∈ (-1, 1) and ϕ ω ∈ H 1 (R N ) be a ground state solution of

-∆ϕ ω + (1 -ω 2 )ϕ ω -|ϕ ω | p-1 ϕ ω = 0. Then S(ϕ ω ) = min{S(v) v ∈ H 1 (R N ) \ {0}, K α,β (v) = 0} where
S(v) := 1 2 ∇v 2 2 + 1 -ω 2 2 v 2 2 - 1 p + 1 v p+1 p+1 . K α,β (v) := 2α-β(N -2) 2 ∇v 2 2 + (2α-βN )(1-ω 2 ) 2 v 2 2 -α(p+1)-βN p+1 v p+1 p+1 . The functional K α,β is based on the rescaling v λ ( • ) := λ α v(λ β • ) for v ∈ H 1 (R N ), precisely, K α,β (v) = ∂ ∂λ S(v λ ) |λ=1 .
The main idea of the proof of Theorem 1 is to use rescaled functions to construct for any v ∈ H 1 (R N ) such that K α,β (v) = 0 a path in Γ attaining his maximum at v.

It is also of interest to consider a limit case of Theorem 1.

Theorem 2. Let α, β ∈ R be such that [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF] β < 0, α(p -1) -2β 0 and 2αβ(N -2) = 0 or β > 0, α(p -1) -2β 0 and 2α -βN = 0.

Let ω ∈ (-1, 1) and ϕ ω be a ground state solution of

-∆ϕ ω + (1 -ω 2 )ϕ ω -|ϕ ω | p-1 ϕ ω = 0. Then S(ϕ ω ) = min{S(v) v ∈ H 1 (R N ) \ {0}, K α,β (v) = 0}.
Remark 1. Looking to the proofs of Theorems 1 and 2 one see that our Theorems remain unchanged when (1ω 2 ) is replaced by any m > 0. We choose however to present our results in the setting of [START_REF] Liu | Strong instability of solitary waves for nonlinear Klein-Gordon equations and generalized Boussinesq equations[END_REF][START_REF] Ohta | Strong instability of standing waves for nonlinear Klein-Gordon equations[END_REF][START_REF] Ohta | Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system[END_REF].

For (α, β) = ( N 2 , 1), Theorem 2 gives a simpler proof of a variational characterization of the ground state proved by Berestycki and Cazenave [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] for 1 + 4 N < p < 1 + 4 N -2 and by Nawa [19, Proposition 2.5] for p = 1 + 4 N . This characterization is at the heart of the classical result of Berestycki and Cazenave [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] dealing with the instability of the ground states of nonlinear Schrödinger equations.

For our second direction of application we consider the instability of the stationary solutions of [START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF] u tt -∆u = g(u).

In 1985, Shatah established in [START_REF] Shatah | Unstable ground state of nonlinear Klein-Gordon equations[END_REF] that under the conditions (g0)-(g3) the radial ground states solutions associated with the standing waves corresponding to ω = 0 are unstable when N 3. Under stronger hypothesis, but in any dimension and for non necessary radial solutions, Berestycki and Cazenave [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] had previously proved that these ground states are unstable by blow up in finite time. In [START_REF] Shatah | Unstable ground state of nonlinear Klein-Gordon equations[END_REF], instability may occur by blow up in infinite time, in the sense that the H 1 (R N )-norm of a solution starting close to a ground state goes to infinity when t → +∞. Here, we show that the same result hold when N = 2.

We make the following hypothesis on the existence and properties of solutions for [START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF].

Assumption H. For all (u 0 , v 0 ) ∈ H 1 rad (R 2 ) × L 2 rad (R 2 ) there exist 0 < T +∞ and u : [0, T ) × R 2 → C such that • (u(0), u t (0)) = (u 0 , v 0 ), • u (resp. u t ) is weakly continuous in H 1 rad (R 2 ) (resp. L 2 rad (R 2 )), • u satisfies (9) in the sense of distributions, • E(u(t), u t (t)) E(u 0 , v 0 ) for all t ∈ [0, T ) ( energy inequality), • if T < +∞, there exists (t n ) ⊂ [0, T ) such that t n → T as n → +∞ and lim tn→T u(t n ) H 1 (R 2 ) = +∞ ( blow-up alternative),
The

energy functional E is defined for u ∈ H 1 (R N ) and v ∈ L 2 (R N ) by E(u, v) := 1 2 v 2 2 + 1 2 ∇u 2 2 - R 2 G(u)dx.
In what follows, as above, we write

H 1 rad (R N ) (resp. L 2 rad (R N )) for the space of radial functions of H 1 (R N ) (resp. L 2 (R N )).
Remark 2. When N 3, Shatah claims that Assumption H holds under (g0)-(g3) without any additional restrictions. For others dimensions, Assumption H is known to hold under stronger assumptions on g, see, for example, [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF]Chapter 6]. From now on a solution of ( 9) with initial data (u 0 , v 0 ) will refer to a solution of [START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF] with initial data (u 0 , v 0 ) as given by Assumption H.

Our third main result is the following Theorem 3. Assume N = 2, (g0)-(g3) and Assumption H. Let ϕ be a radial ground state of (1). Then ϕ viewed as a stationary solution of ( 9) is strongly unstable. Namely for all ε > 0 there exist

u ε ∈ H 1 (R 2 ), T ε ∈ (0, +∞] and (t n ) ⊂ (0, T ε ) such that ϕ -u ε H 1 (R 2 ) < ε and lim tn→Tε u(t n ) H 1 (R 2 ) = +∞,
where u(t) is a solution of ( 9) with initial data (u ε , 0).

It is still an open question to describe what happen in dimension

N = 1.
Indeed, the use of the radial compactness lemma of Strauss (see Lemma 5) restricts our proof to dimensions N 2. A partial answer is given by the work of Berestycki and Cazenave : for nonlinearities satisfying some additional assumptions (see [2, (H.3)]), the stationary solutions are unstable.

We do hope that the methods developed in this paper will find other areas of applications. In that direction, we mention the work [START_REF] Coz | A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations[END_REF] in which the variational characterization c = m derived from [START_REF] Jeanjean | A remark on least energy solutions in R N[END_REF][START_REF] Jeanjean | A note on a mountain pass characterization of least energy solutions[END_REF] is essential to get an alternative, more general proof of the classical result of Berestycki and Cazenave [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] on the instability by blow-up for nonlinear Schrödinger equations. This paper is organized as follows. In Section 2 we prove Theorem 1 and Theorem 2. In Section 3 we prove Theorem 3. The proof that the results of [START_REF] Jeanjean | A remark on least energy solutions in R N[END_REF][START_REF] Jeanjean | A note on a mountain pass characterization of least energy solutions[END_REF] extend to the complex case along with a technical lemma are given in the Appendix.

Acknowledgments. The authors wish to thank Masahito Ohta and Grozdena Todorova for the interest they have taken in this work and for fruitful discussions. They are also grateful to Mariana Hȃrȃguş for fruitful discussions.

