Céline Hidalgo
email: célinehidalgo@univ-amu.fr

Isaïh Mohamed

Christelle Zielinski

Daniele Schön

The effect of speech

Keywords: conversation, ERPs, eye-tracking, prediction, turn-taking

Introduction

Speech perception is not the result of a real-time decoding of audio-visual information. It is a dynamic process of an anticipatory nature building on 1) the use of the linguistic and extralinguistic context to generate hypotheses about the upcoming signal, i.e. we make predictions about what we are likely to hear and 2) a comparison of our predictions against what is actually perceived. Thus, when we listen to a conversation between two people, we anticipate, for example, the words that the speakers will produce according to the linguistic context that precedes their utterances. The more we know about this context, the fewer prediction errors we make and the faster and more accurate our understanding of their conversation. According to predictive coding theory [START_REF] Clark | Whatever next? Predictive brains, situated agents, and the future of cognitive science[END_REF], predictions would indeed be a default brain mode of operation for processing sensory information [START_REF] Friston | A theory of cortical responses[END_REF]. To optimize speech perception, the brain would pre-activate representations of the expected speech input [START_REF] Molinaro | Is there a common oscillatory brain mechanism for producing and predicting language? Language[END_REF] at a phonological, lexical, semantic or even syntactic level [START_REF] Pickering | Predicting while comprehending language: A theory and review[END_REF]. With increasingly accurate predictions, the brain would only need to compute the difference between sensory input and prediction, thereby decreasing the cost of perceptual processing and speeding up the understanding of the spoken message. A well-known fact in the comprehension literature is that the amount of contextual information, i.e. the increase in the precision of prediction, positively influences the speed of word recognition [START_REF] Tyler | Quantifying contextual contributions to word-recognition processes[END_REF]. This has been thoroughly studied using event-related potentials, by observing changes in the amplitude of a negative component. Indeed, words with low cloze probability engender a prediction error that is visible in an increase of the N400 amplitude. By contrast the N400 amplitude is reduced when perceiving speech in highly predictive contexts (see for a review [START_REF] Kutas | Thirty years and counting: Finding meaning in the N400 component of the event-related brain potential (ERP)[END_REF]. However, until now, no study had shown a direct link between contextual predictions and the integration of these predictions in the semantic comprehension process in a conversational context. This is due to the fact that, while prediction errors engender observable brain responses (e.g., N400 or Mismtach Negativity), this is not the case of predictions. Recently, Grisoni and colleagues [START_REF] Grisoni | Correlated Brain Indexes of Semantic Prediction and Prediction Error: Brain Localization and Category Specificity[END_REF] described an electrophysiological index of predictions called the "Semantic Prediction Potential", and showed, by measuring the N400 brain response during the perception of more or less contextually constrained sentences, the direct effect of predictions (SPP) on the processing of prediction errors (N400).

Another possibility to infer the ongoing predictions is to use behavioural measures. In the case of language, several studies have used for instance eye movements during reading [START_REF] Rayner | Eye movements in reading and information processing[END_REF][START_REF] Staub | The Effect of Lexical Predictability on Eye Movements in Reading: Critical Review and Theoretical Interpretation[END_REF]. Similarly, in speech perception, some authors relied on the implicit anticipatory gaze behaviour when listening to different people having a conversation [START_REF] Casillas | The development of children's ability to track and predict turn structure in conversation[END_REF][START_REF] Keitel | Perception of conversations: The importance of semantics and intonation in children's development[END_REF]. More precisely, listeners anticipate the end of the turn of the current speaker and switch gaze to the following speaker, before the actual onset of the new turn. This predictive behaviour is particularly important considering that the silence separating speech turns of conversing speakers only lasts a few hundred milliseconds [START_REF] Levinson | Turn-taking in Human Communication -Origins and Implications for Language Processing[END_REF][START_REF] Stivers | Universals and cultural variation in turn-taking in conversation[END_REF]. In considering turn-taking in conversation, predictions build on several linguistic cues such as prosody [START_REF] Roberts | The effects of processing and sequence organization on the timing of turn taking: a corpus study[END_REF], lexical-semantic cues [START_REF] De Ruiter | Projecting the end of a speaker's turn: a cognitive cornerstone of conversation[END_REF], syntactic structures [START_REF] Selting | On the interplay of syntax and prosody in the constitution of turnconstructional units and turns in conversation[END_REF] or pragmatic cues [START_REF] Beňuš | Pragmatic aspects of temporal accommodation in turn-taking[END_REF]. When listening to a dialogue (as a third person), a similar behaviour possibly takes place via the generation of internal models, allowing a listener to anticipate in a precise temporal window the speakers' turn-taking.

The aim of the current study is to explore the links between predictions at the turn level, measured by gaze switch, and both local and global semantic processing when listening to a dialogue, as indexed by N400 to turn onset and comprehension score, respectively.

Importantly we are interested in studying how these relations evolve when adding uncertainty to the context by degrading the input signal. Indeed, degrading the acoustic signal or adding noise or competing speakers, adds uncertainty and challenges speech comprehension [START_REF] Mattys | Speech recognition in adverse conditions: A review[END_REF][START_REF] Peelle | Listening effort: How the cognitive consequences of acoustic challenge are reflected in brain and behavior[END_REF]. To this aim, we asked participants to watch videos of two people having a conversation manipulating different levels of speech degradation. We measured eye movements to estimate predictive gaze behaviour and used this measure to class turns as well predicted (early gaze switch) or poorly predicted (late gaze switch). We also measured EEG and used the N400 as a marker of prediction error at the turn onset and studied the extent to which this response is influenced by gaze behaviour. Finally, all participants responded to several open questions at the end of each dialogue, allowing estimating a global comprehension measure. Most importantly, we explored the relation between these different variables as a function of speech degradation and increasing uncertainty. We hypothesized that poorly predicted turns (late gaze switch) would engender a larger N400 response compared to well anticipated turns (early gaze switch). Importantly, this effect may change as a function of speech degradation and should be reduced at high speech degradation levels when uncertainty is highest. We also hypothesized that, if predictions are fully implicit and based on the understanding of the dialogue, then gaze switches should behave similarly to comprehension score and decrease from normal to moderately degraded and even more for highly degraded speech.

Materials and Methods

We report how we determined our sample size, all data exclusions, all inclusion/exclusion criteria, whether inclusion/exclusion criteria were established prior to data analysis, all manipulations, and all measures in the study. No part of the study procedures or analyses was preregistered prior to the research being conducted.

Participants

Forty-eight participants (27 females) were tested after being informed of the procedure of the study, which was approved by the Sud Méditerranée Ethics Committee (ID RCB: 2015-A01490-49). In absence of a known effect size, the sample size was chosen as rather large with respect to N400 studies (Šoškić et al., 2021). Inclusion and exclusion criteria: Participants were between 18 and 62 years old (mean = 32, SD = 6 years). They were native French speakers. The had normal or corrected to normal vision and normal hearing. years); they had no history of speech disorder or neurological disease based on self-report. Prior the experiment, the hearing thresholds were controlled using a 5dB-step custom screening hearing test made in Python 3.7.9 (Expyriment version 0.10.0). Data from nine participants were excluded because of noisy EEG recordings (high impedances or excessive eye/movement artifacts) or track loss for long temporal windows in the eye-tracking dataset, resulting in 39 participants in the final dataset.

Stimuli

We presented 6 audiovisual stimuli of ~4.5 minutes each, showing a man and a woman having a conversation. The themes of the conversations differed and were joyful to maintain attention during the whole duration of the dialogue. Dialogues contained on average 59 turns (min = 46; max = 66; see details in table 1). Turns lasted on average 3.75s (SD = 2.75) and global speech rate was approximately 5Hz. 60% of these turns were questions/responses adjacency pairs (min = 56 %; max = 66%) in order to elicit predictive behaviours as suggested in [START_REF] Casillas | The development of children's ability to track and predict turn structure in conversation[END_REF][START_REF] Casillas | The development of children's ability to track and predict turn structure in conversation[END_REF]. We used Final Cut Pro X to set the gap duration at speakers' turn to 500ms in order to have homogeneous turn conditions, allow anticipatory gaze behaviour (cf. [START_REF] Foulsham | Gaze allocation in a dynamic situation: Effects of social status and speaking[END_REF][START_REF] Casillas | The development of children's ability to track and predict turn structure in conversation[END_REF] and obtain equivalent EEG baseline for turn onset analyses.

For the clarity of the evoked potentials, each turn onset began with a plosive consonant (although we avoided bilabial to avoid visual cues preceding the plosion). In the stimuli recording session, all dialogues were written and displayed on a desk in front of the speakers during video shooting. Dialogues were all read by the same male and female speaker and videos were recorded in an anechoic room. To avoid body movement when reading the dialogues, the speakers sat on a chair and their head was maintained still by a hard hat fixed to the chair. Each speaker wore sunglasses to hide eye movements and a wig to hide the hard hat (Figure 1A left). Furthermore, in order to discard spurious visual cues, the nonspeaking speaker was "frozen" via video editing whenever a head or lip movement occurred before the turn. This happened in approximately ~20% of the turns. In order to avoid sharp jumps in the images of the upcoming speaker such freezing took advantage of fading and morphing technics (mMorphCut plug-in ; FINAL CUT pro X).

In other words, turns could not be anticipated on the basis of visual information of the next speaker right before the turn. Thus, anticipatory gaze behaviour cannot be interpreted in response to visual cues but as auditory predictive behaviour only. We also created 12 open questions for each dialogue (e.g. "why was Tom walking in the wood?") (see table 2 for more examples for one dialogue). These questions were recorded using an auditory only format, by a female speaker different from the one in the dialogues.

Speech recordings of the six dialogues were degraded using a noise vocoding approach [START_REF] Shannon | Speech recognition with primarily temporal cues[END_REF]custom Matlab script). This allows to parametrically vary the spectral detail, with increasing numbers of channels associated with increasing perceptual clarity (Figure 1A right). The procedure allowed creating a highly degraded condition in which we kept only frequency bands from 120Hz to 237Hz, 405Hz to 538Hz and 919Hz to 1028Hz (i.e.

3 frequency bands left with 5 ERB scales) and a moderately degraded condition in which we removed all frequency bands except from 120Hz to 237Hz, from 674Hz to 805Hz, from 2025Hz to 2112Hz and 5208Hz to 5236Hz (so 4 frequency bands left with 8 ERB scales). The overall RMS amplitude of the audio files was adjusted to be the same across all dialogues in the three conditions (normal, moderately degraded, highly degraded). We also run a pilot test to ensure that comprehension varied as a function of speech degradation and that, in the most challenging condition, participants could still understand part of the conversation.

