Yoann Roth
email: yoann.roth@cea.fr

Jean-Baptiste Doré
email: jean-baptiste.dore@cea.fr

Laurent Ros
email: laurent.ros@gipsa-lab.grenoble-inp.fr

Vincent Berg
email: vincent.berg@cea.fr

Turbo-FSK, a Physical Layer for Low Power Wide Area Networks: Analysis and Optimization

Keywords: Turbo FSK, Low rate, Internet of Things (IoT), Low Power Wide Area (LPWA)

As the Internet-of-Things is becoming a reality, the need for a new Low Power Wide Area (LPWA) network emerged in the last few years. Numerous low cost devices will be connected and this requires an optimization of the link budget: the physical layer needs to be designed highly energy efficient. The combination of M -ary orthogonal Frequency-Shift-Keying (M -FSK) modulation and coding in the same process has been shown to be a promising candidate when associated to an iterative receiver (turbo principle). In this work, we study this new digital transmission scheme, called Turbo-FSK. An EXtrinsic Information Transfer (EXIT) Chart analysis is realized. The influence of the packet length is investigated, and the scheme is shown to stay energy efficiency even with short packet sizes. Comparison with LPWA current technologies is performed, showing the potential of this technology.

Introduction

Energy is a major concern when it comes to embedded systems; some systems will harvest the surrounding sources of energy, other will use low power components to save it. In the context of wireless communication systems, the technique used to transmit the information can be more or less energy efficient: for embedded systems, it becomes a critical aspect. The Internet-of-Things (IoT) has emerged as one of the leading research area for wireless communications as several billions connected devices are forecasted in years to come [START_REF]Verizon, State of the Market: Internet of Things 2016[END_REF]. A significant part of communication transactions in the IoT are expected to be done through Low Power Wide-Area (LPWA) networks [START_REF]LTE-M -Optimizing LTE for the Internet of Things[END_REF][START_REF] Rebbeck | Low-powered wireless solutions have the potential to increase the M2M market by over 3 billion connections[END_REF], for which requirements include low cost and low energy consumption. A link budget improvement of 15/20dB in comparison to existing cellular technologies is expected [START_REF]LTE-M -Optimizing LTE for the Internet of Things[END_REF], as reducing the costs to connect the devices to the wide-area network will encourage IoT deployment. To meet these requirements, the current generation of LPWA technologies dramatically extended the sensitivity performance of its receivers.

The sensitivity of a receiver is defined as the minimal power required to achieve an arbitrary level of error rate. It is commonly expressed as ρ dBm = SNR dB min + 10 log 10 (BN 0 ), [START_REF]Verizon, State of the Market: Internet of Things 2016[END_REF] where B is the bandwidth of the signal (in Hz), SNR min the minimum Signal-to-Noise Ratio (SNR) to reach the level of error rate, and N 0 the spectral density of the noise in mW/Hz (with 10 log 10 (N 0 ) -174dBm/Hz at 20°C). In Equation [START_REF]Verizon, State of the Market: Internet of Things 2016[END_REF], ρ is expressed in dBm, i.e 10 times the log of the ratio of the power expressed in mW. The energy efficiency of a physical layer can be estimated from the energy per bit to noise spectral density ratio, denoted by E b /N 0 . This metric shows how efficiently the energy is used: the lower is the required E b /N 0 , the more energy efficient is the technology. The relationship between the SNR and the energy efficiency is

SNR = R B • E b N 0 = η • E b N 0 . ( 2
)
where R is the data rate and η = R/B is the spectral efficiency of the technology, expressed in bits/s/Hz. A different expression of the sensitivity, regardless of the bandwidth, can then be given as

ρ dBm = E b N 0 dB min
+ 10 log 10 (R) + 10 log 10 (N 0 ).

This expression clearly explains the current trend of LPWA networks towards low data rate: if the value of R is reduced, lower levels of sensitivity are required to guarantee the quality of service and longer range of communication may be provided by the system. Reducing the data rate can be done by reducing the bandwidth B for a constant spectral efficiency, or by reducing the spectral efficiency η for a constant bandwidth B. The first solution leads to narrow-band signaling, the option chosen by the IoT company Sigfox [START_REF]SigFox website[END_REF]. Dealing with narrow-band signals involves some technological issues, as it implies long durations of emitted signals. The second option, reducing the value of η, is often done by the use of the well-known spreading factor, or repetition factor. Indeed, repeating by a factor λ divides both the values of η and SNR min by the same factor, thus lowering the sensitivity level (when the bandwidth is fixed).

However, neither of these techniques change the energy efficiency, as the required E b /N 0 is intrinsic to the modulation used. It is furthermore bounded by the Shannon's limit of the information theory [START_REF] Shannon | A mathematical theory of communication[END_REF], which defines the maximum transmission rate with arbitrarily low bit-error probability, for a given SNR and bandwidth. A formulation of this limit can be [START_REF] Proakis | Digital Communications 3rd Edition, Communications and signal processing[END_REF]

E b N 0 ≥ 2 η -1 η , (4)
thus giving the minimum E b /N 0 (i.e maximum energy efficiency) for a reliable communication as an increasing function of the spectral efficiency, with the ultimate limit E b /N 0 = -1.59dB when η tends toward 0. As having a system's performance close to this ultimate limit would imply an optimal use of the energetic resource for a given spectral efficiency, it is clear that decreasing the required E b /N 0 should be a major concern.

An interesting option to combine both E b /N 0 and η reduction is the use of M -ary orthogonal modulations. Indeed, by increasing the size of the alphabet M , the spectral efficiency (given by log 2 (M )/M ) is reduced and the energy efficiency is increased, eventually reaching the Shannon's limit for an infinite value of M [START_REF] Proakis | Digital Communications 3rd Edition, Communications and signal processing[END_REF]. However, since the spectral efficiency becomes close to 0, this solution is quite unrealistic. It is nonetheless purportedly used by a current commercial off-the-shelf long range solution, supported by the LoRa Alliance [START_REF][END_REF], as suggested by the patent [START_REF] Seller | Low power long range transmitter[END_REF] held by a company member of the alliance. Another option to reduce both E b /N 0 and η is the use of channel coding [START_REF] Costello | Channel coding: The road to channel capacity[END_REF]. In this area, the use of the turbo principle [START_REF] Berrou | Near Shannon limit error-correcting coding and decoding: Turbo-codes[END_REF] has been shown to be particularly efficient, but implies high consumption at the receiver side. Even though the transmitter for this scheme has a low complexity, most of the current LPWA solutions rely on other Forward Error Correction (FEC) codes, and a potential improvement can be done by introducing more sophisticated receiver algorithms.

The use of orthogonal alphabet and coding in the same transmit process combined with a turbo receiver was first presented for the case of the binary Hadamard code in [START_REF] Ping | Low-rate turbo-Hadamard codes[END_REF]. In [START_REF] Roth | Turbo-FSK: A New Uplink Scheme for Low Power Wide Area Networks[END_REF], we proposed the Turbo-FSK scheme as an adaptation of [START_REF] Ping | Low-rate turbo-Hadamard codes[END_REF], replacing the binary Hadamard codewords by the non-binary complex codewords of the orthogonal M -ary Frequency-Shift-Keying modulation (M -FSK). This modulation is an interesting choice as its constant envelope property provides a power efficient solution regarding the transmit power amplifier. The use of pure frequency waveforms also leads to robustness through frequency-selective multipath channel. Demodulation can be performed using the Fast-Fourier Transform (FFT), as in Orthogonal-Frequency-Division-Multiplexing (OFDM) receivers [START_REF] Kikuchi | Highly-sensitive coherent optical detection of M-ary frequency-shift keying signal[END_REF]. M -FSK is widely used for monitoring application, and off-the-shelf optimized chips are available [START_REF]SX1272 from Semtech[END_REF]. Limitations of the Turbo-Hadamard code from [START_REF] Ping | Low-rate turbo-Hadamard codes[END_REF] have been studied in [START_REF] Wu | On the limiting performance of turbo-Hadamard codes[END_REF], using the EXtrinsic Information Transfert (EXIT) Chart analysis [START_REF] Brink | Convergence behavior of iteratively decoded parallel concatenated codes[END_REF] and its extension to multi-dimensional codes [START_REF] Brink | Convergence of multidimensional iterative decoding schemes[END_REF]. The EXIT analysis is used to observe the exchanges of information inside the decoder, and to predict the "waterfall" region of a turbo process, i.e the region where the Bit-Error Rate (BER) curve drops significantly.

In this paper, we propose to extend the initial analysis of the Turbo-FSK done in [START_REF] Roth | Turbo-FSK: A New Uplink Scheme for Low Power Wide Area Networks[END_REF]. After presenting the modulation and demodulation procedures, we perform the EXIT analysis of the scheme, considering the modifications implied by the use of the FSK alphabet. Using this analytical tool, we optimize the parameters of the scheme by finding, for every set of parameters, the best pinch-off value. This value is defined as the E b /N 0 value for which the decoder can recover without error the information word after a certain number of iterations. EXIT chart accurately predicts the convergence behavior of the iterative decoder for large interleaving depth. Considering the LPWA context, where short packet lengths are expected, we propose to perform extensive BER computation to confirm if the general trend obtained from the EXIT analysis is confirmed when the packet length is shortened. The scheme is then compared to state-of-the art modulations, including the Turbo Code standardized and used in the 3G and 4G standards [START_REF]LTE Evolved Universal Terrestrial Radio Access (E-UTRA): Multiplexing and Channel Coding[END_REF]. This channel code is a natural candidate for the next generation of cellular IoT network, the Narrow-Band IoT (NB-IoT) recently released by the 3GPP [START_REF]Introduction of Rel-13 feature of NB-IoT in 36.212[END_REF].

The paper is organized as follows: the system model is introduced in Section 2, while the EXIT analysis and the parameters optimization are performed in Section 3. Performance of the scheme and comparison versus LPWA solutions are presented Section 4 and Section 5 concludes the paper.

System Model

For the following parts, bold lowercase letters denote row vectors, e.g. x, and bold uppercase letters matrices, e.g. X. The Discrete Fourier Transform (DFT) matrix of size 2 r × 2 r is denoted F r . j is the unit imaginary number, z the complex conjugate of z and Z H is the Hermitian transpose of the matrix Z. Re(z) (resp. Im(z)) is the real part (resp. imaginary part) of complex number z.

Transmitter

We denote D the information bit matrix of size P × r, which elements are {b p,n }, with p ∈ {0, ..., P -1} and n ∈ {0, . . . r -1}. We assume that the information bits are independent and equiprobable. The transmitter is shown in Figure 1 (a). It is composed of λ stages, each one having for input an interleaved version of D.

The convolutional-FSK encoder is presented in Figure 1 (b). The encoding process consists in adding one redundancy bit, computed as the accumulated parity of the r input bits. The r + 1 bits are then mapped to the FSK alphabet, which contains 2 r+1 codewords. Because of this encoding, each of the P codewords is linked to the previous one. An extra codeword is appended to set back the memory to 0, leading to a total of P + 1 FSK codewords of length M . The alphabet A can be constructed as orthogonal or bi-orthogonal. In the orthogonal case, A = F r and r = log 2 (M ) -1. In the bi-orthogonal case, A = [F r ; -F r ] and r = log 2 (M ). Each stage output S (k) , k ∈ {0, . . . λ -1}, is then a (P + 1) × M matrix, and all the codewords are concatenated in a serial way to be sent through the transmission channel, giving the (P +1)M λ elements of the vector x. The code is entirely described by the parameters M , λ and N . The spectral efficiency of the scheme is defined as

η = N (N/r + 1)M λ ≈ r M λ , ( 5
)
where N is the information block size with N = P r. The approximation is valid when a large value of N is considered.

