Sergio Aguilar
email: sergio.aguilar.romero@upc.edu

Patrick Maill
email: patrick.maille@imt-atlantique.fr

Laurent Toutain
email: laurent.toutain@imt-atlantique.fr

Carles Gomez
email: carlesgo@entel.upc.edu

Rafael Vidal
email: rafael.vidal@entel.upc.edu

Nicolas Montavont
email: nico-las.montavont@imt-atlantique.fr

Georgios Z Papadopoulos
email: georgios.papadopoulos@imt-atlantique.fr

G Z Papadopou- Los

Performance Analysis and Optimal Tuning of IETF LPWAN SCHC ACK-on-Error Mode

Keywords: LPWAN, SCHC, ACK-on-Error, LoRaWAN, Sigfox, IoT, LoRa, IETF, mathematical model, fragmentation

The Internet Engineering Task Force (IETF) Low Power Wide Area Network (LPWAN) Working Group has developed the Static Context Header Compression (SCHC) framework to enable IPv6 over LPWAN. In order to support 1280-byte packets, as required for IPv6, SCHC includes a fragmentation functionality, since relevant LPWAN technologies offer very short data unit sizes and do not provide native fragmentation mechanisms. SCHC offers 3 fragmentation modes: No-ACK, ACK-Always, and ACK-on-Error, the latter being especially promising due to its reliability and high efficiency. In this article, we develop a mathematical model to compute the most critical performance parameters for the SCHC ACK-on-Error mode, namely, the acknowledgment traffic incurred by a fragment receiver for the successful delivery of a fragmented packet. The model is used to evaluate the SCHC ACKon-Error mode performance, as well as to optimally tune its main parameters when used over LoRaWAN and Sigfox, for different packet sizes.

I. INTRODUCTION

L OW Power Wide Area Networks (LPWANs) refer to network technologies designed for the Internet of Things (IoT) that are characterized by a long-range and low-energy operation [START_REF] Farrell | Low-Power Wide Area Network (LPWAN) Overview[END_REF][START_REF] Davcev | IoT agriculture system based on LoRaWAN[END_REF][START_REF] Grunde | LoPy as a building block for Internet of Things in coastal marine applications[END_REF]. LPWAN technologies are based on star topology deployments, where a potentially high number of Internet of Things (IoT) devices are directly connected to a radio gateway.

In some quintessential LPWAN technologies, such as Lo-RaWAN and Sigfox, the layer 2 Maximum Transmission Unit (L2 MTU) ranges from tens to hundreds of bytes [START_REF] Farrell | Low-Power Wide Area Network (LPWAN) Overview[END_REF]. To carry IPv6 packets over LPWAN despite those limitations, the Internet Engineering Task Force (IETF) has defined a new adaptation layer called Static Context Header Compression (SCHC) [START_REF] Minaburo | LPWAN Static Context Header Compression (SCHC) and fragmentation for IPv6 and UDP," Internet-Draft draft-ietf-lpwan-ipv6-static-context-hc-18.txt[END_REF], [START_REF] Gomez | IPv6 over LPWANs: connecting Low Power Wide Area Networks to the Internet (of Things)[END_REF], which along with a header compression functionality, provides fragmentation mechanisms to transport an IPv6 packet over several LPWAN frames. SCHC defines 3 Fragmentation/Reassembly (F/R) modes called No-ACK, ACK-Always and ACK-on-Error. This paper focuses on the ACK-on-Error mode, which is promising due to its reliability and high efficiency by minimizing the number of acknowledgments (ACKs) compared to ACK-Always. In this mode, ACKs are sent by the fragment receiver only when the latter detects fragment losses. Then, the fragment sender selectively retransmits any lost fragments reported in the ACKs. To maintain consistency, a final ACK is also unconditionally sent at the end of the fragmented packet transmission.

A key constraint for LPWAN is the amount of downlink traffic, i.e., from the gateway to the IoT device. Indeed, in the spectrum band used by unlicensed LPWAN technologies in Europe (e.g. the 868 MHz band), each device must respect a duty-cycle limit, that can be especially binding for gateways, which may manage many flows. Even if the downlink traffic is not limited by the duty-cycle constraint, one may still want to minimize it for economic reasons: the amount of downlink traffic is now used by some operators as a basis for charging IoT users in some plans 1 or is limited [6]. Noting that IoT flows are mostly uplink flows (e.g., from the IoT device, say a sensor, sending its data readings), the focus in this paper is on the ACK traffic incurred by the SCHC F/R processthe main expected type of downlink traffic-which needs to be minimized in order to reduce costs and/or respect gateway duty-cycle constraints [START_REF] Adelantado | Understanding the limits of LoRaWAN[END_REF].

More specifically, in this paper we propose a mathematical model to compute the volume of ACK traffic (relative to the IoT device data volume) based on the quality of the radio link and the SCHC F/R parameters. We then use the model to optimize those parameters in order to minimize the ACK traffic. For a direct practical use of our results, we provide the optimal parameter values for the specifics of LoRaWAN [START_REF]LoRaWAN 1.1 Specification[END_REF] and Sigfox [START_REF] Sigfox | Sigfox connected objects: Radio specifications[END_REF] technologies.

The remainder of this paper is organized as follows. In Section II we first present the existing works related to LPWAN fragmentation and its performance. We then detail the SCHC framework in Section III, and the SCHC F/R modes in Section IV, especially focusing on the ACK-on-Error mode. The mathematical model used to analyze the ACK-on-Error mode is developed in Section V, and is it used in Section VI to evaluate the performance metrics, mostly regarding the amount of ACK messages. Section VII explains how our results can be applied to state-of-the-art LPWAN technologies such as LoRaWAN and Sigfox. Finally, we provide some conclusions in Section VIII.

II. RELATED WORK

Recent attention on LPWAN technologies has partly focused on the evaluation of the physical layers [6], [START_REF] Adelantado | Understanding the limits of LoRaWAN[END_REF], [START_REF] Casals | Modeling the energy performance of LoRaWAN[END_REF], [START_REF] Lavric | Long Range SigFox Communication Protocol Scalability Analysis Under Large-Scale, High-Density Conditions[END_REF]. In [START_REF] Adelantado | Understanding the limits of LoRaWAN[END_REF] the authors analyze LoRaWAN and explore its limitations. The results show that the application and network design must minimize the number of acknowledged frames to avoid capacity drain, because the LPWAN gateway must enforce a time-off following the transmission to comply with dutycycle regulations. Other works provide a mathematical model that characterizes LoRaWAN and Sigfox end-device energy consumption, lifetime and energy involved in data delivery [6], [START_REF] Casals | Modeling the energy performance of LoRaWAN[END_REF]. In [START_REF] Lavric | Long Range SigFox Communication Protocol Scalability Analysis Under Large-Scale, High-Density Conditions[END_REF] a performance evaluation of Sigfox scalability is presented. Together, these studies provide important insights into the physical layer of LPWAN technologies, but do not consider the compatibility with IPv6, nor fragmentation mechanisms.

Several studies compare different LPWAN technologies [START_REF] Mroue | MAC layer-based evaluation of IoT technologies: LoRa, SigFox and NB-IoT[END_REF][START_REF] Osman | Simulation and Modelling of LoRa and Sigfox Low Power Wide Area Network Technologies[END_REF][START_REF] Mekki | Overview of Cellular LPWAN Technologies for IoT Deployment: Sigfox, LoRaWAN, and NB-IoT[END_REF]. While Mroue et al. [START_REF] Mroue | MAC layer-based evaluation of IoT technologies: LoRa, SigFox and NB-IoT[END_REF] perform an evaluation using the packet error rate for Sigfox, LoRa and NB-IoT, a comprehensive and comparative study for a number of performance metrics is presented in [START_REF] Mekki | Overview of Cellular LPWAN Technologies for IoT Deployment: Sigfox, LoRaWAN, and NB-IoT[END_REF]. The study in [START_REF] Osman | Simulation and Modelling of LoRa and Sigfox Low Power Wide Area Network Technologies[END_REF] evaluates, by simulation, the influence of the number of devices, on the packer error rate, collisions and spectrum utilization for Sigfox and LoRa. None of these studies address the problem of transmitting a fragmented IPv6 packet over LPWAN.

Regarding the upper layer functionalities in LPWAN, the study in [START_REF] Suciu | An analysis of packet fragmentation impact in LPWAN[END_REF] evaluates the effect of fragmentation, and its efficiency in terms of energy consumption, throughput, goodput and end time delay in dense networks. Suciu et al [START_REF] Suciu | An analysis of packet fragmentation impact in LPWAN[END_REF] showed that fragmentation increases reliability, especially when sending several fragments instead of only one of the MTU size. Other works focus on IPv6 over LPWAN by means of using SCHC [START_REF] Gomez | IPv6 over LPWANs: connecting Low Power Wide Area Networks to the Internet (of Things)[END_REF], [START_REF] Moons | Using SCHC for an optimized protocol stack in multimodal LPWAN solutions[END_REF][START_REF] Ayoub | Implementation of SCHC in NS-3 Simulator and Comparison with 6LoWPAN[END_REF][START_REF] Abdelfadeel | LSCHC: Layered Static Context Header Compression for LPWANs[END_REF][START_REF] Abdelfadeel | Dynamic Context for Static Context Header compression in LPWANs[END_REF][START_REF] Aguilar | LoRaWAN SCHC Fragmentation Demystified[END_REF]. Some of these propose enhancements to SCHC Header compression, but do not consider SCHC F/R [START_REF] Abdelfadeel | LSCHC: Layered Static Context Header Compression for LPWANs[END_REF], [START_REF] Abdelfadeel | Dynamic Context for Static Context Header compression in LPWANs[END_REF]. On the other hand, Moons et al. [START_REF] Moons | Using SCHC for an optimized protocol stack in multimodal LPWAN solutions[END_REF] compared SCHC and 6LoWPAN compression and fragmentation functionalities. Their results show that SCHC has a smaller footprint, uses less memory and the header overhead is twenty times smaller when compared with 6LoWPAN. Ayoub et al. [START_REF] Ayoub | Implementation of SCHC in NS-3 Simulator and Comparison with 6LoWPAN[END_REF] present an implementation of SCHC using the ns-3 network simulator and also compare SCHC with 6LoWPAN, finding the same performance advantage for SCHC in terms of header overhead. The authors in [START_REF] Gomez | IPv6 over LPWANs: connecting Low Power Wide Area Networks to the Internet (of Things)[END_REF] provided an overview of SCHC and a simple evaluation of the different F/R modes, but it is a superficial study due to its tutorial purpose. In [START_REF] Aguilar | LoRaWAN SCHC Fragmentation Demystified[END_REF], the authors compared the different SCHC F/R modes in terms of total channel occupancy, goodput and total delay at the SCHC layer in an ideal communication channel. The authors showed that, when comparing the reliable SCHC F/R modes, namely, ACK-Always and ACK-on-Error, the latter provided better goodput.

To the best of our knowledge, no previous work provides a mathematical model to estimate the ACK volume and its relation with key configuration parameters of SCHC F/R ACKon-Error mode, nor contribute with configuration guidance based on radio link quality and packet size. We think that this mode is worth analyzing, because it provides reliable communication while minimizing the number of ACKs when compared to ACK-Always.

III. TECHNICAL BACKGROUND: SCHC FRAGMENTATION AND REASSEMBLY

This section provides an overview of SCHC F/R. We first introduce the SCHC adaptation layer, then focus on the main SCHC F/R components and tools.

A. SCHC Adaptation Layer Overview

Flagship LPWAN technologies, such as LoRaWAN and Sigfox, are characterized by a reduced L2 MTU [START_REF] Farrell | Low-Power Wide Area Network (LPWAN) Overview[END_REF]. Furthermore, these technologies do not provide a native fragmentation mechanism for transferring larger packets. The SCHC framework provides header compression and F/R functionalities specifically designed for LPWAN [START_REF] Minaburo | LPWAN Static Context Header Compression (SCHC) and fragmentation for IPv6 and UDP," Internet-Draft draft-ietf-lpwan-ipv6-static-context-hc-18.txt[END_REF]. SCHC defines a set of Rules, each identified with a RuleID, which determine how to perform the compression and fragmentation and allow the sender and receiver to determine the operation mode and configuration parameters.

SCHC is composed of two sublayers, namely the Compression and the Fragmentation sublayers. In SCHC F/R, a SCHC Packet is fragmented into units called tiles. One or more tiles are carried by one SCHC Fragment, which is sent in an LPWAN frame. In some SCHC F/R modes, a determined number of tiles are grouped into a window, and the receiver generates SCHC ACKs to tell the sender which tiles of that window have been received or not. Missing fragments or tiles are retransmitted. Tiles and windows are numbered in a way that each tile can be identified for further retransmissions (see Fig. 2).

SCHC Packet

B. SCHC F/R Messages and Headers

The SCHC framework defines different messages that are used to carry out the SCHC F/R process between the sender and the receiver. The main messages are the SCHC Fragment and the SCHC ACK. Each message has a SCHC F/R Header with the following fields:

• Rule ID: this field identifies whether a SCHC message is a SCHC Fragment. In a SCHC Fragment, it indicates which F/R mode and settings are used. • Datagram Tag (DTag): this field is used to identify-along with the Rule ID-a SCHC Packet. The length of the DTag field is T in bits. 1 if the integrity check of the reassembled SCHC Packet succeeded, and 0 otherwise. A SCHC Fragment carries a part of a SCHC Packet from the sender to the receiver. The FCN field of a SCHC Fragment has all bits set to 1 (it is then called an All-1 SCHC Fragment), to indicate it is the last fragment for the current SCHC Packet; that fragment carries the MIC for this SCHC Packet. Fig. 3 shows a regular and an All-1 SCHC Fragment. L SH is the length of the SCHC Fragment Header in bytes. Padding bits are added at the end of the SCHC Fragment if needed by the LPWAN technology. A SCHC ACK is sent by the receiver to the sender to acknowledge the complete or partial reception of the fragmented SCHC Packet. In the latter case, a SCHC ACK reports whether the tiles of a given window have been received or not, in the form of a bitmap (see section III-E).

C. Tiles

As previously explained (see Fig. 2), a SCHC Packet is fragmented into units called tiles, which have a size of t bytes. In ACK-on-Error mode, each tile must be of the same size, except for the last one, which can be smaller. In the No-ACK and ACK-Always modes, tiles can be of different sizes. If the payload field is present in a SCHC Fragment, it must carry at least one tile. In ACK-on-Error mode, each tile of a SCHC Packet is uniquely identified by the window and tile numbers. The FCN of the SCHC Fragment, together with the window index W , identifies the first tile carried by the SCHC Fragment.