More precisely, 6 native French participants listened to audio excerpts of the dialogues and repeated all words they could recognize. The excerpts were degraded in a parametric manner in the number of channels, the bandwidth and the low frequency cut-off. The parameters were chosen in such a way to yield a condition of moderate difficulty (comprehension level between 30 and 50%) and another condition of great difficulty (comprehension level between 10 and 30%). normal, moderately degraded and highly degraded speech (using a vocoding approach). B. Schematic illustration of the turns in the dialogue, the controlled gap between turns (500ms) and the gaze switch window of analysis. C. On the left, gaze behaviour of a single participant showing the gaze switch for every turn of a single dialogue in the normal speech condition with respect to turn onset (zero). On the right, the gaze switch distribution for a single participant in the three different degradation conditions. A median split is used to class turns as being more or less well anticipated.

Procedure

Participants were equipped with a 64 preamplified Ag-AgCl electrodes (International 10/10 system site, BrainAmp system). The ground electrode was placed at AFz, the reference electrode at FCz, and the EEG signal sampling rate recording was 1000 Hz. Participants were comfortably seated in a Faraday sound-proof booth in front of a computer screen (24'') with a resolution of 1920 x 1080px and a refresh rate of 100Hz), at a distance of approximately 70 cm. A Gazepoint GP3 eye-tracker (sampling rate: 60 Hz; accuracy: 0.5-1° of visual angle) was installed at the bottom of the screen on a tripod to record the participant's gaze positions.

OpenSesame software [START_REF] Mathôt | OpenSesame: An open-source, graphical experiment builder for the social sciences[END_REF] installed on a Dell laptop (Precision T1700) launched the eye-tracker recording through the PyGaze package [START_REF] Dalmaijer | PyGaze: an open-source, crossplatform toolbox for minimal-effort programming of eyetracking experiments[END_REF] and the audio-visual stimulation. The videos were displayed on full screen resolution, and the sound was delivered through a 2040 YAMAHA amplifier and two NS 1020 Studio YAMAHA loudspeakers located on both sides of the screen. Before each video presentation, the participant's eyes were calibrated using the PyGaze's standard 12-points calibration procedure.

The experimenter asked participants to attentively follow the dialogues and informed them that, after each dialogue, they should answer 12 questions on the dialogue content. These questions allowed us to ascertain participants' attention and assess the level of comprehension. Each participant saw 2 dialogues in each of the three experimental conditions (normal speech, moderately degraded speech, highly degraded speech). The level of degradation of each dialogue and the order of the dialogues were counterbalanced across participants (each dialogue was presented at a different degradation level -normal, moderately degraded, highly degraded -to different participants).

Stimuli, eye-tracking and EEG data synchronization

A custom-made sync box based on an Arduino micro-controller ensured the synchronization between the EEG data and the acoustic stimulation that was embedded in the video file. The stereo sound goes from the stimulation PC to the sync box (audio cable). There, the stereo is split and the first channel containing the speech signals of the dialogues goes directly to the loudspeakers. The second channel signal goes through the micro-controller. This channel contains an audio trigger indicating the beginning and the end of each dialogue. The Arduino detects the audio trigger and sends an adapted signal to the EEG system. To ensure a precise synchronization of the eye-tracking on the acoustic stimulation and the EEG data, we used saccadic movements. More precisely, saccades can be easily detected in the eye-tracking data, but they are also clearly visible in the EEG signal, especially using Independant Component Analysis decomposition [START_REF] Makeig | Independent component analysis of electroencephalographic data[END_REF]. Thus, for each dialogue, we detected gaze switch for every turn in both the eye-tracking and EEG data (see supplementary material 1). Then, we computed for each subject and dialogue the median delay between the two signals across all turns and used it to temporally realign the eyetracking to EEG (median delay across participants = 23ms; median sd across participants = 12ms). This procedure ensured a good time-alignment of the three types of data (video, eyetracking and EEG).

Data Analysis

Eye-tracking

In this study we used the gaze switch from one speaker to the other as a proxy of anticipatory behaviour with respect to the upcoming turn onset (see Figure 1A left andB).

However, in contrast with previous studies [START_REF] Casillas | The development of children's ability to track and predict turn structure in conversation[END_REF]) that were mostly interested in anticipatory gaze shift, we were also interested in somewhat late gaze shifts.

We assumed that adult participants need approximately 200ms to plan an eye movement [START_REF] Kamide | The time-course of prediction in incremental sentence processing: Evidence from anticipatory eye movements[END_REF]. Thus, we considered a temporal window around the turn allowing to keep both anticipatory and non-anticipatory gaze switch, resulting in a -1/+1s window around the turn onset (Figure 1B). Note, nonetheless that, as expected, most shifts occurred in the inter-turn gap, that is before the turn onset [START_REF] Foulsham | Gaze allocation in a dynamic situation: Effects of social status and speaking[END_REF][START_REF] Keitel | Perception of conversations: The importance of semantics and intonation in children's development[END_REF] Casillas and Frank 2017, see Figure 2B). Moreover, we also applied three supplementary criteria to filter spurious gaze switches.

First, gaze before and after the switch should fall within an area of interest (AOIs), defined as stationary rectangle surrounding each face (see Figure 1A left). Second, gaze switch should be preceded by at least 100ms fixation on the current speaker. Third, it should be followed by at least 150ms fixation to the next speaker. Concerning the possible back and forth gaze behaviour preceding turn (4.5% in our data), we kept only the first gaze switch.

In short, we computed a gaze switch latency, that is relative to the upcoming turn onset time. A positive indicates thus a gaze switch following the turn onset, while a negative value indicates a gaze switch before the turn onset (Figure 1C left). Then, we used a median split to class turns as early or late, separately for each subject and condition (Figure 1C right).

EEG

Signal processing was done using EEGLAB [START_REF] Delorme | EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics[END_REF]) and custom scripts written in MATLAB. We high-pass filtered (0.5Hz) continuous data and rejected major artifacts (> 300 μV). For every participant, we systematically used Independent Component Analysis to remove eye blinks and saccadic movements and, when needed, muscular activity.

We then low-pass filtered (45Hz) and segmented continuous data into epochs of 900ms starting at 100ms prior to turn onset. We zero-mean normalized epochs to the baseline ([-100, 0]ms) and re-referenced to the algebraic average of all electrodes. Finally, we averaged epochs according to the three conditions, also separating epochs according to the latency of the gaze shift corresponding to each turn (early or late, see above).

Statistical analysis

We computed all statistical analysis using R [START_REF] Team | R: A language and environment for statistical computing[END_REF] and the lme4 package [START_REF] Bates | Fitting Linear Mixed-Effects Models Using {lme4}[END_REF]. We evaluated participants' comprehension scores (1 point attributed to each good answer; range 0 -12) as a function of speech degradation by fitting linear models on the mean of correct responses for each condition (2 dialogues by condition) and each subject.

Similarly, we modelled the mean of gaze switch latencies for each dialogue as a function of speech degradation by fitting linear model on latency data. We then used a general linear model (glm) in modelling the time spent gazing at the current speaker (ratio of the total fixation time). All theses models were tested against the null model before being interpreted. We computed a linear mixed model with interaction to explain comprehension score according to gaze switch latencies and speech degradation, with subject as random effect (comprehension ~ gaze switch latency * speech degradation + 1|subject). This model was compared 1) to the null model, 2) to the model with only gaze switch latencies as predictor and 3) to the additive model (gaze switch latencies and speech degradation as independent predictors). Statistical significance of the fixed effects was assessed by model comparison using the Akaike Information Criterion, thus arbitrating between complexity and explanatory power of the models. Normality and homoscedasticity of the residuals of all the models were systematically visually inspected. Reported p values are Satterthwhaite approximations obtained with the lmerTest package [START_REF] Kuznetsova | lmerTest Package: Tests in Linear Mixed Effects Models[END_REF].

The statistical significance of the differences between conditions for the ERP data was evaluated by a cluster-based random permutation approach for the full set of 64 electrodes (2-tailed test with 500 permutations over the whole time range of the ERP epoch, i.e. between -100 and 800ms). This statistical approach handles the multiple-comparisons problem. More precisely, the approach controls for the Type-1 error rate in multiple testing across channels and time points by identifying clusters of significant differences between conditions in the time and space dimensions [START_REF] Maris | Nonparametric statistical testing of EEG-and MEG-data[END_REF]. Finally, in order to assess the relation between gaze switch and ERPs, we used simple linear models with the amplitude difference between late and early trials in the N400 window explained as a function of the interaction between gaze switch latencies and speech degradation (N400 effect ~ gaze switch latency * speech degradation). As for the other analyses, this model was compared to the additive model.

Results

As expected, the comprehension score decreases as a function of increasing speech degradation (β = -2.551, SE = 0.22, t = -11.55, p < 0.001, Figure 2A). Compared to the normal condition, both the highly degraded and degraded speech conditions altered the comprehension score (β = -1.23, SE = 0.42, t = -2.929 p = 0.004 ; β = -5.102, SE = 0.42, t = -12.145, p < 0.001, respectively).

It is important to note that participants' score are well above zero and this is also the case in the highly degraded condition (range: 1-10, median 5.5). Considering that the score is based on response to open questions, it is extremely unlikely that a single correct answer is given by chance. As such, these results show that participants of the current experiment can partly understand the dialogues even in the most difficult condition of speech perception.

When turning to eye-movements, participants gaze data are available on average for 87% of the total duration of the dialogues (range: [82][83][84][85][86][87][88][89][90]). The loss of data is possibly due to a gaze off the screen and/or to a temporary loss of the eye from the tracker. From these available data, it appears that participants gazed at the current speaker most of the time during video playback (93%; range: [87][88][89][90][91][92][93][94][95][96][97]). Compared to the normal condition, time spent on current speaker is greater in the degraded (β = 0.580, SE = 0.073, t = 7.881, p < 0.001) and highly degraded conditions (β = 0.616, SE = 0.074, t = 8.317, p < 0.001). Importantly, not only participants gazed most of the time at the current speaker, but when considering gaze switch towards the current speaker, these took place most often in a -1/+1sec window around the turn onset (mean 80 %, range: [0.56-0.96]), the rest being gaze switches far away from the turn or missing data. Normalizing by time unit this gives a number of gaze switch per second of 0.47 during the turn window and 0.08 during the rest of the dialogue (that is switches from one AOI to the other that were performed outside the -1/+1s window around the turn, see supplementary material 2). Overall, these results clearly show that gaze and gaze switches are not randomly distributed across the dialogues. On the contrary, gaze behaviour to current speakers demonstrates that participants are tracking turns during the videos, as previously reported [START_REF] Casillas | The development of children's ability to track and predict turn structure in conversation[END_REF]. When moving to gaze switch latencies, one can see that these are influenced by speech degradation (β = 43.382, SE = 9.52, t = 4.553, p < 0.001, see Figure 2B). This is due to longer latencies to turns with highly degraded speech compared to normal and degraded speech (always p < 0.001) while there are no latency differences between degraded and normal conditions (p > 0.1). Moreover, latencies globally move from negative values in the normal condition to positive values in the highly degraded condition. Because latencies are estimated with respect to turn onset, this means that while in the normal condition participants realize the gaze switch before the turn, switches mostly take place after the turn onset in the highly degraded condition.