For illustration purpose, a typical mapping of the FSK alphabet is represented on the Figure 2, along with the oversampled time representation and the spectrum of each codeword.

AWGN channel and its parameters

The discrete vector x is transformed into an analog signal by using chip pulse at chip-time T c = η/R = 1/B. After up transposition to carrier frequency, the RF real signal is transmitted over an AWGN channel with monolateral bandwidth Binary information word FSK codeword Oversampled waveform Shaped spectrum

0 0 1 1 1 1 0 1 1 -j -1 j 1 0 1 -1 1 -1 1 1 1 j -1 -j -1 1 t Re Im -1 1 t -1 1 t -1 1 t f |S| f f f
y (0)
y (1) y (λ-1)

Y (0)
Y ( 1)

Y (λ-1) L FFT (a) State 1 State 0 t t + 1 c 4 c 7 c 0 c 3 c 1 c 2 c 5 c 6 (b)
where n is a zero-mean circular complex white noise. All vectors are composed of (P + 1)M λ chips. Each element of n 50 are independant and have a variance equal to σ 2 = N 0 B, and the power of the signal x is equal to P x .

Receiver

The Turbo-FSK receiver is depicted in Figure 3 (a). After a Serial-to-Parallel conversion, the λ vectors y (k) (with k ∈ {0, . . . λ}) of P + 1 orthogonal codewords are fed to the FSK detector. This block consists in performing the projection of each codewords in y (k) over the alphabet of all possible codewords, i.e performing the matrix operation y (k) A H . Since a FSK alphabet has been chosen, this step can be performed using the Fast-Fourier Transform (FFT) algorithm. The output Y (k) is a (P + 1) × 2 r+1 matrix, which will be used as channel observations by its associated decoder. The turbo principle consists in exchanging iteratively information between the decoders. After the appropriate deinterleaving, a decoder will use both the channel observation and the extrinsic information from all the other decoders (the a priori information) to perform a new estimation of the information bits. The exchanged information between the decoders, denoted as L in Figure 3, is the Log Likelihood Ratio (LLR) of the information bits, defined as

L = L(b p,n ) = log p(b p,n = 0) p(b p,n = 1) p∈{0,...,P -1} n∈{0,...,r-1} , (7)
where p(b p,n = 1) (resp. p(b p,n = 0)) is the probability that the bit b p,n equals 1 (resp. 0). The initial value for matrix L is all-zero, and does not include information about the trellis termination bits. While the sign of L(b p,n ) gives the hard value of the bit, its magnitude gives the reliability. At the end of every iteration, the matrix L includes the information 55 from all the decoders, and a decision can be made on the information bits.

Probabilistic Decoding

Each decoder performs an estimation of the information bits by computing the A Posteriori Probabilities (APP) of the bits. We consider the received noisy FSK codeword y p , p ∈ {0, . . . P -1} from the k-th stage. d = {d n } n∈{0,...r-1} is the decoded information word. c i is a codeword from the FSK alphabet A, and b i = {b i n } n∈{0,...r-1} its associated information word (i ∈ {0, . . . 2 r+1 -1}). The APP of a codeword is the probability of having the codeword knowing the observation, and

p(c i |y p ) = p(y p |c i ) p(c i ) p(y p ) , (8)
by applying Bayes' theorem. p(c i ) is the a priori probability of having the codeword c i , or that the decoded information word d is b i . Introducing Y p i the i-th component of the DFT of y p , the APP becomes [START_REF] Roth | Turbo-FSK: A New Uplink Scheme for Low Power Wide Area Networks[END_REF] p(c

i |y p ) = C exp 1 σ 2 Re (Y p i ) + r-1 n=0 1 -2b i n 2 L(d n ) , (9)
with C a constant eliminated in further computations. This expression conveniently uses the DFT of the received codeword, i.e the channel observation, and a particular combination of the LLR L(d n ). The use of a memory in the encoding process, as described Figure 1 (b), induces the construction of a trellis. The Bahl, Cocke, Jelinek and Raviv (BCJR) [START_REF] Bahl | Optimal decoding of linear codes for minimizing symbol error rate (corresp.)[END_REF] algorithm can be used to decode the trellis, as suggested in [START_REF] Ping | Low-rate turbo-Hadamard codes[END_REF]. Denoting c i , i ∈ {0, 2 r+1 -1} the codewords of the alphabet, the trellis for the special case of an orthogonal alphabet and M = 8 is depicted on the Figure 3 (b). Each state transition is encoded by 2 r-1 codewords (i.e 2 is this case). Due to the orthogonality of the alphabet, every transition is orthogonal to each other. This orthogonality between the transitions is a fundamental characteristic of the code. For one particular section of the trellis, the BCJR algorithm will update the codewords probabilities with the knowledge of all the other sections. The updated probabilities are denoted P (c i |y p ).

The LLR of the information bits can be computed with [START_REF] Hagenauer | Iterative decoding of binary block and convolutional codes[END_REF] L(d n |y p ) = log

i∈B n 0 P (c i |y p ) -log i∈B n 1 P (c i |y p ), (10)
where B n 0 (resp B n 1 ) are the codewords indexes associated to an information word for which bit n equals 0 (resp 1), with n ∈ {0, . . . r -1}.

Complexity of the Maximum A-Posteriori (MAP) algorithm presented here can be reduced using the max log approximation

log i e xi max i (x i ). ( 11
)
This simplification will induce some performance loss, but reduces the computations to sum and max operations. Under AWGN consideration, it also allows the system to work without any Channel State Information (CSI), as the term σ 2 from (9) will be eliminated during the computations. 60

EXIT Chart Analysis

The Turbo-FSK scheme offers two adjustable parameters, M and λ. They can be optimized for better energy efficiency. However, a theoretical approach must be done to find the best set of parameters. A useful tool for iterative process analysis is the EXIT Chart, or the tracking of the exchanges of information between the decoders. It was first introduced by ten Brink [START_REF] Brink | Convergence behavior of iteratively decoded parallel concatenated codes[END_REF], for the particular case of Parallel Concatenated Convolutional Codes (PCCC), or Turbo Codes (TC).

The idea of the EXIT Chart is to compute the extrinsic information from the output of the decoder for a certain amount of a priori information, which is simulated using a model. The metric used to measure the quantity of information is the Mutual Information (MI). Considering two sources of information X and Y and assuming that X is a binary source with equiprobable possible values +1 or -1, the MI between these two sources is

I(X, Y ) = 1 2 x=±1 y f Y |X (y | X = x). log 2f Y |X (y | X = x) f (y|x = +1) + f (y|x = -1)
dy, [START_REF] Roth | Turbo-FSK: A New Uplink Scheme for Low Power Wide Area Networks[END_REF] with f the probability density functions. The value of the MI is between 0 (independence between the two sources) and 1 (complete correlation).

A priori model

The a priori information model chosen is the standard Gaussian model N (µ A , σ A ), as suggested in [START_REF] Brink | Convergence behavior of iteratively decoded parallel concatenated codes[END_REF]. The observation of the exchanged LR in the decoder allows us to validate this hypothesis. Assuming the consistency of the LR, we have

µ A = σ 2
A /2, such that the model is now described by only one parameter. The a priori LR are expressed

L A = µ A • x + n A ( 13
)
where x is the information (x = ±1) and n A is a real white Gaussian noise with zero-mean, with standard deviation σ A .

The MI between the information source X and the a priori source can be expressed with the function J defined in [START_REF] Brink | Convergence behavior of iteratively decoded parallel concatenated codes[END_REF], giving

I A = J(σ A ) = 1 - +∞ -∞ 1 √ 2πσ A exp - 1 2σ 2 A z - σ 2 A 2 x 2 log (1 + e -z )dz. (14)

Extrinsic information

The output of a decoder can be separated into three types of information: the a priori information L A , the channel observation L ch and the extrinsic information L E generated by the decoder. The output LR is thus expressed

L = L A + L ch + L E . (15)
Extracting L E can be realized by simply substracting the two other informations from the output LR L. The classic TC construction of a codeword includes a systematic part, i.e the uncoded information bits, and a parity part, or the redundancy bits added by the encoder. Obtaining the channel observation of the systematic bits and computing the extrinsic LR is then straightforward. However, in the Turbo-FSK case, the construction of the codewords does not include any systematic part: the codewords are elements of the DFT matrix, composed of complex symbols. The channel observation L ch cannot be retrieved: decoding only the FSK modulation (without decoding the trellis) would still include the extrinsic information of the orthogonal code itself. This residual information from the channel must be taken into account for further interpretations.

Multi-dimensional EXIT Chart

When two decoders are used, the exchange of information between the decoders during the successive iterations can be represented in a two dimension plot. The case where the iterative process includes more than two decoders has been studied in [START_REF] Brink | Convergence of multidimensional iterative decoding schemes[END_REF]. If λ decoders are considered, the representation of the exchanges needs to be done in λ dimensions. Graphical interpretation becomes impossible for λ > 3, but for the special case where all decoders are the same (which is the case for the Turbo-FSK), a two-dimensional projection may be computed. If this projection does not intersect with the line going from (0, 0) to (1, 1), it indicates that convergence toward the maximum of MI is possible. The formula of the a priori MI must be changed [START_REF] Brink | Convergence of multidimensional iterative decoding schemes[END_REF] to

I A = J √ λ -1 • σ A . (16)
This formula can be interpreted as the fact that each decoder receives information from the λ -1 other decoders.

J( √ λ -1σ A ) I E E b /N 0 = 1dB E b /N 0 = 0dB E b /N 0 = -1.6dB 0

EXIT Chart Computation

The process to compute the EXIT Chart is depicted Figure 4. As previously mentioned, the code is not systematic, implying the impossibility to extract the channel observation and the necessity to convert the a priori LR to codewords probabilities. The subtraction after the decoder will give an extrinsic information that still contains the channel observation. The extrinsic MI block computes the integrals summation from [START_REF] Roth | Turbo-FSK: A New Uplink Scheme for Low Power Wide Area Networks[END_REF]. The a priori MI is computed using function J from ( 14) and the correction of [START_REF] Brink | Convergence behavior of iteratively decoded parallel concatenated codes[END_REF]. For a specific value of σ ch , the extrinsic MI is computed for several values of σ A , so that the a priori MI spans between 0 and 1. The EXIT Chart computation does not depend on the interleaving function, and needs to be done using very large block sizes, to ensure a good statistic for the density probability functions estimations (this assumption also implies statistical independence of all incoming messages into the decoder).

The EXIT Chart computations result for three values of E b /N 0 are depicted on Figure 5 (a), along with the middle line (0, 0) → (1, 1). Turbo-FSK scheme with parameters M = 128, λ = 4 is selected, using an orthogonal alphabet, an information block size of N = 100000 is chosen and the MAP algorithm. For E b /N 0 = -1.6dB, the EXIT Chart intersects the middle line around I A 0.1. It can be concluded that whatever is the number of iterations, the decoding process cannot converge towards the error-free information word for this E b /N 0 value. However, for E b /N 0 values of 0 and 1dB, there is no intersection with the middle line: the process will then, after a certain number of iterations, successfully retrieve the exact information word. An exhaustive search using these parameters shows that the smallest value of E b /N 0 for which the EXIT Chart does not intersect with the middle line, or the "pinch-off" value, is E b /N 0 = -1.12dB. The EXIT Chart for this value is illustrated Figure 5 (b), along with the BER performance of the Turbo-FSK scheme with the same parameters, and 100 decoding iterations. The EXIT Chart predicts accurately the beginning of the "waterfall" region, i.e the E b /N 0 region where the BER drops significantly. Indeed, after the pinch-off value, the BER decreases to reach a value of 10 -5 at E b /N 0 = -1.03dB.