D. Windows

A group of w successive tiles is called a window. Each window in a fragmented SCHC Packet transmission, except the last one, must have the same number of tiles. Windows are numbered from 0 upwards. The window field (W ) has a size of M bits and the window size (window size) has to be less than 2 N -1 (in each window, the tiles are numbered from window size-1 downwards). Fig. 2 shows the fragmentation of a SCHC Packet in 4 windows, with 5 tiles per window.

E. Bitmap

A bitmap is a sequence of bits where each bit indicates the received status of a tile within a specific window. The bitmap has a size of 2 N -1. The rightmost and leftmost bits will correspond to tile numbers 0 and window size -1, respectively. The receiver will set a 1 in the bitmap when the corresponding tile is received successfully and a 0 when the tile was not received, as exemplified in Fig. 5. The C field is set to 0, indicating the packet reassembly was not successful, since the SCHC Packet was not received completely.

SCHC ACK Header Bitmap

Rule ID DTag W C=0 0 1 1 1 1 1 1 padding(as needed) tile number = 6 5 4 3 2 1 0 Fig. 5. Example of a SCHC ACK, where a window of 7 tiles (w = 7) was sent and tile FCN # 6 was not received successfully.

IV. SCHC FRAGMENTATION AND REASSEMBLY MODES

SCHC offers 3 SCHC F/R modes to perform the SCHC F/R process: No-ACK, ACK-Always and ACK-on-Error. If a reliable communication is required, ACK-Always and ACKon-Error use ACKs to support potential retransmission upon failure. This section provides a brief description of No-ACK and ACK-Always modes, and a more detailed explanation of the ACK-on-Error mode.

A. The No-ACK mode

The No-ACK mode provides a mechanism for in-sequence delivery of SCHC Fragments between the sender and the receiver. This mode does not provide reliability when errors are present, since there is no feedback from the receiver and the sender cannot perform SCHC Fragment retransmissions. Variable L2 MTU size is supported. Tiles can be of different sizes, while windows are not used.

B. ACK-Always mode

The ACK-Always mode is a window-based mechanism for in-sequence delivery of SCHC Fragments that supports reliability. At the end of the transmission of each window, a SCHC ACK is sent by the receiver to the sender to report on the tiles received for the current window. The sender only begins the transmission of the next window, once the receiver confirms the correct reception of all tiles of the current window. Variable L2 MTU size is not supported. Tiles can have different sizes.

C. ACK-on-Error mode

The ACK-on-Error mode is a window-based mechanism that supports reliable and out-of-order delivery of SCHC Fragments and variable L2 MTU. This SCHC F/R mode reduces the number of SCHC ACKs, when compared to ACK-Always, since in all windows except for the last one carrying a SCHC Packet, SCHC ACKs are only sent when at least one tile is lost. For the last window, in order to ensure that the sender can expect to receive feedback on the fragmented SCHC Packet transmission, a SCHC ACK is unconditionally sent. All tiles must be of the same size, except for the last one, which can be smaller. One or more tiles can be carried in a SCHC Fragment and they can be from multiple windows.

In a SCHC Fragment, the window number (W ) needs to be set to identify each window number unambiguously during the transmission of a SCHC Packet. The sender can retransmit the SCHC Fragments for any lost tiles, from previous windows. This allows the sender and the receiver to work in a loosely coupled manner.

Transmission of a fragmented SCHC Packet may end in three ways: first when the integrity check for the SCHC Packet shows a correct reassembly at the receiver, second when too many retransmission attempts were made, finally when an inactivity timer at the receiver indicates that the transmission has been inactive for too long. Fig. 6 shows the transmission of the fragmented SCHC Packet of Fig. 2 with no errors, where the SCHC Fragment size allows 4 tiles per fragment.

If the receiver receives the All-1 SCHC Fragment, it performs the integrity check for the SCHC Packet. This is carried out by comparing the MIC calculated with the MIC received in the All-1 SCHC ACK Fragment. With the result of the Integrity Check, the C field is populated with 1 or 0, indicating success or failure of the SCHC Packet reassembly, respectively. In case some tiles of a window or a complete window were lost, the receiver prepares the bitmap for the lowest-numbered window that was not entirely received.

The sender has to listen for a SCHC ACK from the receiver after sending the All-1 SCHC Fragment. Moreover, a technology-oriented profile specification can establish other times when the sender may need to listen for a SCHC ACK, for example after sending a complete window of tiles. The sender can terminate the transmission of a SCHC Packet when receiving a SCHC ACK with C = 1. If the SCHC ACK carries C = 0, the sender must resend the SCHC Fragments corresponding to the missing tiles indicated in the bitmap. As an example, Fig. 7 shows the transmission of the SCHC Packet presented in Fig. 2 with an error in the 4th SCHC Fragment. Even though the SCHC Fragment carries tiles from 2 windows, the SCHC ACK indicates the window number of the lower-numbered window and the sender is able to identify which SCHC Fragment has to be retransmitted. After the retransmission of the missing SCHC Fragment, the receiver computes the MIC, performs the integrity check and sends a SCHC ACK reporting successful SCHC Packet reassembly.

V. MATHEMATICAL MODEL AND ANALYSIS

This section provides the mathematical model used to calculate the expected number of SCHC ACKs, hereafter called ACKs, required to successfully transfer a fragmented SCHC Packet in ACK-on-Error mode, in presence of losses.

For the analysis we consider an infinite maximum number of fragment retries, all tiles of the same size and no padding in the bitmap. We do not count the unconditional SCHC ACK (see Fig. 4) generated by the receiver to notify that all tiles of a SCHC Packet have been received successfully because it does not provide more information to the analysis.

A. ACK Message Overhead

The ACK message overhead (E k ) is defined as the expected number of ACKs required to successfully transfer a given window. E k will depend on the average number of SCHC Fragments necessary to transmit all the tiles of a window and on the probability that each SCHC Fragment is successfully received.

If we assume that all bits must be received without errors for the transmission to be successful, the probability of success (P success ) for a SCHC Fragment can be related to the Bit Error Rate (BER) through the relation

P success = (1 -BER) 8F , ( 1
)
where F is the fragment size in bytes of the L2 MTU of the underlying LPWAN technology. (Note that we implicitly assume a uniform BER that refers to the residual BER after application of physical layer error correcting techniques; the impact of the frame header size is considered separately, see Section V-C.) Assuming P success fixed and all transmissions independent, the number of transmission attempts N TA needed to successfully deliver a SCHC Fragment from the sender to the receiver follows a geometric distribution, i.e.,

P(N TA = n) = (1 -P success ) n-1 P success . (2)
Recall that at most only one ACK per window is generated: the ACK reports all the tiles not received in that window. Hence, until all SCHC Fragments containing all tiles of a window have been successfully received, a negative ACK is sent for that window and the missing SCHC Fragments are re-sent.

As a result, the number of such negative ACKs that are sent per window is the maximum number of retransmissions over all the SCHC Fragments in the window. The expected value of that number can be computed recursively, using a renewal argument. To do so, denote by A j the expected number of transmission attempt cycles (a cycle meaning that we send all the missing SCHC Fragments of a window once) until successful reception of all SCHC Fragments, when j > 0 SCHC Fragments remain to be sent. Then, consider the situation after a transmission attempt cycle of all those j SCHC Fragments: with probability Å j i ã P j-i success (1-P success ) i , there have been j -i SCHC Fragments successfully received, and i SCHC Fragments that need retransmission. From that situation, the expected number of transmission attempt cycles is simply A i , hence the recursive relation

A j = 1 + j i=0 Å j i ã P j-i success (1 -P success ) i A i , (3)
where the first term accounts for the transmission attempt we considered. Rearranging, we get

A j = 1 1+(1-Psuccess) j ï 1 + j-1 i=0 Å j i ã P j-i success (1 -Psuccess) i A i ò (4
) Using (4) with A 0 = 0, the number of ACKs required per window is simply A k -1, removing the last success transmission attempt cycle, when all the SCHC Fragments with all the tiles of that window were correctly received, with k the number of SCHC Fragments in a window.

A SCHC Fragment can contain tiles from several consecutive windows, hence the number of SCHC Fragments to successfully send a window may vary. For a fixed window size w (in tiles) and fragment size f (in tiles), the number of SCHC Fragments per window (k) can be modeled as a random variable, as follows:

k = ® k 1 = w f , with probability 1 + w f -w f k 2 = w f , with probability w f -w f , (5)
as illustrated in Fig. 8. To compute E k , the expected number of transmission attempts for k 1 and k 2 are first obtained from [START_REF] Minaburo | LPWAN Static Context Header Compression (SCHC) and fragmentation for IPv6 and UDP," Internet-Draft draft-ietf-lpwan-ipv6-static-context-hc-18.txt[END_REF], and E k is the weighted average of A k1 -1 and A k2 -1 with the weights of [START_REF] Gomez | IPv6 over LPWANs: connecting Low Power Wide Area Networks to the Internet (of Things)[END_REF]. This gives us the E k for a given window size, fragment size and tile size, as follows:

E k = 1 + w f -w f • A w f -1 + w f -w f • A w f -1. (6)

B. ACK Bit Overhead

The ACK bit overhead (O ACK ) is defined as the average number of (negative) ACK bits sent for each data bit sent and provides the trade-off between the amount of data that can be transferred in a window and the resulting ACK(s) volume.

O ACK is obtained by dividing E k by the window size and then multiplying the result by the ACK size (L ACK ). The window size can be obtained as the tile size (t) multiplied by the number of tiles in a window (w). L ACK is equal to the bitmap size, that exactly equals w (one bit per tile in a window) plus the ACK header L SH (in bytes) allowing to express the ACK bit overhead as:

O ACK = (8L SH + w) • E k 8wt . (7)
O ACK quantifies the relation between the quantity of data and the quantity of ACKs that need to be sent. The ACK size (L ACK ) is related to the tile size. For the same window size (in bytes), larger tiles produces smaller bitmaps, thus smaller ACKs. Recall that the O ACK is from the point of view of the SCHC framework layer, as it only considers the SCHC Headers.

C. ACK Bit Overhead with L2 Headers

As the SCHC framework sits on top of a LPWAN technology, there is an extra overhead due to L2 Headers. To consider the additional cost involved in sending an ACK, the L2 Header (with size L L2H in bytes) is added to O ACK in the downlink, i.e. a penalty for sending an ACK, and can be calculated as follows:

O ACKL2 = (8 • (L L2H + L SH ) + w) • E k 8wt . (8)
The ACK bit overhead with L2 headers (O ACKL2 ) analyzes the impact of the L2 overhead and its relation with the window and tile sizes. Minimizing O ACKL2 maximizes the uplink data while optimizing the ACK size and volume, reducing the utilization of the LPWAN gateway and leveraging the available resources in duty-cycle-constrained networks.

D. Percentage of used bits per fragment

The percentage of used bits per SCHC Fragment provides information on how efficient a tile size is for a given L2 MTU (F ) considering the SCHC Headers. Due to LPWAN technologies capacity constraints, the tile size must be set to maximize the SCHC Fragment payload. Therefore, the percentage of usage of a SCHC Fragment, which we will denote by P U , equals

P U = f • t F -L SH • 100. (9)
In LPWAN technologies such as LoRaWAN, F can change during an on-going fragmented packet transmission [START_REF]LoRaWAN 1.1 Specification[END_REF] and the tile size previously chosen may not be multiple of the modified F as it was in the previous one, leaving some bytes unused. The number of unused bytes in a SCHC Fragment (f unused ), for a given tile size, is calculated as follows:

f unused = (F -L SH ) -(f • t). ( 10
)
For example, a 9-byte tile (t = 9) will use all the bytes when F = 11, f = 1 with L SH = 2, but when having 5 tiles (f = 5) of 9 bytes and F = 53, 6 bytes per SCHC Fragment will not be used.

E. Maximum window and bitmap sizes

The SCHC framework defines a method for F/R on the uplink to transfer SCHC Packets, but it does not propose a SCHC ACK F/R method. Without a fragmentation method the ACK is limited to one SCHC Fragment, i.e. one L2 MTU. Therefore, there is a maximum bitmap size (MBS) that leads to a maximum window size (MWS). Moreover, the tile size will limit the maximum data on a window. The maximum bitmap size in tiles, i.e., number of bits in the bitmap, is therefore calculated as follows:

MBS = 8 • (F -L SH ), (11)
and the maximum window size in bytes is then:

MWS = (t • MBS) -U. (12)

VI. PERFORMANCE EVALUATION

In this section, we use the models derived in section V to evaluate the ACK-on-Error mode in terms of the performance metrics presented.

The fragment sizes F used in the performance evaluation are the minimum (F = 11) and maximum (F = 242) MTU values in LoRaWAN for US 915 MHz band (US915) and EU 868 MHz band (EU868), respectively. Note that the physical layer configuration in LoRaWAN (including data unit size, and thus fragment size) is determined by the Data Rate (DR) and Spreading Factor (SF) [START_REF]LoRaWAN 1.1 Specification[END_REF]. The SCHC ACK and SCHC Fragment Header size used is 2 bytes (L SH = 2).

A. ACK Message Overhead

Fig. 9a, shows the ACK message overhead, (E k ) as a function of P success for different window sizes for F = 11 and t = 9 bytes so that f = 1 (one tile per fragment), k = w, and no unused bits. The difference in E k between a small window size (e.g., w = 1) and a large window (e.g., w = 143) is larger for lower P success than with higher P success . This happens because E k has a logarithmic behavior as a function of k (see Fig. 9b) and "flattens" when P success increases: for small values of k and P success , a small variation in k produces large changes in E k . The E k variation decreases as k increases. For higher P success , the difference in E k is less dependent on k, since less ACKs are produced.

For P success = 1, only one positive ACK is generated at the end of the fragmented packet transmission because no SCHC Fragments are lost, but E k = 0 because it only counts negative ACKs, i.e. failed window transmission cycles.

B. ACK Bit Overhead

We now consider the ACK bit overhead metric O ACK defined in [START_REF] Adelantado | Understanding the limits of LoRaWAN[END_REF], with the aim of minimizing that metric. 1) Optimal Tile Size: Fig. 10a shows the impact of the tile size on O ACK , for F = 11 and P success = 0.9. For small tiles, O ACK increases faster with the window size than for large tiles; indeed a smaller tile size will require a larger bitmap for each data bit transferred, since the bitmap size equals the number of tiles in a window, hence is inversely proportional to the tile size for a fixed window size (in data volume). As the tile size increases, the ACK size required for the same window size decreases. When the window size increases, the ACK size becomes more relevant in O ACK than the average number of ACKs, because the average number of ACKs has a logarithmic behavior (see Fig. 9b) while the ACK size increases linearly with the window size. The tile size that yields the smallest ACK size and the lowest O ACK ratio is the largest possible tile size.