Variational characterizations of the ground states

In this section, we consider (3) with a power type nonlinearity : [START_REF] Chen | Strong instability of standing waves for a nonlocal Schrödinger equation[END_REF] -

∆ϕ ω + (1 -ω 2 )ϕ ω -|ϕ ω | p-1 ϕ ω = 0
where 1 < p < 1 + 4/(N -2) and |ω| < 1. For this nonlinearity it is known (see [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]Section 8.1] and the references therein) that there exists a unique positive radial ground state ϕ ω ∈ H 1 (R N , R) of ( 10) and that all ground states are of the form e iθ ϕ ω (•y) for some fixed θ ∈ R and y ∈ R N . The standing waves e iωt ϕ ω are solutions of the nonlinear Klein-Gordon equation ( 11)

u tt -∆u + u = |u| p-1 u
and the natural functional associated with [START_REF] Chen | Strong instability of standing waves for a nonlocal Schrödinger equation[END_REF] becomes

S(v) := 1 2 ∇v 2 2 + 1 -ω 2 2 v 2 2 - 1 p + 1 v p+1 p+1 .
Various results of instability for the standing waves of [START_REF] Cingolani | Multi-peak solutions for magnetic NLS equations without non-degeneracy condition[END_REF] were recently proved in [START_REF] Liu | Strong instability of solitary waves for nonlinear Klein-Gordon equations and generalized Boussinesq equations[END_REF][START_REF] Ohta | Strong instability of standing waves for nonlinear Klein-Gordon equations[END_REF][START_REF] Ohta | Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system[END_REF]. For instance, it was proved in [START_REF] Ohta | Strong instability of standing waves for nonlinear Klein-Gordon equations[END_REF] that for any 1 < p < 1 + 4/(N -2) the standing wave associated with a ground state of ( 10) is strongly unstable by blow up if ω 2 (p -1)/(p -3) and N 3. In [START_REF] Ohta | Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system[END_REF], a result of strong instability was showed for the optimal range of parameter ω in dimension N 2 (namely |ω| < ω c , where ω c was determined in [START_REF] Shatah | Instability of nonlinear bound states[END_REF]). In both cases, it is central in the proofs that the ground states can be characterized as minimizers on constraints having all the form

K α,β := {v ∈ H 1 (R N ) \ {0} K α,β (v) = 0} for some α, β ∈ R. Recall that the functional K α,β is defined for v ∈ H 1 (R N ) by K α,β (v) := ∂ ∂λ S(λ α v(λ β • )) |λ=1 = 2α-β(N -2) 2 ∇v 2 2 + (2α-βN )(1-ω 2 ) 2 v 2 2 -α(p+1)-βN p+1 v p+1 p+1 .
For example, it is proved in [START_REF] Ohta | Strong instability of standing waves for nonlinear Klein-Gordon equations[END_REF] that the ground states are minimizer of S on K α,β for (α, β) = (1, 0) and (α, β) = (0, -1/N) (see [21, (2.1)]) whereas in [START_REF] Ohta | Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system[END_REF], the values of (α, β) considered are (α, β) = (N/2, 1) if p 1 + 4/N (see [22, (2.11)]) and (α, β) = (2/(p -1), 1) if p < 1 + 4/N (see [22, (2.18)]). Recently, Liu, Ohta and Todorova [START_REF] Liu | Strong instability of solitary waves for nonlinear Klein-Gordon equations and generalized Boussinesq equations[END_REF] extended the approach of [START_REF] Ohta | Strong instability of standing waves for nonlinear Klein-Gordon equations[END_REF] to the dimensions N = 1, 2. Once more, a main feature of their proof is to minimize S on K α,β , but this time with

α = (p -1) -(p + 3)ω 2 2(p -1)ω 2 , β = -1.
In [START_REF] Liu | Strong instability of solitary waves for nonlinear Klein-Gordon equations and generalized Boussinesq equations[END_REF][START_REF] Ohta | Strong instability of standing waves for nonlinear Klein-Gordon equations[END_REF][START_REF] Ohta | Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system[END_REF], the proofs that the ground states are minimizers of S on K α,β follow similar schemes. First, one has to show the convergence of a minimizing sequence to some function solving a Lagrange equation. After that, the difficulty is to get rid of the Lagrange multiplier. For each choice of (α, β), long computations are involved to prove that the Lagrange multiplier is 0 and to conclude that the obtained function is in fact a solution of [START_REF] Chen | Strong instability of standing waves for a nonlocal Schrödinger equation[END_REF].

Our proof of Theorem 1 relies on the following lemma. We recall that Γ is defined in [START_REF] Berestycki | Nonlinear scalar field equations I[END_REF]. Lemma 3. Let α, β ∈ R satisfy [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]. Then for all v ∈ K α,β we can construct a path γ in Γ such that

max t∈[0,1] S(γ(t)) = S(v). Proof. Let v ∈ K α,β . For all λ ∈ (0, +∞) we define v λ ∈ H 1 (R N ) by v λ ( • ) := λ α v(λ β • ).
The idea is to construct the path such that γ(t) = v Ct for some C > 0.

The first thing to check is that we can extend γ at 0 by continuity. Namely, we must show that under [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] we have lim λ→0 v λ H 1 (R N ) = 0. This is immediate if we remark that

v λ 2 H 1 (R N ) = λ 2α-β(N -2) ∇v 2 2 + λ 2α-βN v 2 2
, and that (7) implies 2αβ(N -2) > 0 and 2α -βN > 0.

The next step is to prove that λ → S(v λ ) increases for λ ∈ (0, 1), attains its maximum at λ = 1 and decreases toward -∞ on (1, +∞). We have

S(v λ ) = λ 2α-β(N -2) 2 ∇v 2 2 + (1 -ω 2 )λ 2α-βN 2 v 2 2 - λ (p+1)α-βN p + 1 v p+1 p+1
and from easy computations it comes

λ -(2α-βN -1) ∂ ∂λ S(v λ ) = λ 2β 2α -β(N -2) 2 ∇v 2 2 + (2α -βN)(1 -ω 2 ) 2 v 2 2 -λ α(p-1) α(p + 1) -βN p + 1 v p+1 p+1 .
Therefore, if α and β satisfy [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] it is clear that (12) hold under [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF].

∂ ∂λ S(v λ ) < 0 for λ ∈ (1, +∞), lim λ→+∞ S(v λ ) = -∞. Since α > 0 when β = 0 in
Finally, choosing C large enough to have S(v C ) < 0 and defining γ : [0, 1] → H 1 (R N ) by γ(0) := 0 and γ(t) := v tC we have a path satisfying the conclusion of the lemma.

Proof of Theorem 1. Let ϕ ω be a least energy solution of (10) for |ω| < 1. From Lemma 14 we know that c = m where m is the least energy level and c the mountain pass value (see ( 4) and ( 6) for the definitions of m and c). Since ϕ ω is a solution of (10), ϕ ω ∈ C 1 and ϕ ω , ∇ϕ ω are exponentially decaying at infinity (see, for example, [7, Theorem 8.1.1]); in particular, x.∇ϕ ω ∈ H 1 (R N ), and

K α,β (ϕ ω ) = ∂ ∂λ S(λ α ϕ ω (λ β • )) λ=1 = S ′ (ϕ ω ), αϕ ω + βx.∇ϕ ω = 0. Thus ϕ ω ∈ K α,β and (13) min{S(v) v ∈ K α,β } S(ϕ ω ) = c.
Conversely, it follows from Lemma 3 that

(14) c min{S(v) v ∈ K α,β }.
To combine ( 13) and ( 14) finishes the proof.

We now turn to the proof of Theorem 2. It follows the same lines as for Theorem 1 : find a path reaching its maximum on the constraint K α,β and use the equality c = m. The main difference is in the way we construct the path : we still want to use the rescaled functions v λ , but their H 1 (R N )-norm does not any more converge to 0 as λ → 0. This difficulty is overcome by gluing to {v λ } λ>λ 0 a path linking 0 to v λ 0 for λ 0 suitably chosen. The lemma is Lemma 4. Let α, β ∈ R satisfy [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF]. Then for all v ∈ K α,β we can construct a path γ in Γ such that max t∈[0,1]

S(γ(t)) = S(v).

Proof. Let v ∈ K α,β and v λ 0 (•) := λ α 0 v(λ β 0 • ) for some λ 0 ∈ (0, 1) whose value will be fixed later. Let C > 0 be such that S(v C ) < 0 and consider the curves

Λ 1 := {v λ λ ∈ [λ 0 , C]}, Λ 2 := {tv λ 0 t ∈ [0, 1]}.
To get a path as desired, we will glue the two curves Λ 1 and Λ 2 . It is clear that as in the proof of Lemma 3, S attained its maximum on Λ 1 at v. Thus the only thing we have to check is that t → S(tv λ 0 ) is increasing on [0, 1].