Parameters optimization

A way to optimize the energy efficiency of the scheme is to reduce the E b /N 0 needed to reach an arbitrary level of error rate; for a given error rate (and a fixed binary information rate), the transmitted power is also reduced when the E b /N 0 is decreased. The performance of the Turbo-FSK scheme directly depends on the value of parameters M , λ and N , the information block size. Since the EXIT Chart predicts the waterfall region, minimizing the pinch-off value will give the most efficient scheme under the assumed hypothesis. An exhaustive search of the pinch-off value is done for every set of parameters (M, λ), using an orthogonal alphabet. Coherent detection of the FSK is assumed and MAP algorithm without any approximation is used. The results are presented in Table 1. For each value M , we denoted with a * symbol the lowest pinch-off value depending on λ, and the overall lowest pinch-off is in bold. Results shows that Shannon (E b /N 0 -1.56dB for this spectral efficiencies) can be approached by only 0.29dB, using the set of parameters (512, 3). A similar result was shown in [START_REF] Wu | On the limiting performance of turbo-Hadamard codes[END_REF], where a Hadamard systematic bi-orthogonal alphabet was used. The performance depending on the value of λ is not predictable, justifying the exhaustive approach for the optimization.

These results show the good asymptotic performance of the code. However, for the considered LPWA scenario, the size of the payload should be limited to a few bytes, up to a few hundreds of bytes. Small information block sizes are expected, but the EXIT Chart analysis is inaccurate when the value of N is small [START_REF] Brink | Convergence behavior of iteratively decoded parallel concatenated codes[END_REF]. We can nonetheless perform BER computations to evaluate the performance depending on the set of parameters. For each set, we choose N = 1000 and N = 100 bits and search the E b /N 0 value for which the BER reaches 10 -4 . 10 decoder iterations and a random interleaver are used for the simulations. To depict the results, the normalized spectral efficiency of the scheme using the different sets of parameters, as defined in ( 5) is proposed. Results can be compared to the ultimate Shannon limit as defined by (4), and are depicted Figure 6. For each alphabet size, λ value from 3 to 6 are represented. Since increasing λ lowers the normalized spectral efficiency, the point with the highest η is λ = 3, then 4, and so on. The figure clearly shows the optimum value for the parameters, for both EXIT and BER analysis. The general trend obtained with the EXIT Chart seems to be confirmed even when the block size is shortened. Reducing the block size to N = 1000 bits (125 bytes) induces a performance loss of approximately 1dB (on average), implying a minimum gap to Shannon's limit equals to 1.35dB. For a block size of N = 100, the performance loss compared to the asymptotic pinch-off is 3dB, with the best set of parameters (128, 4) being 3.42dB away from Shannon's limit.

Performance

As the Turbo-FSK parameters have been optimized to allow high energy efficiency, our interest goes into comparing the performance of the scheme against current published LPWA solutions. Comparisons are done versus three other state-of-the-art Machine-to-Machine (M2M) transmission schemes: the IEEE 802.15.4k standard [22], the LoRa physical layer, as described in the patent [START_REF] Seller | Low power long range transmitter[END_REF], and the serial concatenation of a TC [START_REF]LTE Evolved Universal Terrestrial Radio Access (E-UTRA): Multiplexing and Channel Coding[END_REF][START_REF] Valenti | The UMTS Turbo Code and an Efficient Decoder Implementation Suitable for Software-Defined Radios[END_REF] with a M -FSK modulation.

Proposed solutions

For each solution, the physical layer is first described. Since mostly the transmitter side is discussed in the literature, we assume for each scheme the receiver to be the dual receiver version of the transmitter, with soft decoding (MAP) and demodulation when possible. The parameter λ is introduced, which will allow the spectral efficiency to be modulated for comparison purpose. The IEEE 802.15.4k is a standard designed for local and metropolitan area networks. It is part of the Low-Rate Wireless Personal Area Networks (LR-WPANs). This standard supports three PHY modes: DSSS with BPSK or Offset-Quadrature Phase Shift Keying (O-QPSK), or 2-FSK. Since DSSS with BPSK modulation is the more robust scheme, we select this configuration. The standard specifications allow the use of a spreading factor value λ from 16 to 32768. In addition to the modulation and the spreading factor, the standard uses a Forward-Error-Correction (FEC). It is defined to be the convolutional code of rate 1/2, with generators polynomial [171 133] (in octal) and constraint length k = 7. An interleaving operation is considered between the steps of channel coding and modulation. The spectral efficiency of the scheme is defined as

η 1 = 1 2λ . ( 17
)
Soft decoding using the well-known soft-input soft-output Viterbi decoder is considered.

OSSS with block code

Non-linear modulations based on an alphabet of M-ary orthogonal waveforms are known to be energy efficient, and to reach channel capacity for infinite size of alphabet [START_REF] Proakis | Digital Communications 3rd Edition, Communications and signal processing[END_REF], as discussed in the introduction of this paper. This property can be extended to Orthogonal Sequence Spread Spectrum (OSSS), and is purportedly used by the LoRa technology [START_REF] Seller | Low power long range transmitter[END_REF] with a Hamming block code in addition. For this study, we consider a Hamming block code of rate 4/7. The parity bits can be punctured to increase the rate. We denote by p the number of parity bits (p ∈ {0, 1, . . . 3}). After the encoding, the transmitter consists in an interleaver and an orthogonal modulation of size 2 λ . The spectral efficiency is equal to

η 2 = 4 4 + p λ 2 λ . (18)
The receiver side consists in a soft demodulation, and soft Hamming decoding.

The UMTS/LTE TC

The recent 4G standard specification includes a TC [START_REF]LTE Evolved Universal Terrestrial Radio Access (E-UTRA): Multiplexing and Channel Coding[END_REF]. Turbo codes are known to be very effective FEC schemes, and its use in the Cellular-IoT context is considered. They consist in the parallel concatenation of two identical recursive systematic convolutional codes. In the specification, each code is rate 1/2, of constraint length k = 4 with feedback and generator polynomials [13 15] (in octal). The overall rate of the TC is approximately 1/3 (extra few bits are required to close the trellis). With this FEC, we consider an orthogonal modulation of size M , and a spreading factor λ. The spectral efficiency can be expressed

η 3 = log 2 (M ) 3λM . (19)
The receiver side consists in a turbo receiver [START_REF] Berrou | Near Shannon limit error-correcting coding and decoding: Turbo-codes[END_REF][START_REF] Valenti | The UMTS Turbo Code and an Efficient Decoder Implementation Suitable for Software-Defined Radios[END_REF]. The use of soft iterative decoding implies high consumption at the receiver side, but complexity at the transmitter side is rather low. This scheme, denoted as FSK + TC in the sequel, is an interesting element of comparison against our proposed Turbo-FSK scheme. Indeed, it also implies the use of a powerful channel code and an orthogonal modulation, but in two successive steps, instead of the joint process used in the Turbo-FSK scheme.

Performance comparison

To compare the different schemes, we compute simulations for the AWGN channel, with coherent reception and ideal synchronization. Random interleavers are used, except for the TC UMTS where the interleaver specified by the 3G standard is used. The packet size is set to N = 1000. For the iterative receivers, 10 iterations are performed (TC and Turbo-FSK), and the MAP algorithm without any approximation is used. We choose, for the Turbo-FSK, the parameters M = 128 and λ = 4, i.e the best couple of parameters for this size of alphabet. The spectral efficiency is equal to η = 1.17 • 10 -2 for these parameters. For each scheme, we can choose a different value of λ in order to equalize the spectral efficiency of the schemes. The selected parameters are summarized in Table 2. Figure 7 (a) depicts the BER performance versus E b /N 0 for the selected parameters. The scheme OSSS uses the Hamming code, which is less powerful than the convolutional code of the IEEE 802.15.4k standard, but offers better 2. The packet size is set to 1000 bits.

performance when combined with a relatively large size of orthogonal alphabet (512). The gain using the turbo principle is illustrated by the performance of both the FSK + TC and the Turbo-FSK. The scheme FSK + TC reaches a BER of 10 -5 for a E b /N 0 value of 2.91dB, and the Turbo-FSK outperforms all the other scheme, showing a 4.8dB gain versus the OSSS + Hamming scheme for the same BER. The gain offered by the Turbo-FSK scheme versus the scheme FSK + TC also shows the benefit of jointly optimizing the modulation and the channel coding instead of treating them separately. The Packet Error Rate (PER) versus E b /N 0 is depicted on the Figure 7 (b). For a PER of 10 -3 , the Turbo-FSK with these parameters outperforms the scheme OSSS + Hamming code by 5.5dB. This level of PER is reached for E b /N 0 = 0.12dB. Using Equation (2), with the spectral efficiency given in Table 2, the equivalent SNR is equal to -19.2dB, demonstrating the ability of the system to work at low levels of SNR.

These two figures show the real benefit of turbo processing on the sensitivity gain. Because all the spectral efficiency are normalized, the E b /N 0 gain can be interpreted as the sensitivity gain between two schemes, when the same data rate is considered (see Equation ( 3)). The 5.5dB gain between Turbo-FSK versus the LoRa based scheme for a PER or 10 -3 means that the sensitivity will be lower using our scheme. This can be interpreted as a potential reduction of the transmit power by a factor 3.5 while ensuring the same level of performance, or a distance increase by a factor 1.8 (under the approximation of free path loss exponent equal to 2). Comparison with other scheme could be considered; the use of a turbo code with a linear modulation also offers a good tradeoff between performance and spectral efficiency.

The sensitivity performance improvements are done at the expense of an increase of complexity at the receiver side.

As we focus on the complexity at the node level, having a more complex receiver at the base-station is acceptable. However, a recent study showed that the Turbo-FSK physical layer can be implemented on off-the-shelf components [START_REF] Estavoyer | Implementation and Analysis of a Turbo-FSK transceiver for a new Low Power Wide Area Physical Layer[END_REF], demonstrating that the complexity increase can be handled by components with low computation capacity.

Conclusion

While global LPWA solutions aims at lowering the sensitivity, improving the energy efficiency should remain a major concern. The Turbo-FSK scheme proves to be a serious candidate for the PHY layer of LPWA networks, offering high energy efficiency with reasonable spectral efficiency. The optimization realized using the EXIT analysis and the performance versus State-of-the-Art solutions show that very interesting gain in sensitivity can be achieved. The best set of parameters is shown to be only 1.35dB away for Shannon's limit for a packet size of 128 bytes, a fair size in the M2M context. The Turbo-FSK demonstrates the positive impact of mixing orthogonal modulations and turbo processing, and achieves promising performance while ensuring low complexity and a constant envelope at the transmitter. Further studies should however be considered about the complexity increase, and the synchronization, which can reveal itself arduous for the considered levels of SNR.

π λ- 1 ParallelFigure 1 :

 11 Figure 1: (a) The Turbo-FSK transmitter with λ stages. (b) The convolutional-FSK encoder

Figure 2 :

 2 Figure 2: Example of the mapping of the FSK codewords for the case M = 4, and representation of the oversampled time waveform and the spectrum S after time-domain rectangular shaping.