When looking at the relation between comprehension scores and gaze latency, this changes as a function of speech degradation (β = -5.740, SE = 0.001, t = -3.936, p < 0.001). More precisely, in the highly degraded condition shorter gaze switch latencies to turn onsets are associated to good comprehension scores (β = -0.008, SE = 0.003, t = -2.989, p < 0.01). This effect was not significant in the moderately degraded nor in normal conditions (p > 0.5, Figure 2C). This model with gaze switch latencies and speech degradation as interaction factors was the best compared to the additive model (AIC = 1038.5 and 1043.6 respectively ; p < 0.01 and p < 0.001).

The classification of turns on the basis of the latency of gaze switches (late vs early) shows a clear negative component for turns with late switches evolving between 350 and 700ms post-stimulus onset over a fronto-central region (Figure 3B). This effect is visible in the three different levels of speech degradation (p < 0.05, FDR corrected). However, this effect seems to differ across the three levels of speech degradation (Figure 3A) insofar as the clusters start at different latencies. Because it is not appropriate to make an inferential claim on the time extension of the cluster [START_REF] Sassenhagen | Cluster-based permutation tests of MEG/EEG data do not establish significance of effect latency or location[END_REF], we run a further analysis using a jackknife approach in five subsequent 50ms latency windows from 150 to 400ms. This approach is considered appropriate to make inferences on the latency of an ERP effect [START_REF] Miller | Jackknife-based method for measuring LRP onset latency differences[END_REF]. Results show that differences between early and late turns are significant starting 200ms post turn onset in the normal condition, 150ms in the moderately degraded condition and 300ms in the highly degraded condition (p < 0.05, FDR corrected).

Finally, when looking at the relation between the ERP effect (late vs early gaze switches) in the 300-550ms window and the global gaze switch latency of participants, we find a significant interaction between gaze switch latencies and speech degradation levels (F = 7.135, p = 0.001). More precisely, one can notice, that, in the highly degraded condition, participants with overall shorter gaze switch latencies to turn onsets show a larger effect (β = 0.007, SE = 0.001, t = 3.753, p < 0.001). A similar but not significant trend is also visible in the moderately degraded (β = 0.003, SE = 0.001, t = 1.912, p = 0.058) but not in the normal conditions (β = 0.000, SE = 0.001, t = 0.204, p = 0.838, Figure 2D). Cluster-based statistical analyses on all electrodes showing the main effect of gaze behaviour (early vs late).

The topography illustrates the difference between late and early turns across all levels of degradation. The red dots represent electrodes with significant differences (cluster corrected).

Discussion

In the present experiment, participants watched a series of videos showing two interlocutors having a conversation. We parametrically degraded speech and measured eye-movements, EEG and dialogue comprehension to gather a deeper understanding of the mechanisms underlying speech comprehension under challenging conditions. We build on the fact that participants switch gaze at the current speaker and use this measure as a proxy of prediction to class dialogue turns as more or less well predicted and study ERPs accordingly. Results

show that 1) Participants switch gaze towards the current speaker in a limited temporal window around the turn. 2) Speech degradation reduces the anticipatory gaze behaviour. 3) Globally, participants with a low comprehension score tend to have a later gaze behaviour when speech is strongly degraded. 4) ERPs to turn onsets vary as a function of the latency of the gaze switch. 5) Finally, participants with overall shorter switch gaze latencies show a better comprehension score and a stronger anticipatory effect in their neural responses; this relation between behavioural and neural variables is significant only in the most challenging (degraded) condition. We discuss these findings with respect to the relation between predictions, neural predictions errors and speech comprehension under challenging conditions. Of course, these findings are limited by the constraints inherent to the present experiment, wherein predictions are possible purposely on the basis of auditory cues only because we controlled all extralinguistic visual cues (e.g. lip preparatory movement) that undoubtedly play an important role in conversation.

Our study replicates previous findings [START_REF] Casillas | The development of children's ability to track and predict turn structure in conversation[END_REF][START_REF] Foulsham | Gaze allocation in a dynamic situation: Effects of social status and speaking[END_REF][START_REF] Keitel | Perception of conversations: The importance of semantics and intonation in children's development[END_REF], showing that gaze switch takes place in a precise temporal window centered before turn onset. Considering the short period between turns and the time needed to prepare and realize a saccade, gaze switch, in this context, can be interpreted as anticipatory behaviour [START_REF] Keitel | The use of intonation for turn anticipation in observed conversations without visual signals as source of information[END_REF]. The new result, in this respect, is that gaze switch behaviour is affected by the more or less challenging speech comprehension, here obtained by spectrally degrading speech signal. The reduction of anticipatory gaze behaviour in presence of stronger speech degradation seems to indicate that increasing the cost of speech processing has a detrimental effect on the prediction of the upcoming turn. This is in line with results on hearing-impaired individuals showing a delay (greater effort) in processing sentences with more or less contextual cues, compared to normal-hearing [START_REF] Winn | Rapid Release From Listening Effort Resulting From Semantic Context, and Effects of Spectral Degradation and Cochlear Implants[END_REF]. Importantly, our results show a relation between comprehension score and turn anticipatory behaviour, under challenging conditions. This result is not trivial. Indeed, the comprehension score reflects the global understanding of the whole dialogues, because it corresponds to questions that were asked at the end of each dialogue. By contrast, gaze switch and EEG dynamics reflect anticipatory behaviour at precise temporal windows surrounding conversational turns. This means, that, under challenging conditions, participants that are able to understand can take advantage of an anticipatory gaze behaviour. Alternatively, participants with an anticipatory gaze behaviour are better able to understand. While the present design does not allow for a causal interpretation between gaze behaviour and global understanding of the dialogues, we will see that ERP results play here a very important role.

However, before discussing this point one has to interpret the ERP components, in particular the negative component peaking around 400ms.

First, one may raise the possibility that residual gaze-related artefact may still be present in the EEG, even following ICA procedures (see methods). However, 1) the systematic difference in latency between late and early switches was on average 480ms, while we do not observe such a delay in ERPs to early and late turns; 2) while highly degraded speech induces overall later gaze switches, the N400 peak in this condition has the same latency as in the other conditions (in Figure 3A the N400 peak is always 400ms). Thus, overall, one can consider that differences in ERPs are due to cognitive processes and not to residual physiological artefacts due to eye movements.

The first and most likely interpretation of the larger negativity to both degraded turns and late turns is in terms of an N400 like component. Indeed, the amplitude of the N400 is strongly affected by expectations: the more a target word is unexpected within a sentence context, the larger the N400 amplitude [START_REF] Kutas | Thirty years and counting: Finding meaning in the N400 component of the event-related brain potential (ERP)[END_REF][START_REF] Kutas | Brain potentials during reading reflect word expectancy and semantic association[END_REF].

While the N400 topography we report is more anterior compared to the classic centroparietal distribution, this may be due to the audiovisual modality (Kelly et al., 2004) or to the presence of complex scenes [START_REF] Kutas | Thirty years and counting: Finding meaning in the N400 component of the event-related brain potential (ERP)[END_REF] inducing a topography more frontal than for written words. Importantly, the N400 has been proposed to be a proxy of prediction errors [START_REF] Bornkessel-Schlesewsky | Toward a neurobiologically plausible model of language-related, negative event-related potentials[END_REF][START_REF] Rabovsky | Simulating the N400 ERP component as semantic network error: Insights from a feature-based connectionist attractor model of word meaning[END_REF], that play a key role within the predictive coding framework [START_REF] Friston | The free-energy principle: A unified brain theory[END_REF][START_REF] Schultz | Neuronal coding of prediction errors[END_REF].

As stated above, in order to assess the relevance of predictive coding theories with respect to speech comprehension, it is necessary to have access to both prediction and prediction error indicators. Our results, as well as previous studies, seem quite clear in showing that gaze switch behaviour can be considered as a proxy of predictions. Previous research using an electrophysiological marker of prediction (the semantic prediction potential, SPP) showed that smaller SPP amplitude (weak predictions) result in a large N400 [START_REF] Grisoni | Correlated Brain Indexes of Semantic Prediction and Prediction Error: Brain Localization and Category Specificity[END_REF].

Similarly, our results show that turns preceded by a late gaze switch, indicating poor predictions, give rise to a larger N400 like complex compared to turns preceded by an early gaze switch. An interesting advantage of the current design is that the target words (here turn onsets) do not need to differ in content or in context in order to engender a N400. This is different for instance, from previous studies manipulating the semantic and temporal turn relation to induce an N400 effect [START_REF] Bögels | Never say no . . . How the brain interprets the pregnant pause in conversation[END_REF]. In other words, in the current study, the word and the context eliciting or not the N400 are strictly identical and we take advantage of the natural variability in the accuracy of participants' predictions across turns.

The interpretation in terms of N400 also fits quite well with the literature on weaker expectancies under degraded speech. Previous work showed that adverse listening conditions narrow the expectancies about the upcoming speech. This is visible in a reduced N400 effect in response to incongruent or less likely words under acoustic degradation [START_REF] Aydelott | Effects of acoustic distortion and semantic context on event-related potentials to spoken words[END_REF][START_REF] Obleser | Multiple brain signatures of integration in the comprehension of degraded speech[END_REF][START_REF] Strauß | Narrowed expectancies under degraded speech: Revisiting the N400[END_REF]. The present findings show a somewhat similar result, with an N400 effect that is significant in a later latency window in the most challenging condition (strong speech degradation) compared to the two other conditions. While the typical N400 effect concerns the amplitude of the negative wave, several works reported later N400 latencies in context presenting more difficult semantic access [START_REF] Deacon | Variation in the latencies and amplitudes of N400 and NA as a function of semantic priming[END_REF][START_REF] Moreno | Processing semantic anomalies in two languages: An electrophysiological exploration in both languages of Spanish-English bilinguals[END_REF]. Interestingly, the temporal window analysis shows that traces to early and late turns diverge first in the degraded condition, then in the normal and last in the highly degraded conditions. It thus seems as if the advantage of an early gaze switch was integrated faster in moderately challenging conditions. There, the benefit of an anticipatory gaze behaviour and temporally precise (local) predictions may be maximal compared to the two other conditions wherein uncertainty may either be too low (normal speech) or too high (highly degraded speech). However, while the N400 latency is earliest in the degraded condition (Figure 3A), the N400 amplitude best correlates with turn anticipatory behaviour in the highly degraded condition (Figure 2D). In considering this inconsistency one should keep in mind that 1) the N400 effect, in terms of amplitude, is also significant in the highly degraded condition; 2) while the correlation between N400 amplitude and average gaze behaviour does not reach the significance threshold (p = .06), the trend of the correlation is similar across the two degraded conditions. Importantly, Figure 2C and 2D clearly show that there is a similar relation between gaze behaviour and the semantic access of dialogues. Differently from global comprehension score, the N400 amplitude does not reflect a global understanding of the conversation, but it is a rather local measure, as it is the case with gaze switch behaviour. Moreover, the two measures are serially ordered in time, with gaze switch taking place earlier ~400ms before the N400 peak. Thus, one can hypothesize that participants with an overall early gaze switch, by having a better prediction of the turn, have an easier lexical-semantic integration of the word starting the turn in the conversational context. This relation between gaze and both local (N400 effect) and global (comprehension scores) understanding is only shown under challenging conditions, wherein indeed participants can benefit of a predictive behaviour to improve their understanding of the dialogues.