We have

∂ ∂t S(tv λ 0 ) = t( ∇v λ 0 2 2 + (1 -ω 2 ) v λ 0 2 2 -t p-1 v λ 0 p+1 p+1 ).
If β > 0 and α = βN/2 (see ( 8)), then λ 0 → v λ 0 2 is constant. If β < 0 and α = β(N -2)/2 then λ 0 → ∇v λ 0 2 is constant. Moreover, we have in any case lim

λ 0 →0 v λ 0 p+1 p+1 = 0. Therefore, if λ 0 ∈ (0, 1) is small enough we have ∂ ∂t S(tv λ 0 ) > 0 for t ∈ (0, 1). To define γ : [0, 1] → H 1 (R N ) by γ(t) = Ct λ 0 v λ 0 for t ∈ [0, λ 0 C ) γ(t) = v Ct for t ∈ [ λ 0 C , 1]
gives us the desired path.

Proof of Theorem 2. The proof is identical to the proof of Theorem 1 with Lemma 3 replaced by Lemma 4.

Orbital instability for a generalized nonlinear Klein-Gordon equation

In this section, we consider the nonlinear Klein-Gordon equation with a general nonlinearity [START_REF] Coz | A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations[END_REF] u tt -∆u = g(u).

In [START_REF] Shatah | Unstable ground state of nonlinear Klein-Gordon equations[END_REF], Shatah proved that for N 3, under (g0)-(g3), the radial ground states solutions of ( 16)

-∆ϕ = g(ϕ), u ∈ H 1 (R N )
viewed as stationary solutions of ( 15) are unstable in the sense of Theorem 3.

The restriction to N 3 has its origin in, at least, two reasons.

First, one needs to control the decay in |x| of u(t, x) uniformly in t. This appears in the proofs of Proposition 12 and Lemma 15. For this control, the following compactness lemma due to Strauss [START_REF] Strauss | Existence of solitary waves in higher dimensions[END_REF] is used.

Lemma 5. Let N 2 and v ∈ H 1 rad (R N ). Then |v(x)| C|x| 1-N 2 u H 1 (R N ) a.e.
with C independent of x and u. In particular, the following injection is compact

H 1 rad (R N ) ֒→ L q (R N ) for 2 < q < 2 ⋆ , where 2 ⋆ = 2N N -2 if N 3 and 2 ⋆ = +∞ if N = 2.
Actually, to use this lemma only N 2 is necessary.

A second reason for the restriction N 3 in [START_REF] Shatah | Unstable ground state of nonlinear Klein-Gordon equations[END_REF] is found in the use of a constraint based on Pohozaev's identity to derive a variational characterization of the ground states, to define an invariant set, and, most important, to choose suitable initial data close to the ground states. Thanks to our approach, we arrive on this second point to require only N 2.

Our proof will make use of the following variational characterization of the ground states. Lemma 6. Let ϕ ∈ H 1 (R 2 ) be a ground state of [START_REF] Liu | Strong instability of solitary-wave solutions to a Kadomtsev-Petviashvili equation in three dimensions[END_REF]. Then where

P := {v ∈ H 1 (R 2 ) \ {0} P (v) = 0} with P (v) := R 2 G(v)dx for v ∈ H 1 (R N ).
This lemma was proved in [START_REF] Berestycki | Équations de champs scalaires euclidiens non linéaires dans le plan[END_REF] [START_REF] Cingolani | Multi-peak solutions for magnetic NLS equations without non-degeneracy condition[END_REF].

when v ∈ H 1 (R N , R). It can trivially be extended to v ∈ H 1 (R N ), see
Remark 7. The functional P is related to the so-called Pohozaev identity (see [5, Proposition 1], [START_REF] Strauss | Existence of solitary waves in higher dimensions[END_REF]): for N 1, any solution v ∈ H 1 (R N ) of ( 16) satisfies

N -2 2 ∇v 2 2 -N R N G(v)dx = 0.
A main feature of the dimension N = 2 is that we lose the control on the L 2 (R N )-norm of ∇v.

Remark 8. For N 3, Shatah also showed that the radial ground states are minimizers of S among all non trivial functions satisfying Pohozaev identity (see [START_REF] Shatah | Unstable ground state of nonlinear Klein-Gordon equations[END_REF]Proposition 1.5]). His method consists in proving that the minimization problem has a solution and then to eliminate the Lagrange multiplier.

In fact, as it is done in [13, Lemma 3.1], a shorter proof can be performed by simply establishing a correspondence with a minimization problem already solved in [START_REF] Berestycki | Nonlinear scalar field equations I[END_REF].

The scheme of the proof is the following : first, define a set

I ⊂ H 1 rad (R 2 ) × L 2 rad (R 2
) such that any solution of [START_REF] Coz | A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations[END_REF] with initial data in I stays in I for all time and blows up, then prove that the ground states can be approximated by functions in I.

Let I be defined by

I := {u ∈ H 1 rad (R 2 ) \ {0}, v ∈ L 2 rad (R 2 ) E(u, v) < m, P (u) > 0}.
We begin by proving an equivalence between two variational problems.

Lemma 9. We have

m = min v∈P S(v) = min{T (v) v ∈ H 1 (R 2 ) \ {0}, P (v) 0}, where T (v) := 1 2 ∇v 2 2 . Proof. Let v ∈ H 1 (R 2 ). If v ∈ P, then v satisfies T (v) = S(v)
and thanks to Lemma 6, T (v) m. Suppose that P (v) > 0. For λ > 0, define v λ ( • ) := λv(λ • ). We claim that there exists λ 0 < 1 such that P (v λ 0 ) = 0. Indeed, by (g1)-(g2), for all α > 0 there exists C α > 0 such that for s > 0

g(s) -ρs 2 + 2sαC α e αs 2 .
We recall that ρ > 0 is given in (g1) by lim s→0 g(s)s -1 = -ρ. Therefore, for s > 0 we have

G(s) -ρs 2 4 + C α (e αs 2 - 1) and (18)
R 2 G(v λ ) -ρ v λ 2 2 4 + C α R 2
(e αv 2 λ -1)dx.

We remark that v λ

2 2 = v 2 2 and R 2 (e αv 2 λ -1)dx = λ -2 R 2 (e αλ 2 v 2 -1)dx.
For λ < 1 we have

λ -2 (e αλ 2 v 2 (x) -1) < e αv 2 (x) -1 for all x ∈ R 2 ,
and by Moser-Trudinger inequality (see [START_REF] Adams | Sobolev spaces[END_REF]Theorem 8.25]) there exists α > 0 such that (e αv 2 -1) ∈ L 1 (R 2 ). Hence, Lebesgue's Theorem gives

R 2
(e αv 2 λ -1)dx → 0 when λ → 0.

Coming back to [START_REF] Liu | Instability of standing waves of the Schrödinger equation with inhomogeneous nonlinearity[END_REF] this means that

R 2
G(v λ ) < 0 for λ > 0 small enough, and by continuity of P this proves the claim. Now, we have

inf u∈P S(u) S(v λ 0 ) = T (v λ 0 ) = λ 2 0 T (v) < T (v),
and the lemma is proved.

Next we prove that the set I is invariant under the flow of [START_REF] Coz | A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations[END_REF].

Lemma 10. Let (u 0 , v 0 ) ∈ I, 0 < T +∞ and u(t) a solution of ( 15) on [0, T ) with initial data (u 0 , v 0 ). Then (u(t), u t (t)) ∈ I for all t ∈ [0, T ).