The results for F = 51 and F = 242 for P success = 0.9 are shown in Fig. 10c and Fig. 10d, respectively. As in the previous case, a tile size equal to the SCHC Fragment payload size minimizes O ACK . As expected, there is a large difference between using a 9-byte and 49-byte tile, when compared to a 240-byte tile, as Fig. 10d shows.

Fig. 10b shows O ACK for P success = 0.5 and F = 11, evaluated for different window sizes. For larger tile sizes, as the window size increases, O ACK decreases because larger windows are more efficient when reporting many lost fragments.

For all the displayed settings, the optimal tile size is the largest possible tile, i.e., a tile size of the SCHC Fragment payload size, independently of P success . Hence, for technologies with fixed MTU such as Sigfox, the optimal tile size is simply the SCHC Fragment payload size. For technologies with variable MTU size, such as LoRaWAN, it is not possible to use a tile of the SCHC Fragment payload size in the larger fragment sizes because the tile has to fit in the smallest fragment size to support variable MTU. In the case the SCHC Fragment size is known beforehand, the Rule can be chosen with the optimal tile and windows sizes. If the MTU changes, then the tile size can change accordingly.

2) Optimal Window Size: The tile size is set to a fixed value, i.e., equal to the SCHC Fragment payload size. Fig. 11a and Fig. 11b illustrate the O ACK as a function of P success for different window sizes, for F = 11, t = 9 and F = 51, t = 49, respectively. When P success is low, many fragments are lost and a larger window size leads to lower O ACK .

The interest of having large windows lies in what we can call ACK pooling, that is the fact that when several fragments (of the same window) fail, the information of the failures is pooled into a single ACK message. Large window sizes benefit from ACK pooling when P success is low because one large ACK can report many lost fragments. Conversely, smaller windows sizes will generate a greater number of ACKs leading to higher overhead.

Henceforth, the tradeoff is between ACK pooling (larger windows mean less ACKs) and ACK size (larger windows mean larger ACKs), that we manage through the total ACK volume metric O ACK .

As P success increases, there exists one point (see zoom in Fig. 11a and Fig. 11b) beyond which smaller windows outperform larger window sizes. From this point forward, having smaller ACKs becomes more important since the number of losses is low. For example, the window size of 1 tile performs worst for lower P success but as P success increases, it becomes the one yielding the lowest O ACK . When P success is greater than 0.9, the window size of just 1 tile is optimal because rarely a SCHC Fragment will get lost, in which case a smaller ACK will be sent (almost no ACK pooling, and smaller ACKs). In contrast, large windows require a large ACK to report the few lost fragments.

As a consequence, there exists an optimal window size that minimizes O ACK , which depends not only on P success , but also on F and t. Fig. 11c and Fig. 11d show the values for O ACK for different windows sizes, and for F = 11, t = 9 and F = 51, t = 49, respectively. When P success is 0.8, ACK pooling benefits the larger windows and the optimal window size is large, as for example, 342 bytes (w = 38) in Fig. 11c. As P success increases, O ACK increases with the window size and for P success ≥ 0.9 the benefit of ACK pooling is lost, leading to an optimal window size of one tile (w = 1). Fig. 12a and Fig. 12b illustrate the results for F = 240 with t = 49 and t = 240, respectively. When the tile size is small compared with the SCHC Fragment payload size (f > 1) there is no gain from ACK pooling for P success > 0.5. Hence smaller windows perform better, as Fig. 12a shows. As the window size increases, more small tiles are required when compared with larger tile sizes (see Fig. 12b), making the larger windows size yield a greater O ACK when using smaller tiles. Fig. 12c shows how O ACK is minimum for small window sizes when P success > 0.80 for F = 240 and t = 49: the optimal window size is the smaller window, i.e. w = 4. By contrast, Fig. 12d illustrates for F = 242 and t = 240 how a larger tile size will yield a larger optimal window size and benefit from ACK pooling for P success < 0.9.

Hence, depending on the parameters and the channel conditions, the window size minimizing O ACK varies. We now focus on that optimal window size.

Fig. 13a presents the optimal window size as a function of P success , for different F values. When P success is below 0.8, in most of the cases, the optimal window size is large. The optimal window size decreases as P success increases, as expected the importance of ACK pooling decreasing with respect to the ACK size. Fig. 13b presents the optimal window sizes for different P success and for larger F values. As for the case of smaller F , as P success becomes higher, the optimal window size becomes smaller.

C. ACK Bit Overhead with L2 Headers

Previous figures were considering the ACK overhead as defined in [START_REF] Adelantado | Understanding the limits of LoRaWAN[END_REF], i.e., ignoring L2 headers in the size of ACKs. This may be justified if the operator charges the user based on the volume of L2 payloads. In this section we investigate the impact of counting the whole L2 ACK size by counting those headers, as suggested in [START_REF]LoRaWAN 1.1 Specification[END_REF]. The L2 headers of Sigfox according to [6] is 21 bytes in downlink with a payload of 8 bytes. The L2 headers of LoRaWAN according to [START_REF]LoRaWAN 1.1 Specification[END_REF] are 13 byte long, hence we will especially focus on those values.

For clarity in the figures, we define P failure as 1 -P success . Smaller windows outperform larger ones for P success > 0.99 (see the zoom in Figs. 14a and14b), whereas in Section VI-B it is for P success > 0.90. This happens because the additional constant length added to the ACK size favors the ACK pooling effect of large windows over the additional ACK length, making it more efficient to send one large ACK than several smaller ones (and their large L2 headers). Hence, only with very large success probability P success > 0.99, i.e., P failure < 10 -2 , does the ACK length effect take over the ACK pooling effect, as Fig. 14a shows. For P success < 0.99, the best window size is 1 tile (w = 1), but as P success decreases, the optimal window size increases (e.g. w = 143 for P success > 0.99).

D. Percentage of used bits per fragment

The percentage of used bits per fragment (P U ) provides an overview of how efficient a tile size is for a given SCHC Fragment size. Fig. 15a illustrates P U for F = 11, 12, 51, 53 and different tiles sizes. When fragment sizes are a multiple of the tile size, P U is 100%, e.g. t = 1 ∀ F , t = 3 for F = 11, t = 5 for F = 12, t = 7 for F = 51 and t = 17 F = 53. Furthermore, there are tile sizes that have a low P U . For example, t = 5 for F = 11 only uses the 55.56%, t = 6 for F = 12 with only 60%, t = 25 for F = 51 with 51.02% and t = 26 for F = 53 with 50.98%. Fig. 15b shows the percentage of used bits for F = 115, 125, 242. As with smaller SCHC Fragments, some tile sizes have a better P U than others. Moreover, as the tile size gets larger and only one tile can be fitted in the SCHC Fragment payload, the percentage of used is reduced significantly. For example, t = 57 for F = 115 with 50.44%, t = 62 for F = 125 with 50.41% and t = 121 for F = 242 with 50.42%.

VII. TECHNOLOGY-ORIENTED EVALUATION

This section provides configuration guidance for the main parameters of SCHC F/R ACK-on Error mode when used over LoRaWAN (EU868 and US915) and Sigfox, with respect to the ACK bit overhead O ACK and O ACKL2 metrics. Furthermore, we present the maximum window and bitmap size for each fragment and tile size.

Considering that the ACK size has to be a byte multiple, we obtained the optimal window size value as the next integer value that will not require padding. For the SCHC ACK Header size we consider L SH = 2. This makes 3 bytes the smallest negative ACK size possible without padding, and with a bitmap of 8 bits. The values presented in sections VII-A and VII-B are for P success ≥ 0.85. The FCN field size (N ) is set to handle the number of tiles in the optimal or maximum window size. The length of the Window field (M ) is calculated according to the number of windows required to transmit a 320-byte and 1280-byte SCHC Packet.

In SCHC ACK-on-Error mode, several F/R parameters, such as window size, tile size, N and M must be selected when starting a transmission of a SCHC Packet. However, instead of using a fixed configuration, a device may select the most suitable ACK-on-Error mode configuration values by estimating the BER, therefore P success , and considering the available SCHC Fragment payload size.

A. ACK Bit Overhead

Table I presents the optimal window sizes for the ACK bit overhead O ACK and SCHC F/R ACK-on-Error configuration parameters for LoRaWAN fragment sizes (F ) in EU868 and US915 bands. Table II presents the optimal window size and SCHC F/R ACK-on-Error mode configuration parameters for the fragment size of Sigfox (only the uplink is considered).

B. ACK Bit Overhead with L2 Headers

Tables III and IV present the optimal window sizes from the perspective of the ACK bit overhead with L2 headers O ACKL2 and SCHC F/R ACK-on-Error configuration parameters, for LoRaWAN in EU868 and US915 bands, and for Sigfox, respectively.

C. Maximum window and bitmap sizes

Table V shows the maximum bitmap and window sizes for LoRaWAN and Sigfox with L SH = 2. Different tile sizes are presented considering different SCHC F/R ACK-on-Error parameters configurations. Recall that the optimal tile size is the size that entirely fills a SCHC Fragment. The maximum bitmap size does not depend on the tile size, but on the SCHC Fragment and SCHC Header sizes as shown in [START_REF] Lavric | Long Range SigFox Communication Protocol Scalability Analysis Under Large-Scale, High-Density Conditions[END_REF]. Note that, Sigfox has a downlink payload size of 8 bytes [6], hence with L SH = 2 at most 6 bytes can be used for the bitmap.

VIII. CONCLUSIONS AND PERSPECTIVES

The SCHC framework enables LPWAN technologies to comply with IPv6 by performing a static context header compression and fragmentation. ACK-Always and ACK-on-Error modes, provide reliable F/R between the sender and the receiver. In this article, we have developed a mathematical model to analyze the ACK-on-Error mode, focusing on the number of ACKs required to transmit data. We were then able to perform an analysis that yielded to transmission parameters minimizing the ACK burden imposed on the (more sensitive to duty-cycle constraints) downlink. Finally we have applied our mathematical model to state-of-the-art LPWAN technologies such as LoRaWAN and Sigfox, to minimize the ACK bit overhead taking into account the retransmission process subtleties of the ACK-on-Error mode.

Fig. 1 Fig. 1 .

 11 Fig.1. Protocol stack illustrates the location of the SCHC sublayers between the IPv6 layer and the underlying LPWAN technology[START_REF] Minaburo | LPWAN Static Context Header Compression (SCHC) and fragmentation for IPv6 and UDP," Internet-Draft draft-ietf-lpwan-ipv6-static-context-hc-18.txt[END_REF].

3 Fig. 2 .

 32 Fig. 2. Example of a SCHC Packet fragmented into 20 tiles, with 5 tiles per window. The tile index in the window, called the Fragment Compressed Number (FCN) is indicated for each tile.

Fig. 4 Fig. 3 .Fig. 4 .

 434 Fig. 4 shows the SCHC ACK format. A SCHC ACK carries the C field. SCHC Frag Header Rule ID DTag W FCN Fragment Payload padding(as needed) T M N Rule ID DTag W 11..11 MIC Fragment Payload padding(as needed) T M N U L SH Fig. 3. Illustration of a regular SCHC Fragment, and an All-1 SCHC Fragment with the MIC field. The length in bits of each header field is indicated below each field.

Fig. 6 .

 6 Fig. 6. Example of a transmission of the SCHC Packet shown in Fig. 2 with no errors. The transmission ends with a SCHC ACK that indicates successful SCHC Packet reassembly.

Fig. 7 .

 7 Fig. 7.Example of a transmission of a SCHC Packet with one lost SCHC Fragment in ACK-on-Error mode. Once the All-1 SCHC Message is received by the receiver, a SCHC ACK with C = 0 and its corresponding bitmap is generated. The sender performs the SCHC Fragment retransmission. When the lost SCHC Fragment is received, the receiver performs the Integrity Check and sends the corresponding SCHC ACK Success indicating the end of the transmission of the SCHC Packet.

w 0 w 1 Rule ID DTag w1 001 1 0 4 3 Rule ID DTag w2 010 2 1 0 4 w 1 w 2 w 2 w 3 Rule ID DTag w3 011 3 4 Rule ID DTag w4 000 0 4 3 2 w 4 w 5 Fig. 8 .= 1 / 4

 13433425814 Fig. 8. Example of a SCHC Packet and SCHC Fragments. The SCHC Packet is fragmented in 5 tiles per window (w = 5) and the SCHC Fragments can transport 4 tiles per fragment (f = 4). For a successful transmission of the packet, w 0 needs k 2 = 2 successful SCHC Fragment transmissions, then w 1 , w 2 , and w 3 need each k 1 = 1 more successful SCHC Fragment transmission, in compliance with (5): a proportion w f -w f = 1/4 of windows need k 2 = w f new SCHC Fragment transmissions, while the other windows need only k 1 = w f = 1 new SCHC Fragment transmission.

Fig. 9 .

 9 Fig. 9. E k for F = 11, t = 9 and L SH = 2 for different windows sizes as a function of Psuccess (a) and a function of k (b).

9 Fig. 10 .

 910 Fig. 10. O ACK vs window size. Impact of the tile size t on the O ACK . L SH = 2.

Fig. 11 .

 11 Fig. 11. Impact of the window size w (in tiles) or 8wt (in bits) on the ACK overhead O ACK . The window size is a multiple of the tile size using w. L SH = 2.

Fig. 12 . 3 Psuccess 2 10 3 PsuccessFig. 13 .

 1232313 Fig. 12. Impact of the window size w (in tiles) or 8wt (in bits) on the ACK overhead O ACK . The window size is a multiple of the tile size using w. L SH = 2.

  Fig 14a and Fig. 14b illustrate O ACKL2 for F = 11, t = 9 and for F = 242, t = 49, with L L2H = 13, respectively.

  Fig. 14c and Fig. 14d confirm our conclusions, by showing O ACKL2 for F = 11, t = 9 and for F = 242, t = 49 with L L2H = 13 for different P success , respectively.

Fig. 14 .

 14 Fig. 14. Impact of the window size w (in tiles) or 8wt (in bits) on the ACK overhead O ACK L2 . The window size is a multiple of the tile size using w. L SH = 2 and L L2H = 13.

Fig. 15 .

 15 Fig. 15. Percentage of usage P U vs t. L SH = 2.

TABLE I OPTIMAL

 I WINDOW SIZES (O ACK ) FOR LORAWAN LoRaWAN physical layer options are defined and identified by the Spreading Factor (SF) and Data Rate (DR) parameters.

