Fanny Dégeilh

Miriam H Beauchamp

Élizabel Leblanc

Véronique Daneault

Annie Bernier
email: annie.bernier@umontreal.ca

Socioeconomic status in infancy and the developing brain: Functional connectivity of the hippocampus and amygdala

Keywords: Early environment, Brain development, Parental education, Family income, Resting-state fMRI, Subcortical-cortical functional connectivity 1

The development of the hippocampus and amygdala is particularly sensitive to environmental factors, including socioeconomic status (SES). Studies that have investigated associations between SES and brain development markers have rarely focused on connectivity. Accordingly, this longitudinal study examined whether SES in infancy (parental education and income-to-needs ratio) predicts the functional connectivity of the hippocampus and amygdala in late childhood, and in turn whether functional connectivity is associated with child socioemotional adjustment in a middle-class sample. SES indices were measured when children (N = 28) were 7 months old. When children were 10 years of age, they underwent a resting-state functional magnetic resonance imaging exam, and their school teachers completed a questionnaire assessing child socioemotional adjustment. Wholebrain regression analyses, including left and right hippocampi and amygdalae as seeds and SES indices as predictors, revealed that higher parental education predicted stronger functional connectivity between the left and right hippocampi and the right amygdala with the dorsal anterior cingulate cortex, and between the left amygdala and bilateral angular gyrus, after accounting for child age and sex. In turn, the connectivity of these regions was associated with higher child prosocial behavior. These findings contribute to the emerging literature suggesting that SES is associated with variability in the neural substrates of social abilities in children.

Introduction

Childhood socioeconomic status (SES) is a multidimensional construct that reflects both economic and social resources available to children and their families, and is typically indexed by parental education and family income [1][2][3]. While poverty implies a threshold that dichotomizes families as being poor or not, SES represents a gradient, and indeed the effects of SES on child outcomes operate across the whole SES continuum [2][3][4]. Although many effects of SES on child outcomes are attributable to the lower end of the continuum, small variations of SES within the middle-class range can be meaningfully related to child development [2]. The present report investigates whether variation in SES predicts differences in child brain functional connectivity in a low-risk sample of mostly middle-class families.

SES is associated with early experiences likely to shape child development and brain maturation [1,5]. Children growing up in higher-SES families are more likely to be exposed to stimulating environmental input (e.g., linguistic, cognitive and social stimulation) that is favorable to child development [5,6]. In line with this, numerous studies report that socioeconomic differences in childhood are sources of variation in child development at both the behavioral and neural levels [5][6][7][8].

Early exposure to higher-SES environments is associated with better child affective and cognitive functioning, including fewer behavioral problems, lower rates of psychopathology, and better cognitive performance and socioemotional adjustment [1,[8][9][10][11][12]. Neural differences may underlie these observed behavioral differences. SES-related differences in the morphometry and activity of brain regions important for affective and cognitive processing are reported [see 2,6 for reviews], notably in the hippocampus and the amygdala [13][14][15], two regions centrally involved in socioemotional processes [16,17].

Prior research has provided evidence for the links between SES and individual brain regions, but the relation between SES and brain connectivity is still under-investigated. Yet, complex cognitive and emotional functions depend on the dynamic interplay and exchange of information between distributed brain regions [18][19][20][21]. Brain connectivity development is modulated by early experiences that determine to a large degree which synaptic connections persist and are strengthened [22][23][24].

The development of the hippocampus and amygdala is sensitive to early experiences and environmental factors, including SES [1,16,25]. During childhood, the connectivity networks associated with the hippocampus and the amygdala develop and likely underlie the refinement of child socioemotional functioning [25][26][27][28][29]. Consequently, differences in the connectivity networks of the hippocampus and the amygdala may be a mechanism for SES-related variation in child development.

Two studies report a positive association between childhood SES and structural connectivity in white matter tracts involved in affective and cognitive processes, including the parahippocampal cingulum bundle and the uncinate fasciculus tracts, which connect the hippocampus and the amygdala with the frontal cortex [30,31]. Structural connectivity between brain regions is presumed to provide the anatomical basis for functional interactions between these regions [32,33]. However, the associations between SES and functional connectivity in children are just beginning to receive empirical attention.

The brain is organized into multiple resting-state functional networks that reflect correlated spontaneous activities between two or more brain regions (called resting-state functional connectivity [rsFC]) measured with resting-state functional magnetic resonance imaging (rs-fMRI). RsFC is believed to reflect a history of repeated co-activation between brain regions [18,34,35]. Thus, rsFC reflects functional brain organization and when measured during childhood can provide insight into fundamental organizing principles underlying brain development [35,36].