Proof. Let

t 0 := inf   {t ∈ [0, T ) P (u(t)) 0} ∪ {+∞}   .
Assume by contradiction that t 0 = +∞ and consider (t n ) ⊂ (t 0 , T ) such that t n ↓ t 0 with P (u(t n )) 0. By Assumption H, u(t n ) ⇀ u(t 0 ) weakly in H 1 (R 2 ). Thus we have ( 19)

T (u(t 0 )) lim inf n→+∞ T (u(t n )) lim inf n→+∞ [T (u(t n )) -P (u(t n ))] . Moreover (20) lim inf n→+∞ [T (u(t n )) -P (u(t n ))] = lim inf n→+∞ S(u(t n )) lim inf n→+∞ E(u(t n ), u t (t n ))
and by the energy inequality in Assumption H we get

(21) lim inf n→+∞ E(u(t n ), u t (t n )) E(u 0 , v 0 ).
Recalling that (u 0 , v 0 ) ∈ I, we have

(22) E(u 0 , v 0 ) < m.
Combining ( 19)-( 22) gives ( 23)

T (u(t 0 )) < m.
Now, take ( t n ) ⊂ (0, t 0 ) such that t n ↑ t 0 . By Lemma 16, v → P (v) is upper weakly semi-continuous, thus [START_REF] Shatah | Instability of nonlinear bound states[END_REF] P (u(t 0 )) lim sup n→+∞ P (u( t n )) 0. Now together ( 23) and ( 24) lead to a contradiction with Lemma 9.

The following lemma is a key step in the proof.

Lemma 11. Let (u 0 , v 0 ) ∈ I and u(t) an associated solution of [START_REF] Coz | A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations[END_REF] in [0, T ).

Then there exists δ > 0 such that P (u(t)) > δ for all t ∈ [0, T ).

Proof. Indeed, assume by contradiction that there exists a sequence (t n ) such that P (u(t n )) → 0 as n → +∞. Then

T (u(t n )) = S(u(t n )) + P (u(t n )) E(u(t n ), u t (t n )) + P (u(t n )).
By the energy inequality in Assumption H this implies

T (u(t n )) E(u 0 , v 0 ) + P (u(t n ))
and thus ( 25)

T (u(t n )) < m + P (u(t n )) -ν with ν := m -E(u 0 , v 0 ) > 0 since (u 0 , v 0 ) ∈ I.
For n large enough we have

0 P (u(t n )) < ν/2
and thus [START_REF] Strauss | Existence of solitary waves in higher dimensions[END_REF] gives

T (u(t n )) < m - ν 2 ,
which contradicts the result of Lemma 9.

The proof of Theorem 3 relies on the following proposition.

Proposition 12. Let (u 0 , v 0 ) ∈ I and u(t) an associated solution of ( 15) on [0, T ). Then there exists (t n ) ⊂ (0, T ) such that lim tn→T u(t n ) H 1 (R But, thanks to the energy inequality u t (t) 2 is bounded, and ∇u(t) 2 is bounded by assumption, therefore, for t large enough we reach a contradiction in (30).

In dimension N 3, it is easily seen that for λ < 1 the dilatation of a ground state ϕ λ ( • ) := ϕ( • λ ) gives a sequence of initial data in I converging to this ground state. This property, combined with the equivalent of Proposition 12, gives immediately the instability of the ground states in [START_REF] Shatah | Unstable ground state of nonlinear Klein-Gordon equations[END_REF]. This is not the case any more in dimension N = 2 where the dilatation ϕ λ ( • ) := ϕ( • λ ) leaves P and T invariant. To overcome this difficulty, we borrow and adapt an idea of [6, Proposition 2] which consists in using separately (and successively) a dilatation and a rescaling to get initial data in I close to the ground states.

Lemma 13. Let ϕ ∈ H 1 (R 2 ) be a ground state of [START_REF] Liu | Strong instability of solitary-wave solutions to a Kadomtsev-Petviashvili equation in three dimensions[END_REF]. For all ε > 0 there exists ϕ ε such that

ϕ -ϕ ε H 1 (R 2 ) < ε, S(ϕ ε ) < S(ϕ), P (ϕ ε ) > 0. Proof. For λ, µ > 0 consider ϕ λ,µ ( • ) := λϕ( • µ ). Then ∂ ∂λ S(ϕ λ,µ ) = λ 2 ∇ϕ 2 2 -µ 2 R 2
g(λϕ)ϕdx.

To multiply ( 16) by ϕ and integrate gives us

∇ϕ 2 2 = R 2 g(ϕ)ϕdx.
Hence, for λ = 1 we get ∂ ∂λ S(ϕ λ,µ ) λ=1 = (1µ 2 ) ∇ϕ 2 2 . Thus, for all µ > 1, there exists λ µ > 0 such that (H2) H(s) = o(s 2 ) when s → 0.

Let (u n ) ⊂ H 1 rad (R 2 ) be a sequence bounded in H 1 (R 2 ) such that u n → u a.e. Then we have

H(u n ) → H(u) in L 1 (R 2 ).
This lemma was proved in [4, Lemma 5.2], the extended version of [START_REF] Berestycki | Équations de champs scalaires euclidiens non linéaires dans le plan[END_REF]. We recall it here for the sake of completeness.

Proof of Lemma 15. From the continuity of H we have H(u n ) → H(u) a.e. By a theorem of Vitali (see, for example, [20, p 380]), it is enough to prove (i) for each ε > 0 there exists R > 0 such that Therefore, (i) is satisfied.

For (ii), we first remark that, by (H1) and Moser-Trudinger inequality, there exists α > 0 and M > 0 such that Proof. It is standard to show that under (g2), P ∈ C 1 (H 1 (R N ), R). Now let v n ⇀ v in H 1 (R N ). Using (g1)-(g2), we can decompose G in

G(s) = -ρs 2 + H(s)
where H satisfies the hypothesis of Lemma 15. Hence

R N H(v n )dx → R N H(v)dx when n → +∞.
Since v →v 2 is upper weakly semicontinuous, this conclude the proof.

N = 2 ,

 2 ∀α > 0 ∃C α > 0 such that |g(s)| C α e αs 2 ∀s > 0. (g3) there exists ξ 0 > 0 such that G(ξ 0 ) > 0.

  {γ ∈ C([0, 1], H 1 (R N )), γ(0) = 0, S(γ(1)) < 0} one has Γ = ∅ and

( 12 )

 12 β = 0 and α(p -1) 2β or β = 0 and α(p -1) v λ ) > 0 for λ ∈ (0, 1),

( 17 )

 17 S(ϕ) = m = min v∈P S(v)

2 R 2 g

 22 λ,µ ) < 0 for λ ∈ (1λ µ , 1 + λ µ ) and therefore (31) S(ϕ λ,µ ) < S(ϕ) for λ ∈ (1, 1 + λ µ ). Now, ∂ ∂λ P (ϕ λ,µ ) λ=1 = µ (ϕ)ϕdx = µ 2 ∇ϕ 2 2 > 0.

R 2 2 2α u n 2 2 < ε 2 .

 2222 \{|x|<R} H(u n )dx < ε for all n ∈ N, (ii) for each ε > 0 there exists δ > 0 such that {|x-y|<δ}H(u n )dx < ε for all y ∈ {x ∈ R 2 such that |x| < R} (equiintegrability).Let ε > 0 be arbitrary chosen. From (H1)-(H2), for α > 0 there existsC α > 0 such that for all s ∈ R |H(s)| αs 2 + C α (e s 2 -1). Thus, for any R > 0 {|x|>R} |H(u n )| α u n + C α {|x|>R} (e u 2n -1)dx.On one hand, since (u n ) is bounded in L 2 (R N ) we can take α > 0 small enough such thatOn the other hand, from Lemma 5 there exists C such thatC α {|x|>R} (e u 2 n -1)dx C α {|x|>R} (e C|x| -1-1)dx and for R > 0 chosen large enough we have C α {|x|>R} (e C|x| -1 -1)dx < ε 2 .