	Region	F (bytes)	SF	DR	t (bytes)	Psuccess	Optimal Window Size (tiles)	N (bits)	M (bits) SCHC Packet Size 320 1280 (bytes) (bytes)
		51	12, 11,10	DR0, DR1,DR2	49	0.85 ≤ Psuccess < 0.90 0.90 ≤ Psuccess ≤ 1	40 8	6 4	1 1	1 3
					49	0.85 ≤ Psuccess ≤ 1	8	4	1	3
	Europe EU868	115	9	DR3	113 49	0.85 ≤ Psuccess < 0.90 0.90 ≤ Psuccess ≤ 1 0.85 ≤ Psuccess ≤ 1	40 8 8	6 4 4	1 1 1	1 2 3
		242	8, 7	DR4, DR5	113 240	0.85 ≤ Psuccess ≤ 1 0.85 ≤ Psuccess ≤ 0.90 0.90 < Psuccess ≤ 1	8 24 8	4 5 4	1 1 1	2 1 1
		11	10	DR0	9	0.85 ≤ Psuccess < 0.90 0.90 ≤ Psuccess ≤ 1	48 40	6 6	1 1	3 3
					9	0.85 ≤ Psuccess ≤ 1	8	4	3	5
		53	9	DR1	51	0.85 ≤ Psuccess ≤ 0.90 0.90 < Psuccess ≤ 1	40 8	6 4	1 1	1 3
					9	0.85 ≤ Psuccess ≤ 1	16	5	2	4
	USA US915	125	8	DR2	51 123	0.85 ≤ Psuccess ≤ 1 0.85 ≤ Psuccess ≤ 0.90 0.90 < Psuccess ≤ 1	8 40 16	4 6 5	1 1 1	3 1 1
					9	0.85 ≤ Psuccess ≤ 1	32	6	2	3
					51	0.85 ≤ Psuccess ≤ 1	8	4	1	3
		242	7, 8	DR3, DR4	123	0.85 ≤ Psuccess < 0.90 0.90 ≤ Psuccess ≤ 1	40 8	6 4	1 1	1 2
					240	0.85 ≤ Psuccess ≤ 0.90 0.90 < Psuccess ≤ 1	24 8	5 4	1 1	1 1
	LSH = 2.

TABLE III OPTIMAL

 III WINDOW SIZES WITH L2 HEADERS (O ACK L2 ) FOR LORAWAN LoRaWAN physical layer options are defined and identified by the Spreading Factor (SF) and Data Rate (DR) parameters.

	Region	F (bytes)	SF	DR	t (bytes)	Psuccess	Optimal Window Size (tiles)	N (bits)	M (bits) SCHC Packet Size 320 1280 (bytes) (bytes)
		51	12, 11,10	DR0, DR1,DR2	49	0.85 ≤ Psuccess ≤ 0.98 0.98 < Psuccess ≤ 1	392 8	9 4	1 1	1 3
						0.85 ≤ Psuccess < 0.90	512	10	1	1
						0.90 ≤ Psuccess < 0.95	480	9	1	1
					49	0.95 ≤ Psuccess < 0.97	400	9	1	1
						0.97 ≤ Psuccess < 0.98	584	10	1	1
		115	9	DR3		0.98 ≤ Psuccess ≤ 1 0.85 ≤ Psuccess < 0.90	8 512	4 10	1 1	2 1
						0.90 ≤ Psuccess < 0.95	480	9	1	1
					113	0.95 ≤ Psuccess < 0.97	400	9	1	1
	Europe					0.97 ≤ Psuccess < 0.98	584	10	1	1
	EU868					0.98 ≤ Psuccess ≤ 1	8	4	1	2
					49	0.85 ≤ Psuccess ≥ 1	8	4	1	2
						0.85 ≤ Psuccess < 0.90	512	10	1	1
						0.90 ≤ Psuccess < 0.95	480	9	1	1
					113	0.95 ≤ Psuccess < 0.97	400	9	1	1
		242	8, 7	DR4, DR5		0.97 ≤ Psuccess < 0.98 0.98 ≤ Psuccess ≤ 1 0.85 ≤ Psuccess < 0.90	584 8 576	10 4 10	1 1 1	1 2 1
					240	0.90 ≤ Psuccess < 0.95 0.95 ≤ Psuccess < 0.99	624 368	10 9	1 1	1 1
						0.99 ≤ Psuccess ≤ 1	8	4	1	1
		11	10	DR0	9	0.85 ≤ Psuccess < 0.98 0.98 ≤ Psuccess ≤ 1	72 8	7 4	1 3	2 5
		53	9	DR1	9 51	0.85 ≤ Psuccess ≤ 0.90 0.90 < Psuccess ≤ 1 0.85 ≤ Psuccess ≤ 0.98 0.98 < Psuccess ≤ 1	328 8 408 8	9 4 9 4	1 3 1 1	1 5 1 3
					9	0.85 ≤ Psuccess ≤ 1	16	5	2	4
					51	0.85 ≤ Psuccess < 0.98 0.98 ≤ Psuccess ≤ 1	72 8	7 4	1 1	1 3
						0.85 ≤ Psuccess < 0.90	576	10	1	1
		125	8	DR2		0.9 ≤ Psuccess < 0.95	624	10	1	1
					123	0.95 ≤ Psuccess < 0.97 0.97 ≤ Psuccess < 0.98	360 456	9 9	1 1	1 1
	US US915				9	0.98 ≤ Psuccess < 0.99 0.99 ≤ Psuccess ≤ 1 0.85 ≤ Psuccess ≤ 1	624 8 32	10 4 6	1 1 2	1 2 3
						0.85 ≤ Psuccess < 0.90	376	9	1	1
					51	0.90 ≤ Psuccess < 0.95 0.95 ≤ Psuccess < 0.96	328 488	9 9	1 1	1 1
						0.96 ≤ Psuccess ≤ 1	8	4	1	3
						0.85 ≤ Psuccess < 0.90	576	10	1	1
		242	7, 8	DR3, DR4	123	0.90 ≤ Psuccess < 0.95 0.95 ≤ Psuccess < 0.98	624 360	10 9	1 1	1 1
						0.98 ≤ Psuccess < 0.99	624	10	1	1
						0.99 ≤ Psuccess ≤ 1	8	4	1	2
						0.85 ≤ Psuccess < 0.90	576	10	1	1
					240	0.90 ≤ Psuccess < 0.95 0.95 ≤ Psuccess < 0.99	624 368	10 9	1 1	1 1
						0.99 ≤ Psuccess ≤ 1	8	4	1	1
	LSH = 2.

TABLE IV OPTIMAL

 IV WINDOW SIZES WITH L2 HEADERS (O ACK L2 ) FOR SIGFOX

				Optimal		M
	F (bytes)	t (bytes)	Psuccess	Window Size	N (bits)	(bits) SCHC Packet Size
				(tiles)		320	1280
						(bytes)	(bytes)
	12	10	0.85 ≤ Psuccess ≤ 0.95 0.95 < Psuccess ≤	48 8	6 4	1 3	3 5
	L SH = 2.

See, e.g., https://iotmarket.orange.com/connectivity. html (accessed on 30/06/2019)

ACKNOWLEDGMENT

This research was done during Sergio Aguilar visit at IMT-Atlantique and is supported in part by the Spanish Government through project TEC2016-79988-P, AEI/FEDER, EU.

by the Spanish Government through project TEC2016-79988-P, AEI/FEDER, EU.


Ayoub Harrat

El Hassan Zerouali

Lech Zielinski

ASYMPTOTIC EXPANSION OF LARGE EIGENVALUES FOR A CLASS OF UNBOUNDED JACOBI MATRICES

Keywords: tridiagonal matrix, band matrix, unbounded self-adjoint operator, discrete spectrum, large eigenvalues, asymptotics Mathematics Subject Classification: 47B25, 47B36, 15A18

We investigate a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. Our purpose is to establish the asymptotic expansion of large eigenvalues and to compute two correction terms explicitly.

1. Introduction 1.1. General remarks. Infinite Jacobi matrices have appeared in many recent papers related to various questions of pure and applied mathematics (see [START_REF] Boutet De Monvel | The asymptotic behaviour of eigenvalues of a modified Jaynes-Cummings model[END_REF], [START_REF] Djakov | Simple and double eigenvalues of the Hill operator with a two term potential[END_REF], [START_REF] Janas | Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach[END_REF], [START_REF] Teschl | Jacobi operators and completely integrable nonlinear lattices[END_REF], [START_REF] Volkmer | Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation[END_REF]). In this paper we consider a Hermitian tridiagonal matrix such that (d(k)) ∞ k=1 is an increasing sequence going to infinity and (a(k)) ∞ k=1 is a complex valued sequence dominated by (d(k)) ∞ k=1 . Then (1.1) defines in 2 a self-adjoint operator J with discrete spectrum (see [START_REF] Cojuhari | Discreteness of the spectrum for some unbounded Jacobi matrices[END_REF]). Moreover, J is bounded from below and the eigenvalue sequence of J is defined as the non-decreasing sequence (λ n (J)) ∞ n=1 such that Jv n = λ n (J)v n for n ∈ N * and (v n ) ∞ n=1 is an orthonormal basis of 2 . We begin the discussion of known results by the fundamental paper of J. Janas, S. Naboko [START_REF] Janas | Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach[END_REF]. In this paper the authors describe a method of approximative diagonalization and its application to the analysis of large eigenvalues in certain quantum models. The paper [START_REF] Janas | Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach[END_REF] gives also explanations why this type of analysis is important in Quantum Physics.

       d(1) a(1) 0 0 • • • a(1) d(2) a(2) 0 • • • 0 a (
Concerning non self-adjoint problems, we refer to [START_REF]Trace formula and spectral Riemann surfaces for a class of tri-diagonal matrices[END_REF] and [START_REF] Malejki | Asymptotics of the discrete spectrum for complex Jacobi matrices[END_REF]. Concerning the self-adjoint problem, we remark that the papers [START_REF] Boutet De Monvel | The asymptotic behaviour of eigenvalues of a modified Jaynes-Cummings model[END_REF]- [START_REF]Asymptotic behaviour of large eigenvalues for Jaynes-Cummings type models[END_REF], [START_REF] Janas | Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach[END_REF] and [START_REF] Malejki | Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices[END_REF]

use the following

Hypothesis. There exist µ, ρ, C, c, k 0 ∈ (0; ∞) such that for k ≥ k 0 one has (H1)

ck µ ≤ d(k) ≤ Ck µ , (H2) d(k + 1) -d(k) ≥ ck µ-1 , (H3) |a(k)| ≤ Ck µ-ρ .
Date: January 28, 2020.

1

The co-author was supported by CNRST excellence grant number 44UM5R2017

1 It turns out that asymptotic estimates crucially depend on whether ρ > 1 or not. Case ρ > 1. The papers [START_REF] Janas | Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach[END_REF] and [START_REF] Boutet De Monvel | The asymptotic behaviour of eigenvalues of a modified Jaynes-Cummings model[END_REF] treat a type of coefficients satisfying (H1)-(H3) with µ = 2 and ρ = 3 2 . An asymptotic expansion of large eigenvalues is constructed in [START_REF] Malejki | Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices[END_REF] using (H1)-(H3) with µ ≥ 1, ρ > 1 and in [START_REF] Boutet De Monvel | Explicit error estimates for eigenvalues of some unbounded Jacobi matrices, Spectral Theory, Mathematical System Theory[END_REF] using (H1)-(H3) with µ > 0, ρ > 1.

Case ρ ≤ 1. It appears that ρ = 1 2 is the most important value for the Quantum Optics and the asymptotic behaviour of large eigenvalues for models considered in [START_REF] Boutet De Monvel | Explicit error estimates for eigenvalues of some unbounded Jacobi matrices, Spectral Theory, Mathematical System Theory[END_REF]- [START_REF]On the spectrum of the quantum Rabi Models[END_REF], [START_REF] Tur | Jaynes-Cummings model: solution without rotating wave approximation[END_REF]- [START_REF] Yanovich | Asymptotics of eigenvalues of an energy operator in a problem of quantum physics[END_REF] turned out to have a quite special form. However, it is an open problem to describe eigenvalue asymptotics without additional regularity of entries. Other special models are investigated in papers [START_REF] Edward | Spectra of Jacobi matrices, differential equations on the circle, and the su(1, 1) Lie algebra[END_REF], [START_REF] Janas | Alternative approaches to asymptotic behaviour of eigenvalues of some unbounded Jacobi matrices[END_REF] and [START_REF] Masson | Spectral theory of Jacobi matrices in l 2 (Z) the su(1,1) Lie algebra[END_REF].

In this paper we construct an asymptotic expansion of large eigenvalues under additional regularity conditions imposed on the entries and we give explicit expressions for correction terms in the asymptotic formula with error O(n µ-6ρ ). In Theorem 1.1 we assume that the entries have a classical expansion at infinity. The case ρ = 1 2 gives a nice surprise by ensuring a classical expansion of eigenvalues. A similar construction still works if entries satisfy regularity conditions of symbol type (see Section 1.3). The restriction to tridiagonal matrices is not essential in our approach and we can work all the time with band matrices, but tridiagonal matrices make formulas and calculus more simple. We mention as well that this paper can be viewed as a development of [START_REF] Boutet De Monvel | Asymptotics of large eigenvalues for a class of band matrices[END_REF] (see also [START_REF] Malejki | Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices[END_REF]).