Figure 3 : 8 B

 38 Figure 3: (a) The Turbo-FSK receiver with λ stages. (b) Trellis of the Turbo-FSK, for an orthogonal alphabet and M = 8 B = 1/T c , noise monolateral spectral density N 0 and useful signal average power P x = E b η/B. At the receiver side, after frequency down conversion, chip matched filter, and sampling at every T c , the discrete model of the baseband signal can be expressed in vector form as y = x + n,(6)

Figure 4 :

 4 Figure 4: The EXIT Chart computation

Figure 5 :

 5 Figure 5: (a) EXIT Charts of the Turbo-FSK decoder with parameters M = 128, λ = 4, for various values of E b /N 0 . The information block size was set to N = 100000. (b) EXIT Chart and BER performance of the Turbo-FSK scheme with parameters M = 128, λ = 4. The EXIT Chart is computed for E b /N 0 = -1.12dB and the information block size was set to N = 100000. BER performance is computed using N = 100000 and the MAP algorithm.

Figure 6 :

 6 Figure 6: Values from Table 1 (EXIT Chart analysis with N = 100000) and BER simulation for N = 1000 and N = 100 in the plot normalized spectral efficiency versus energy efficiency.

4. 1 . 1 .

 11 IEEE 802.15.4k

Figure 7 :

 7 Figure 7: (a) BER and (b) PER performance comparison versus E b /N 0 , using the parameters given Table2. The packet size is set to 1000 bits.

Table 1 :

 1 ultimate capacity Pinchoff values in E b /N 0 for every M and λ tested. EXIT Chart are computed using an information block size N = 100000. Best value for each M is denoted with a symbol * , and best couple of parameters is in bold.

	λ	M	16	32	64	128	256	512	1024	2048
	2		2.23	1.61	1.15	0.86	0.53	0.30	-0.13	-0.61
	3		0.33	-0.17	-0.59	-0.94	-1.23 *	-1.30 *	-1.19 *	-1.07 *
	4		0.02	-0.46	-0.84	-1.12 *	-1.20	-1.04	-0.83	-0.59
	5		-0.06 *	-0.51 *	-0.86 *	-1.09	-1.03	-0.77	-0.47	-0.18
	6		-0.06 *	-0.51 *	-0.84	-1.02	-0.84	-0.52	-0.14	0.19

Table 2 :

 2 Parameters used for comparison.

	PHY-layer	802.15.4k	OSSS	FSK+TC	Turbo-FSK
	Modulation	DBPSK	512-Orthog	128-FSK	128-FSK
	FEC	CC [171 133]	Hamming	TC [13 15]	Turbo-FSK
	Binary code-rate	1/2	4/6	1/3	-
	λ	43	9	2	4
	η (•10 -2 )	1.163	1.172	0.907	1.170


A Ammar

E Cueto
email: ecueto@unizar.es

F Chinesta

Separated Representations: A Powerful Strategy for Model Reduction

Résumé -In this paper we discuss about the features of a novel numerical method based on the use of separated representations for the variables of interest. This separated representation allows for an easy treatment of problems defined in a space of high number of dimensions, or in which some parameters are unnown or uncertain. As an example of such a problem we study the problem of cell signalling under a stochastic framework. The method also enables to solve problems in which one of the variables within is not known, by considering these unknowns as new model dimensions.

Mots clés -Separated representation, curse of dimensionality, cell signalling.

1 Introduction. A method of separation of variables.

It is frequent in Science and Engineering that models are defined in spaces with a high number of dimensions. This characterisctic makes the numerical solution of these problems an intricate procedure, since a standard finite element discretisation of them involves an exponentially-growing number of degrees of freedom. This is a well-known phenomenon that has received the name of curse of dimensionality.

Several attempts have been made to overcome this curse. The employ of Monte Carlo procedures somawhat alleviates this burden, but it is well known that it converges with low velocity. Sparse grid techniques [START_REF] Bungartz | Sparse grids[END_REF] also help to overcome this difficulty, but are limited to a number of dimensions on the order of 20.

In this paper we overview a new attempt to overcome this difficulty that has proven to be very efficient for several problems [START_REF] Ammar | A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids[END_REF]. The method is based upon the assumption that the essential variable of the problem can be approximated as a finite sum of separable functions, i.e.,

Ψ(x 1 , x 2 , ..., x N ) = n F ∑ i=1 α i F i 1 (x 1 ) • F i 2 (x 2 ) • . . . • F i N (x N ) (1)
It is readily seen that since each function F j can be discretised in a one-dimensional space by standard finite elements, the approximation grows only linearly -and not exponentially-with the number of space dimensions.

In order to find the final approximation of the solution, the method is composed of two steps : -A projection stage in which the best set of α coefficients is sought.

-An enrichment procedure, in which new terms are added to Eq. ( 1).

Ψ = n F ∑ j=1 α j F j 1 ⊗ F j 2 ⊗ . . . ⊗ F j N Ψ F + R 1 ⊗ R 2 ⊗ . . . ⊗ R N Ψ R (2)
We usually employ an alternated directions approach, in which we are looking at each iteration to only one vector R j assuming known all the other vectors. Thus, in the weak formulation the test functions given by

Ψ * = R 1 ⊗ ... ⊗ R j-1 ⊗ R * j ⊗ R j+1 ⊗ ... ⊗ R N . (3)
For a detailed derivation of the method, the interested reader can consult [START_REF] Ammar | A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids[END_REF].

Modelling of gene regulatory networks

In this contribution we are concerned, as an example of potential use of the proposed technique, with a problem of great scientific interest and social relevance. Gene regulatory networks are usually described in terms of the number of molecules of each biochemical species (e.g., A, B, C and D in Eq. ( 4)) that are present at a given location

Z(t) = (#A, #B, #C, #D) T , with initial state Z(t 0 ) = z 0 . ( 4
)
This approach is usually preferred for the simulation of cell signaling processes since the number of molecules of interest present at a given region of a cell is usually on the order of dozens or hundreds. In this framework, the random ingredient of the reaction becomes evident and a stochastic differential equation is preferred to describe the process.

For any reaction R j , the propensity function will be given by a j (z)dt ≡ the probability, given

Z(t) = z, that R j occurs in [t,t + dt]. (5)
Once reaction R j has occurred, a change in the number of molecules of the state variables follows, in the amount given by the stoichiometry of the reaction :

v j = s in i, j + s out i, j , (6)
where s i, j repesents the stoichiometric coefficients of molecules of species i involved in reaction j. Equivalently,

z -v j a j (z -v j )
GGGGGGGGGGGGGAz,

and

z a j (z) G GGGGGGGG Az+v j . (8)
Let us define the probability that each species exists in z number of molecules at any time t :

P(z,t|z 0 ,t 0 ) ≡ Prob{Z(t) = z, given Z(t 0 ) = z 0 }. (9)
The Chemical Master Equation describes the time evolution of the probability taking into account each propensity a j :

∂P(z,t|z 0 ,t 0 ) ∂t = ∑ j [a j (z -v j )P(z -v j ,t|z 0 ,t 0 ) -a j (z)P(z,t|z 0 ,t 0 )] .
As explained before, the main problem with the simulation of cell signaling processes by the Chemical Master Equation (CME) is that the CME grows exponentially with the number of species [START_REF] Sreenath | Modelling the dynamics of signalling pathways[END_REF]. Consider, for instance, a situation in which there are N species with n copies of each specie. The number of possible states in that case is n N . In an eukaryotic cell, for instance, the number of species, on averge, can be of the order of 10 4 different proteins, while there will be 10 6 copies of each protein on average. This gives a situation in which we have 10 6000 possible states of the cell. But note that the presumed total number of elementary particles in the universe is on the order of 10 80 , thus making impossible to deal with such a problem with any -existing or not-computer. In the case of mRNA, or proteines, however, the most common situation is to have on the order of 10 copies per cell of each specie. Even in the case of such a simplification, the problem is of a tremendous magnitude.

The most extended algorithm for the simulaiton of chemical reaction under the assumption of few copies of each molecule is that of D.T. Gillespie [START_REF] Gillespie | Exact stochastic simulation of coupled chemical reactions[END_REF]. It is based on the calculation of individual trajectories of the species, instead of solving for the individual state transition probabilities. The method has a slow convergence rate and no guaranteed error bounds. It also requires a large number of realizations [START_REF] Munsky | The finite state projection algorithm for the solution of the chemical master equation[END_REF].

On the contrary, following the separation of variables algorithm described before, consider the CME as given by ∂P(z,t) ∂t

= AP(z,t), (10)
where operator A contains the propensities :

A = ∑ i A i (11)
One can write each operator A i in terms of tensorial products applied on each direction of the state space z :

A = n A ∑ j=1 A j 1 ⊗ A j 1 • • • ⊗ A j N (12)
and, equivalently, the probability :

P ≈ n F ∑ j=1 α j F j 1 ⊗ F j 2 • • • ⊗ F j N . ( 13
)
This algorithm provides a means of overcoming the curse of dimensionality associated with the CME. In the next secions we include some examples to illustrate the performance of the method.

3 Numerical examples

λ-phage switch

This constitutes the simplest model in cell signalling processes. When a bacteriophage λ infects a cell, either stays dormant or it reproduces until the dead of the cell. The resulting behaviour depends on two competing proteins that inhibit mutually each other, see Fig. 1. The so-called toggle switch is composed of a two-gene co-repressive network. The constitutive P L promoter The operator for this example is composed by two terms : A = A 1 + A 2 :

A 1 P(z 1 , z 2 ) = αβ β + γz 2 p(z 1 -1, z 2 ) + δ(z 1 + 1) • P(z 1 + 1, z 2 )- αβ β + γz 2 + δ • z 1 P(z 1 , z 2 ).
and A 2 equivalent with z 1 and z 2 interchanged. We computed the solution for δ = 0.05, α = 1.0, γ = 1.0 and β = 0.4. It can be shown that the corresponding deterministic model in this case leads to a monostable point. The simulations show that after t = 5s one has a case where both average values of both proteins and small levels of the one protein combined with higher level of the other protein are quite likely, and this remains the case for the stationary distribution as well [START_REF] Hegland | A solver for the stochastic master equation applied to gene regulatory networks[END_REF], Fig. 2.

Simulations in the presence of uncertainty

The true problem when modeling the kinetics of gene networks is the determination of the propensities for all the reactions involved in the network. Although many databases are nowadays available (see, for instance [START_REF] Sreenath | Modelling the dynamics of signalling pathways[END_REF]), for complex reactions, such as the TGF-β, for instance, whose activity has been related to cancer, around half of the propensities have not been determined experimentally.

For this cases, the separation of variables can provide a valuable insight into some problems, by just considereing the unknow propensities as new space dimensions. A way to overcome this difficulty is to take the unknown propensity as a new coordinate belonging to the uncertainty interval. The transient solution for a particular value of the propensity can then be computed by restricting the general solution to each particular value of this extra-coordinate. Obviously, the price to be paid is the increase of the model dimensionality. However, this is not a serious issue when one proceeds within the separated representation framework just described.

To illustrate this feature, we have simulated a cascade of two terms. A cascade occurs when a gene produces proteins that enahnce the expression of the succeding gene, see Fig.

A = k ∑ i=1 A i (14)
where A 1 is given by : A 1 (z)P(z) = α • p(ze 1 ) + δ • (z 1 + 1) • P(z + e 1 ) -(α + δ • z 1 )p(z) (15)

In order to check the proposed technique, and for the ease of illustration, we have considered a cascade of only 2 terms, with the parameter δ as an unknown. Note that the solution (obtained in one execution of the program), see Fig. 4, provides the solution for different values of δ, that repoduce the ones in the literature [START_REF] Hegland | A solver for the stochastic master equation applied to gene regulatory networks[END_REF].