{|x|<R}H(u n )dx {|x|<R} e αu 2 nLemma 16 .

 216 dx < M for all n ∈ N In particular, then H(u n ) is bounded in L r (|x| < R) for any 1 < r < +∞. Hence (ii) holds by de La Vallée Poussin equiintegrability lemma. The functional P (v) = R N G(v)dx is of class C 1 and upper weakly semi-continuous in H 1 (R N ).

Thus, for all µ > 0, there exists Λ µ such that

and therefore (32)

Finally, from (31)-(32), for λ, µ > 1 close enough to 1 we get the desired result.

Proof of Theorem 3. Let ε > 0 and ϕ ε given in Lemma 13. Then (ϕ ε , 0) satisfies E(ϕ ε , 0) = S(ϕ ε ) < m and P (ϕ ε ) > 0, namely (ϕ ε , 0) ∈ I. Theorem 3 follows now from Proposition 12.

Appendix

Lemma 14. Let m denote the least energy level defined in (4) and c the mountain pass level defined in [START_REF] Byeon | Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimensional cases[END_REF]. Then m = c.

S(γ(t)) = m

where m is the least energy level among real valued solutions of (1). From [START_REF] Berestycki | Équations de champs scalaires euclidiens non linéaires dans le plan[END_REF][START_REF] Berestycki | Nonlinear scalar field equations I[END_REF][START_REF] Cingolani | Multi-peak solutions for magnetic NLS equations without non-degeneracy condition[END_REF] we know that m = m. Also trivially c c. Now for each γ ∈ Γ we observe that setting γ(t) = |γ(t)| one has

Thus γ ∈ Γ and S(γ) S(γ). This show that c c and ends the proof. Now we prove the upper weakly semicontinuity of P . We begin by a convergence lemma Lemma 15. Let H ∈ C(R, R) be such that (H1) For all α > 0 there exists C α > 0 such that |H(s)| C α (e αs 2 -1) for all s 1,


P Van Isacker

S Heinze

Bose-Einstein condensates of atoms with arbitrary spin

Keywords: numbers: 03, 75, Fi, 03, 65, Fd

Bose-Einstein condensates of atoms with arbitrary spin

If atoms in a Bose-Einstein condensate (BEC) are trapped by optical means [1], their hyperfine spins (or spins) are not frozen in one particular direction but are essentially free but for their mutual interactions. As a result, the atoms do not behave as scalar particles but each of the components of the spin is involved in the formation of the BEC. This raises interesting questions concerning the structure of the condensate and how it depends on the spin exchange interactions between the atoms. Such questions were addressed in a series of theoretical papers by Ho and coworkers [2] who obtained solutions based on a generating function method. In the case of spin-1 atoms the problem of quantum spin mixing was analyzed by Law et al. [3] who proposed an elegant solution based on algebraic methods. It is the purpose of this paper to point out that a wide class of many-body hamiltonians appropriate for the problem of interacting bosons with spin can be solved through algebraic techniques which have found fruitful applications in nuclear physics [START_REF] Iachello | The Interacting Boson Model[END_REF] as well as in other fields of physics (see, e.g. Ref. [START_REF] Iachello | Algebraic Theory of Molecules[END_REF]). The main result derived in this paper is that an exact solution is available for spin values f = 1 and f = 2 (for any value of the number of atoms N) which allows the analytic determination of the structure of the ground state of the condensate. For spin values f > 2 solvable classes of hamiltonians give insights into the generic properties of the phase diagram.

We consider a one-component dilute gas of trapped bosonic atoms with arbitrary (integer) hyperfine spin f . In second quantization the hamiltonian of this system has a one-body and a two-body piece that can be written as (we follow the notation of Ref. [3])

H ≡ H 1 + H 2 = m Ψ † m - ∇ 2 2M a + V trap Ψm d 3 x + m i Ω m 1 m 2 m 3 m 4 Ψ † m 1 Ψ † m 2 Ψm 3 Ψm 4 d 3 x, (1)
where h = 1, M a is the mass of the atom, and Ψm and Ψ † m are the atomic field annihilation and creation operators associated with atoms in the hyperfine state |f m with m = -f, . . . , +f , the possible values of all summation indices in (1). The trapping potential V trap is assumed to be the same for all 2f + 1 components. The coefficients Ω m 1 m 2 m 3 m 4 follow from the interaction between atoms which is assumed to be of shortrange, two-body character,

U( x i , x j ) = δ( x i -x j ) F M ν ′ F |f 2 ; F M f 2 ; F M|, (2)
where |f 2 ; F M is the combined state of the atoms i and j with total spin F , and ν ′ F ≡ 4πh 2 a F /M a with a F being the s-wave scattering length in the F channel. The assumption underpinning the form ( 2) is rotational invariance of the hamiltonian in hyperfine-spin space. We assume in this paper that the scattering lengths in the different F channels are comparable and that, in first approximation, the interaction strength between the bosons is independent of F . In that case the dominant part of the hamiltonian (1) is of the form

H s = H 1 + λ ′ m 1 m 2 Ψ † m 1 Ψ † m 2 Ψm 1 Ψm 2 d 3 x, (3)
and is symmetric under any interchange of the spin-component indices. Under this assumption the condensate wave functions for each spin component φ m ( x) (m = -f, . . . , +f ) can be approximated by a single wave function φ( x) which satisfies the Gross-Pitaevskii equation associated with the dominant hamiltonian [3]. Furthermore, the atomic field creation and annihilation operators at zero temperature can be approximated by

Ψ † m ≈ b † m φ( x), Ψm ≈ b m φ( x), m = -f, . . . , +f, (4)
where b m and b † m are annihilation and creation operators associated with the entire condensate, satisfying the usual boson commutation rules

[b m , b † m ′ ] = δ mm ′ , [b m , b m ′ ] = [b † m , b † m ′ ] = 0. ( 5
)
In this approximation the entire hamiltonian (1) can be rewritten as

H ≈ Ĥ ≡ ǫ b † • b + 1 2 F ν F [b † × b † ] (F ) • [ b × b] (F ) , (6)
where the coefficients ǫ and ν F are related to those in the original hamiltonian through integration over x, viz.

ν F = ν ′ F |φ( x)| 4 d 3 x.
The notation × in Eq. ( 6) implies the coupling to a given spin F and projection M,

[b † × b † ] (F ) M = mm ′ f m f m ′ |F M b † m b † m ′ , (7)
where [START_REF] Edmonds | Angular Momentum in Quantum Mechanics[END_REF]. Furthermore, the dot • denotes a scalar product,

• • • • | • • is a Clebsch-Gordan coefficient
T F • T F ≡ (-) F √ 2F + 1[ T F × T F ] (0) 0 , (8)
for tensor operators T F M of rank F . The definition of the adjoint operator bm ≡ (-) f -m b -m ensures that bm is an annihilation operator with transformation properties under rotations that are the same as those for the creation operator b † m [START_REF] Iachello | Lie algebras and applications[END_REF]. With the above definitions we have that b

† • b = m b †
m b m is the number operator N which counts the total number of atoms in the condensate.