1.2. Asymptotic expansion of large eigenvalues. We denote by 2 the Hilbert space of square summable complex valued sequences x : N * → C with the norm

x = ∞ k=1 |x(k)| 2 1/2
and the scalar product x, y = ∞ k=1

x(k)y(k). For any θ > 0 we denote

2,θ := {x ∈ 2 : ∞ k=1 |k θ x(k)| 2 < ∞}. (1.2) Let (d(k)) ∞ k=1 , (a(k)) ∞
k=1 satisfy the hypothesis (H1)-(H3) for a fixed µ > 0 and ρ > 0. For x ∈ 2,µ we define Jx ∈ 2 by the formula

Jx(k) = d(k)x(k) + ā(k)x(k + 1) + a(k -1)x(k -1) for k ∈ N * , (1.3)
where we assume x(0) = 0 and a(0) = 0. The formula (1.3) defines the operator J on the domain of definition D(J) = 2,µ . Then J is self-adjoint in 2 , bounded from below and has compact resolvent (see [START_REF] Cojuhari | Discreteness of the spectrum for some unbounded Jacobi matrices[END_REF]). We will prove the following Theorem 1.1. Let µ > 0, ρ > 0 be fixed and assume

d(k) ∼ k µ ∞ i=0 δ i k i as k → ∞, (1.4) a(k) ∼ k µ-ρ ∞ i=0 α i k i as k → ∞, (1.5)
where α i ∈ C, δ i ∈ R for i ∈ N and δ 0 > 0. Then (H1)-(H3) hold and (1.3) defines in 2 the self-adjoint operator J on the domain D(J) = 2,µ . If (λ n (J)) ∞ n=1 is the eigenvalue sequence of the operator J, then

λ n (J) = d(n) + r(n), (1.6)
where r(n) obeys the asymptotic expansion of the form

r(n) ∼ n µ-2ρ ∞ i,j=0 c i,j n i+2jρ as n → ∞ (1.7)
and c i,j are real coefficients obtained by the induction scheme in Section 5. The explicit values of c 0,0 , c 1,0 , c 2,0 are computed in Section 6.4 and the value of c 0,1 in Section 6.5 (see also (1.16) and

(1.19)-(1.20)).
The operator J is a relatively compact perturbation of the diagonal operator diag(d(n)) ∞ n=1 and the special form of this perturbation allows one to deduce

λ n (J) = d(n) + O(n µ-ρ ) (1.8)
from the min-max principle (see Corollary 3.3). It appears that the assumptions of Theorem 1.1 allow one to replace (1.8) by the stronger estimate

λ n (J) = d(n) + O(n µ-2ρ ) (1.9)
and the remainder r(n) = λ n (J) -d(n) obeys the asymptotic formula

r(n) = r 1 (n) + O(n µ-4ρ ), (1.10)
where r 1 (n) is given by (1.14). The quantity r 1 (n) is of order n µ-2ρ and we call it the first correction term. We mention that the expression (1.14) for the first correction term was obtained in [START_REF] Malejki | Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices[END_REF] under the assumptions (H1)-(H3) with µ ≥ 1, ρ > 1 (and with weaker remainder estimates). The next step of precision is attained in Theorem 6.2 which describes the asymptotic behaviour of r(n) modulo O(n µ-6ρ ). We show the formula (1.17) and give explicit expressions (1.18)-(1.20) for the leading coefficient of the second correction term, which is of order O(n µ-4ρ ).

In order to obtain a complete asymptotic expansion, we use a method of approximative diagonalization. The idea of our approach is similar to the method presented in [START_REF] Janas | Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach[END_REF], but we do not need the assumptions µ ≥ 1 and ρ > 1 used in [START_REF] Janas | Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach[END_REF].

At the end of this discussion we remark that the result of Theorem 1.1 is inspired by the paper [START_REF] Rozenbljum | Near-similarity of operators and the spectral asymptotic behaviour of pseudodifferential operators on the circle (Russian), Trudy Maskov[END_REF]. The main differences between [START_REF] Rozenbljum | Near-similarity of operators and the spectral asymptotic behaviour of pseudodifferential operators on the circle (Russian), Trudy Maskov[END_REF] and our work are the following: -we consider the Hilbert space 2 = 2 (N * ) instead of 2 (Z), -we give a simple algorithm of computing the coefficients c i,j adopted to our problem, -our approach allows us to see that (1.7) contains only even powers of n -ρ . 1.3. Further results. We construct an asymptotic expansion for large eigenvalues under assumptions slightly weaker than used in Theorem 1.1.

Notation 1.2. For any

f : N * → C we write ∂ 0 f (k) := f (k), ∂f (k) = f (k + 1) -f (k),
and, by using induction, we define

∂ m+1 f := ∂(∂ m f ) for m = 0, 1, 2, . . . . If ν ∈ R then S ν 1 denotes the set of p : N * → C such that the estimate ∂ m p(k) = O(k ν-m )
holds for every m ∈ N.

Throughout the paper µ > 0, ρ > 0 are fixed and we centre our analysis on the Jacobi operators (1.3)

such that d ∈ S µ 1 , (1.11)
a ∈ S µ-ρ 1

(1.12) and (H1)-(H2) hold. These conditions could be weakened if one wants to obtain the remainder estimate O(n -η ) for a fixed value of the exponent η (see Theorem 4.1), but for simplicity we use (1.11)-(1.12) throughout Section 5 and 6.

As before J is the self-adjoint operator in 2 with D(J) = 2,µ and (λ n (J)) ∞ n=1 denotes its non-decreasing eigenvalue sequence (counted with multiplicities). We consider the asymptotic formula for (λ n (J)) ∞ n=1 with three different degrees of precision. (i) The asymptotic formula with one correction term. We show that

λ n (J) = d(n) + r 1 (n) + O(n µ-4ρ ) (1.13) holds with r 1 (n) := |a(n -1)| 2 d(n) -d(n -1) - |a(n)| 2 d(n + 1) -d(n) . (1.14)
If (1.4)-(1.5) hold, then (1.11)-(1.12) hold as well. Under the assumptions (1.4)-(1.5), it is easy to see that the quantity r 1 (n) defined by (1.14) satisfies

r 1 (n) ∼ n µ-2ρ ∞ i=0 c i,0 n i as n → ∞.
(1.15)

In Section 6.4 we show that in the case δ 0 = 1 one has (1.15) with

c 0,0 = -µ-2ρ+1 µ |α 2 0 |, c 1,0 = µ-2ρ µ µ -ρ + δ 1 1 -1 µ |α 2 0 | -2Re(α 0 α 1 ) . (1.16)
It is clear that the general case can be reduced to the case δ 0 = 1 by means of the multiplication of J by a suitable constant.

(ii) The asymptotic formula with two correction terms. In Theorem 6.2 we prove

λ n (J) = d(n) + r 1 (n) + r 2 (n) + O(n µ-6ρ ), (1.17)
where r 1 is given by (1.14) and r 2 ∈ S µ-4ρ 1 . Let us assume that (1.4) holds and (1.5) holds with δ 0 = 1. Then (1.15)-(1.16) are still true and in Section 6.5 we show that

r 2 (n) = c 0,1 n µ-4ρ + O(n µ-4ρ-1 ) (1.18) holds with c 0,1 = c 1 + c 2 + c 3 , (1.19) where        c 1 = -(µ -2ρ)(µ + 1 -2ρ)(µ + 1 -4ρ) |α 4 0 | 2µ 3 , c 2 = (µ -1)(µ + 2 -4ρ)(µ + 1 -4ρ) |α 4 0 | 4µ 3 , c 3 = -(µ -1) 2 (µ + 1 -4ρ) |α 4 0 | 4µ 3 .
(1.20) (iii) A complete asymptotic expansion. In Section 5 we construct r l ∈ S µ-2 l ρ 1

, where l = 1, 2, . . . and for every m ∈ N * one has

λ n (J) = d(n) + m l=1 r l (n) + O(n µ-2 m+1 ρ ).
This construction is used to deduce Theorem 1.1.

Preliminaries

2.1. Basic notations and properties. Throughout the paper {e k } k∈Z denotes the canonical basis in 2 (i.e. e k (j) = δ k,j ) and the shift operator is denoted by S (i.e. S is the linear bounded operator in 2 satisfying Se k = e k+1 for k ∈ N * ). We denote by Λ the self-adjoint operator in 2 defined by the formula (Λx)(k) = kx(k) on the domain of definition D(Λ) = 2,1 . Using the functional calculus, we can define f (Λ) as the closed linear operator satisfying f (Λ)e k = f (k)e k for all k ∈ N * . These notations allow us to write down the Jacobi operator (1.3) in the form

J = d(Λ) + Sa(Λ) + ā(Λ)S * , (2.1)
where S * is the adjoint of S. Below we introduce more notations.

Notation 2.1. Let i ∈ Z. We write

S (i) = S i if i ∈ N and S (i) = S * |i| if i ∈ Z \ N.
According to this notation, the operator S (i) is the linear mapping on 2 satisfying (i)

S (i) e k = e k+i if k + i ≥ 1, (ii)
S (i) e k = 0 if k + i ≤ 0. Let i ∈ Z and f : N * → C. Then τ i f denotes the sequence N * → C satisfying (i) τ i f (k) = f (k + i) if k + i ≥ 1, (ii) τ i f (k) = 0 if k + i ≤ 0.
We denote by ∂ i f the sequence N * → C given by the formula

∂ i f (k) := τ i f (k) -f (k). (2.2)
In particular

∂ 0 f = 0 and ∂ 1 f (k) = ∂f (k) = f (k + 1) -f (k)
is the derivative introduced in Section 1. We observe that for any fixed i ∈ Z,

∂f (k) = O(k ν ) =⇒ ∂ i f (k) = O(k ν ). (2.3)
Remark 2.2. The above notations allow us to write S (i) x = τ -i x and

f (Λ) S (i) = S (i) (τ i f )(Λ), (2.4)
where

(τ i f )(Λ) is the shifted diagonal operator satisfying (τ i f )(Λ)e k = τ i f (k)e k . Notation 2.3. If ν ∈ R and N ∈ N, then S ν 1 (N ) denotes the set of all p : N * → C such that ∂ m p(k) = O(k ν-m ) holds for m ∈ {0, . . . , N }. Lemma 2.4. (i) If N ≥ 1, i ∈ Z and p ∈ S ν 1 (N ) then ∂ i p ∈ S ν-1 1 (N -1). (ii) If p ∈ S ν 1 (N ) and q ∈ S η 1 (N ) then pq ∈ S ν+η 1 (N ). (iii) Assume that p ∈ S ν 1 (N ) and there exists c > 0 such that |p(k)| ≥ ck ν for all k ∈ N * . Then 1/p ∈ S -ν 1 (N ).
Proof. (i) It suffices to use (2.3) and the definition of

S ν 1 (N ). (ii) We obtain ∂ m (pq)(k) = O(k ν+η-m
) by induction with respect to m using

∂(f g) = f ∂g + g∂f + ∂f ∂g. (iii) We obtain ∂ m 1/p (k) = O(k -ν-m
) by induction with respect to m using

∂ 1/p (k) = - ∂p(k) p(k)p(k + 1)
.

2.2.

Operator valued errors. In this section we introduce notations used throughout the paper to control errors of large eigenvalues. As before S denotes the shift operator in 2 , i.e. Se k = e k+1 . If θ ≥ 0, Λ θ denotes the self-adjoint operator in 2 such that

Λ θ e k = k θ e k and D(Λ θ ) = 2,θ . If θ ≤ 0, then Λ θ is the inverse of Λ |θ| .
In order to deal with these operators, we introduce the subspace of fast decaying sequences

S -∞ = θ>0 2,θ = {x ∈ 2 : x(k) = O(k -N ) for any N ≥ 0}. (2.5)
The space S -∞ defined by (2.5) will be assumed to be invariant for all operators considered further on and we use the fact that this type of operators form an algebra.

Clearly S -∞ is invariant for the shifts S, S * and for

f (Λ) if f : N * → C is polynomially bounded.
We adopt the following convention: for a linear mapping P : D(P ) → 2 we write P ∈ B( 2) if and only if the closure of P is a bounded operator 2 → 2 .

Let η ∈ R. We introduce the notation

P = O(Λ η ) ⇐⇒ ∀ θ ∈ R, Λ θ P Λ -θ-η ∈ B( 2 ). (2.6)
Then (2.6) immediately ensures the properties

P = O(Λ η ) =⇒ S -∞ is an invariant subspace of P, (2.7
)

f (k) = O(k η ) =⇒ f (Λ) = O(Λ η ), (2.8)
P = O(Λ η ), Q = O(Λ η ) =⇒ P Q = O(Λ η+ η ).
(2.9)

The notation P = O(Λ -∞ ) means that P = O(Λ -θ ) holds for all θ ≥ 0 and the notation

P = Q + O(Λ η ) (2.10) means that P -Q = O(Λ η )
and S -∞ is invariant for both operators P , Q.

2.3. Finite difference operators. Any complex matrix (P (i, j)) ∞ i,j=1 can be written in the form

       p 0 (1) p -1 (2) p -2 (3) p -3 (4) . . . p 1 (1) p 0 (2) p -1 (3) p -2 (4) . . . p 2 (1) p 1 (2) p 0 (3) p -1 (4) . . . p 3 (1) p 2 (2) p 1 (3) p 0 (4) . . . . . . . . . . . . . . . . . .        , (2.11)
where (p i (k)) ∞ k=1 is a complex valued sequence (for any i ∈ Z).

Definition 2.5. We say that P is a band matrix if and only if P has the form (2.11) and there is i 0 ∈ N such that |i| > i 0 implies p i (k) = 0 for all k ∈ N * . We say that P is off-diagonal if and only if p 0 (k) = 0 holds for all k ∈ N * . If P is given by (2.11), then p 0 (Λ) is called the diagonal part of P .

Lemma 2.6. We fix ν ∈ R and i 0 ∈ N. Assume that p i (k) = O(k ν ) for all i ∈ Z and |i| > i 0 implies p i (k) = 0 for all k ∈ N * . Then the band matrix P given by (2.11) can be identified with the linear mapping acting on S -∞ according to the formula

(P x)(k) = min{i0, k-1} i=-i0 p i (k -i)x(k -i).
(2.12)

Moreover P = O(Λ ν ).
Proof. We observe that (2.12) can be written in the form

P x = -i0≤i≤i0 S (i) p i (Λ)x for x ∈ S -∞ . (2.13)
The assumptions

p i (k) = O(k ν ) ensure p i (Λ) = O(Λ ν
) and due to (2.9) it remains to show that S = O(Λ 0 ) and S * = O(Λ 0 ). In order to show

Λ θ SΛ -θ ∈ B( 2 ) for any θ ∈ R, we use (2.4) to write Λ θ SΛ -θ = S(Λ + 1) θ Λ -θ and observe that (Λ + 1) θ Λ -θ ∈ B( 2 ).
Taking the adjoint, we get

Λ -θ S * Λ θ ∈ B( 2 ). 2.4. Class of operators FDO ν 1 (N ). Notation 2.7. If ν ∈ R and N ∈ N, then FDO µ 1 (N )
denotes the set of all band matrices P given by (2.11), where p i ∈ S ν 1 (N ) for all i. Further on the elements of FDO ν 1 (N ) are always identified with linear mappings given by (2.13) where p i ∈ S ν 1 (N ). Lemma 2.8. If P ∈ FDO ν 1 (N + 1) and Q ∈ FDO η 1 (N + 1), then the commutator

[P, Q] := P Q -QP ∈ FDO ν+η-1 1 (N ).
Proof. It suffices to prove the formula

B i,j := [S (i) p i (Λ), S (j) q j (Λ)] = S (i+j) (q j ∂ j p i -p i ∂ i q j )(Λ).
(2.14) Indeed, if p i ∈ S ν 1 (N + 1) and q j ∈ S µ 1 (N + 1), then b i,j := q j ∂ j p i -p i ∂ i q j ∈ S ν+µ-1 1 (N ), which ensures that the right hand side of (2.14) belongs to FDO ν+µ-1 1 (N ). In order to prove (2.14), we observe that (2.4) allows us to express

B i,j = S (i) p i (Λ)S (j) q j (Λ) -S (j) q j (Λ)S (i) p i (Λ) = S (i) S (j) (τ j p i )(Λ)q j (Λ) -S (j) S (i) (τ i q j )(Λ)p i (Λ).
Since b i,j = q j τ j p i -p i τ i q j , it remains to check the equalities

S (i) S (j) (τ j p i )(Λ)q j (Λ) = S (i+j) (τ j p i )(Λ)q j (Λ), (2.15)
S (j) S (i) (τ i q j )(Λ)p i (Λ) = S (i+j) (τ i q j )(Λ)p i (Λ).
(2.16)

Case j ≥ 0. In this case S (i) S (j) = S (i+j) holds for any i ∈ Z. Case j < 0, i < 0. In this case S (i) S (j) = S (i+j) holds as well. Case j < 0, i ≥ 0. In this case S (i) S (j) e k = S (i+j) e k holds if k > -j. Assume now that k ≤ -j. Then S (i) S (j) e k = 0 and (τ j p i )(Λ)e k = τ j p i (k) = 0, hence (2.15) holds.