Figure 1 -

 1 Figure 1 -Schematic mechanism of the bacteriophage λ-switch.

Figure 2 -

 2 Figure 2 -Solution at steady state by separation of variables. Axes denote the number of protein 1 (abscissa) and protein 2 (ordinate).

Figure 3 -

 3 Figure 3 -Schematic representation of a cascade.

3

 3

  . The operator related to a cascade of length k writes :

Figure 4 -

 4 Figure 4 -Solution for the cascade problem with unknown propensities. Solution at time t = 0, t = 30s, and t = 600s (approx. steady state).


Denis Bonheure

AND Silvia Cingolani

Jean Van Schaftingen

THE LOGARITHMIC CHOQUARD EQUATION : SHARP ASYMPTOTICS AND NONDEGENERACY OF THE GROUNDSTATE

Keywords: 2010 Mathematics Subject Classification. 35J50, 35Q40 Logarithmic Choquard equation, planar Schrödinger-Newton system, positive ground state solution, asymptotics, nondegeneracy. 1

We derive the asymptotic decay of the unique positive, radially symmetric solution to the logarithmic Choquard equation

and we establish its nondegeneracy. For the corresponding three-dimensional problem, the nondegeneracy property of the positive ground state to the Choquard equation was proved by E. Lenzmann (Analysis & PDE, 2009).

Introduction

We consider the nonlocal model equation

(1.1) -∆u + au = Φ N * |u| 2 u in R N
where a is a constant and Φ : R N → R is the Newton kernel, that is the fundamental solution of the Laplace equation in R N , namely

Φ N (x) = Γ( N -2 2 ) 4π N/2 |x| N -2 if N ≥ 3, and Φ(x) = 1 2π ln 1 |x| if N = 2.
In dimension N = 3, the integro-differential equation (1.1) has been introduced to study the quantum physics of electrons in an ionic crystal (Pekar's polaron model) [START_REF] Pekar | Untersuchungen über die Elektronentheorie der Kristalle[END_REF]. It has later also been proposed as a coupling of quantum physics with Newtonian gravitation [START_REF] Diósi | Gravitation and quantum-mechanical localization of macro-objects[END_REF][START_REF] Jones | Newtonian quantum gravity[END_REF][START_REF] Penrose | On gravity's role in quantum state reduction[END_REF]. E. H. Lieb has proved the existence of a unique ground state solution of (1.1) in dimension N = 3, which is positive and radially symmetric [START_REF] Lieb | Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation[END_REF] (see also [START_REF] Cingolani | Multiple solutions to a magnetic nonlinear Choquard equation[END_REF][START_REF] Lions | Solutions of Hartree-Fock equations for Coulomb systems[END_REF][START_REF] Ma | Classification of positive solitary solutions of the nonlinear Choquard equation[END_REF][START_REF]Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics[END_REF][START_REF] Wang | Uniqueness of positive solutions of the Choquard type equations[END_REF]). Successively, E. Lenzmann has shown the nondegeneracy of the unique positive ground state solution to the three-dimensional equation (1.1) [START_REF] Lenzmann | Uniqueness of ground states for pseudorelativistic Hartree equations[END_REF].

In this paper we focus on the planar integro-differential equation corresponding to (1.1)

(1.2) -∆u + au = 1 2π ln 1 | • | * |u| 2 u in R 2 .
We refer to it as the logarithmic Choquard equation (or planar Schrödinger-Newton system). This two-dimensional problem has remained for a long time a quite open field of study. While Lieb's existence proof has a straightforward extensions to the higher dimensions N = 4 and N = 5 and the existence of finite energy solutions is forbidden for N ≥ 6 by a Pohozhaev identity (see for example [6, Lemma 2.1; 10, (56); 19, (2.8); 21, Proposition 3.1]), the situation is less clear for lower dimensions due to the lack of positivity of the Coulomb interaction energy term. For N = 1, this difficulty has been overcome recently and the existence of a unique ground state has been shown by solving a minimization problem [START_REF] Choquard | The one dimensional Schrödinger-Newton equations[END_REF].

Back to our planar case N = 2, after numerical studies suggesting the existence of bound states [11, §6], Ph. Choquard, J. Stubbe and M. Vuffray have proved the existence of a unique positive radially symmetric solution to (1.2) by applying a shooting method to the associated system of two ordinary differential equations [START_REF] Choquard | Stationary solutions of the Schrödinger-Newton model-an ODE approach[END_REF].

In contrast with the higher-dimensional case N ≥ 3, the applicability of variational methods is not straightforward for N = 2. Although (1.2) has, at least formally, a variational structure related to the energy functional

u → I(u) = 1 2 R 2 |∇u| 2 + au 2 + 1 8π R 2 R 2 ln(|x -y| 2 )|u(x)| 2 |u(y)| 2 dx dy
this energy functional is not well-defined on the natural Sobolev space H 1 (R 2 ). J. Stubbe has tackled that problem [START_REF] Stubbe | Bound states of two-dimensional Schrödinger-Newton equations[END_REF] by setting a variational framework for (1.2) within the functional space where, for each s ∈ (0, +∞) ln + s = (ln s) + . This functional I is well-defined and continuously differentiable on the space X. Critical points u ∈ X of I are strong solutions in W 2,p (R 2 ), for all p ≥ 1, and classical solutions in

X := u ∈ H 1 (R 2 ) :
C 2 (R 2 ) of (1.2).
Even if X provides a variational framework for (1.2), some difficulties arise. First, the norm of X is not invariant under translations whereas the functional I is invariant under translations of R 2 . Second, the quadratic part of the functional I is never coercive on X, whatever the value of a ∈ R.

By using strict rearrangement inequalities, J. Stubbe has proved that there exists, for any a ≥ 0, a unique ground state, which is a positive spherically symmetric decreasing function [START_REF] Stubbe | Bound states of two-dimensional Schrödinger-Newton equations[END_REF]. In addition, he proved that there exists a negative number a * < 0 such that for any a ∈ (a * , 0) there are two ground states with different L 2 norm and that in the limiting case a = a * , there is again a unique ground state. T. Weth and the second author [START_REF] Cingolani | On the planar Schrödinger-Poisson systems[END_REF] recently constructed a sequence of solution pairs (±u n ) n∈N ⊂ X of the equation (1.2) such that I(u n ) → ∞ as n → +∞. They also provided a variational characterization of the least energy solution. Namely, they proved that the restriction of the functional I to the associated Nehari manifold N := {u ∈ X \ {0} : I ′ (u)u = 0} attains a global minimum and that every minimizer u ∈ N of I| N is a solution of (1.2) which does not change sign and obeys the variational characterization

I(u) = inf u∈X sup t∈R I(tu).
In addition, the following uniqueness result was proved by T. Weth and the second author.

Theorem 1 ([7, Theorem 1.3]). For every a > 0, every positive solution u ∈ X of (1.2) is radially symmetric up to translation and strictly decreasing in the distance from the symmetry center. Moreover u is unique up to translation in R 2 .

Our first result is a description of the asymptotic behaviour of this unique positive solution of the logarithmic Choquard equation (1.2).

Theorem 2. If a > 0 and if u ∈ X is a radially symmetric positive solution of (1.2), then there exists µ ∈ (0, +∞) such that, as |x| → ∞,

u(x) = µ + o(1) |x|(ln|x|) 1/4 exp - √ M e -a/M e a/M |x| 1 √ ln s ds , where M = 1 2π R 2 |u| 2 .
The integral does not seem to have an explicit asymptotic equivalent at the order o(1) as |x| → ∞ in terms of elementary functions; roughly speaking it behaves as We obtain this decay rate by studying the decay rate of solutions to the linear problem -∆u + V u = 0, when V (x) ≡ M ln|x| as |x| → ∞.

The asymptotic behaviour of u is a key ingredient to derive the precise description of the kernel of the linear operator L(u) defined by

(1.3) L(u) : X → L 2 (R 2 ) : ϕ → -∆ϕ + (a -w)ϕ + 2u ln 2π * (uϕ) ,
where

(1.4) w : R 2 → R : x → 1 2π R 2 ln 1 |x -y| |u(y)| 2 dy and (1.5) X := ϕ ∈ X : there exists f ∈ L 2 (R 2 ) such that for every ψ ∈ C ∞ c (R 2 ) R 2 ϕL(u)ψ = R 2 f ψ .
By standard arguments, one easily shows that L(u) is a self adjoint operator acting on L 2 (R 2 ) with domain X. Also, differentiating the equation (1.2), it is clear that γ • ∇u ∈ ker L(u) for every γ ∈ R 2 . Our main result is the nondegeneracy of the positive solution u of Theorem 1. Namely, the kernel of the operator L(u) is exactly the vector space spanned by the partial derivatives of u.

Theorem 3. If a > 0 and u ∈ X is a positive solution of (1.2), then

ker L(u) = γ • ∇u : γ ∈ R 2 .
The paper is organized as follows. In Section 2, we set up the variational framework and establish useful preliminary estimates. In Section 3, we study the asymptotic decay and prove Theorem 2. Section 4 is devoted to the proof of Theorem 3. Assuming without loss of generality that u is radial, we prove, as a first step, the nondegeneracy of the linearized operator L(u) restricted to the subspace of radial functions of X, that is, we show the triviality of its kernel on that subspace. As a second step, using the fact that u and w are radial, we describe by means of an angular decomposition, how the operator L(u) acts on each subspace X ∩ L 2 k (R 2 , C), where

L 2 k (R 2 ; C) := f ∈ L 2 (R 2 ; C) : for almost every z ∈ R 2 ≃ C and θ ∈ R, f (e iθ z) = e ikθ f (z) .
Our proof relies on the multipole expansion of the logarithm kernel [25, §IV.5.7], which is an identity related to the generating function of the Chebyshev polynomials and is also known as the cylindrical multipole expansion (see formula (4.1)). The corresponding multipole expansion of the Newtonian kernel was already used in the proof of the nondegeneracy of the groundstate solution for the three-dimensional Choquard equation, see [START_REF] Lenzmann | Uniqueness of ground states for pseudorelativistic Hartree equations[END_REF].

Finally we emphasize that the nondegeneracy of the groundstate is an important spectral assumption in a series of papers on effective solitary waves motion and semi-classical limit for Hartree type equations (see for instance [START_REF] Bonanno | Soliton dynamics for the generalized Choquard equation[END_REF][START_REF] Squassina | Soliton dynamics for the Schrödinger-Newton system[END_REF][START_REF] Wei | Strongly interacting bumps for the Schrödinger-Newton equations[END_REF]). In a forthcoming paper we use our nondegeneracy result for proving existence result of semiclassical states for the planar Schrödinger-Newton system.

Acknowledgements. The research of the authors was supported by GNAMPA project 2016 "Studio variazionale di fenomeni fisici non lineari", the Projet de Recherche (Fonds de la Recherche Scientifique-FNRS) T.1110.14 "Existence and asymptotic behavior of solutions to systems of semilinear elliptic partial differential equations", the Mandat d'Impulsion Scientifique (FNRS) F.4508.14 "Patterns, Phase Transitions, 4NLS & BIon" and the ARC AUWB-2012-12/17-ULB1-IAPAS.

Variational framework

We begin by showing that the planar Choquard equation (1.2) can be derived from the Newton-Schrödinger system by a formal inversion.