It seems appropriate here to evoke an alternative (although not necessarily the most likely) interpretation of the negative component in terms of a phonological mismatch negativity (pMMN). Several studies have addressed the timing of audio-visual integration. For instance, in McGurk illusion, the temporal window allowing modality fusion and allusion ranges between 30ms and 170ms of asynchrony [START_REF] Van Wassenhove | Temporal window of integration in auditory-visual speech perception[END_REF]. When looking at the electrophysiological response to the incongruent audio-visual stimulation, several authors report a pMMN [START_REF] Colin | Mismatch negativity evoked by the McGurk-MacDonald effect: A phonetic representation within short-term memory[END_REF][START_REF] Eskelund | Face configuration affects speech perception: Evidence from a McGurk mismatch negativity study[END_REF][START_REF] Stekelenburg | Electrophysiological correlates of predictive coding of auditory location in the perception of natural audiovisual events[END_REF]. The latency of this response is later than the classic MMN and is described approximately 500ms after the voice onset [START_REF] Proverbio | Electrophysiological Indexes of Incongruent Audiovisual Phonemic Processing: Unraveling the McGurk Effect[END_REF]. In our study, the classification of turns as being accompanied by late or early gaze switch implies that the audio-visual relation of the stimuli differs in the two classes. For early gaze switch, the gaze is well anticipated and thus the audio-visual information is perceived simultaneously. By contrast, in turns with a late gaze switch, often occurring after the voice onset, the audiovisual integration has a higher level of uncertainty. In other words, if a gaze switch takes place later than a turn onset, there may be an audiovisual mismatch because the participant is looking at the previous speaker while listening to the upcoming speaker's voice. In this context, looking at the closed lips of the previous speaker and hearing the word pronounced by the following speaker, may engender a phonological audiovisual MMN.

We would like to end the discussion with a consideration on the implicit and explicit role of predictions. It is known that the influences of the visual system on auditory perception can be so strong as to override under certain conditions the original input of the auditory system [START_REF] Mcgurk | Hearing lips and seeing voices[END_REF]. More commonly, viewing the lips provides relevant complementary information and can augment and improve auditory capabilities [START_REF] Calvert | Activation of auditory cortex during silent lipreading[END_REF][START_REF] Drijvers | Visual context enhanced: The joint contribution of iconic gestures and visible speech to degraded speech comprehension[END_REF][START_REF] Grant | The use of visible speech cues for improving auditory detection of spoken sentences[END_REF][START_REF] Sumby | Visual Contribution to Speech Intelligibility in Noise[END_REF]. In this context, it is interesting to note that, compared to the normal condition, the moderately degraded condition yields a poorer comprehension but similar gaze switch latencies and an earlier N400 effect. These preserved latencies, in a context of poorer comprehension that should rather induce poorer predictions, may be mediated by an active compensatory strategy, explicitly making greater use of visual cues to improve predictive processes [START_REF] Sohoglu | Perceptual learning of degraded speech by minimizing prediction error[END_REF]. Such a compensatory strategy may not hold when speech is highly degraded, yielding to slower gaze switches due to a poor understanding of the semantic and syntactic context. The respective role of implicit and explicit predictions and their link to conversation strategies [START_REF] Hadley | Conversation in small groups: Speaking and listening strategies depend on the complexities of the environment and group[END_REF] will require further work addressing this specific question. Another interesting aspect that will require further work is the inter-relation between local predictions (gaze), local semantic processing (N400) and global comprehension (here assessed via specific questions on the dialogues). While on one side gaze switches temporally precede auditory and semantic processing of the upcoming turn, this anticipatory behaviour can only build on the global comprehension of the dialogue, that, in turn, depends on the integration of local semantic processing.

To conclude, this study confirms that gaze switch can be used as a proxy of predictions in a conversational context. It shows that these predictions are related to the lexico-semantic processing of the turn start, as estimated by a neural marker of prediction error. It also shows that predictions are less accurate in more challenging listening conditions, but that they are also most useful in that specific context to make sense of the upcoming turn and, more generally, are a good indicator of global dialogue comprehension.

Figure 1 .

 1 Figure 1. Schematic illustration of the task and analyses. A. On the left, snapshot of the video illustrating the fixation points of the participant and the gaze switch. On the right, the three different speech conditions with

Figure 2 .

 2 Figure 2. Behavioural results and correlations. A. Effect of speech degradation level on comprehension. Dots represent the average comprehension score for each participant in each condition. B. Effect of speech degradation level on gaze switch latency. Latency at zero milliseconds corresponds to the onset of the turn. Dots represent the average latency for each participant in each condition. C. Scatter plots and linear regression between comprehension scores and gaze switch latency in the three different conditions. D. Scatter plots and linear regression between N400 effect size (late minus early, in the 300-550ms latency window) and gaze switch latency in the three different conditions.

Figure 3 .

 3 Figure 3. A. event-related potentials (ERPs) in a fronto-central region of interest time-locked to the turn onset for early and late gaze switch latencies. Shaded areas indicate the standard error of the mean. The black line below the ERPs indicates the FDR corrected significant difference between early and late conditions. B.

Acknowledgements

We would like to thank Laura Leone, Romane Pradels and Benjamin Morillon for their participation in the design of the dialogues and the production of videos. Funding sources: Work supported by APA foundation (RD-2016-9), ANR-16-CONV-0002 (ILCB) and the Excellence Initiative of Aix-Marseille University (A*MIDEX).

Est-ce que Damien veut attraper une bête avec sa cage ?

Does Damien want to catch an animal with his cage?

Cette cage est faite pour qui ?

Who is this cage for?

Comment Damien arrive à rentrer dans sa cage ?

How does Damien get into his cage?

Dans quoi peut rentrer le génie d'Aladin ?

What can Aladdin's genie fit into?

Pourquoi Damien veut se cacher dans une cage ?

Why does Damien want to hide in a cage?

Comment se déplace l'étrange fée ?

How does the strange fairy move?

Qui porte un grand chapeau ?

Who is wearing a big hat?

Pourquoi la sorcière veut attraper Damien ?

Why does the witch want to catch Damien?

Camille va chercher qui pour aider Damien à se débarrasser de la méchante sorcière ?

Who will Camille look for to help Damien get rid of the wicked witch?

Dans cette histoire, où habite Harry Potter ?

In this story, where does Harry Potter live?

Pourquoi Camille pense qu'une bête s'est échappée ?

Why does Camille think an animal has escaped?

Est-ce que Harry Potter aime les vilaines sorcières ?

Does Harry Potter like bad witches?

Table 2. Exemple of questions for one dialogue. Supplementary material 2. Number of gaze switch per second in and out of the turn window.


V Vougalter

V Volpert

On the Solvability Conditions for Some non Fredholm Operators

Keywords: solvability conditions, non Fredholm operators, elliptic problems AMS subject classification: 35J10, 35P10, 35P25

We obtain solvability conditions for some elliptic equations involving non Fredholm operators, which are sums of second order differential operators with the methods of spectral theory and scattering theory for Schrödinger type operators.

Introduction

Fredholm type solvability conditions, which affirm that an operator equation is solvable if and only if its right-hand side is orthogonal to solutions of the homogeneous adjoint equation, are used directly or indirectly in the most methods of linear and nonlinear analysis. If the operator does not satisfy the Fredholm property, applicability of these solvability conditions is not established. In this work we study solvability conditions for some class of non Fredholm operators. We consider the equation

Hu ≡ -∆u + W (x)u -au = f (1.1)
with a non-negative constant a and with some conditions on the potential W (x) which will be specified below. If W (x) → 0 as |x| → ∞, the case studied in our previous work [START_REF] Vougalter | Solvability conditions for some non Fredholm operators[END_REF], then the essential spectrum of the operator H consists of the half-axis [-a, ∞) [START_REF] Mukhamadiev | Normal solvability and the noethericity of elliptic operators in spaces of functions on R n . Part I[END_REF][START_REF] Mukhamadiev | Normal solvability and noethericity of elliptic operators in spaces of functions on R n . II. Zap[END_REF][START_REF] Volpert | Fredholm property of elliptic operators in unbounded domains[END_REF][START_REF] Jonsson | Applied Analysis[END_REF]. Since it contains the origin, the operator does not satisfy the Fredholm property, and the usual solvability conditions for equation (1.1) are not applicable.

There are two distinct cases, a = 0 and a > 0. Let us first discuss the case a = 0. It is known that homogeneous elliptic operators with constant coefficients satisfy the Fredholm property if considered in some specially chosen polynomial weighted spaces. For the operator H given by (1.1) this is the case if a = 0 and W (x) ≡ 0, that is for the Laplace operator. Lower order terms with the coefficients decaying at infinity represent compact operators if the decay rate is sufficiently high. Therefore the operator H in the weighted spaces remains Fredholm under some conditions on the potential. This allows one to make some conclusions about its index and solvability conditions. This approach is based on a priori estimates of solutions obtained in [START_REF] Nirenberg | The null spaces of elliptic partial differential operators in R n[END_REF]. The Fredholm property of this class of operators in Sobolev spaces is studied in [START_REF] Lockhart | Fredholm property of a class of elliptic operators on non-compact manifolds[END_REF], [START_REF] Lockhart | On elliptic systems in R n[END_REF]. Similar problems for elliptic operators in Hölder spaces are investigated in [START_REF] Benkirane | Propriété d'indice en théorie Holderienne pour des opérateurs elliptiques dans R n[END_REF], [START_REF] Bolley | Propriété d'indice en théorie Holderienne pour des opérateurs différentiels elliptiques dans R n[END_REF]. Exterior problems for the Laplace operator in weighted Sobolev spaces are considered in [START_REF] Amrouche | Dirichlet and Neumann exterior problems for the ndimensional Laplace operator. An approach in weighted Sobolev spaces[END_REF], [START_REF] Amrouche | Mixed exterior Laplace's problem[END_REF], and for more general operators in Hölder spaces in [START_REF] Bolley | Propriété dindice en théorie Hölderienne pour le problème extérieur de Dirichlet[END_REF]. The dimension of the kernel and the Fredholm property of elliptic operators of the first order are studied in [START_REF] Walker | On the null-space of first-order elliptic partial differential operators in R n[END_REF], [START_REF] Walker | A Fredholm theory for a class of first-order elliptic partial differential operators in R n[END_REF].