Similarly we check (2.16).

Main ingredients

3.1. Consequences of the min-max principle. In this section we prove Proposition 3.2 which is the first fundamental tool of our approach. Its purpose is to detect perturbations which give small errors for large eigenvalues. The importance of Proposition 3.2 lies in the fact that it is used as the ingredient of our approach replacing Lemma 2.1 of J. Janas, S. Naboko [START_REF] Janas | Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach[END_REF]. We notice that Lemma 2.1 from [START_REF] Janas | Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach[END_REF] is used in [START_REF] Boutet De Monvel | The asymptotic behaviour of eigenvalues of a modified Jaynes-Cummings model[END_REF], [START_REF] Malejki | Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices[END_REF]- [START_REF] Malejki | Asymptotics of the discrete spectrum for complex Jacobi matrices[END_REF], [START_REF]Jaynes-Cummings model without rotating wave approximation[END_REF], but we prefer using Proposition 3.2 for several reasons. One of them is the fact that Lemma 2.1 from [START_REF] Janas | Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach[END_REF] needs the additional assumption µ ≥ 1 in the hypothesis (H1).

Notation 3.1. If L is a self-adjoint, bounded from below operator with compact resolvent in 2 , then λ 1 (L) ≤ • • • ≤ λ n (L) ≤ λ n+1 (L) ≤ .
. . are eigenvalues of L, enumerated in non-decreasing order, counting multiplicities. Proposition 3.2. We fix µ, ρ, η ∈ (0; ∞). Let D = d(Λ) be the diagonal operator with d satisfying (H1). Let A be a symmetric operator such that D(A) ⊃ 2,µ and

A = O(Λ µ-ρ ).
(3.1)

Let J be the operator defined by

J = D + A (3.2) and D(J) = 2,µ . Then J is self-adjoint, bounded from below and has compact resolvent in 2 . If R is a symmetric operator such that D(R) ⊃ 2,µ and R = O(Λ µ-η ), (3.3)
then the operator J + R is self-adjoint in 2 on the domain D(J + R) = 2,µ . Moreover, J + R is a bounded from below operator with compact resolvent and one has is self-adjoint, bounded from below, has compact resolvent and

λ n (J + R) = λ n (J) + O(n µ-η ). ( 3
(2C) -1 Λ µ ≤ J ≤ 2CΛ µ (3.6) holds if C is large enough. Since λ n ( J) = λ n (J)+ C and λ n ( J +R) = λ n (J +R)+ C, it remains to prove λ n ( J +R) = λ n ( J) + O(n µ-η ). (3.7) Step 1. We prove that (3.7) holds if η ≤ 2µ. Since λ n ((2C) ±1 Λ µ ) = (2C) ±1 n µ , the min-max principle used in (3.6) gives (2C) -1 n µ ≤ λ n ( J) ≤ 2Cn µ . (3.8)
The well known result concerning the powers of positive operators ensures that

(2C) -|s| Λ µs ≤ Js ≤ (2C) |s| Λ µs (3.9) follows from (3.6) if -1 ≤ s ≤ 1. Since 0 ≤ η ≤ 2µ, we can use (3.9) with s = 1 -η/µ ensuring existence of positive constants C 1 , C 2 such that ±R ≤ C 1 Λ µ-η ≤ C 2 J1-η/µ . (3.10)
Let f ± : (0; ∞) → R be given by the formula f ± (t) := t ± C 2 t 1-η/µ . Then (3.10) ensures

f -( J) ≤ J + R ≤ f + ( J) (3.11)
and the min-max principle ensures

λ n (f -( J)) ≤ λ n ( J + R) ≤ λ n (f + ( J)). (3.12) Since f ± is increasing on (t 0 ; ∞) if t 0 is fixed large enough, there is n 0 ∈ N such that n ≥ n 0 =⇒ λ n (f ± ( J)) = f ± (λ n ( J)). (3.13)
Combining (3.12) and (3.13), we find that for n ≥ n 0 one has

λ n ( J) -C 2 λ n ( J) 1-η/µ ≤ λ n ( J + R) ≤ λ n ( J) + C 2 λ n ( J) 1-η/µ . (3.14)
Using once more (3.9) with s = 1 -η/µ, we conclude that (3.14) ensures

|λ n ( J + R) -λ n ( J)| ≤ C 2 λ n ( J) 1-η/µ = O(n µ-η ). (3.15)
Step 2. Let k ∈ N * . We claim that the assertion of Proposition 3.2 holds if η ≤ 2 k µ. Indeed, the case k = 1 was proved in Step 1 and using induction with respect to k ∈ N * , we fix k ∈ N * and assume that the assertion of Proposition 3.2 holds if η ≤ 2 k µ.

As before C > 0 is fixed large enough, D = d(Λ) + C I and d satisfies (H1) for a given µ > 0. Let J = D + A where A = O(Λ µ-ρ ) and ρ > 0. Then

J2 = D2 + A holds with A = DA + A D + A 2 = O(Λ 2µ-ρ ), (3.16)
i.e. J := J2 satisfies the hypotheses of Proposition 3.2 with µ := 2µ instead of µ.

Assume now that R = O(Λ µ-η ) and 0 < η ≤ 2 k+1 µ = 2 k µ . Then

( J + R) 2 = J2 + R holds with R = JR + R J + R 2 = O(Λ 2µ-η ) (3.17)
and by using Proposition 3.2 with J := J2 , µ := 2µ instead of J, µ, we obtain

λ n (( J + R) 2 ) = λ n ( J2 ) + O(n 2µ-η ). (3.18)
Let C be large enough to ensure J + R ≥ 0. Then (3.18) implies

λ n ( J + R) 2 = λ n ( J) 2 + O(n 2µ-η ) = λ n ( J) 2 1 + O(n -η ) , (3.19)
where the last estimate follows from (3.8). Thus To complete the proof, we observe that λ n (d(Λ)) = d(n) for n ≥ n 0 .

λ n ( J + R) = λ n ( J) 1 + O(n -η ) 1/2 = λ n ( J) 1 + O(n -η ) (3.
3.2. Special self-adjoint operators.

Lemma 3.4. Let P be a symmetric operator on S -∞ given by the formula

P x = 1≤i≤i0 S i p i (Λ) + p i (Λ)S (-i) x for x ∈ S -∞ . (3.22)
If p i (k) = O(k) for i = 1, . . . , i 0 , then P is essentially self-adjoint on S -∞ .

Proof. Let θ ≥ 0. Then [iP, Λ 2θ ] is symmetric on S -∞ and we claim that

[iP, Λ 2θ ] = O(Λ 2θ ). (3.23) Indeed, if f θ (k) = k 2θ , then ∂ i f θ (k) = O(k 2θ-1 ), hence (2.4) ensures [P, Λ θ ] = 1≤i≤i0 S i (p i ∂ i f θ )(Λ) + hc with (p i ∂ i f θ )(k) = O(k 2θ ).
By (3.23) we can find a constant C θ > 0 such that

± y, Λ -θ [iP, Λ 2θ ]Λ -θ y ≤ C θ ||y|| 2 for y ∈ 2 .
Writing y = Λ θ x, we get

± x, [iP, Λ 2θ ]x ≤ C θ ||Λ θ x|| 2 for x ∈ 2,θ . (3.24)
Using (3.24) with θ = 1/2 and the fact that

||P x|| ≤ C||Λx|| for x ∈ S -∞ , (3.25)
we deduce the assertion by Nelson's Commutator Theorem. Indeed, the properties (3.25) and (3.24) with θ = 1 2 allow us to apply Corollary 1.1 in [START_REF] Faris | Commutators and self-adjointness of Hamiltonian operators[END_REF] (using P , Λ and S -∞ instead of H, N and C in [START_REF] Faris | Commutators and self-adjointness of Hamiltonian operators[END_REF]). Notation 3.5. If P ∈ FDO ν 1 (N ) is given by (2.13), then the adjoint matrix (P † (i, j)) ∞ i,j=1

= ( P (j, i) ) ∞ i,j=1 satisfies

P † x = -i0≤i≤i0 p i (Λ)S (-i) x = -i0≤i≤i0 S (-i) (τ -i p i )(Λ)x for x ∈ S -∞ (3.26)
and it is clear that P † ∈ FDO ν 1 (N ). We will use the notation P + hc := P + P † .

(3.27)

If moreover p i (k) = O(k) for i = 1, . . . , i 0 , then P = 1≤i≤i0 S i p i (Λ) + hc (3.28)
will always denote the self-adjoint operator satisfying (3.22).

Proposition 3.6. Assume that p i (k) = O(k) for i = 1, . . . , i 0 and P is the self-adjoint operator given by (3.28). If θ ≥ 0 then 2,θ is an invariant subspace of e itP for any t ∈ R. Moreover there is a constant C θ such that

||Λ θ e itP x|| ≤ e C θ |t| ||Λ θ x|| (3.29)
holds for all x ∈ S -∞ and t ∈ R.

Proof. Let D(P ) denote the domain of the self-adjoint operator P equipped with its graph norm. Let ε > 0 and denote Λ (θ) ε := Λ 2θ (εΛ 2θ + I) -1 = ε -1 I -ε -1 (εΛ 2θ + I) -1 and we define the quadratic form

q (θ) ε (x) = iP x, Λ (θ) ε x + Λ (θ) ε x, iP x for x ∈ D(P ). We claim that there is a constant C θ > 0 such that ± q (θ) ε (x) ≤ C θ ||Λ θ (εΛ 2θ + I) -1/2 x|| 2 = C θ x, Λ (θ) ε x (3.30)
holds for all x ∈ D(P ). Indeed, if x ∈ S -∞ then we compute

q (θ) ε (x) = x, [Λ (θ) ε , iP ]x = (εΛ 2θ + I) -1 x, [iP, Λ 2θ ](εΛ 2θ + I) -1 x (3.31)
and by using (εΛ 2θ + I) -1 x instead of x in (3.24), we find that (3.30) holds for x ∈ S -∞ . Since S -∞ is dense in D(P ), the estimate (3.30) holds for x ∈ D(P ) as well. Let x ∈ D(P ) and denote x t := e itP x for t ∈ R. By using d dt x t , Λ (θ)

ε x t = q (θ)
ε (x t ) and (3.30), we obtain

± d dt x t , Λ (θ) ε x t ≤ C θ x t , Λ (θ)
ε x t and the Gronwall's inequality ensures

x t , Λ (θ) ε x t ≤ e C θ |t| x, Λ (θ) ε x . (3.32)
If x ∈ S -∞ , then the limit ε → 0 in (3.32) gives x t ∈ 2,θ and (3.29) holds. Since S -∞ is dense in 2,θ , we still have (3.29) for all x ∈ 2,θ and x t ∈ 2,θ if x ∈ 2,θ .

Corollary 3.7. Let P be as in Lemma 3.4. Then (i) S -∞ is an invariant subspace of e itP for any t ∈ R, (ii) t → e itP x is a smooth function R → 2,θ for any θ ≥ 0 if x ∈ S -∞ .

Proof. (ii) Let x ∈ S -∞ , ε ≥ 0 and denote x ε,θ t := Λ θ (εΛ θ + I) -1 e itP x. It is clear that t → x ε,θ t is smooth R → 2 if ε > 0. Let us fix N ∈ N * and t 0 > 0. Then the family (x ε,θ t ) 0<ε≤1 is bounded in C N ([-t 0 , t 0 ]; 2 ) due to (3.29), hence using the Ascoli Theorem we can find a sequence (ε n ) ∞ n=0 which converges to 0 and (x εn,θ t ) ∞ n=0 is convergent in C N -1 ([-t 0 , t 0 ]; 2 ). We conclude that the pointwise limit t → Λ θ e itP x is a function belonging to C N -1 ([-t 0 , t 0 ]; 2 ).

3.3.