To derive the solvability properties of the hamiltonian (6), we first determine its algebraic structure by introducing the bilinear operators b † m b m ′ . From Eq. ( 5) one finds the commutation relations

[b m 1 b † m 2 , b m 3 b † m 4 ] = b m 1 b † m 4 δ m 2 m 3 -b m 3 b † m 2 δ m 1 m 4 , (9)
which can be identified as those of the unitary (Lie) algebra U(2f + 1) [START_REF] Iachello | Lie algebras and applications[END_REF]. Exactly solvable hamiltonians with rotational or SO(3) invariance are now found by the determination of all Lie algebras

G satisfying U(2f + 1) ⊃ G ⊃ SO(3). The canonical reduction of U(2f + 1) is of the form U(2f + 1) ⊃ SO(2f + 1) ⊃ SO(3). (10)
[For f = 3 there is an additional exceptional G 2 algebra between SO(2f + 1) and SO(3) which for the symmetric representations of U(2f + 1) considered here does not add anything to the discussion.] The relevance of a chain of nested algebras of the type [START_REF] Talmi | Simple Models of Complex Nuclei.The Shell Model and Interacting Boson Model[END_REF] is that it defines a set of commuting operators and with it a class of solvable hamiltonians. Consider in particular the hamiltonian

Ĥ′ = a 1 Ĉ1 [U(2f + 1)] + a 2 Ĉ2 [U(2f + 1)] + b Ĉ2 [SO(2f + 1)] + c Ĉ2 [SO(3)], (11)
where a 1 , a 2 , b, and c are numerical coefficients and Ĉn [G] is the n th -order Casimir operator of the algebra G which satisfies the property that it commutes with all generators of G [START_REF] Wybourne | Classical Groups for Physicists[END_REF]. Solvability of the hamiltonian [START_REF] Gheorghe | [END_REF] follows from the fact that it is written as a sum of commuting operators, a property which indeed is valid for the Casimir operators associated to any chain of nested algebras such as [START_REF] Talmi | Simple Models of Complex Nuclei.The Shell Model and Interacting Boson Model[END_REF]. The Casimir operators appearing in Eq. ( 11) are known in closed form,

Ĉ1 [U(2f + 1)] = N, Ĉ2 [U(2f + 1)] = N( N + 2f ), Ĉ2 [SO(2f + 1)] = -(2f + 1) T 0 + • T 0 -+ N ( N + 2f -1), Ĉ2 [SO(3)] = F 1 2 F (F + 1) -f (f + 1) T F + • T F -+ f (f + 1) N, ( 12
)
in terms of the operators

T F +,M ≡ [b † × b † ] (F ) M and T F -,M ≡ [ b × b] (F )
M . Equations (12) show that the solvable hamiltonian ( 11) is a special case of the general hamiltonian [START_REF] Edmonds | Angular Momentum in Quantum Mechanics[END_REF] with coefficients ǫ and ν F that are linear combinations of a 1 , a 2 , b, and c according to

ǫ = a 1 + (2f + 1)a 2 + 2f b + f (f + 1)c, ν F = 2a 2 + 2b + [F (F + 1) -2f (f + 1)]c, F = 0, ν 0 = 2a 2 -4f b -2f (f + 1)c, ( 13
)
The eigenvalues of the hamiltonian ( 11) are

E ′ (N, v, F ) = a 1 N + a 2 N(N + 2f ) + b v(v + 2f -1) + c F (F + 1). ( 14
)
The allowed values of v are v = N, N -2, . . . , 1 or 0, as can be obtained from the U(2f + 1) ⊃ SO(2f + 1) branching rule [START_REF] Wybourne | Classical Groups for Physicists[END_REF]. The quantum number v corresponds to the number of bosons not in pairs of bosons coupled to F = 0, and is known as seniority [START_REF] Racah | [END_REF][START_REF] Talmi | Simple Models of Complex Nuclei.The Shell Model and Interacting Boson Model[END_REF]. The allowed values of the total spin F are obtained from the SO(2f + 1) ⊃ SO(3) branching rule which is rather complicated but known in general [START_REF] Gheorghe | [END_REF]. The f = 2 example is discussed below.

The generic solvability properties of the original hamiltonian (6) now follow from a simple counting argument. For atoms with spin f = 1 the solvable hamiltonian [START_REF] Gheorghe | [END_REF] has three coefficients a 1 , a 2 , and c [since SO(2f +1)=SO(3)] while the general hamiltonian (6) also contains three coefficients ǫ, ν 0 , and ν 2 . [Note that the coupling of two spins to odd F is not allowed in the approximation (4) of a common spatial wave function, so no ν 1 term occurs.] For atoms with spin f = 2 both the solvable and general hamiltonian contains four coefficients (a 1 , a 2 , b, and c versus ǫ, ν 0 , ν 2 , and ν 4 ) which can be put into one-to-one correspondence. Hence the general hamiltonian ( 6) is solvable for f = 2. The same counting argument shows that it is no longer solvable for f > 2.

The case of interacting f = 1 atoms was discussed by Law et al. [3] who identified the existence of two possible condensate ground states: one with all atoms aligned to maximum spin F = N and a second with pairs of atoms coupled to F = 0. Whether the condensate is aligned or paired depends on a single interaction parameter which in our notation is c. With the technique explained above we can also derive the phase diagram for atoms with spin f = 2. The results are exact and valid for arbitrary N. The entire spectrum is determined by the eigenvalue expression [START_REF] Frank | Algebraic Methods in Molecular and Nuclear Structure Physics[END_REF] together with the necessary branching rules. In particular, the allowed values of total spin F for a given seniority v are derived from the SO(5) ⊃ SO(3) branching rule [START_REF] Iachello | The Interacting Boson Model[END_REF] given by F = 2τ, 2τ -2, 2τ -3, . . . , τ + 1, τ with τ = v, v -3, v -6, . . . and τ ≥ 0.

It is now possible to determine all possible ground-state configurations of the condensate. This problem has been considered in the study of the spectral features of quantal systems with random interactions [12]. We note that the character of the ground state does not depend on the coefficients a i since the first two terms in the expression [START_REF] Frank | Algebraic Methods in Molecular and Nuclear Structure Physics[END_REF] give a constant contribution to the energy of all states. Although this contribution is dominant according to our earlier assumptions, the spectrum generating perturbation of the hamiltonian is confined to the last two terms and depends solely on the coefficients b and c which are related to the original interactions ν F according to with spin f = 2 characterized by a ground state (v 0 , F 0 ) where v 0 is the seniority of the ground state and F 0 is its total spin. The total number of atoms N is even in (a) and odd in (b). The grey area corresponds to a ground state with (v 0 , F 0 ) = (N -3 + δ, 0) which only occurs for δ = ±1 and disappears in the limit N → ∞.

b = 1 70 (-7ν 0 + 10ν 2 -3ν 4 ), c = 1 14 (-ν 2 + ν 4 ), (15)
The following exact finite-N results are found where the ground state of the condensate is characterized by a seniority v 0 and a total spin F 0 .

(i) N is even. We introduce N = 6k + 2δ with k integer and δ = -1, 0, +1. The possible ground-state configurations have (v 0 , F 0 ) = (0, 0), (N, 2N), (N, 2|δ|), or (N -3 + δ, 0), the latter existing only for δ = ±1, (ii) N is odd. We introduce N = 6k + 3 + 2δ with k integer and δ = -1, 0, +1. The possible ground-state configurations have (v 0 , F 0 ) = (1, 2), (3, 0), (N, 2N), (N, 2|δ|), or (N -3 + δ, 0), the latter existing only for δ = ±1.