The case of positive a is qualitatively different. The method described above is not applicable. In the 1D case we can introduce exponential weighted spaces where the operator will satisfy the Fredholm property [START_REF] Volpert | Fredholm property of elliptic operators in unbounded domains[END_REF]. However, in R n with n ≥ 2 this method is not applicable neither. The reason for this can be already seen for the case W (x) ≡ 0. If the equation is solvable, then the right-hand side is orthogonal to all functions e ipx 1 e iqx 2 , where p 2 + q 2 = a, x = (x 1 , x 2 ) ∈ R 2 . Hence there is a continuous family of solvability conditions while the Fredholm property implies only a finite number of them.

The method developed in our previous work is based on the spectral theory of self-adjoint operators [START_REF] Vougalter | Solvability conditions for some non Fredholm operators[END_REF]. Similar to the case W (x) ≡ 0 where we can use the Fourier transform and explicitly find the solution, in the case of nonzero potential we use spectral decomposition with respect to the functions of the continuous spectrum of the operator H 0 u = -∆u + W u. This allows us to obtain solvability conditions as orthogonality to the functions of the continuous spectrum. To the best of our knowledge, this is the first result on solvability conditions for this class of operators. This method is applicable both for a = 0 and a > 0. Though these solvability conditions are similar to the usual ones, we should not forget that the operator does not satisfy the Fredholm property which limits their application to nonlinear problems.

In this work we continue the investigation of equation (1.1) under different assumptions on the potential. We will assume that it can be represented as W (x) = W 1 (x ′ ) + W 2 (x ′′ ), where x = (x 1 , ..., x n ), x ′ = (x 1 , ..., x k ), x ′′ = (x k+1 , ..., x n ). Though the potential does not converge to zero as |x| → ∞, we can apply the method of [START_REF] Vougalter | Solvability conditions for some non Fredholm operators[END_REF] using separation of variables. A particular case of such equations with W 1 (x ′ ) ≡ 0 arises in reaction-diffusion problems [START_REF] Volpert | Solvability conditions for elliptic problems with non-Fredholm operators[END_REF].

Formulation of the results

For the sake of convenience we will denote independent variables by x and y and put W (x, y) = V (x) + U (y). We begin with the operator H a on L 2 (R 6 ), such that

H a u = -∆ x u + V (x)u -∆ y u + U (y)u -au
with the Laplace operators ∆ x and ∆ y in x = (x 1 , x 2 , x 3 ) and y = (y 1 , y 2 , y 3 ), a ≥ 0 is a parameter, the potentials V (x) and U (y) decay to zero as x, y → ∞. We investigate the conditions on the function f (x, y) ∈ L 2 (R 6 ) under which the equations

H a u = f (2.1)
and

H 0 u = f, (2.2)
the second one is the limiting case of the first one as a → 0, have the unique solution in L 2 (R 6 ).

Thus the case of a single Schrödinger operator studied in [START_REF] Vougalter | Solvability conditions for some non Fredholm operators[END_REF] is being generalized to the case of the sum of two such operators. We will use the spectral decomposition of self-adjoint operators.

For a function ψ(x) belonging to a L p (R d ) space with 1 ≤ p ≤ ∞, d ∈ N its norm is being denoted as ψ L p (R d ) . As technical tools for estimating the appropriate norms of functions we will be using, in particular Young's inequality

f 1 * f 2 L ∞ (R 3 ) ≤ f 1 L 4 (R 3 ) f 2 L 4 3 (R 3 ) , f 1 ∈ L 4 (R 3 ), f 2 ∈ L 4 3 (R 3 ) ,
where * stands for the convolution. The inner product of functions on L 2 (R d ), d ∈ N is being denoted as

(f 1 (x), f 2 (x)) L 2 (R d ) := R d f 1 (x) f2 (x)dx; for a vector function A(x) = (A 1 (x), A 2 (x), A 3 (x)), x ∈ R 3 the inner product (f 1 (x), A(x)) L 2 (R 3 )
is the vector with the coordinates R 3 f 1 (x) Āi (x)dx, i = 1, 2, 3. Note that with a slight abuse we use the same notation even when functions may not be square integrable, for instance the functions ϕ k (x) and η q (y) of the continuous spectrum of the operators -∆ x + V (x) and -∆ y + U (y) respectively are normalized to Dirac delta-functions (see (3.1) and (3.2) in Section 3).

We make the following technical assumptions on the potential functions involved in the equations (2.1) and (2.2) and on the right sides of these equations. 3.5+ε with some ε > 0 and x, y ∈ R 3 a.e. such that

4 1 9 9 8 ( 4π
) -2 3 V 1 9 L ∞ (R 3 ) V 8 9 L 4 3 (R 3 ) < 1, 4 1 9 9 8 ( 4π
) -2 3 U 1 9 L ∞ (R 3 ) U 8 9 L 4 3 (R 3 ) < 1 √ c HLS V L 3 2 (R 3 ) < 4π, √ c HLS U L 3 2 (R 3 ) < 4π
Here c HLS is the constant in the Hardy-Littlewood-Sobolev inequality

R 3 R 3 f 1 (x)f 1 (y) |x -y| 2 dxdy ≤ c HLS f 1 2 L 3 2 (R 3 ) , f 1 ∈ L 3 2 (R 3 ) (2.3)
and given on p.98 of [START_REF] Lieb | Loss. Analysis[END_REF].

Assumption 2. The function f (x, y) ∈ L 2 (R 6 ) and |x|f (x, y) ∈ L 1 (R 6 ), |y|f (x, y) ∈ L 1 (R 6 ).
Here and further down C denotes a finite positive constant. Due to our assumptions on the potentials the essential spectrum σ ess (H a ) of the Schrödinger type operator H a = H 0a fills the interval [-a, ∞) (see e.g. [START_REF] Jonsson | Applied Analysis[END_REF] ), the Fredholm alternative theorem does not work. The problem can be trivially handled by the method of the Fourier transform when the potential terms V (x) and U (y) vanish. We prove that the method can be generalized in the presence of shallow, short-range V (x) and U (y) replacing the standard Fourier harmonics by the functions ϕ k (x)η q (y), k, q ∈ R 3 of the continuous spectrum of the operator H 0 . These functions satisfy the Lippmann-Schwinger equations (see (3.1), (3.2)) in Section 3 and the explicit formulas (3.3).

While the wave vectors k, q attain all the possible values in R 3 , the functions ϕ 0 (x) and η 0 (y) correspond to k, q = 0 in the formulas (

(f (x, y), ϕ k (x)η q (y)) L 2 (R 6 ) = 0 f or (k, q) ∈ S 6 √ a a.e. b) Problem (2.2) has a unique solution u(x, y) ∈ L 2 (R 6 ).
Remark 4. Note that as distinct from the single Schrödinger operator case studied in [START_REF] Vougalter | Solvability conditions for some non Fredholm operators[END_REF], part b) of Theorem 3 does not require any orthogonality conditions at all.

In the second part of the article we study the operator

L a on L 2 (R n+3 ), n ∈ N, such that L a u = -∆ x u -∆ y u + U (y)u -au with x = (x 1 , x 2 , ..., x n ) ∈ R n , y = (y 1 , y 2 , y 3 ) ∈ R 3
, a ≥ 0 and the potential U (y) satisfying the same assumptions as before. We establish the solvability conditions in L 2 (R n+3 ) for the inhomogeneous elliptic problems

L a u = φ (2.4) and L 0 u = φ , (2.5)
where L 0 is the limiting case of the operator L a when a → 0 and φ(x, y) ∈ L 2 (R n+3 ). We impose the following technical conditions on the right side of the equations (2.4) and (2.5).

Assumption 5. We have φ(x, y) ∈ L 2 (R n+3 ), |x|φ(x, y) ∈ L 1 (R n+3 ) and |y|φ(x, y) ∈ L 1 (R n+3 ).
As for the potential term U (y) involved in the left sides of the equations (2.4) and (2.5), we assume the same technical conditions to hold as in the first Theorem. Thus for the operator L a the essential spectrum coincides with the semi-axis [-a, ∞) and the Fredholm alternative theorem fails to work.

Theorem 6. Let Assumption 5 hold and the function U (y) satisfy Assumption 1. Then a) Equation (2.4) possesses a unique solution

u(x, y) ∈ L 2 (R n+3 ), n ∈ N if and only if (φ(x, y), e ikx (2π) n 2 η q (y)) L 2 (R n+3 ) = 0 f or (k, q) ∈ S n+3 √ a a.e. b) When n = 1 equation (2.5) admits a unique solution u(x, y) ∈ L 2 (R 4 ) if and only if (φ(x, y), η 0 (y)) L 2 (R 4 ) = 0 c) When n ≥ 2 equation (2.5) admits a unique solution u(x, y) ∈ L 2 (R n+3 ).
Remark 7. Note that the solvability conditions for equation (2.5) depend on the dimensions of the problem, such that we have the explicit one in case b) and no orthogonality conditions are required at all in higher dimensions as stated in case c).

In the third part of the article we consider the operator

L = -∆ x +V (x)-∆ y +V(y) on L 2 (R 3+m )
with the Laplacian operators ∆ x and ∆ y such that x = (x 1 , x 2 , x 3 ) ∈ R 3 , y = (y 1 , y 2 , ..., y m ) ∈ R m , m ∈ N and prove the necessary and sufficient conditions for the solvability in L 2 (R 3+m ) of the following inhomogeneous problem

Lu = g(x, y), (2.6)
where g(x, y) ∈ L 2 (R 3+m ). Note that problem (2.6) is the extension of the elliptic equation studied in the second part of [START_REF] Vougalter | Solvability conditions for some non Fredholm operators[END_REF] to the case of the potential function V (x) explicitly present in it, which is assumed to satisfy the same assumptions as in the first part of our article and x ∈ R 3 . The assumptions on the second potential function V(y) are analogous to those in [START_REF] Vougalter | Solvability conditions for some non Fredholm operators[END_REF].

Assumption 8. The function V(y) : R m → R is continuous and lim y→∞ V(y) = V + > 0.

Thus for the operator

h := -∆ y + V(y) on L 2 (R m ) the essential spectrum σ ess (h) = [V + , ∞).
Let us denote the eigenvalues of the operator h located below V + as e j , e j < e j+1 , j ≥ 1 and the corresponding elements of the orthonormal set of eigenfunctions as ψ l j , such that hψ l j = e j ψ l j ,

1 ≤ l ≤ m j , (ψ l i , ψ s j ) L 2 (R m ) = δ i,j δ l,s
, where m j stands for the eigenvalue multiplicity, which is finite since the essential spectrum starts only at V + and δ i,j denotes the Kronecker symbol. We make the following key assumption on the discrete spectrum of the operator h relevant to the problem (2.6).