Let us consider a classical solutions (u, w) of the planar Schrödinger-Newton system (2.1)

-∆u + au = wu in R 2 , -∆w = u 2 in R 2 ,
where a is positive constant, subject to the conditions

(2.2) u ∈ L ∞ (R 2 ) and w(x) → -∞ as |x| → ∞.
By Agmon's Theorem (see [START_REF] Agmon | Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N -body Schrödinger operators[END_REF]), (2.1) and (2.2) imply that

(2.3) u(x) = o(e -α|x| )
as |x| → ∞ for every α > 0.

Moreover, since every semibounded harmonic function R 2 → R is constant, we have

(2.4) w(x) = c + 1 2π R 2 ln 1 |x -y| |u(y)| 2 dy,
for every x ∈ R 2 and some constant c ∈ R.

We recall that the solutions for which u is positive are known to enjoy symmetry properties [7, Theorem 6.1] up to the symmetries of the problem. Precisely, the following result holds. We write

u λ (•) to denote λ 2 u(λ•) for λ = 0. Theorem 4. Let a > 0. If (u, w) is a classical solution of (2.1) and (2.2) with u > 0 in R 2 ,
then, up to translation, the functions u and w are radially symmetric and strictly radially decreasing. Moreover, if (ũ, w) is another classical solution of (2.1) and (2.2) with ũ > 0 in R 2 , then the exists

x 0 ∈ R 2 and λ > 0 such that for each x ∈ R 2 , ũ(x) = u λ (x -x 0 ) w(x) = w λ (x -x 0 ) + a 1 -λ 2 .
It follows from Theorem 4, that the solution (u, w) is unique up to translations, under a suitable additional condition at infinity on w. Indeed, we know from (2.4) that there exists c ∈ R such that

1 2π R 2 ln 1 |x -y| |u(y)| 2 dy -w(x) = c.
Therefore, if ρ > 0 and (ũ, w) is another classical solution of (2.1) and (2.2), then, with λ > 0 given by Theorem 4,

1 2π R 2 ln ρ |x -y| |ũ(y)| 2 dy -w(x) = λ 2 1 2π R 2 ln 1 |λx -y| |u(y)| 2 dy -w(λx) + λ 2 ln λρ 2π R 2 |u| 2 + a λ 2 -1 = λ 2 c + λ 2 ln λρ 2π R 2 |u| 2 + a λ 2 -1 .
Since the right hand side is an increasing continuous function of λ that takes -a as a limit at 0 and diverges to +∞ at +∞, there exists a unique solution ũ such that

1 2π R 2 ln ρ |x -y| |ũ(y)| 2 dy -w(x) = 0.
In particular, the asymptotic boundary condition

lim x→∞ w(x) - 1 2π R 2 ln ρ |x -y| |u(y)| 2 dy = 0
implies uniqueness of the solution up to translations. Because of the decay of u at infinity, this is equivalent with requiring the asymptotic condition

lim x→∞ w(x) - 1 2π ln ρ |x| R 2 |u| 2 = 0.
Fixing thus ρ > 0, we are reduced to consider the integro-differential equation

(2.5) -∆u + au = 1 2π ln 1 | • | * |u| 2 u in R 2 .
Solutions of (2.5) are formally critical points of the functional

1 2 R 2 |∇u| 2 + a|u| 2 - 1 8π R 2 ln 1 |x -y| |u(x)| 2 |u(y)| 2 dx dy.
The first integral is the norm on the Sobolev space H 1 (R 2 ) induced by the scalar product [START_REF] Cingolani | On the planar Schrödinger-Poisson systems[END_REF][START_REF] Stubbe | Bound states of two-dimensional Schrödinger-Newton equations[END_REF]. However, an adequate functional setting that we now recall has been introduced by J. Stubbe [START_REF] Stubbe | Bound states of two-dimensional Schrödinger-Newton equations[END_REF] who used a smaller space with a stronger norm. One first defines for the functions f, g : R 2 → R, the three symmetric bilinear forms

(u|v) = R 2 ∇u • ∇v + a uv , for u, v ∈ H 1 (R 2 ), by u 2 := (u|u) for each u ∈ H 1 (R 2 ). The second integral is not continuous on H 1 (R 2 )
B + (f, g) = 1 2π R 2 ×R 2 ln + 1 |x -y| f (x)g(y) dx dy, B -(f, g) = 1 2π R 2 ×R 2 ln + |x -y|f (x)g(y) dx dy, B(f, g) = 1 2π R 2 ×R 2 ln 1 |x -y| f (x)g(y) dx dy,
whenever the integrand is Lebesgue measurable, so that in particular,

B(f, g) = B + (f, g) -B -(f, g).
The classical Young convolution inequality, see for example [15, theorem 4.2], implies

|B + (f, g)| ≤ 1 2π R 2 ln + 1 |x| p p-2 dx 2-2 p f L p (R 2 ) g L p (R 2 )
for every p ∈ (2, +∞). On the other hand, we have for every x, y ∈ R 2

ln + |x -y| ≤ ln + (|x| + |y|) ≤ ln + |x| + ln + |y|, so that |B -(f, g)| = 1 2π R 2 ×R 2 ln + |x -y|f (x)g(y) dx dy ≤ 1 2π g L 1 (R 2 ) R 2 |f (x)| ln + |x| dx + f L 1 (R 2 ) R 2 |g(x)| ln + |x| dx .
We have thus proved Proposition 2.1. For every p > 2, the bilinear form B is well defined and bounded on Y × Y , where the space

Y = f : R 2 → R : R 2 |f (x)| p + |f (x)|(1 + ln + |x|) dx < ∞ , is endowed with the norm defined for f ∈ Y by f Y = f L p (R 2 ) + f L 1 (R 2 ) + R 2 |f (x)| ln + |x| dx.
In order to go back to our original functional, we first note that the multiplication map (u, v) → uv is a bilinear map which is bounded from Z × Z to Y , where the space Z is defined by

Z = u : R 2 → R : R 2 |u(x)| 2p + (1 + ln + |x|)|u(x)| 2 dx < ∞ is endowed with the norm defined for u ∈ Z by u Z = R 2 |u| p 1 2p + R 2 (1 + ln + |x|)|u(x)| 2 dx 1 2 .
We now define the functional space

X := u ∈ H 1 (R 2 ) : R 2 ln + |x| |u(x)| 2 dx < ∞ endowed with norm defined through u 2 X := R 2 |∇u(x)| 2 + 1 + ln + |x| |u(x)| 2 dx,
on which we consider the functional

I(u) = R 2 (|∇u| 2 + a|u| 2 ) dx - 1 4 B(u 2 , u 2 ).
Since the second term of the functional I is the composition of the continuous linear embedding of X into Z, a continuous bilinear map from Z × Z to Y and a continuous bilinear map from Y ×Y to R, it follows that the functional I is smooth on the space X. Moreover, its first two derivatives are given by

I ′ (u)[ϕ] = R 2 ∇u • ∇ ϕ + a uϕ -B(u 2 , uϕ),
and

I ′′ (u)[ϕ, ψ] = R 2 ∇ϕ • ∇ψ + a ϕψ -B(u 2 , ϕψ) -2B(uϕ, uψ),
for each u, ϕ, ψ ∈ X.

Asymptotic behaviour of the groundstate solution

The goal of the present section is to study the asymptotics of the groundstate solution to (1.2) and prove Theorem 2.

Rough asymptotics for linear

Schrödinger operators with a logarithmic potential. We first construct upper and lower solutions to a linear problem related to (1.2). These estimates are too rough to deduce Theorem 2. We state them because the proof is quite elementary and might help the reader to understand the more sophisticated construction in the proof of Lemma 3.3 below. Moreover, the reader can verify that these rough estimates would be sufficient to obtain the proof of Theorem 2 concerning the nondegeneracy which is given in Section 4.

Lemma 3.1. Let V ∈ C(R N ) be such that lim |x|→∞ V (x) ln|x| = λ > 0.
For every ε > 0, there exist W ε , W ε and R ε > 0 such that

-∆W ε + V W ε ≤ 0 in R 2 \ B Rε , -∆W ε + V W ε ≥ 0 in R 2 \ B Rε , lim |x|→∞ W ε (x) exp (-(1 + ε)|x| λ ln|x| ) = 1, lim |x|→∞ W ε (x) exp (-(1 -ε)|x| λ ln|x| ) = 1.
Proof. We define, for every τ ∈ R, the function

w τ : (1, +∞) → R : r → exp(-τ r √ ln r).
We compute directly for each r ∈ (1 + ∞),

w ′ τ (r) = -τ √ ln r + 1 2 √ ln r w τ (r)
and

w ′′ τ (r) = τ τ √ ln r + 1 2 √ ln r 2 - 1 2r √ ln r + 1 4r (ln r) 3 w τ (r).

If we define the function

W τ : R 2 \B 1 for each x ∈ R 2 \B 1 by W τ (x) = w τ (|x|), we have -∆W τ (x) + V (x)W τ (x) = ln|x| V (x) ln|x| -τ 2 + O(1) W τ (x).
We obtain the conclusion by taking R ε > 1 sufficiently large and by choosing

W ε = W √ λ (1-ε) and W ε = W √ λ (1+ε)
. We immediately deduce from Lemma 3.1 some rough asymptotic decay estimates on the solutions of linear equations with a potential growing logarithmically at infinity. Corollary 3.2. Assume that V ∈ C(R N ) and that there exists λ > 0 such that

lim |x|→∞ V (x) ln|x| = λ.
If u is a positive solution of

-∆u + V u = 0, x ∈ R 2 , then lim |x|→∞ ln u(x) |x| ln|x| = - √ λ.
3.2. Refined asymptotics for linear Schrödinger operators with a logarithmic potential. In order to obtain fine asymptotics, we rely on the following construction of upper and lower solutions.

Lemma 3.3. Assume V ∈ C(R N ) is such that for some β ∈ (0, 1], V ′ (r) V (r) = 1 r ln r + o 1 r β+1
as r → ∞. Then, for R > 0 sufficiently large, there exist radial functions

W ± ∈ C 2 (R 2 \ B R ) and W ± ∈ C 2 (R 2 \ B R ) such that W ± > 0 and W ± > 0 in R 2 \ B R , -∆W ± (x) + V (|x|)W ± (x) ≤ 0 for x ∈ R 2 \ B R , -∆W ± (x) + V (|x|)W ± (x) ≥ 0 for x ∈ R 2 \ B R , W ± (x) = 1 + O(|x| -β ) exp ± |x| R V (s) ds |x| 1/2 (ln|x|) 1/4 W ± (x) = 1 + O(|x| -β ) exp ± |x| R V (s) ds |x| 1/2 (ln|x|) 1/4 ,
and

W ± (x) = W ± (x) 1 + O 1 |x| β+1 , as |x| → ∞.
Proof of Lemma 3.1. We define the functions w τ,+ and w τ,-for every τ ∈ R and r ∈ (0, +∞) by

w τ,± (r) = exp ± r 0 V (s) ds r -1/2 (ln(r/R)) -1/4 1 - τ r β . We compute directly w ′ τ,± (r) = ± V (r) - 1 2r - 1 4r ln r + βτ r β+1 -τ r w τ,± (r)
and

w ′′ τ,± (r) = ± V (r) - 1 2r - 1 4r ln r + βτ r β+1 -τ r 2 ± V ′ (r) 2 V (r) + 1 2r 2 + 1 4r 2 ln r + 1 4r 2 (ln r) 2 - βτ (β + 1)r β -τ ) (r β+1 -τ r) 2 w τ,± (r).
If we set W τ,± (x) = w τ,± (|x|), we have

-∆W τ,± (x) + V (x)W τ,± (x) = ± V (r) V ′ (r) 2V (r) - 1 2r ln r + 2βτ r β+1 -τ r + O 1 r 2 W τ,± (x).
We therefore conclude by taking W ± = w ∓1,± and W ± = w ±1,± .