The phase diagram displays a richer structure than in the f = 1 case as is shown in Fig. 1. We observe first of all the presence of the aligned phase where the seniority is maximal, v 0 = N, and all spins are aligned, F 0 = 2N. Secondly, we have a low-seniority (paired) and consequently low-spin phase. For even N this corresponds necessarily to (v 0 , F 0 ) = (0, 0). For odd N there must be at least one unpaired atom leading to the ground-state configuration (v 0 , F 0 ) = (1, 2); alternatively, however, it might consist of a triplet of atoms which is coupled to total spin F 0 = 0 leading to the ground-state configuration (v 0 , F 0 ) = (3, 0). The

= - 2N(2N + 1) N(N + 3) c, b = - (2N -2)(2N + 3) (N -1)(N + 4) c, (16)
for N even or odd respectively, which in both cases tends to b = -4c for N → ∞. So far we have recovered the aligned and paired phases also encountered for interacting f = 1 atoms (although the paired phase is somewhat more intricate for f = 2 due to the possible presence of a triplet of atoms coupled to F = 0). For f = 2 a third phase occurs for negative b and positive c characterized by high seniority (i.e. unpaired) and low total spin, (v 0 , F 0 ) = (N, 2|δ|). Finally, for δ = ±1 there exists a pathological region in the phase diagram characterized by (v 0 , F 0 ) = (N -3 + δ, 0) (see Fig. 1). It is separated from the high-seniority, low-spin region by the line

b = - |δ(δ + 3)| 4(2N + δ) c, (17)
which tends to b = 0 for N → ∞. Hence this region disappears in the large-N limit.

We conclude that the ground state of a BEC consisting of atoms with spin f = 2 can be of three different types: (i) a maximum-seniority spin-aligned, (ii) a low-seniority low-spin, or (iii) a maximum-seniority low-spin configuration. Note that 'seniority' in this context refers to number of atoms that are not in pairs coupled to F = 0.

Since the hamiltonian ( 11) is solvable for f = 2, all eigenstates, and in particular the three different ground states, can determined analytically. The general expressions given by Chacón et al. [13] reduce to

|v = N, F = M = 2N ∝ d † +2 N |0 , |v = 0, F = M = 0 ∝ d † • d † N/2 |0 , |v = N, F = M = 0 ∝ [a † × a † ] (2) • a † N/3 |0 , (18)
where the f = 2 atoms are denoted as d bosons. In the second of these expressions it is assumed that N is even and in the third that N = 3k; other cases are obtained by adding a single boson or an F = 0 pair. The a † are the so-called traceless boson operators [13] which are defined as (see also Chapt. 8 of Ref. [START_REF] Frank | Algebraic Methods in Molecular and Nuclear Structure Physics[END_REF])

a † m = d † m - d † • d † 2N + 5 dm . (19)
We emphasize that (18) are the exact finite-N expressions for the eigenstates of the hamiltonian [START_REF] Gheorghe | [END_REF]. Since in the large-N limit the traceless boson operators a † m become identical to d † m , we arrive at a simple interpretation of the three types of configurations: (i) spin-aligned, (ii) condensed into pairs of atoms coupled to F = 0, and (iii) condensed into triplets of atoms coupled to F = 0.

How will these features evolve with increasing spin f of the atoms? For arbitrary interaction strengths ν F in the different F channels the hamiltonian ( 6) is not solvable. By imposing f -2 conditions on ν F it can be brought into the form [START_REF] Gheorghe | [END_REF] and this gives an idea of the structure of the general phase diagram by constructing a two-dimensional slice of it. For example, for atoms with spin f = 3 the elimination of a 1 , a 2 , b, and c from Eq. ( 13) yields the condition 11ν 2 -18ν 4 + 7ν 6 = 0. For f > 3 more conditions on ν F are found. If all conditions are satisfied, the phase diagram in b and c with b

= -7ν 0 + 10ν 2 -3ν 4 14(2f + 1) , c = 1 14 (-ν 2 + ν 4 ), (20)
has properties similar to those in the f = 2 case. The analysis requires the knowledge of the multiplicity d (f ) v (F ), (i.e., the number of spin-f atom states with seniority v coupled to total spin F ) which can be derived from the SO(2f + 1) ⊃ SO(3) branching rule [START_REF] Gheorghe | [END_REF]. We find that for sufficiently large even N there are four competing ground states with (v 0 , F 0 ) = (N, f N), (N, 0), (0, 0), and (2, 2), the latter of which disappears as a ground state in the large-N limit. For sufficiently large odd N the four competing ground states have (v 0 , F 0 ) = (N, f N), (N, 0), (1, f ), and (3, f 2 ≡ f mod 2), the latter two being separated by the line b = [f (f + 1) -f 2 (f 2 + 1)]c/(4f + 6). The results correspond to what is found in the f = 2 case and lead to an essentially identical (b, c) phase diagram.

Finally, we point out that the appearance of exact seniority ground states requires weaker conditions on ν F than those that have been discussed so far. In fact, the spinaligned configuration (N, f N) is always an eigenstate of the general hamiltonian [START_REF] Edmonds | Angular Momentum in Quantum Mechanics[END_REF] because the F = f N state is unique. Furthermore, it can be shown [START_REF] Talmi | Simple Models of Complex Nuclei.The Shell Model and Interacting Boson Model[END_REF] that seniority is a good quantum number if the interaction strengths ν F satisfy ⌊f /3⌋ conditions only (where ⌊x⌋ is the largest integer smaller than or equal to x). For all cases of any conceivable interest for BECs, this reduces to no condition on the strengths ν F for f = 1, 2 or just a single one for f = 3, 4, 5. So there is at most a single condition required for all eigenstates to carry exact seniority and for the results of this paper to be valid. Nevertheless, the determination of the complete phase diagram for f > 2 with unconstrained interaction strengths ν F remains a problem worthy of further investigation.

Figure 1 .

 1 Figure1. Diagrams of the different phases of a Bose-Einstein condensate of atoms with spin f = 2 characterized by a ground state (v 0 , F 0 ) where v 0 is the seniority of the ground state and F 0 is its total spin. The total number of atoms N is even in (a) and odd in (b). The grey area corresponds to a ground state with (v 0 , F 0 ) = (N -3 + δ, 0) which only occurs for δ = ±1 and disappears in the limit N → ∞.

  (1,2) and (3,0) phases are divided by the line b = 3c/7. The paired and aligned phases are separated by the line b

Acknowledgments

We acknowledge a conversation with H.T. Stoof that prompted our interest in this problem.


Wolf Marc
email: marc.wolf@tsoftemail.com

Wolf François
email: francois.wolf@dbmail.com

Keywords: composite odd numbers, indices, sequences with arithmetic differences, finite sequence, differences of two squares, odd primes, not appear randomly, reference points, remarkable index

We study the composite odd numbers via their indices with a function ranging through all of them. This mapping allows us to demonstrate that the set of these indices are described with two families of finite sequences with arithmetic differences. Composite odd numbers are then shown to be obtained as differences of two squares. We then conjecture odd primes do not appear randomly.

Contents

Introduction

The distribution of prime numbers may seem random. However, in 2016, Robert Lemke Oliver and Kannan Soundararajan [START_REF] Lemke | Unexpected biases in the distribution of consecutive primes[END_REF] of Stanford University (California) discovered that links exist between consecutive prime numbers. Prime numbers might be distributed according to a structure, as well as composite odd numbers. We will demonstrate in this paper that the set of composite odd numbers is characterized by two families of finite sequences with arithmetic differences. The study is performed by indexing the odd numbers greater than 1 with the family , .

The integers are represented by quadratic forms [START_REF] Berkovich | On representation of an integer as the sum of three squares and ternary quadratic forms with the discriminants p 2 , 16p 2[END_REF] [3] [4] [START_REF] Ramanujan | On the expression of a number in the form ax2 + by2 + cz2 + du2[END_REF]. From the two precedent sequences, we deduce that every composite odd number is a difference of two squares.

I-Indices of composite odd numbers: the set

We will define the function mapping to the set of indices of composite odd numbers. These indices form the set . We will show that these indices are obtained with a sequence of arithmetic sequences.