Assumption 9.

The eigenvalues e j < 0 for all 1 ≤ j ≤ N -1, N ≥ 1 and e N = 0.

Hence under our assumptions the operator L is not Fredholm. The bottom of the essential spectrum of the operator -∆ x + V (x), which is unitarily equivalent to the free Laplacian in our problem (see Section 3) is zero and h has the square integrable zero modes. Furthermore, the operator h has the negative eigenvalues e j , j = 1, ..., N -1 and -∆ x + V (x) has the functions of the continuous spectrum ϕ k (x), such that k ∈ S 3 √ -e j . However, equation (2.6) can be solved on the proper subspace with the orthogonality conditions stated precisely below. Our third main result is as follows.

Theorem 10. Let the potential function V (x) satisfy Assumption 1, Assumptions 8 and 9 hold, g(x, y) ∈ L 2 (R 3+m ) and |x| α 2 g(x, y) ∈ L 2 (R 3+m ) with some α > 5. Then equation (2.6) admits a unique solution u(x, y) ∈ L 2 (R 3+m ) if and only if:

(g(x, y), ϕ 0 (x)ψ l N (y)) L 2 (R 3+m ) = 0, 1 ≤ l ≤ m N and (g(x, y), ϕ k (x)ψ l j (y)) L 2 (R 3+m ) = 0, 1 ≤ l ≤ m j , k ∈ S 3 √ -e j a.e., 1 ≤ j ≤ N -1
Establishing solvability conditions for linear elliptic problems involving non Fredholm operators plays the significant role in various applications including the existence of travelling wave solutions of reaction-diffusion systems (see e.g. [START_REF] Volpert | Solvability conditions for elliptic problems with non-Fredholm operators[END_REF]).

3. Properties of the operator H 0 and proof of Theorem 3

The functions of the continuous spectrum of the operator -∆ x + V (x) are solutions to the Lippmann-Schwinger equation (see e.g. [START_REF] Reed | Methods of modern mathematical physics, III. Scattering theory[END_REF] p.98)

ϕ k (x) = e ikx (2π) 3 2 - 1 4π R 3 e i|k||x-y| |x -y| (V ϕ k )(y)dy (3.1)
with the orthogonality conditions

(ϕ k (x), ϕ k 1 (x)) L 2 (R 3 ) = δ(k -k 1 ), k, k 1 ∈ R 3 . Let us define the integral operator (Qϕ)(x) := - 1 4π R 3 e i|k||x-y| |x -y| (V ϕ)(y)dy, ϕ ∈ L ∞ (R 3 )
Similarly the functions of the continuous spectrum of the operator -∆ y + U (y) satisfy η q (y) = e iqy (2π)

3 2 - 1 4π R 3 e i|q||y-z| |y -z| (U η q )(z)dz (3.2)
and the orthogonality relations (η q (y), η q 1 (y)) L 2 (R 3 ) = δ(qq 1 ), q, q 1 ∈ R 3 . Therefore, it makes sense to define the integral operator

(P η)(y) := - 1 4π R 3 e i|q||y-z| |y -z| (U η)(z)dz, η ∈ L ∞ (R 3 )
For the potentials V (x) and U (y) satisfying Assumption 1 we have the estimates for the operator norms of

Q : L ∞ (R 3 ) → L ∞ (R 3 ) and P : L ∞ (R 3 ) → L ∞ (R 3
) denoted as Q ∞ and P ∞ respectively. These k and q independent bounds are proven in Lemma 2.1 of [START_REF] Vougalter | Solvability conditions for some non Fredholm operators[END_REF], namely

Q ∞ ≤ 4 1 9 9 8 (4π) -2 3 V 1 9 L ∞ (R 3 ) V 8 9 L 4 3 (R 3 ) < 1
and

P ∞ ≤ 4 1 9 9 8 (4π) -2 3 U 1 9 L ∞ (R 3 ) U 8 9 L 4 3 (R 3 ) < 1 ,
via Assumption 1 which enables us to express the functions of the continuous spectrum as

ϕ k (x) = (I -Q) -1 e ikx (2π) 3 2
, η q (y) = (I -P ) -1 e iqy (2π)

3 2 , k, q ∈ R 3 (3.3)
and obtain the trivial estimates on their norms (see Corollary 2.2 of [START_REF] Vougalter | Solvability conditions for some non Fredholm operators[END_REF])

ϕ k (x) L ∞ (R 3 ) ≤ 1 1 -Q ∞ 1 (2π) 3 2 , η q (y) L ∞ (R 3 ) ≤ 1 1 -P ∞ 1 (2π) 3 2
We revise the argument of Lemma 2.3 of [START_REF] Vougalter | Solvability conditions for some non Fredholm operators[END_REF]. By means of inequality (2.3) and Assumption 1 the Rollnik norms for both potentials V (x) and U (y) (see e.g. [START_REF] Simon | Quantum mechanics for Hamiltonians defined as quadratic forms[END_REF]) given by

V 2 R := R 3 R 3 |V (x)||V (y)| |x -y| 2 dxdy, U 2 R := R 3 R 3 |U (x)||U (y)| |x -y| 2 dxdy are bounded above, such that V R < 4π, U R < 4π
which yields the self-adjointness of the operators -∆ x +V (x) and -∆ y +U (y) on L 2 (R 3 ) and their unitary equivalence to -∆ x and -∆ y respectively (see [START_REF] Kato | Wave operators and similarity for some non-selfadjoint operators[END_REF], also [START_REF] Rodnianski | Time decay for solutions of Schrödinger equations with rough and time-dependent potentials[END_REF]) via the wave operators

Ω ± V := s -lim t→∓∞ e it(-∆+V ) e it∆ , Ω ± U := s -lim t→∓∞ e it(-∆+U ) e it∆
with the limits understood in the strong L 2 sense (see e.g. [START_REF] Reed | Methods of modern mathematical physics, III. Scattering theory[END_REF] p.34, [START_REF] Cycon | Schrödinger operators with application to quantum mechanics and global geometry[END_REF] p.90). The spectral theorem implies that any function in L 2 (R 6 ) can be expanded through the products of the functions of the continuous spectrum ϕ k (x)η q (y), k, q ∈ R 3 which form the complete system in L 2 (R 6 ).

Let f (k, q) denote the generalized Fourier transform of the function f (x, y) with respect to these functions , namely f (k, q) := (f (x, y), ϕ k (x)η q (y)) L 2 (R 6 ) , k, q ∈ R 3

We have the following auxiliary statement.

Lemma 11. Let Assumptions 1 and 2 hold. Then

(∇ k + ∇ q ) f (k, q) ∈ L ∞ (R 6 )
Proof. Clearly we need to estimate the sum

(f (x, y), ∇ k ϕ k (x)η q (y)) L 2 (R 6 ) + (f (x, y), ϕ k (x)∇ q η q (y)) L 2 (R 6 ) (3.4)
From the Lippmann-Schwinger equations (3.1) and (3.2) we easily obtain

∇ k ϕ k (x) = e ikx (2π) 3 2 ix + (I -Q) -1 Q e ikx (2π) ix + (I -Q) -1 (∇ k Q)(I -Q) -1 e ikx (2π) 3 2
(3.5) and ∇ q η q (y) = e iqy (2π) iy + (I -P ) -1 (∇ q P )(I -P ) -1 e iqy (2π)

3 2 (3.6)
with the operators

∇ k Q : L ∞ (R 3 ) → L ∞ (R 3 ; C 3 ) and ∇ q P : L ∞ (R 3 ) → L ∞ (R 3 ; C 3 ) having the integral kernels ∇ k Q(x, y, k) = - 1 4π e i|k||x-y| i k |k| V (y)
and ∇ q P (y, z, q) = -1 4π e i|q||y-z| i q |q| U (z)

respectively. A trivial computation gives the upper bounds for their norms

∇ k Q ∞ ≤ 1 4π V L 1 (R 3 ) < ∞ and ∇ q P ∞ ≤ 1 4π U L 1 (R 3 ) < ∞
by means of the rate of decay of the potentials V (x) and U (y) stated explicitly in Assumption 1 By means of representation (3.5) the first term in (3.4) can be written as the sum of the three terms T 1 (k, q) + T 2 (k, q) + T 3 (k, q), where the first one is

T 1 (k, q) := (f (x, y), e ikx (2π)
3 2 ixη q (y)) L 2 (R 6 )
Clearly we have the bound

|T 1 (k, q)| ≤ 1 (2π) 3 1 1 -P ∞ R 3 dx R 3 dy|x||f (x, y)| < ∞
due to Assumption 2. The second term

T 2 (k, q) := (f (x, y), (I -Q) -1 Q e ikx (2π) 3 2 ixη q (y)) L 2 (R 6 )
can be estimated as

|T 2 (k, q)| ≤ 1 (2π) 3 1 1 -P ∞ 1 1 -Q ∞ f L 1 (R 6 ) Qe ikx x L ∞ (R 3 )
We obtain the upper bound using the definition of the operator Q and Young's inequality

|Qe ikx x| ≤ 1 4π R 3 |V (y)||y| |x -y| dy = 1 4π { 1 |x| χ {|x|≤1} * |V (x)||x| + 1 |x| χ {|x|>1} * |V (x)||x|} ≤ ≤ 1 4π { V (y)y L ∞ (R 3 ) 1 0 4πrdr + χ {|x|>1} 1 |x| L 4 (R 3 ) V (x)x L 4 3 (R 3 ) } < ∞ 8 and k-independent since V (x)x ∈ L ∞ (R 3 ) ∩ L 4 3 (R 3
) by means of the rate of decay of the potential function V (x) given in Assumption 1. Here and further down χ A stands for the characteristic function of a set A. By means of Assumption 2 and Fact 2 of the Appendix f (x, y) ∈ L 1 (R 6 ). Hence T 2 (k, q) ∈ L ∞ (R 6 ). The third term is given by

T 3 (k, q) := (f (x, y), (I -Q) -1 (∇ k Q)(I -Q) -1 e ikx (2π) 3 2 η q (y)) L 2 (R 6 )
We estimate from above

|T 3 (k, q)| ≤ 1 4π(2π) 3 f L 1 (R 6 ) V L 1 (R 3 ) 1 (1 -Q ∞ ) 2 1 1 -P ∞ < ∞ uniformly for all k, q ∈ R 3 . Thus T 3 (k, q) ∈ L ∞ (R 6
). Similarly we express the second term in (3.4) as the sum of the terms R

1 (k, q) + R 2 (k, q) + R 3 (k, q) with R 1 (k, q) := (f (x, y), ϕ k (x) e iqy (2π) 3 2 iy) L 2 (R 6 ) Obviously |R 1 (k, q)| ≤ 1 (2π) 3 1 1 -Q ∞ R 3 dx R 3 dy|y||f (x, y)| < ∞
The next term is given by

R 2 (k, q) := (f (x, y), ϕ k (x)(I -P ) -1 P e iqy (2π) 3 2 iy) L 2 (R 6 ) , such that |R 2 (k, q)| ≤ 1 (2π) 3 1 1 -Q ∞ 1 1 -P ∞ f L 1 (R 6 ) P e iqy y L ∞ (R 3 )
The definition of the integral operator P along with Young's inequality yield

|P e iqy y| ≤ 1 4π R 3 |U (z)| |y -z| |z|dz = 1 4π { χ {|y|≤1} 1 |y| * |U (y)||y| + χ {|y|>1} 1 |y| * |U (y)||y|} ≤ ≤ 1 4π { U (y)y L ∞ (R 3 ) 1 0 4πrdr + χ {|y|>1} 1 |y| L 4 (R 3 ) U (y)y L 4 3 (R 3 ) } < ∞ and q-independent since U (y)y ∈ L ∞ (R 3 ) ∩ L 4 3 (R 3
) due to the rate of decay of the potential function U (y) stated explicitly in Assumption 1. Thus R 2 (k, q) ∈ L ∞ (R 6 ). Therefore, it remains to estimate the term

R 3 (k, q) := (f (x, y), ϕ k (x)(I -P ) -1 (∇ q P )(I -P ) -1 e iqy (2π) 3 2
)

L 2 (R 6 )
for which we easily obtain the bound uniform in k, q ∈ R 3

|R 3 (k, q)| ≤ 1 4π(2π) 3 f L 1 (R 6 ) U L 1 (R 3 ) 1 (1 -P ∞ ) 2 1 1 -Q ∞ < ∞
Armed with the technical lemma above we proceed to prove Theorem 3.