A longitudinal rs-fMRI study reported that higher SES (i.e., higher family income and greater maternal education) was related to more optimal maturation of cortical functional networks during the first year of life [37]. Another study reported that change in family income during adolescence was associated with changes in rsFC [38]. In a study focused on the connectivity of the amygdala and hippocampus, Barch and colleagues [39] reported that higher family income during early childhood predicted stronger rsFC between these regions and the superior frontal cortex and lingual gyrus, which in turn predicted lower negative mood/depression at school age. In addition, higher family income during late childhood predicts greater connectivity between the hippocampus and the posterior cingulate cortex in early adulthood [40].

The study by Barch and colleagues (2016) indirectly suggested an association between childhood SES, amygdala and hippocampus rsFC and child socioemotional functioning, in that the authors examined psychological dysfunction (i.e., negative mood/depression). However, they focused their analyses on negative mood/depression in a sample enriched for depression symptoms; hence, the generalization of the results to low-risk samples is unknown. In addition, only one index of SES (family income) was used, and the role of other indices (such as parental education) remains unclear. Family income and parental education, although correlated, represent distinct facets of SES and may have different links to childhood experiences and child neural and behavioral development [1,2,14,41,42]. Higher family income is associated with greater access to resources, including more nutritious foods, enriched home learning environments, higher quality child care and safer neighborhoods [5,41]. Independent of family income, more educated parents spend more time with their children, have higher quality parent-child interactions, use more elaborate language, and are more likely to engage in parenting activities that promote socioemotional development [5,41]. Thus, both family income and parental education may influence children's development, including their neural development.

The goal of the present report was to examine whether SES in infancy, indexed by parental education and family income separately, predicts the rsFC of left and right hippocampi and amygdalae in late childhood, and in turn whether differences in rsFC are associated with child socioemotional adjustment in a middle-class sample of typically-developing children. Based upon previous studies [30,31,39], we hypothesized that higher SES (i.e., higher parental education and/or family income) would predict stronger hippocampus and amygdala connectivity with frontal regions. In turn, we expected these patterns of connectivity to be associated with child socioemotional adjustment. As parental education and family income have differential links with child neural and behavioral development [1,41], it was further expected that parental education and family income would predict different patterns of hippocampus and amygdala connectivity (i.e., the brain regions with which the hippocampus and amygdala would be most strongly connected). Given the lack of previous research including the two SES indices, no hypothesis was formulated regarding the exact differences in patterns of connectivity.

Materials and Methods

Participants

Participants included in the present study (n = 28) were followed annually as part of a larger longitudinal prospective cohort project (N = 272 at baseline recruited between 2005 and 2011) investigating early relational predictors of child development [43]. In 2016-2017, all families who were still active in the project and whose child was at least 10 years old (n = 64 -most children were younger) were invited to take part in a magnetic resonance imaging (MRI) exam. Younger children were not invited to participate because we wanted to maintain a narrow age window for the MRI exam to avoid confounds related to ongoing brain development. Here, we report on SES indices (parental education and family income) assessed when children were 7 months of age and rs-fMRI data as well as child socioemotional adjustment measured around 10-11 years of age. The larger project and current study were approved by the local human research ethics committee and all families provided written informed consent for participation.

Families were recruited from birth lists randomly generated by the Ministry of Health and Social Services. Inclusion criteria were full-term pregnancy (i.e., at least 37 weeks) and the absence of any known disability or severe delay in the infant. When children were 10 years of age, they were invited to undergo an MRI exam including an rs-fMRI sequence. Inclusion criteria for the MRI exam were the absence of neurological or psychiatric disorders, traumatic brain injury, psychoactive medication, and standard MRI contraindications. Of the 64 families approached for the MRI exam, 39 (60.94%) agreed to participate. However, four (6.25%) of these children were not eligible due to security reasons (e.g., wearing braces); thus, 35 children underwent the MRI exam (54.69%). There were no sociodemographic differences between the 35 children who underwent the MRI exam and the 29 who did not in terms of family income, parental age, education, ethnicity, language, or child sex (see Table 1, all ps ≥ .21).

Of the 35 children who took part in the MRI exam, four were subsequently excluded because of a diagnosis of anxiety disorder (n = 1) or attention deficit hyperactivity disorder treated with psychoactive medication (n = 3). In addition, the rs-fMRI data of three children were excluded due to significant motion (translation > 2.5 mm or rotation > 2.5 degrees). Therefore, data from 28 children (17 girls and 11 boys, χ 2 (1) = 1.29; p = .26) were used in the analyses. There were no sociodemographic differences between the final sample (n = 28) and the other families (n = 36) in terms of family income, parental age, education, ethnicity, language, or child sex (all ps ≥ .25).