If V (x) = Ṽ (x) ln|x|, the assumption of Lemma 3.3 can be written

Ṽ ′ (x) Ṽ (x) = o 1 |x| β+1
as |x| → ∞. As we derived Corollary 3.2 from Lemma 3.1, we are able to improve the asymptotics of the solutions of linear equations thanks to Lemma 3.3.

Corollary 3.4. Assume

V ∈ C(R N ) is such that for some β ∈ (0, 1], V ′ (|x|) V (|x|) = 1 |x| ln|x| + o 1 |x| β+1
as |x| → ∞. Let u be a positive radial solution of

-∆u + V u = 0. If u(x) = o    exp |x| 0 V (s) ds |x| 1/2 (ln|x|) 1/4    ,
then there exists µ ∈ (0, +∞) such that, as |x| → ∞,

u(x) = µ + O r -β exp - |x| 0 V (s) ds |x| 1/2 (ln|x|) 1/4 .
Proof. Let R > 0, W ± and W ± be given by Lemma 3.3. We now consider the function

v ε,r = (1 + ε) u(r) W -(r) W --ε u(r) W + (r) W + -u,
for every r ≥ R. By the growth assumptions on u and by the known growth of W -and W + , the set

Ω ε,r = x ∈ R 2 \ B r : v ε,r (x) > 0 is bounded. Moreover -∆v ε,r + V v ε,r ≤ 0,
and v ε,r ≤ 0 on ∂B r , so that we infer from the weak maximum principle for second order linear operators on bounded sets that v ε,r ≤ 0 in Ω ε,r . Henceforth, we conclude that v ε,r ≤ 0 in R 2 \ B r . Since ε > 0 is arbitrary, we have

u(s) ≥ W -(s) u(r) W -(r) for all s ≥ r ≥ R.
Arguing similarly with the function

w ε,r = (1 + ε)u - u(r) W -(r) W --ε u(r) W + (r) W + , we deduce that u(s) ≤ W -(s) u(r) W -(r) ,
for all s ≥ r ≥ R. By the asymptotic equivalence of W -and W -, we have thus proved

1 ≤ u(s)/W -(s) u(r)/W -(r) ≤ 1 + O 1 r β+1 ,
and it follows that the function u/W -has a limit at infinity by the Cauchy criterion of convergence.

Asymptotics on the logarithmic potential.

In order to use our previous asymptotic estimates, we need to understand the logarithmic term ln * |u| 2 . A naïve approach gives the following inequality. Proof of Theorem 2. We first observe that since lim |x|→∞ ( ln 2π * |u| 2 )(x) = -∞, we have for each λ > 0,

Proposition 3.5. If f is radial and f ∈ L 1 (R 2 ), then for each x ∈ R 2 R 2 ln|x -y| f (y) dy -ln|x| R 2 f (y) dy ≤ R 2 \B
-∆u + λu ≤ 0, x ∈ R 2 \ B R
M = 1 2π R 2 |u| 2 .
This implies that, as r → ∞, The asymptotic estimate now follows from the linear asymptotics of Corollary 3.4.

Remark 3.1. The integral appearing in the statement of Theorem 2 can be expressed in terms of classical special functions. Indeed, by integration by parts and by a change of variable s = exp(σ 2 ),

λ 1 √ ln s ds = λ √ ln λ - λ 1 1 2 √ ln s ds = λ √ ln λ - √ ln λ 0 e σ 2 dσ = λ √ ln λ -F ( √ ln λ) = λ √ ln λ - √ π 2 erfi √ ln λ , = λ √ ln λ - γ 1 2 , -ln λ 2(-1) 1/2 = γ 3 2 , -ln λ (-1) 3/2 = √ ln λ Γ( 3 2 )γ * ( 3 2 , -ln λ) .
where F is Dawson's integral, erfi is the imaginary error function (defined for z ∈ C by erfi(z) = -i erfi(zi)) and γ is the lower incomplete gamma function (using the same branch in its computation than for (-1) 1/2 ) [START_REF] Temme | Error functions, Dawson's and Fresnel integrals[END_REF] and γ * is its entire part [START_REF] Paris | Incomplete gamma and related functions, NIST Handbook of Mathematical Functions[END_REF].

Nondegeneracy of the positive solutions

Let u ∈ X be a solution of the planar logarithmic Choquard equation (1.2) which does not change sign and satisfies the variational characterization

I(u) = inf u∈X sup t∈R I(tu).
Let w : R 2 → R be the function defined for each x ∈ R 2 by

w(x) = 1 2π R 2 ln 1 |x -y| |u(y)| 2 dy.
By Theorem 4, up to translation, this solution u is unique and radially symmetric. We can therefore assume without loss of generality that the functions u and w are radially symmetric. We consider the linear operator L(u) defined by (1.3), that is,

L(u)ϕ = -∆ϕ + (a -w)ϕ + 2u ln 2π * (uϕ)
on the space X defined by (1.5). In the sequel, we just write L to shorten the notation.

4.1. Closedness of the operator L. We show that the operator L is closed.

Proposition 4.1. The operator L is a closed operator from L 2 (R 2 ) to L 2 (R 2 ).
The proof will rely on the following estimate.

Lemma 4.2. For each ε > 0, there exists

C ε > 0 such that for each ϕ, ψ ∈ H 1 (R 2 ), R 2 R 2 ln|x -y|u(x)ϕ(x)u(y)ψ(y) dx dy ≤ ε R 2 |∇ϕ| 2 1 2 R 2 |∇ψ| 2 1 2 + C ε R 2 |ϕ| 2 1 2 R 2 |ψ| 2 1 2 .
Proof. Let δ > 0, and let

A δ = {(x, y) ∈ R 2 × R 2 : |x -y| ≤ δ}.
Using the Young convolution inequality and the Sobolev inequality for some fixed p ∈ (2, +∞), we have then

A δ ln|x -y|u(x)ϕ(x)u(y)ψ(y) dx dy ≤ |z|≤δ |ln|z|| p 2(p-1) 2-2 p R 2 |u| p |ϕ| p 1 2 R 2 |u| p |ψ| p 1 2 ≤ ε R 2 |∇ϕ| 2 + |ϕ| 2 1 2 R 2 |∇ψ| 2 + |ψ| 2 1 2 ,
provided that δ > 0 is sufficiently small. On the other hand, we observe that if |x -y| ≥ δ, |x| ≥ δ and |y| ≥ δ, then

0 ≤ ln |x -y| δ ≤ ln + |x| δ + ln + |y| δ
so that, because of the exponential decay of u (Theorem 2),

R 2 ×R 2 \A δ ln|x -y|u(x)ϕ(x)u(y)ψ(y) dx dy ≤ R 2 ×R 2 |ln δ| + ln + |x| δ + ln + |y| δ |u(x)| |u(y)| |ϕ(x)| |ψ(y)| dx dy ≤ C R 2 |ϕ| 2 1 2 R 2 |ψ| 2 1 2 .
Proof of Proposition 4.1. Let (ϕ) n∈N be a sequence in X that converges strongly in L 2 (R 2 ) to ϕ ∈ L 2 (R 2 ) and such that (Lϕ n ) n∈N converges strongly in L 2 (R 2 ) to some f ∈ L 2 (R 2 ). We observe that, in view of Lemma 4.2, for each n ∈ N,

R 2 ϕ n Lϕ n ≥ 1 2 R 2 |∇ϕ| 2 + R 2 (-w)|ϕ| 2 -C R 2 |ϕ| 2 .
By the convergences of the sequences (ϕ n ) n∈N and (Lϕ n ) n∈N and by the asymptotic behaviour of the function w, it follows that the sequence (ϕ n ) n∈N is bounded in X, and thus it converges weakly to ϕ in the space X.

If we now take ψ ∈ C 1 c (R 2 ), we have

R 2 ψf = lim n→∞ R 2 ψLϕ n = lim n→∞ R 2 ϕ n Lψ = R 2 ϕLψ.
Since ϕ ∈ X, we have by the Hölder inequality and by the exponential decay of u we have for

x ∈ R 2 large enough, |ln * (uϕ)(x)| ≤ C ln|x|. In particular u (ln * (uϕ)) ∈ L 2 (R 2 ).
4.2. Angular splitting of the operator L. Since the solution u is radial, the operator L commutes with rotations acting on L 2 (R 2 ). This suggests to use the orthogonal splitting [25, §IV.2]

L 2 (R 2 ; C) = k∈Z L 2 k (R 2 ; C),
where the summands

L 2 k (R 2 ; C) = f ∈ L 2 (R 2 ; C) : for almost every z ∈ R 2 ≃ C and θ ∈ R, f (e iθ z) = e ikθ f (z) ,
are closed mutually orthogonal subspaces of L 2 (R 2 ; C). In particular L 2 0 (R 2 ; C) is the subspace of radial functions of L 2 (R 2 ; C). In general, if f ∈ L 2 k (R 2 ; C), then there exists a function g : (0, +∞) → C such that

∞ 0 |g(r)| 2 r dr < ∞ and for each r ∈ (0, +∞) and θ ∈ R f (re iθ ) = g(r)e ikθ .
In order to describe how the linear operator L acts on each of the subspaces L 2 k (R 2 ; C), we rely on the multipole expansion of the logarithm kernel [25, §IV.5.7]: for each

x = re iθ ∈ R 2 ≃ C and y = se iη ∈ R 2 ≃ C, if r > s, then (4.1) ln|x -y| = ln r + ∞ k=1 s r k cos(k(θ -η)) k .
This identity is related to the generating function of the Chebyshev polynomials and is also known as the cylindrical multipole expansion. The formula (4.1) follows directly from the convergence of the Taylor series of the complex logarithm, that is,

ln|x -y| = ln r + Re ln 1 - s r e i(η-θ) = ln r + ∞ k=1 s r k cos(k(θ -η)) k .
The corresponding multipole expansion of the Newtonian kernel was used in the proof of the nondegeneracy of the groundstate solution for the threedimensional Choquard equation [START_REF] Lenzmann | Uniqueness of ground states for pseudorelativistic Hartree equations[END_REF].

If ϕ ∈ X ∩ L 2 k (R 2 ; C), then we can write ϕ(re iθ ) = ψ(r)e ikθ for some ψ : (0, +∞) → C, and

(Lϕ)(re iθ ) = (L k ψ)(r)e ikθ ,
where for each k ∈ Z \ {0}, the operator L k is defined by

L k ψ(r) = -ψ ′′ (r) - 1 r ψ ′ (r) + a + k 2 r 2 -w(r) ψ(r) - u(r) |k| ∞ 0 ψ(s)u(s) min(r, s) max(r, s) k s ds,
whereas for k = 0, the operator L 0 is defined by

L 0 ψ(r) = -ψ ′′ (r) - 1 r ψ ′ (r) + a -w(r) ψ(r) -2u(r) ∞ 0 ψ(s)u(s) ln 1 min(r, s) s ds.
This last formula can also be obtained by Newton's shell theorem. We also have

w(r) = ∞ 0 |u(s)| 2 s ln 1 max(r, s) ds.
Observe in particular that if ϕ ∈ X ∩ L 2 k (R 2 ; C), we have Lϕ ∈ L 2 k (R 2 ; C) and therefore ker L = k∈Z ker L ∩ L 2 k (R 2 ; C) .

This allows to study separately the kernels of the operators L k which is our aim in the next subsections.