Application of the Taylor formula. Let P be the operator satisfying the assumptions of Lemma 3.4 and assume Q = O(Λ µ ). We will denote

F tP (Q) := e -itP Qe itP for t ∈ R. Let x ∈ S -∞ and η ∈ R. Then the function t → Λ η F tP (Q)x is smooth R → 2 due to Corollary 3.7. Denoting ad 1 iP Q = ad iP Q := [Q, iP ] and ad m+1 iP Q = [ad m iP Q, iP ],
we can express the m-th derivative with respect to t ∈ R in the form

d m dt m F tP (Q)x = F tP (ad m iP Q)x for x ∈ S -∞
and by using the Taylor formula in t = 1, we obtain on S -∞ the equality

e -iP Qe iP = Q + N -1 m=1 1 m! ad m iP Q + R N (ad N iP Q), (3.33)
where

R N (T ) := 1 (N -1)! 1 0 e -itP T e itP (1 -t) N -1 dt. (3.34)
It is clear that Proposition 3.6 ensures (1) for i = 1, . . . , i 0 , and define the operator

T = O(Λ ν ) =⇒ R N (T ) = O(Λ ν ). ( 3
A = d(Λ) + 1≤i≤i0 S i a i (Λ) + hc (4.1)
on the domain D(A) = 2,µ . Then A is a self-adjoint and bounded from below operator with compact resolvent in 2 . Moreover (i) the eigenvalue sequence of A satisfies

λ n (A) = d(n) + O(n µ-2ρ ), (4.2)
(ii) if d ∈ S µ 1 (3) and a i ∈ S µ-ρ 1 (2) for i = 1, . . . , i 0 , then λ n (A) = d(n) + r 1 (n) + O(n µ-3ρ ) (4.3)
holds with where k 0 ∈ N is fixed. Since any modification of {d(k) : k < k 0 } can be viewed as a perturbation R satisfying R = O(Λ -∞ ), the corresponding error of the n-th eigenvalue is O(n -∞ ) due to Proposition 3.2. Neglecting such errors we may assume that (H2) holds with k 0 = 1, i.e.

r 1 (n) := i0 i=1 |a i (n -i)| 2 d(n) -d(n -i) - |a i (n)| 2 d(n + i) -d(n) . ( 4
∃ c > 0 ∀ k ∈ N * , d(k + 1) -d(k) ≥ ck µ-1 . (4.5)
Lemma 4.2. Let µ > 0, ρ > 0 and consider a symmetric operator on S -∞ given by

A 0 = 1≤i≤i0 S i a i (Λ) + hc . (4.6) Assume that a i (k) = O(k µ-ρ ) for i = 1, . . . , i 0 , (d(k)) ∞
k=1 satisfies (4.5) and denote

p i (k) = i a i (k) d(k + i) -d(k) = i a i (k) ∂ i d(k) . ( 4
(ii) If d ∈ S µ 1 (N + 1) and a i ∈ S µ-ρ 1 (N ), then p i ∈ S 1-ρ 1 (N ).
Proof. (i) By (2.4), we have

[D, iP ] = 1≤i≤i0 [d(Λ), S i p i (Λ)] + hc = 1≤i≤i0 S i ∂ i d(Λ)ip i (Λ) + hc (4.9)
and (4.8) holds if and only if

∂ i d(k) ip i (k) + a i (k) = 0 for all k ∈ N * .
(ii) It suffices to use Lemma 2.4.

Proof of Theorem 4.1(i).

Step 1. Let A be given by (4.1). Then

A = D + A 0 (4.10)
holds with D = d(Λ) and A 0 given by (4.6). Let P ∈ FDO 1-ρ 1

(1) be as in Section 4.2. According to Section 3.2 the operator P is self-adjoint and we claim that e -iP A 0 e iP = A 0 + O(Λ µ-2ρ ).

(4.11)

Indeed, A 0 ∈ FDO µ-ρ 1 (1) and P ∈ FDO 1-ρ 1 (1) ensure [A 0 , P ] ∈ FDO µ-2ρ 1 (0) due to Lemma 2.8, hence [A 0 , P ] = O(Λ µ-2ρ
) and the Taylor formula gives

e -iP A 0 e iP = A 0 + R 1 ([A 0 , iP ]), where R 1 ([A 0 , iP ]) = O(Λ µ-2ρ ) follows from [A 0 , iP ] = O(Λ µ-2ρ ) due to (3.35).
Step 2. We claim that

e -iP De iP = D -A 0 + O(Λ µ-2ρ ). (4.12)
Indeed, by using [D, iP ] = -A 0 in the Taylor formula

e -iP De iP = D + [D, iP ] + R 2 ([[D, iP ], iP ]),
we obtain

e -iP De iP = D -A 0 -R 2 ([A 0 , iP ]) and R 2 ([A 0 , iP ]) = O(Λ µ-2ρ ) follows from [A 0 , iP ] = O(Λ µ-2ρ
) due to (3.35).

Step 3. Denote A := e -iP Ae iP . Summing up (4.11) and (4.12), we obtain

A = e -iP (D + A 0 )e iP = D + O(Λ µ-2ρ ). (4.13)
Using (4.13) and Proposition 3.2, we obtain

λ n (A) = λ n ( A) = λ n (D) + O(n µ-2ρ ) = d(n) + O(n µ-2ρ ).

Proof of Theorem 4.1(ii).

Step 1. As before we write A = D + A 0 and P is defined as in Section 4.2. Then

A 0 ∈ FDO µ-ρ 1 (2), P ∈ FDO 1-ρ 1 (2) ⇒ [A 0 , P ] ∈ FDO µ-2ρ 1 (1)
follows from Lemma 2.8 and we claim that the diagonal part of [A 0 , iP ] satisfies

[A 0 , iP ](k, k) = 2r 1 (k), (4.14)
where r 1 (k) is given by (4.4). Indeed, for i = 1, . . . , i 0 , we have

[S i a i (Λ), ip i (Λ)(S i ) * ] = S i (ia i pi )(Λ)(S i ) * -(ia i pi )(Λ) = ∂ -i (ia i pi )(Λ) (4.15)
and the diagonal part of

[A 0 , iP ] equals 1≤i≤i0 [S i a i (Λ), ip i (Λ)(S i ) * ] + hc = 1≤i≤i0 q i (Λ), (4.16)
where q i (Λ) = ∂ -i (ia i pi )(Λ) + hc, i.e.

q i = 2Re ∂ -i (ia i pi ) = -2∂ -i Im (a i pi ) (4.17)
and

[A 0 , iP ](k, k) = 1≤i≤i0 q i (k) = 1≤i≤i0 2∂ -i |a i (k)| 2 ∂ i d(k) = 2r 1 (k).
Step 2. We claim that

e -iP A 0 e iP = A 0 + [A 0 , iP ] + O(Λ µ-3ρ ). (4.18)
Indeed, Lemma 2.8 ensures

[A 0 , iP ] ∈ FDO µ-2ρ 1 (1), P ∈ FDO 1-ρ 1 (2) ⇒ ad 2 iP A 0 ∈ FDO µ-3ρ 1 (0), hence ad 2 iP A 0 = O(Λ µ-3ρ
) and the Taylor formula gives

e -iP A 0 e iP = A 0 + [A 0 , iP ] + R 2 (ad 2 iP A 0 ), where R 2 (ad 2 iP A 0 ) = O(Λ µ-3ρ ) follows from ad 2 iP A 0 = O(Λ µ-3ρ ) and (3.35). Step 3. We claim that e -iP De iP = D -A 0 -1 2 [A 0 , iP ] + O(Λ µ-3ρ ). (4.19)
A = e -iP (D + A 0 )e iP = D + 1 2 [A 0 , iP ] + O(Λ µ-3ρ ) (4.20)
and Proposition 3.2 ensures

λ n (A) = λ n ( A) = λ n (A 1 ) + O(n µ-3ρ ), (4.21)
where we denoted

A 1 := D + 1 2 [A 0 , iP ]. Let d 1 (k) := d(k) + 1 2 [A 0 , iP ](k, k) = d(k) + r 1 (k). Then D 1 = d 1 (Λ)
is the diagonal part of A 1 and we can write

A 1 = d 1 (Λ) + A 0 1 , (4.22)
where

A 0 1 ∈ FDO µ-2ρ 1 (1) is off-diagonal. Thus Theorem 4.1(i) can be applied to A 1 with d 1 (k) = d(k) + r 1 (k) instead of d(k) and 2ρ instead of ρ. We obtain λ n (A 1 ) = d 1 (n) + O(n µ-4ρ ), (4.23)
which completes the proof due to (4.21).

5.

A method of approximative diagonalization 5.1. Introduction. In this section J satisfies (H1)-(H3) and assuming (1.11)-(1.12), we describe a special sequence of operators (J

(l) N ) ∞ l=0 such that J (l) N is self-adjoint in 2 on the domain D(J (l) N ) = 2,µ for any l, N ∈ N. Moreover J (l)
N is a bounded from below operator with compact resolvent and the eigenvalue sequence (λ n (J

(l) N )) ∞ n=1 satisfies λ n (J) = λ n (J (l) N ) + O(n µ-ρN ). (5.1)
Besides (5.1), the operators (J

N ) ∞ l=0 have a special structure of band matrices (see (5.4), (5.5)) and in Section 5.2 we describe its construction. Using (J

(l) N ) ∞
l=0 and suitable classes of operators defined in Section 5.3, we complete the proof of Theorem 1.1 in Section 5.4.

Notation 5.1. If ν ∈ R then FDO ν
1 denotes the set of all linear operators on S -∞ of the form P = -i0≤i≤i0

S (i) p i (Λ) with p i ∈ S ν 1 for i ∈ [-i 0 , i 0 ] ∩ Z. (5.2)
Clearly Lemma 2.4 holds with S ν 1 instead of S ν 1 (N ) and the proof of Lemma 2.8 ensures

P ∈ FDO ν 1 , Q ∈ FDO η 1 =⇒ [P, Q] ∈ FDO ν+η-1 1 . (5.3)
The structure of (J

N ) ∞ l=0 is the following. We write ρ l := 2 l ρ and construct J

N such that J (0) N = J and J (l)

N = d (l) N (Λ) + A (l) N with A (l) N ∈ FDO µ-ρ l 1 , (5.4)
where A

(l)

N is off-diagonal and d

N (k) ≥ 1 2 ck µ for k ≥ k (l) N , (5.6) ∂d (l)
N (k) ≥ 1 2 ck µ-1 for k ≥ k (l) N , (5.7)
where k

(l)
N ∈ N * is fixed large enough. Remark 5.3. It is clear that (5.4) implies

J (l) N = d (l) N (Λ) + O(Λ µ-ρ l ).
(5.8)

Thus the operator J (l)

N has the domain 2,µ and is a bounded from below self-adjoint operator with compact resolvent in 2 due to (5.5)-(5.6).

Remark 5.4. The estimate (5.1) ensures

λ n (J) = d (l) N (n) + O(n µ-N ρ ) if l ≥ log 2 N.
(5.9) Indeed, using (5.8) and Proposition 3.2, we obtain

λ n (J (l) N ) = λ n (d (l)
N (Λ)) + O(n µ-ρ l ).

(5.10)

Combining (5.10), (5.1) with ρ l = 2 l ρ ≥ N ρ, we obtain

λ n (J) = λ n (d (l) N (Λ)) + O(n µ-N ρ ) if l ≥ log 2 N.
(5.11)

To deduce (5.9) from (5.11), we observe that due to (5.6) and (5.7) we can find n (l)

N ∈ N * large enough to ensure λ n (d (l) N (Λ))) = d (l) N (n) for n ≥ n (l) N . 5.2. Construction of the sequence (J (l) N ) ∞ l=0 .
We fix N and skip this index writing J (l) instead of J (l) N . If l = 0, then J (0) = J and (5.4) takes the form

J (0) = d(Λ) + A (0) with A (0) = Sa(Λ) + hc, (5.12)
i.e. A (0) ∈ FDO µ-ρ0 1 holds with ρ 0 = ρ. Next we fix l ∈ N and make the induction hypothesis that J (l) has the properties given in Section 5.1. Our purpose is to construct J (l+1) having analogical properties for l+1 instead of l. At the beginning we assume that A (l) ∈ FDO µ-ρ l 1 is off-diagonal, i.e.

A (l) = 1≤i≤i l S i a (l) i (Λ) + hc with a (l) i ∈ S µ-ρ l 1 .
(5.13)

Let k (l) ∈ N * be fixed large enough. Then ∂d (l) (k) ≥ 1 2 ck µ-1 holds for k ≥ k (l) (see (5.7)), hence we can define p

(l) i ∈ S 1-ρ l 1 by the formula p (l) i (k) = i a (l) i (k) d (l) (k + i) -d (l) (k) for k ≥ k (l) (5.14) and p (l)
i (k) = 0 if k < k (l) . Similarly as in Section 4.2 we find that the operator

P (l) = 1≤i≤i l S i p (l) i (Λ) + hc (5.15)
satisfies [d (l) (Λ), iP (l) ] + A (l) = O(Λ -∞ ).

(5.16) We observe that (5.3) ensures and it is easy to see that using induction, we obtain ad m-1 iP (l) A (l) ∈ FDO µ-mρ l 1 holds for m ≥ 2.

P (l) ∈ FDO 1-ρ l 1 , A (l) ∈ FDO µ-ρ l 1 =⇒ [P (l) , A (l) ] ∈ FDO µ-2ρ l 1 (5.
(5.18)

Next we choose m l ∈ N * large enough and write the Taylor formula

e -iP (l) D (l) e iP (l) = D (l) + m l -1 m=1 1 m! ad m iP (l) D (l) + R m l (ad m l iP (l) D (l) ), (5.19)
where

D (l) := d (l) (Λ). Since ad m iP (l) D (l) = -ad m-1 iP (l) A (l) + O(Λ -∞
) holds due to (5.16), we can rewrite (5.19) in the form

e -iP (l) D (l) e iP (l) = D (l) -A (l) - m l -2 m=2 1 m! ad m-1 iP (l) A (l) + O(Λ µ-ρ l m l ), (5.20)
where the estimate of last term follows from ad m l -1 iP (l) A (l) = O(Λ µ-ρ l m l ) and (3.35). Writing m -1 instead of m in the Taylor's formula (3.33), we get

e -iP (l) A (l) e iP (l) = A (l) + m l -2 m=2 1 (m -1)! ad m-1 iP (l) A (l) + O(Λ µ-ρ l m l ). ( 5

.21)

If J is given by (1.3), then its eigenvalue sequence

(λ n (J)) ∞ n=1 satisfies λ n (J) = d(n) + r 1 (n) + r 2 (n) + O(n µ-6ρ ), (6.5)
where r 1 is given by (6.4) and

r 2 = r 2,1 + r 2,2 + r 2,3 (6.6
)

holds with r 2,1 := -1 2 ∂ -1 (|p| 2 ∂r 1
), (6.7)

r 2,2 := 1 4 ∂ -1 Im (p {{a, p}, τ -1 p} 2,-1 ), (6.8)
r 2,3 := 1 4 ∂ -2 |{a, p}| 2 ∂ 2 d . ( 6
.9) 6.2. Analysis of commutators. As in Section 4 we assume that (d(k)) ∞ k=1 is strictly increasing. We also assume that p is given by (6.3), r 1 is given by (6.4) and the operators

A (0) ∈ FDO µ-ρ 1 , P ∈ FDO 1-ρ 1 are defined by A (0) = Sa(Λ) + hc, (6.10)
P = Sp(Λ) + hc. (6.11) Lemma 6.3. If A (0) and P are given by (6.10) and (6.11), then

ad iP A (0) = q 0 (Λ) + (S 2 q 2 (Λ) + hc), (6.12
)

ad 2 iP A (0) = (Sq 1 (Λ) + S 3 q 3 (Λ)) + hc, (6.13)
ad 3 iP A (0) = q 0 (Λ) + (S 2 q 2 (Λ) + hc) + (S 4 q 4 (Λ) + hc), (6.14)
ad 4 iP A (0) = (Sq 1 (Λ) + S 3 q 3 (Λ) + S 5 q 5 (Λ)) + hc (6.15) hold with q 0 = 2r 1 , (6.16) q 2 = {a, ip}, (6.17) q 1 := 2ip∂r 1 -{{a, p}, τ -1 p} 2,-1 , (6.18) q 0 = -2∂ -1 Im (q 1 p) (6.19) and

q i ∈ S µ-2ρ 1 , q i ∈ S µ-3ρ 1 , q i ∈ S µ-4ρ 1
, q i ∈ S µ-5ρ 1 (6.20) hold for any i.