I.1 Preliminaries

Introduction of some notations:

1. Let be the set of odd integers greater than 1, i.e.:

2. Let be the set of prime numbers.

3. Let be the set of composite odd integers greater than 1, i.e.:

The function is bijective. The inverse function is

. The preimage of the set is named :

I.2 The map as the image of the function

By definition, there is a surjection of on:

We match this surjection with a surjection of over the set of indices of composite odd numbers:

Proposition/definition 1.2: Let be defined by: . This map returns the index of a composite odd number . More precisely, is surjective on :

In other words, when and when . Moreover, we have:

. Proof: Let us note first that et . Proof: with definition 1.3.1 we get for all :

So is the union of ranges of sequences for varying in .

Some definitions

Illustration 1: In a coordinate system ( ), sequences are plotted for = 0 to 3 and values are represented by points with positive integer abscissa on the line . 3. For a given , the set is the range of a sequence of indices associated with a particular set of points. In the set , each index is associated with a unique point with second coordinate .

I.4 Remarkable index of the sets of indices

Among the composite odd numbers, the squares will be said to be remarkable. By extension, their index and the matching point are also said to be remarkable. They will be used in the proof of the main theorem as reference points.

Proposition/definition 1.4 : In the set , there is an unique index corresponding to the number that is:

The matching remarkable point is .

Proof: We have: So is the remarkable index associated with remarkable point . We compute:

II-Representation theorem of composite odd numbers indices

We will show that the set of indices of composite odd numbers is described by two families of finite sequences with arithmetic differences via two families of reference points.

II.1 Some definitions

Definition 2.1:

a-We name finite sequence with arithmetic difference any finite sequence verifying:

b-A reference point of the first, second and third type is a point of the form , and respectively. We simply call reference point a reference point of any of these three types. c-We say that the point precedes the point (provided that ), while the point succeeds it (provided that ). Any given point has a finite number of predecessors and successors.

Remark 1: reference points of the first type are therefore remarkable points.

Remark 2: we can notice that a sequence with arithmetic difference can also be described as a quadratic progression.

II.2 Sequences with arithmetic difference and symmetries

We will demonstrate in this section that the indices of points that precede or succeed a remarkable point form a finite sequence with arithmetic difference.

II.2.1 Sequence of indices came from the remarkable index

Proposition 2-2-1: Let be the sequence of predecessors of . Then indices form a finite sequence with arithmetic difference. Moreover, if is a remarkable point, then:

Proof: Predecessors of can be written as: . So

In particular, one gets: . So the sequence of predecessors is a sequence with arithmetic difference. Moreover, if the point is remarkable then and:

Corollary 2-2-1: The sequence of the indices of the successors of a point is also a sequence with arithmetic difference. If this point is remarkable, then the sequence is actually the same sequence as that given by predecessors:

Remark: There is therefore a symmetrical relationship between the predecessors and successors of a remarkable point. This is shown in the graph below for :

II.2.2 Second and third type reference points Proposition 2-2-2:

The sequence of predecessors indices of a reference point of the second type is given by: This sequence coincides with that of the successors of a reference point of the third type:

Proof: We first check that . Indeed, by developing these two expressions, on one hand we get: And on the other hand:

Using the calculations from the prior proposition, we get: hence, with a reference point of second type :

Symmetry axis

Successors & predecessors of k4 [START_REF] Ramanujan | On the expression of a number in the form ax2 + by2 + cz2 + du2[END_REF] and with a reference point of third type :

Remark: Similarly, indices of second type reference points successors coincide with those of third type reference points predecessors. There is a symmetry as previously, but with noninteger axis.

II.3 Representation theorem of composite odd numbers indices

Proposition 2.3: Any element from the set is the index of a predecessor of a reference point of the first or second type.

Proof: Let be an element of . We will first assume . If is odd then there exists an integer such that . In particular, if we let , then we get: so is the index of the -th predecessor of the remarkable point with index . On the other hand, if is even (and therefore non-zero since ) then we write and we let . Then: so is the index of the -th predecessor of the reference point associated with the index . Now, let us assume . If then it is straightforward to get that is the index of a reference point of the third type. It is therefore the index of a reference point of the second type as well as shown in proposition 2-2-2. If then we get that the point succeeds the remarkable point , its index is therefore the same as that of the predecessor of this remarkable point according to proposition 2-2-1. More generally, if we can use (cf. proposition 1.2) to fall back on the initial case. Remark: Thanks to the symmetries described in the previous section, we obtain in the same way that every element of is the index of a successor of a reference point of the first or third type.

Representation theorem of composite odd numbers indices:

The set is thus obtained as the ranges of two families of finite sequences with arithmetic difference. More precisely: with:

Conclusion

We indexed odd numbers to describe properties of composite odd numbers. First, we described the set of indices of composite odd numbers as the image of a function with two integer variables, or equivalently as the countable union of arithmetic sequence ranges.

Then, we improved this description with finite sequences with arithmetic difference. Thanks to symmetries underlying function and reference points which correspond to the centers of these symmetries, we showed that we needed only two families of sequences: one for odd indices and the other for even indiceswhich constitutes our representation theorem of composite odd numbers indices. We then observed that this representation was equivalent to writing the composite odd numbers as differences of two squares.

Composite odd numbers indices are therefore fully described by two families of finite sequences with arithmetic differences. It therefore seems to us that the set of odd primes, complement of in , presents a structure that is not random.

  Introduction .............................................................................................................................. I-Indices of composite odd numbers: the set .................................................................... I.1 Preliminaries ...................................................................................................................... I.2 The map as the image of the function ....................................................................... I.3 The map as a union of arithmetic sequence supports ................................................... 1.3.1 The sets of indices ...................................................................................................... 1.3.2 A sequence of arithmetic sequences ......................................................................... 1.3.3 Some definitions ........................................................................................................ I.4 Remarkable index of the sets of indices ............................................................................ II-Representation theorem of composite odd numbers indices .......................................... II.1 Some definitions .............................................................................................................. II.2 Sequences with arithmetic difference and symmetries .................................................... II.2.1 Sequence of indices came from the remarkable index .............................................. II.2.2 Second and third type reference points ..................................................................... II.3 Representation theorem of composite odd numbers indices ............................................ II.4 Composite odd numbers as differences of two squares ................................................... Conclusion ............................................................................................................................... References: ..............................................................................................................................

Definition 1 .

 1 3.3 : 1. A point designates a pair of integer values . 2. We call the index of the point . Each index is associated with one or more points.

Acknowledgments: We would like to thank François-Xavier Villemin for his attentive comments and suggestions.

The indices of first type reference points (or remarkable points) are thus the initial terms of sequences ranging in . Similarly, the indices of second type reference points are the initial terms of sequences ranging in .

II.4 Composite odd numbers as differences of two squares

Proposition 2-4: is the set of odd indices from .

On the contrary, is the set of even indices from .

In particular .

Proof: It is sufficient to check that the indices of the remarkable points are odd, and those of the reference points of the second type are even. Let us remember that . In particular, the parity of is the same as that of , which completes the proof.

Remark: The index is even if and only if the odd number is congruent to 3 modulo 4, otherwise it is congruent to 1.

Proposition 2-5: Any composite odd number can be written as a difference of two squares. Proof: Let . So .

If then with and . Thus . If then whence .

Corollary 2-5:

There is an infinite number of odd squares that can be written as the sum of two squares.

Proof: Let consider a non-remarkable point with same index as a remarkable point . To prove that such a point exists, it is enough to take since we can then write: thus: which corresponds to the point with and .

The proof of proposition 2.3 then allows us to state that is the strict predecessor of a reference point, and the proposition 2.4 allows us to specify that the reference point is necessarily that of the first type and so a remarkable point. Therefore, using the proof of proposition 2.5, we can conclude that there are two positive integers and such that .

Hence: .

Since we have and we could choose any , we deduce that there is also an infinite number of possible values for .

Example:

. Thus is the smallest square equal to a sum of two squares we can obtain with this method.