Proof of Theorem 3. Assume first that equation (2.1) admits two solutions u 1 , u 2 ∈ L 2 (R 6 ). Then their difference w(x, y) := u 1 (x, y)u 2 (x, y) ∈ L 2 (R 6 ) satisfies the homogeneous equation

H a w = 0
Since the operator H a on L 2 (R 6 ) is unitarily equivalent to -∆ x -∆ y -a, it has no nontrivial square integrable zero modes, and therefore w(x, y) vanishes a.e. The analogous uniqueness argument works for the square integrable solutions of equation (2.2). By applying the generalized Fourier transform with respect to the functions of the continuous spectrum ϕ k (x) and η q (y) to equation (2.1) we easily obtain

ũ(k, q) = f (k, q)
k 2 + q 2a For technical purposes we will be using the spherical layer in the space of six dimensions

A σ := {(k, q) ∈ R 6 : √ a -σ ≤ k 2 + q 2 ≤ √ a + σ}, 0 < σ < √ a ,
such that χ Aσ and χ Aσ c denote the characteristic functions of the layer and its complement respectively. Thus we arrive at

ũ(k, q) = f (k, q) k 2 + q 2 -a χ Aσ + f (k, q) k 2 + q 2 -a χ Aσ c (3.7)
For the second term in the right side of the identity above we have the estimate

f (k, q) k 2 + q 2 -a χ Aσ c ≤ | f (k, q)| σ √ a ∈ L 2 (R 6 )
Clearly, we have the representation

f (k, q) = f ( √ a, ω) + √ k 2 +q 2 √ a ∂ f (|s|, ω) ∂|s| d|s|.
Here and further down ω stands for the angle variables on the sphere and dω denotes integration with respect to these variables. This enables us to split the first term in the right side of (3.7) into the sum ũ1 (k, q) + ũ2 (k, q) with

ũ1 (k, q) = f ( √ a, ω) k 2 + q 2 -a χ Aσ , ũ2 (k, q) = √ k 2 +q 2 √ a ∂ f (|s|,ω) ∂|s| d|s| k 2 + q 2 -a χ Aσ By means of Lemma 11 | ũ2 (k, q)| ≤ C χ Aσ √ a ∈ L 2 (R 6 )
Therefore, it remains to estimate the norm ũ1 (k, q) 2 L 2 (R 6 ) equal to which is finite if and only if (f (x, y), ϕ k (x)η q (y)) L 2 (R 6 ) = 0 for (k, q) ∈ S 6 √ a a.e, which completes the proof of Part a) of the Theorem. Similarly for equation (2.2)

ũ(k, q) = f (k, q) k 2 + q 2
Again we split the expression into the nonsingular and the singular parts, such that

ũ(k, q) = f (k, q) k 2 + q 2 χ {k 2 +q 2 >1} + f (k, q) k 2 + q 2 χ {k 2 +q 2 ≤1} Obviously f (k, q) k 2 + q 2 χ {k 2 +q 2 >1} ≤ | f (k, q)| ∈ L 2 (R 6 )
The remaining expression will be estimated with the help of the identity

f (k, q) = f (0) + √ k 2 +q 2 0 ∂ f (|s|, ω) ∂|s| d|s| Lemma 11 yields √ k 2 +q 2 0 ∂ f (|s|,ω) ∂|s| d|s| k 2 + q 2 χ {k 2 +q 2 ≤1} ≤ C k 2 + q 2 χ {k 2 +q 2 ≤1} ∈ L 2 (R 6 )
Thus it remains to estimate f (0) The operator L 0 is the sum of two operators. The first one is -∆ x acting on L 2 (R n ), n ∈ N and the functions of its continuous spectrum are the Fourier harmonics e ikx (2π)

k 2 + q 2 χ {k 2 +q 2 ≤1} 2 L 2 (R 6 ) = | f (0)| 2 1 0 rdr|S 6 | < ∞ since | f (k, q)| ≤ f L 1 (R 6 ) ϕ k L ∞ (R 3 ) η q L ∞ (R 3 ) < ∞, k, q ∈ R 3 which
n 2
, k ∈ R n . The second one is the Schrödinger operator -∆ y + U (y) on L 2 (R 3 ) unitarily equivalent to -∆ y via the wave operators which is discussed in Section 3 and the functions of its continuous spectrum are η q (y), q ∈ R 3 . Let ˜ φ(k, q) denote the generalized Fourier transform with respect to the products of the functions of the continuous spectrum

e ikx (2π) n 2 η q (y) forming the complete system in L 2 (R n+3 ), namely ˜ φ(k, q) := (φ(x, y), e ikx (2π) n 2 η q (y)) L 2 (R n+3 ) , k ∈ R n , q ∈ R 3
We establish the following auxiliary statement. Lemma 12. Let Assumptions 1 and 5 hold. Then

(∇ k + ∇ q ) ˜ φ(k, q) ∈ L ∞ (R n+3 ), n ∈ N
Proof. Obviously we need to take care of the following sum

(φ(x, y), ∇ k e ikx (2π) n 2 η q (y)) L 2 (R n+3 ) + (φ(x, y), e ikx (2π) n 2 ∇ q η q (y)) L 2 (R n+3 ) (4.1)
For the first term by means of Assumption 5 we have the finite and k, q independent upper bound

(φ(x, y), e ikx ix (2π) n 2 η q (y)) L 2 (R n+3 ) ≤ 1 (2π) n+3 2 1 1 -P ∞ xφ L 1 (R n+3 )
The second term in (4.1) can be written as the sum of three expressions S 1 (k, q) + S 2 (k, q) + S 3 (k, q), such that the first one

S 1 (k, q) := (φ(x, y), e ikx (2π) n 2 e iqy (2π) 3 2 iy) L 2 (R n+3 ) can be easily estimated as |S 1 (k, q)| ≤ 1 (2π) n+3 2 yφ L 1 (R n+3 ) < ∞ for all k ∈ R n , q ∈ R 3 due
to Assumption 5. For the second expression S 2 (k, q) := (φ(x, y), e ikx (2π) n 2 (I -P ) -1 P e iqy (2π)

3 2 iy) L 2 (R n+3 )
we obtain the upper bound

|S 2 (k, q)| ≤ 1 (2π) n+3 2 φ L 1 (R n+3 ) 1 1 -P ∞ P e iqy y L ∞ (R 3 )
The function φ(x, y) ∈ L 1 (R n+3 ) via Assumption 5 and Fact 2 of the Appendix. The estimate on the term P e iqy y L ∞ (R 3 ) established in the proof of Lemma 11 yields S 2 (k, q) ∈ L ∞ (R n+3 ). We complete the proof of the Lemma with the estimate on the third expression

S 3 (k, q) := (φ(x, y), e ikx (2π) n 2 (I -P ) -1 (∇ q P )(I -P ) -1 e iqy (2π) 3 2
)

L 2 (R n+3 ) such that |S 3 (k, q)| ≤ 1 4π 1 (2π) n+3 2 U L 1 (R 3 ) 1 (1 -P ∞ ) 2 φ L 1 (R n+3 ) Hence S 3 (k, q) ∈ L ∞ (R n+3 ).
Establishing Lemma 12 enables us to prove Theorem 6.

Proof of Theorem 6. Assume that equation (2.4) admits two solutions u 1 , u 2 ∈ L 2 (R n+3 ) such that their difference w(x, y) := u 1 (x, y)u 2 (x, y) ∈ L 2 (R n+3 ) is a solution to the homogeneous equation L a w = 0

Since the operator L a on L 2 (R n+3 ) is unitarily equivalent to -∆ x -∆ ya, it possesses no nontrivial zero modes w ∈ L 2 (R n+3 ), we arrive at w(x, y) = 0 a.e. Analogously the uniqueness argument works for the square integrable solutions of problem (2.5). We apply the generalized Fourier transform with respect to the functions of the continuous spectrum e ikx (2π)

n 2
, k ∈ R n and η q (y), q ∈ R 3 to equation (2.4) and obtain

˜ u(k, q) = ˜ φ(k, q) k 2 + q 2 -a
Let us introduce the auxiliary spherical layer in the space of n + 3 dimensions

B σ = {(k, q) ∈ R n+3 : √ a -σ ≤ k 2 + q 2 ≤ √ a + σ}, 0 < σ < √ a
such that χ Bσ and χ B c σ denote the characteristic functions of the layer and of its complement respectively. This enables us to split the right side of the equality above into the singular and the regular parts, such that

˜ u(k, q) = ˜ φ(k, q) k 2 + q 2 -a χ Bσ + ˜ φ(k, q) k 2 + q 2 -a χ B c σ (4.2) Obviously ˜ φ(k, q) k 2 + q 2 -a χ B c σ ≤ | ˜ φ(k, q)| √ aσ ∈ L 2 (R n+3 )
We will make use of the representation

˜ φ(k, q) = ˜ φ( √ a, ω) + √ k 2 +q 2 √ a ∂ ˜ φ(|s|, ω) ∂|s| d|s|.
Thus the first term in the right side of (4.2) can be written as ˜ u

1 (k, q) + ˜ u 2 (k, q) with ˜ u 1 (k, q) := ˜ φ( √ a, ω) k 2 + q 2 -a χ Bσ , ˜ u 2 (k, q) := √ k 2 +q 2 √ a ∂ > φ(|s|,ω) ∂|s| d|s| k 2 + q 2 -a χ Bσ By means of Lemma 12 | ˜ u 2 (k, q)| ≤ C χ Bσ √ a ∈ L 2 (R n+3 ). Finally we estimate ˜ u 1 (k, q) 2 L 2 (R n+3 ) = √ a+σ √ a-σ dr r n+2 (r - √ a) 2 (r + √ a) 2 S n+3 dω| ˜ φ( √ a, ω)| 2 < ∞ if and only if ˜ φ( √ a, ω) = 0 a.e. on S n+3
√ a which is equivalent to the orthogonality condition (φ(x, y), e ikx (2π) n 2 η q (y)) L 2 (R n+3 ) = 0 for (k, q) ∈ S n+3 √ a a.e., n ∈ N as stated in part a) of the theorem.