Data collection

SES indices

Parental education and family income were reported by mothers upon enrollment in the larger project, when children were 7 months old (M = 7.04 months, SD = 0.71, range = 6.0 -8.5). For the final sample (n = 28), mean years of education was 15.79 (SD = 1.95, range = 11 -18) for mothers and 15.71 (SD = 1.88, range = 11 -17) for fathers. Parental education (M = 15.75, SD = 1.70, range = 11.5 -17.5) was calculated by averaging both parents' total years of education completed (r = .58, p = .001). In Quebec (Canada), where this study was conducted, a high school degree corresponds to 11 years of education. Thus, all parents in the sample had at least a high school degree.

Family income (M = 4.48, SD = 1.48, range = 1 -6) corresponded to the total annual raw family income on a 1-6 scale: 1 = less than CAD$20,000 (n = 1); 2 = CAD$20,000 to CAD$39,000 (n = 3); 3 = CAD$40,000 to CAD$59,000 (n = 2); 4 = CAD$60,000 to CAD$79,000 (n = 5); 5 = CAD$80,000 to CAD$99,000 (n = 8); 6 = CAD$100,000 and up (n = 8). One family declined to report family income. The income-to-needs ratio (M = 2.37, SD = 0.83, range = 0.7 -3.5) was calculated for each family by dividing reported annual income by the Canadian low income threshold (https://cdn.carra.gouv.qc.ca/général/pages/IN99KXXX00A001.aspx) for a same-size family in the year the data were collected. Because family income was reported in ranges, the median value of each increment was used in the income-to-needs ratio calculation (e.g., CAD$29,500 for the CAD$20,000 to CAD$39,000 category).

Parental education and income-to-needs ratio (r = .40, p = .04) were used as SES indicators in subsequent analyses.

MRI acquisition at 10-11 years

When children were 10-11 years (M = 10.57 years, SD = 0.46, range = 10 -11.67) neuroimaging data were collected using a 32-channel head coil on a Siemens 3-Tesla scanner (MAGNETOM Trio, Siemens, Erlangen, Germany). Structural data were acquired using two high-resolution anatomic sequences: a three-dimensional T1-weighted 4-echo magnetization-prepared rapid gradient-echo sequence and a two-dimensional T2-weighted turbo spin echo sequence (see [44] for parameter details). Then, rs-fMRI data were acquired with a two-dimensional T2-star echo planar image sequence (repetition time: 2430 ms; echo time: 30 ms; flip angle: 70°; 40 slices; slice thickness: 3.5 mm; gap: 3.5 mm; matrix: 64 × 64; field of view: 224 mm; in-plane resolution: 3.50 × 3.50 mm; 120 volumes; duration: 298 sec) during which children were asked to fixate on a dot on a screen. This short 5-minute sequence allowed for a suitable balance between feasibility and reliability of rs-fMRI in children. Short acquisition times such as this are likely to reduce the risk of motion artefacts [45,46], which are especially salient in pediatric imaging [47]. Many of the brain networks that include the hippocampus and amygdala show very good stability even with acquisition times as short as 3 minutes [46].

Child socioemotional adjustment at 10-11 years

Shortly after the MRI exam (M = 10.99 years, SD = 0.32, range = 10.33 -11.68), children's school teachers were asked to complete the Strengths and Difficulties Questionnaire (SDQ; [START_REF] Goodman | The strengths and difficulties questionnaire: A research note[END_REF]. The SDQ is a behavioral screening questionnaire describing positive and negative attributes of children aged between 2 and 17 years. This 25-item questionnaire yields scores for five subscales: (1) Emotional symptoms, (2) Conduct problems, (3) Hyperactivity/inattention, (4) Peer problems, and (5) Prosocial behavior. The respondent rates each item on a 3-point scale (0 = ''not true'', 1 = ''sometimes true'' and 2 = ''certainly true''). In low-risk or general population samples (as in the present study), an alternative three-subscale division of the SDQ is recommended [49]: 'Internalizing roblems' (Emotional symptoms + Peer problems), 'Externalizing problems' (Conduct problems + Hyperactivity/inattention) and 'Prosocial behavior'. Given the small sample size, this three-subscale division is used here, so as to reduce the number of comparisons. A higher score is indicative of more child difficulties, except for the Prosocial behavior subscale for which a higher score is indicative of more prosocial behavior. The score for each subscale is the sum of item scores, generating a subscale score from 0-20 for Internalizing problems and Externalizing problems and from 0-10 for Prosocial behavior (see Table 2 for means, SDs, observed ranges).