4.3. Radial eigenfunctions. We show that the kernel of the operator L 0 is trivial. As in [START_REF] Lenzmann | Uniqueness of ground states for pseudorelativistic Hartree equations[END_REF], we first decompose the operator L 0 as follows

L 0 ψ(r) = L0 ψ(r) + 2u(r) ∞ 0 u(s)ψ(s) s ln s ds,
where the operator L0 is defined by (4. Proof. As u is a solution of (1.2), it satisfies for each r ∈ (0, +∞), (4.3) u ′′ (r) -1 r u ′ (r) + aw(r) u(r) = 0.

Since L0 ψ = 0, it follows from (4.3) that for every r ∈ (0, +∞),

d dr r(u(r)ψ ′ (r) -u ′ (r)ψ(r)) = r u(r)ψ ′′ (r) + 1 r u(r)ψ ′ (r) -u ′′ (r)ψ(r) - 1 r u ′ (r)ψ(r) = 2r|u(r)| 2 r 0 ψ(s)u(s) s ln r s ds.
Integrating, this implies that if η = ψ/u,

η ′ (r) = r u(r)ψ ′ (r) -ψ(r)u ′ (r) r|u(r)| 2 = 2 r 0 s|u(s)| 2 r|u(r)| 2 s 0 η(t)|u(t)| 2 t ln s t dt ds.
Integrating again, we finally obtain, by exchanging the order of integration η(r) = η(0) + 2 2 dq, from which the conclusion follows.

1 q|u(q)| 2 dq ≥ c r 1 1 q|u(q)|
To go on, in view of (4. We are now in position to prove that there are no radial eigenfunctions.

Lemma 4.4. We have

ker L ∩ L 2 0 (R 2 ; C) = ker L 0 = {0}. Proof. Let us assume that ϕ ∈ L 2 0 (R 2 ; C) \ {0} and L 0 ϕ = 0.
We then deduce that

L0 ϕ = -2 ∞ 0 u(s)ϕ(s) s ln s ds u We define ψ = ϕ -ζ ∞ 0 u(s)ϕ(s) s ln s ds ∞ 0 |u(s)| 2 s ln s ds + a .
By construction, ψ ∈ L 2 0 (R 2 ; C). By Lemma 4.3, this implies ψ(r) = 0 for each r ∈ (0, +∞), that is, for every r ∈ (0, +∞),

ϕ(r) = ζ(r) ∞ 0 u(s)ϕ(s) s ln s ds ∞ 0 |u(s)| 2 s ln s ds + a . Integrating, we infer that ∞ 0 u(s)ϕ(s) s ln s ds = ∞ 0 u(s)ζ(s) s ln s ds ∞ 0 u(s)ϕ(s) s ln s ds ∞ 0 |u(s)|
λ 0 L k = inf Q k (ψ) : ψe ikθ ∈ X and ∞ 0 |ψ(r)| 2 r dr = 1 ,
where

Q k (ψ) = ∞ 0 |ψ ′ (r)| 2 + (1 -w(r))|ψ(r)| 2 r dr + k 2 ∞ 0 |ψ(r)| 2 r dr -Re ∞ 0 ∞ 0 ψ(r)ψ(s)K k (r, s) ds dr, with K k (r, s) = 1 |k| u(r)u(s) min(r, s) max(r, s) k rs, is defined for ψ ∈ W 1,1 loc ((0, +∞); C) such that ∞ 0 |ψ ′ (r)| 2 + |ψ(r)| 2 r + |ψ(r)| 2 r + |ψ(r)| 2 r ln + (r) dr < ∞.
Thanks to the logarithmic weight in the definition of the functional space X and the logarithmic growth of w at infinity, the eigenvalue λ 0 (L k ) is achieved. We now observe that

Q k (|ψ|) = Q k (ψ) - ∞ 0 ∞ 0 |ψ(r)||ψ(s)| -Re(ψ(r)ψ(s)) K k (r, s) ds, Since K k (r, s) > 0 on (0, +∞) × (0, +∞), it follows that Q k (|ψ|) ≥ Q k (ψ),
with equality if and only if Re(ψ(r)ψ(s)) = |ψ(s)||ψ(r)| for almost every (r, s) ∈ (0, +∞) × (0, +∞), which implies that there exists α ∈ C such that |α| = 1 and for each r ∈ (0, +∞), ψ(r) = α|ψ(r)|.

In particular, the space of eigenvectors corresponding to λ 0 (L k ) is spanned by nonnegative eigenvectors. If ψ 1 and ψ 2 are two nonnegative eigenvectors, then the quantity D : (0, +∞) 2 → R D(r, s) = ψ 1 (r)ψ 2 (s)ψ 2 (r)ψ 1 (s) has constant sign, since D(•, s) and D(r, •) are real eigenvectors corresponding to the eigenvalue λ 0 (L k ). Since D(r, s) = -D(s, r), it follows that D(r, s) = 0. As a consequence, we deduce that λ 0 (L k ) is a simple eigenvalue and we can assume that the associated eigenfunction is a nonnegative function ψ k : (0, +∞) → R.

Note that if |ℓ| ≤ |k|, then K k ≤ K ℓ pointwize. Therefore, we have, since This implies that u ′ is an eigenfunction of the operator L 1 . Assume by contradiction that λ 0 (L 1 ) < 0. Then by orthogonality of eigenfunctions, we have

ψ k ≥ 0, Q k (ψ k ) = Q ℓ (ψ k ) + k 2 -ℓ 2

R 2 ln( 1 + 2 X := R 2 |∇u(x)| 2 +

 21222 |x|)|u(x)| 2 dx < ∞ , endowed with a norm defined for each function u ∈ X by u |u(x)| 2 a + ln + |x| dx,

e a/M |x| 1 √

 1 ln s ds = |x| ln|x|(1 + o(1)), as |x| → ∞. This integral can be reexpressed in terms of classical special functions (imaginary error function or Dawson function, see Remark 3.1 below).

  for some R > 0 large enough depending on λ. It follows therefrom that lim |x|→∞ e λ|x| u(x) = 0 for every λ > 0. In view of Proposition 3.5, this leads to lim |x|→∞ w(x) + M ln|x| e λ|x| = 0, where w has been defined in (1.4) and

a 1 √

 1 w(s)a + M ln(s) ds+o[START_REF] Agmon | Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N -body Schrödinger operators[END_REF], where the second integral on the right-hand side is finite and does not depend on the variable r. We observe now by the elementary change of variable s = e -a/M t that r e -a/M √ a + M ln s ds = √ M e -a/M e a/M r ln t dt.

Lemma 4 . 3 . 2 r 1 1

 4321 2) L0 ψ(r) = -ψ ′′ (r) -1 r ψ ′ (r) + aw(r) ψ(r) + 2u(r) r 0 u(s)ψ(s) s ln r s dsand we prove the exponential growth of solutions v to the linear equationL0 v = 0. If ψ ∈ C 2 ([0, +∞); C) satisfies L0 ψ = 0, then, either ψ = 0 or there exists C 1 , C 2 > 0 such that |ψ(r)| ≥ C 1 u(r) exp C s|u(s)| 2 ds ,for each r ≥ 1. In this last case, we have lim r→∞ |ψ(r)| = ∞.

4 )

 4 ) q|u(q)| 2 ln t q dq dt ds = η(0) + r 0 η(s)b(s) ds (4.4) where the function b : (0, +∞) → R is defined for s ∈ (0, +∞) by b(s) is in fact the integral form of a first-order homogeneous linear differential equation. Such an equation has an explicit solution, given by η(r) = η(0) exp

  3), we compute L0 u(r) = 2u(r) r 0 |u(s)| 2 s ln r s ds, for each r ∈ (0, +∞). If we now set z(r) = ru ′ (r), we get L0 z(r) = -ru ′′′ (r) -3u ′′ (r) -1 r u ′ (r) + aw(r) ru ′ (r) + 2u(r) r 0 u(s)u ′ (s) s 2 ln r s ds.On the other hand, by differentiating the equation (4.3) satisfied by u, we have, for each r ∈ (0, +∞)(4.6) u ′′′ (r) -1 r u ′′ (r) + 1 r 2 u ′ (r) + aw(r) u ′ (r)w ′ (r)u(r)If we now define the function ζ : (0, +∞) → R for each r ∈ (0, +∞) byζ(r) = 2u(r) + z(r) = 2u(r) + ru ′ (r), we obtain L0 ζ(r) = -2 a + ∞ 0 |u(s)| 2 s ln s ds u(r). )| 2 s ln s ds = 0.Otherwise, since by integration by parts, we have for each r ∈ (0, +∞), ) + su ′ (s) ds = ru(r)+ r 0 u(s) ds,by the decay properties of u and by Lemma 4.3, this would only be possible if ζ = 0 on (0, +∞). This in turn would imply that for each r ∈ (0, +∞), u(r) = u(1)/r 2 , which is impossible since the function u also has to satisfy the equation (4.3).

4 . 4 .

 44 2 s ln s ds + a and thus, by the definition ofζ ∞ 0 |u(s)| 2 s ln s ds + a = ∞ 0 2|u(s)| 2 s ln s ds + ∞ 0 u(s)u ′ (s) s 2 ln s ds = ∞ 0 |u(s)| 2 s ln s ds -1 2 ∞ 0 |u(s)| 2 s ds,which is impossible since a > 0. Nonradial eigenfunctions. For every k ∈ Z \ {0}, we have the variational definition of the eigenvalues

ψ

  k (r)ψ k (s) K ℓ (r, s) -K k (r, s) ds dr> Q ℓ (ψ k ),from which it follows that λ 0 (L k ) > λ 0 (L ℓ ). We can now conclude our proof of Theorem 3 by showing that λ 0 (L 1 ) = 0.Proof of Theorem 3. We observe thatL 1 (u ′ )(r) = -u ′′′ (r) -u ′′ (r) r + u ′ (r) r 2 + aw(r) u ′ (r) u(r) ∞ 0 u(s)u(s) min(r, s) max(r, s) s ds.By(4.6) and by integration by parts, we deduce thatL 1 (u ′ )(r) = -u(r)

∞ 0 ψ 1 Remark 4 . 1 . 2 ∞ 0 K 1

 0141201 (r)u ′ (r) r dr = 0, which cannot hold since the function u is decreasing and the function ψ 1 is nonnegative. Therefore we have shown λ 0 (L 1 ) = 0. Since we already proved that the eigenvalue λ 0 (L 1 ) is simple, it follows that ker L 1 = span u ′ . By the discussion preceding the proof, we also know that ker L k = {0} whenever |k| > 1 and Lemma 4.4 implies ker L 0 = {0}. We have thus proved thatker L = k∈Z ker L k = L 1 = {u ′ (r)e iθ : θ ∈ [0, 2π)}.Alternatively, we could have used a nonlocal groundstate representation formula [20, Proposition 2.1], to writeQ 1 (ψ) = ∞ 0 (ψ/u ′ ) ′ r dr + 1 (r, s)u ′ (r)u ′ (s) the eigenvalue λ 0 (L 1) is equal to 0 and is simple. Denis Bonheure, Département de Mathématique, Université Libre de Bruxelles, CP 214, Boulevard du Triomphe, B-1050 Bruxelles, Belgium, and INRIA -Team MEPHYSTO. E-mail address: denis.bonheure@ulb.ac.be Silvia Cingolani, Politecnico di Bari, Dipartimento di Meccanica, Matematica e Management, Via E. Orabona 4, 70125 Bari, Italy E-mail address: silvia.cingolani@poliba.it Jean Van Schaftingen, Université catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium E-mail address: Jean.VanSchaftingen@UCLouvain.be