Then we analogously apply the generalized Fourier transform with respect to the functions e ikx (2π) n 2

, k ∈ R n and η q (y), q ∈ R 3 to equation (2.5) which yields

˜ u(k, q) = ˜ φ(k, q) k 2 + q 2 χ {k 2 +q 2 >1} + ˜ φ(k, q) k 2 + q 2 χ {k 2 +q 2 ≤1}
The first term in the right side of the identity above is nonsingular and can be easily estimated as

˜ φ(k, q) k 2 + q 2 χ {k 2 +q 2 >1} ≤ | ˜ φ(k, q)| ∈ L 2 (R n+3 ) We write ˜ φ(k, q) = ˜ φ(0) + √ k 2 +q 2 0 ∂ ˜ φ(|s|, ω) ∂|s| d|s|.
Hence it remains to study the square integrability of the sum of two terms ˜ u

3 (k, q) + ˜ u 4 (k, q) such that ˜ u 3 (k, q) := ˜ φ(0) k 2 + q 2 χ {k 2 +q 2 ≤1} , ˜ u 4 (k, q) := √ k 2 +q 2 0 ∂ > φ(|s|,ω) ∂|s| d|s| k 2 + q 2 χ {k 2 +q 2 ≤1}
Using Lemma 12 we arrive at

| ˜ u 4 (k, q)| ≤ C k 2 + q 2 χ {k 2 +q 2 ≤1} ∈ L 2 (R n+3 ). Finally we compute ˜ u 3 2 L 2 (R n+3 ) = | ˜ φ(0)| 2 |S n+3 | 1 0 r n-2 dr Note that | ˜ φ(k, q)| ≤ 1 (2π) n 2 η q L ∞ (R 3 ) φ L 1 (R n+3 ) < ∞, k ∈ R n , q ∈ R 3 . The L 2 norm above is
finite in all dimensions n ≥ 2. When dimension n = 1 this norm is finite if and only if ˜ φ(0) = 0 which is equivalent to the orthogonality condition (φ(x, y), η 0 (y)) L 2 (R 4 ) = 0. ✷

Properties of the operator L and proof of Theorem 10

By means of the spectral theorem the identity operator on L 2 (R m ) can be written as I = P + + P 0 + P -, where P ± and P 0 denote the orthogonal projections onto the positive, negative and zero subspaces of the operator h. Then problem (2.6) can be easily related to the equivalent system of three equations L + u + = g + (5.1)

L -u -= g - (5.2) L 0 u 0 = g 0 (5.3)
with the operators L ± = P ± LP ± and L 0 = P 0 LP 0 applied to the functions u ± = P ± u and u 0 = P 0 u respectively and the right sides of these equations g ± = P ± g and g 0 = P 0 g. For the first of the equations above we have the following statement.

Lemma 13. Let assumptions of Theorem 10 hold. Then equation ( 5.1) admits a solution

u + ∈ L 2 (R 3+m ), m ∈ N.
Proof. The orthogonal decomposition g = g + + g 0 + g -yields g + L 2 (R 3+m ) ≤ g L 2 (R 3+m ) . For the operator L + on L 2 (R 3 )⊗Ran(P + ) we have the following lower bound in the sense of quadratic forms L + ≥ P + hP + ≥ e N +1 > 0 where e N +1 stands for either the smallest positive eigenvalue of the operator h or the bottom of its essential spectrum, whichever is smaller. Ran(P + ) denotes the range of the projection operator P + . Hence the self-adjoint operator L + has the inverse L -1 + : L 2 (R 3 )⊗Ran(P + ) → L 2 (R 3+m ) with the norm estimated from above by 1 e N +1

. Therefore, equation (5.1) admits the solution u + = L -1

+ g + such that u + L 2 (R 3+m ) ≤ 1 e N +1 g L 2 (R 3+m ) < ∞
Next we turn our attention to equation (5.3). Without loss of generality it is being assumed that g 0 (x, y) = v 0 (x)ψ 1 N (y) = (g(x, y), ψ 1 N (y)) L 2 (R m ) ψ 1 N (y).

Lemma 14. Let assumptions of Theorem 10 hold. Then equation (5.3) possesses a solution u 0 ∈ L 2 (R 3+m ), m ∈ N if and only if (g(x, y), ϕ 0 (x)ψ k N (y)) L 2 (R 3+m ) = 0, 1 ≤ k ≤ m N Proof. Since the operator -∆ x +V (x) has no nontrivial square integrable zero modes (see Section 3), we have u 0 (x, y) = ξ(x)ψ 1 N (y). This allows us to relate equation (5.3) to the problem

(-∆ x + V (x))ξ(x) = v 0 (x) (5.4)
For its right side using Schwarz inequality we establish the square integrability

v 0 2 L 2 (R 3 ) = R 3 |(g(x, y), ψ 1 N (y)) L 2 (R m ) | 2 dx ≤ g 2 L 2 (R 3+m ) < ∞
For the right side of the equation above by means of Schwarz inequality

v j 2 L 2 (R 3 ) = R 3 |(g(x, y), ψ 1 j (y)) L 2 (R m ) | 2 dx ≤ g 2 L 2 (R 3+m ) < ∞
To estimate the norm below Schwarz inequality is being used as well Proof of Theorem 10. The solution of equation (2.6) is being constructed as u := u + + u 0 + u -, such that the existence of u + , u 0 , u -∈ L 2 (R 3+m ) is proven in Lemmas 13, 14 and 15 respectively. Suppose problem (2.6) possesses two solutions u 1 , u 2 ∈ L 2 (R 3+m ). Then the function v := u 1u 2 ∈ L 2 (R 3+m ) is a solution of the homogeneous problem with separation of variables Lv = 0 which solutions are linear combinations of functions c l j (x)ψ l j (y), 1 ≤ l ≤ m j , 1 ≤ j ≤ N since equation ( 5.1) with vanishing right side admits only the trivial solution in L 2 (R 3+m ) (see Lemma 13). But c l j (x), 1 ≤ l ≤ m j , 1 ≤ j ≤ N vanish a.e. since the operator -∆ x + V (x) does not have nonnegative eigenvalues with corresponding eigenfunctions in L 2 (R 3 ) (see Section 3). ✷

xv j (x) L 1 (R 3 ) =

Assumption 1 .

 1 The potential functions V (x), U (y) : R 3 → R satisfy the bounds |V (x)| ≤ C 1 + |x| 3.5+ε and |U (y)| ≤ C 1 + |y|

✷ 4 .

 4 means that no orthogonality conditions are required for Part b) of the theorem. Properties of the operator L 0 and proof of Theorem 6

R 3 dx|x||

 3 (g(x, y), ψ 1 j (y)) L 2 (R m ) y)| 2 dy with some α > 5 such that |x| α 2 g(x, y) ∈ 2 (R 3+m ). Schwarz inequality yields the following upper bound for the expression above∞ 0 4πr 4 1 + r α dr g 2 L 2 (R 3+m ) + |x| α 2 g 2 L 2 (R 3+m ) < ∞Therefore, Assumption 1.1 of[START_REF] Vougalter | Solvability conditions for some non Fredholm operators[END_REF] is satisfied for equation(5.6) and Theorem 1 of[START_REF] Vougalter | Solvability conditions for some non Fredholm operators[END_REF] yields the necessary and sufficient solvability condition in L 2 (R 3 ) for it, namely(v j (x), ϕ k (x)) L 2 (R 3 ) = 0 for k ∈ S 3 √ -e ja.e. which implies the statement of the lemma.

  3.1), (3.2) and (3.3). The sphere of radius r in R d , d ∈ N centered at the origin is being designated as S d r , the unit one as S d and |S d | denotes its Lebesgue measure. Our first main theorem is as follows.

	Theorem 3. Let Assumptions 1 and 2 hold. Then
	a) Problem (2.1) admits a unique solution u(x, y) ∈ L 2 (R

6

) if and only if

√ a+σ √ a-σ d( k 2 + q 2 ) ( k 2 + q 2 ) 5 ( k 2 + q 2 -√ a) 2 ( k 2 + q 2 + √ a) 2 S 6 dω| f ( √ a, ω)| 2 ,

To show the finiteness of another norm Schwarz inequality is being used as well

with some α > 5 such that |x| α 2 g(x, y) ∈ L 2 (R 3+m ). By means of Schwarz inequality this can be estimated from above as

Hence Assumption 1.1 of [START_REF] Vougalter | Solvability conditions for some non Fredholm operators[END_REF] holds for equation ( 5.4) and therefore by means of Theorem 1 of [START_REF] Vougalter | Solvability conditions for some non Fredholm operators[END_REF] we obtain the necessary and sufficient solvability conditions in L 2 (R 3 ) for it, namely (v 0 (x), ϕ 0 (x)) L 2 (R 3 ) = 0, yields the statement of the lemma.

Lemma 15. Let assumptions of Theorem 10 hold. Then equation ( 5.2) possesses a solution

Proof. We denote the orthogonal projections onto the eigenspaces correspondent to the negative eigenvalues {e j } N -1 j=1 of the operator h as {P -,j } N -1 j=1 , such that

P -,j , P -,j P -,m = P -,j δ j,m , 1 ≤ j, m ≤ N -1

Application of these projections to equation ( 5.2) easily yields the system of equations equivalent to it

with P -,j u -= u -,j and P -,j g -= g -,j such that u -= N -1 j=1 u -,j and g -= N -1 j=1 g -,j . We assume that g -,j (x, y) = v j (x)ψ 1 j (y) = (g(x, y), ψ 1 j (y)) L 2 (R m ) ψ 1 j (y), 1 ≤ j ≤ N -1. Then equation (5.5) becomes

We write u -,j (x, y) = ξ j (x)ψ 1 j (y) since the operator -∆ x + V (x) has no positive eigenvalues with corresponding eigenfunctions in L 2 (R 3 ) (see Section 3), which yields

Proof. We estimate the norm f L 1 (R n ) from above using Schwarz inequality by

where |B n | denotes the Lebesgue measure of a unit in the space of n dimensions.

Proof. It can be easily estimated that ✷