Proof. In this proof we deal with operators of the form S (i) p i (Λ), which are written in the shortened way as S (i) p i . In particular, the formula (2.14) expressed in the shortened way, takes the form [S (i) p i , S (j) q j ] = S (i+j) {p i , q j } i,j . (6.21)

Step 1. Analysis of

Q := ad iP A (0) . The diagonal part of Q = [Sa + hc, iP ] is [Sa, ip S * ] + hc = ∂ -1 (iap) + hc = 2∂ -1 Re (iap) = -2∂ -1 Im (ap) = 2r 1 (6.22)
and the off-diagonal part of [Sa + hc, iP ] is [Sa, iSp] + hc = S 2 {a, ip} + hc. (6.23) Thus (6.12) holds with q 0 , q 2 given by (6.16) and (6.17).

Step 2. Analysis of Q := ad 2 iP A (0) . Combining (6.12) with Q

= [Q, iP ], we get Q = [q 0 , iP ] + [S 2 q 2 + hc, iS * τ -1 p] + [S 2 q 2 + hc, iSp] = Q 1 + Q 1 + Q 3 , where Q 1 = [q 0 , iP ] = [q 0 , iSp] + hc = S ip∂q 0 + hc, Q 1 = [S 2 q 2 , iS (-1) τ -1 p] + hc = S i{q 2 , τ -1 p} 2,-1 + hc, Q 3 = [S 2 q 2 ,
iSp] + hc = S 3 i{q 2 , p} 2,1 + hc = S 3 q 3 + hc. Thus (6.13) holds with q 1 = ip∂q 0 +i{q 2 ,τ -1 p} 2,-1 and we get (6.18) from (6.16)-(6.17).

Step 3. Analysis of Q := ad 3 iP A (0) . Using (6.13), (6.18), we find that the diagonal part of Q = [Q , iP ] can be obtained, similarly as in (6.22), by taking q 1 instead of a, i.e.

[Sq

1 , ipS * ] + hc = 2∂ -1 Re (iq 1 p) = -2∂ -1 Im (q 1 p) = q 0 . The off-diagonal part of Q is the sum of [Sq 1 , iSp] + hc = S 2 i{q 1 , p} + hc, [S 3 q 3 , iS (-1) τ -1 p] + hc = S 2 i{q 3 , τ -1 p} 3,-1 + hc, [S 3 q 3 , iSp] + hc = S 4 i{q 3 , p} 3,1 + hc.
Step 4. End of the proof. Denote Q := ad 4 iP A (0) . By using (6.14), we find easily that the commutator Q = [Q , iP ] has the form given in (6.15). To complete the proof, we observe that the properties (6.20) follow from (5.18). 6.3. End of the proof of Theorem 6.2. Writing J = J (0) = D (0) +A (0) = d(Λ) + (Sa(Λ) + hc) in (5.22)-(5.23) with l = 0, ρ 0 = ρ and m 0 = 6, we obtain e -iP Je iP = J + O(Λ µ-6ρ ), (

where J = d(Λ) + 1 2 ad iP A (0) + 1 3 ad 2 iP A (0) + 1 8 ad 3 iP A (0) + 1 30 ad 4 iP A (0) . (6.25)

Taking into account the assertion of Lemma 6.3, we find

J = d (Λ) + A , (6.26)
where d = d + 1 2 q 0 + 1 8 q 0 (6.27) and A = 1≤i≤5 (S i a i (Λ) + hc) (6.28) holds with a 1 = 1 3 q 1 + 1 30 q 1 , (6.29) a 2 = 1 2 q 2 + 1 8 q 2 , (6.30)

a i ∈ S µ-3ρ 1 if i = 2. (6.31)
We observe that by using the expressions for q 0 , q 0 given by (6.16), (6.19), we can write (6.27) in the form d = d + r 1 + r 2,1 + r 2,2 , (6.32)

where r 1 , r 2,1 , r 2,2 are given by (6.4), (6.7), (6.8) respectively. We observe that A ∈ FDO µ-2ρ (6.34) Since (6.24) ensures λ n (J) = λ n (J ) + O(n µ-6ρ ), (6.35) the assertion of Theorem 6.2 will follow from (6.32)-(6.33), if we check that the quantity r 1 introduced in (6.34) satisfies the estimate

r 1 (n) = r 2,3 (n) + O(n µ-6ρ ), (6.36)
where r 2,3 is given by (6.9). In order to check (6.36), we first observe that (6.31) ensures

r 1 (n) = ∂ -2 |a 2 (n)| 2 ∂ 2 d (n) + O(n µ-6ρ
). (6.37)

We observe that

∂ 2 d (n) = ∂ 2 d(n)(1 + O(n -2ρ )) ensures r 1 (n) = ∂ -2 |a 2 (n)| 2 ∂ 2 d + O(n µ-6ρ
). (6.38)

Using a 2 -1 2 q 2 ∈ S µ-4ρ 1 , it is easy to check that (6.38) still holds if a 2 is replaced by 1 2 q 2 = i 2 {a, p}, i.e. (6.36) holds with r 2,3 given by (6.9). 6.4. Expressions for coefficients c 0,0 , c 1,0 , c 2,0 in (1.15). Notation 6.4. We write p(k) = q(k) + S ν 1 if and only if p, q : N * → C satisfy p -q ∈ S ν 1 .

Step 1. Expansion of ∂d(k). Assume that (1. Step 3. Expansion of |a(k)| 2 /∂d(k). Since

|α 0 | 2 + β 1 k -1 + β 2 k -2 1 + δ 1 k -1 + δ 2 k -2 = |α 0 | 2 + γ 1 k -2 + γ 2 k -2 + S 2µ-2ρ-3 1 holds with γ 1 := β 1 -|α 0 | 2 δ 1 , γ 2 := β 2 -β 1 δ 1 + |α 0 | 2 (δ 1 2 -δ 2 ), we find |a(k)| 2 ∂d(k) = k µ-2ρ+1 µ |α 0 | 2 + γ 1 k -1 + γ 2 k -2 + S µ-2ρ-2 1 .
Step 4. Expansion of r 1 (k). We conclude that r 1 (k) = ∂ -1 |a| 2 ∂d (k) = k µ-2ρ (c 0,0 + c 1,0 k -1 + c 2,0 k -2 ) + S µ-2ρ-3 1 (6.41)

. 4 )

 4 Proof. Let C > 0 and denote D := D + CI. Assume that C is fixed large enough. Then (H1) ensures C -1 Λ µ ≤ D ≤ CΛ µ with C > 0, hence D-1 is compact. Since A has zero relative bound with respect to D (due to (3.1)), the operaror J := J + CI = D + A (3.5)

20 )Corollary 3 . 3 .

 2033 and using(3.8) in the last estimate, we deduce (3.7). Let µ > 0 and ρ > 0. Assume that (H1) and (H3) hold and there existsk 0 such that d(k + 1) ≥ d(k) for k ≥ k 0 . If the operator J is given by (1.3), then λ n (J) = d(n) + O(n µ-ρ ). (3.21) Proof. Denote D = d(Λ) and R = Sa(Λ) + ā(Λ)S * . Since (H3) ensures R = O(Λ µ-ρ ), we can write (3.4) with D instead of J and find λ n (D + R) = λ n (D) + O(n µ-ρ ).

.35) 4 . 4 . 1 . 4 . 1 .

 44141 Asymptotic formula with one correction term for band matrices Statement of the result. In this section we will prove Theorem Let µ > 0 and ρ > 0 be fixed. Assume that d ∈ S µ 1 (2) satisfies (H1) and (H2). Assume that a i ∈ S µ-ρ 1

. 4 ) 4 . 2 .

 442 Solution of an auxiliary matrix equation. We observe that the assumptions (H1) and (H2) concern the sequence (d(k)) ∞ k=k0

. 7 )

 7 (i) If P is expressed by (3.22) with p i given by (4.7), then [D, iP ] + A 0 = 0. (4.8)

  Indeed, it suffices to use [D, iP ] = -A 0 in the Taylor formula e -iP De iP = D + [D, iP ] + 1 2 [[D, iP ], iP ] + R 3 (ad 3 iP D) and R 3 (ad 3 iP D) = O(Λ µ-3ρ ) follows from ad 3 iP D = -ad 2 iP A 0 = O(Λ µ-3ρ ). Step 4. Denote A := e -iP Ae iP . Summing up (4.18) and (4.19), we obtain

N 1 . ( 5 . 5 ) 5 . 2 . 1 ,

 155521 (k) = d(k) + 1≤m≤l r m,N (k) with r m,N ∈ S µ-ρm Remark Since (5.5) ensures d (l) N -d ∈ S µ-2ρ

17

 17

  )

1 and applying Theorem 4 . 1

 141 to d , A , 2ρ instead of d, A 0 , ρ, we obtain λ n (J ) = d (n) + r 1 (n) + O(n µ-6ρ ),

1 ( 6 2 + 1 ( 6

 16216 4) holds. Then for any N ∈ N * one has∂d(k) = N -1 i=0 δ i k µ-i (1 + k -1 ) µ-i -1 + O(k µ-N ).Let us assume δ 0 = 1. Then∂d(k) = µk µ-1 (1 + δ 1 k -1 + δ 2 k -2 ) + S µ-4 (µ -1)δ 1 , δ 2 := 1 µ µ 3 + µ-1 2 δ 1 + (µ -2)δ 2 .Step 2. Expansion of |a(k)| 2 . We assume that (1.5) holds. Then|a(k)| 2 = k 2µ-2ρ (|α 0 | 2 + β 1 k -1 + β 2 k -2 ) + S 2µ-2ρ-3 .40) holds with β 1 := 2Re(α 0 α 1 ), β 2 := 2Re(α 0 α 2 ) + |α 1 | 2 .

1

  Center of mathematical research of Rabat, P.O. Box 1014, Department of Mathematics, Mohammed V University of Rabat, Marocco and Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville EA 2597, Université du Littoral Côte d'Opale, F-62228 Calais, France, E-mail ayoub.harrat@etu.univ-littoral.fr 2 Center of mathematical research of Rabat, P.O. Box 1014, Department of Mathematics, Mohammed V University of Rabat, Marocco, E-mail zerouali@fsr.ac.ma 3 Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville EA 2597, Université du Littoral Côte d'Opale, F-62228 Calais, France, E-mail: lech.zielinski@univlittoral.fr

Combining (5.21) with (5.20), we obtain e -iP (l) (D (l) + A (l) )e iP (l) = D (l) + Ã(l) + O(Λ µ-ρ l m l ), (5.22) where

(5.23)

Let us denote J (l+1) := D (l) + Ã(l) .

(5.24)

Using (5.24) and J (l) = D (l) + A (l) , we find that (5.22) gives e -iP (l) J (l) e iP (l) = J (l+1) + O(Λ µ-m l ρ l ).

(5.25)

Applying Proposition 3.2, we find that (5.25) ensures λ n (J (l) ) = λ n (e -iP (l) J (l) e iP (l) ) = λ n (J (l+1) ) + O(n µ-m l ρ l ).

(5.26)

Let A (l+1) be the off-diagonal part of Ã(l) and let r l+1 (Λ) be its diagonal part. Then

where

due to (5.23) and ρ l+1 = 2ρ l . Thus setting

we find

i.e. (5.4), (5.5) hold with l + 1 instead of l. Finally (5.1) and (5.26) ensure

hence (5.1) holds with l + 1 instead of l if m l ρ l ≥ N ρ, i.e. if we choose m l ≥ 2 -l N . γ i,j k ν-2ρj-i (5.29) holds for a certain complex valued sequence (γ i,j ) ∞ i,j=0 . The formula (5.29) means that

holds for any N ∈ N * . One checks easily the following properties

(5.32)

(5.34) Notation 5.6. We denote by FDO ν [1,2ρ] the set of all linear operators on S -∞ of the form P = -i0≤i≤i0

Proof. It suffices to check that the right hand side of (2.14) belongs to FDO ν+µ-1 [1,2ρ] , i.e.

.

(5.36)

and

.

(5.37)

. We deduce the last inclusion using (5.31) with k = 1 2 (|i| + |j| -|i + j|), m = 0 and ν + η -1 instead of ν. Indeed, it is easy to see that 1 2 (|i| + |j| -|i + j|) ∈ N for every i, j ∈ Z. 5.4. Proof of Theorem 1.1. The assertion of Theorem 1.1 follows from Theorem 5.8. Let µ > 0, ρ > 0 and let J be given by (1.3) with entries satisfying

Proof. Let N ∈ N * and as in Section 5.2 denote J (l) instead of J

N . We claim that J (l) ∈ FDO µ . Assume now that (5.38) holds for a given l ∈ N. Then a

and [P (l) , A (l) ] ∈ FDO µ [1,2ρ] due to Lemma 5.7. We observe that ad m iP (l) A (l) ∈ FDO µ [1,2ρ] follows by induction with respect to m, hence Ã(l) ∈ FDO µ [1,2ρ] and (5.38) still holds with l + 1 instead of l. In order to deduce that n → λ n (J) belongs to S µ [1,2ρ] , it suffices to observe that for any N ∈ N * one has the estimate (5.9) and d

, let d ∈ S µ 1 be such that (H1), (H2) hold for all k ∈ N * and define p := i a ∂d , (6.3)

(6.42)

We deduce (1.16) using r 1 expressed in (6.42) with γ 1 given in Step 3, β 1 given in Step 2 and δ 1 given in Step 1.

6.5. Expression of c 0,1 in (1.18). Due to (6.6), it suffices to show that

holds with r 2,1 , r 2,2 , r 2,3 given by (6.7)-(6.9) and c 1 , c 2 , c 3 given by (1.20).

Case i = 1. By using (6.41)-( 6.42), we obtain

Then we observe that (6.44) and p(k

In order to deduce (6.43) for i = 1, it remains to remark that (6.45) gives

).

Case i = 2. Denote q 2 := {a, ip}. By using

In order to deduce (6.43) for i = 2, it remains to remark that the last estimate gives in order to deduce (6.43) for i = 3, it remains to remark that (6.47) gives

).