Head motion

Motion has deleterious effects on resting-state analyses [45,47]. Thus, the following steps of fMRI data pre-processing and analysis were performed to reduce potential motion effects on the results:

(1) All images were inspected for motion artefacts, and only participants who showed no more than 2.5 mm in translation or 2.5 degrees in rotation were included in the analyses; (2) pre-processing included volumes realignment to correct for head motions, noise-correction using the 'aCompCor' strategy, and a band-pass filter between 0.008 and 0.09 Hz that improves control of motion [50]; and

(3) the first-level models included the six motion parameters (3 translations, 3 rotations) estimated during realignment as nuisance regressors. Note that the mean framewise displacement (M = 0.16 mm, SD = 0.06 [47]) was not significantly correlated with parental education (r = -.18, p = .35), income-to-needs ratio (r = -.13, p = .52) or SDQ scores (Internalizing problems: r = -.03, p = .90; Externalizing problems: r = -.34, p = .12; Prosocial behavior: r = -.05, p = .81;).

Resting-state fMRI pre-processing

After all images were visually inspected for motion artefacts and image quality, data pre-processing was performed with SPM12 (Wellcome Trust Centre for Neuroimaging, London, UK) and the CONN Functional Connectivity SPM Toolbox version 17f [51] running on MATLAB version R2017b (MathWorks, Inc., Natick, MA, USA). Using the SPM platform: (1) the echo planar imaging (EPI) volumes were corrected for slice timing and realigned to the first volume to correct for head motions; (2) the T1, T2 and mean EPI (calculated during realignment) images were co-registered; (3) tissues (grey matter [GM], white matter [WM] and cerebrospinal fluid [CSF]) were segmented and normalized using T1 and T2 images and an age-appropriate stereotaxic template (NIHPD 7.5-13.5 asymmetric: www.bic.mni.mcgill.ca/ServicesAtlases/NIHPD-obj1; Fonov et al., 2011); (4) resulting parameters were applied to normalize the co-registered T1 image and EPI volumes with a voxel size of 2 x 2 x 2 mm; and (5) the normalized EPI images were smoothed at 6 mm full width at halfmaximum (FWHM). Finally, the noise reduction step was performed to remove unwanted motion, physiological and other artefactual effects from the blood-oxygen-level dependent (BOLD) signal using the anatomical principal component-based noise-correction 'aCompCor' strategy [51][52][53] implemented in the CONN toolbox. The aCompCor strategy accounts for motion, physiological and other artefactual effects based on principal components of the signals from the white matter and the cerebrospinal fluid voxels, along with the six motion parameters estimated during realignment. Then, a band-pass filter between 0.008 and 0.09 Hz was applied.

Statistical analyses

ROI-to-Voxels resting-state fMRI analyses

ROI-to-Voxels rsFC analyses were performed using the CONN toolbox. Amygdala and hippocampus ROIs (right and left) were selected from the FSL Harvard-Oxford Atlas maximum likelihood subcortical atlas [54,55] implemented in the CONN toolbox to create connectivity maps between ROIs and all brain voxels.

At the first level of analysis, Pearson's correlations were calculated between the mean BOLD signal time-courses of each ROI and each voxels. This provided a ROI-to-Voxels rsFC map for each participant, in which positive and negative correlation coefficients defined positive and negative rsFC, respectively. The six motion parameters (3 translations, 3 rotations) estimated during the realignment step of pre-processing were added as nuisance regressors. Then, Fisher's r-to-z transformation was applied to allow for parametric testing.

For the second-level analysis, multiple regression was performed between each ROI-to-Voxels rsFC map (left and right hippocampus and amygdala) and each SES index (parental education and incometo-needs ratio) separately, controlling for child age and sex. Analyses were masked using a GM mask based on the means of the GM, WM and CSF normalized images (see [44] for calculation details). Resulting statistical maps were thresholded at p < .001 uncorrected at the voxel-level, and p < .05 corrected by family-wise error rate (FWE) at the cluster level using random-field-theory (RFT) assumptions [56,57]. These thresholds ensure that parametric assumptions are valid and the rate of false positives is acceptable [57,58].

Correlations between functional connectivity and child socioemotional adjustment

To examine the associations between rsFC and child socioemotional adjustment, we performed correlations between relevant connectivity coefficients and SDQ scores. To do so, we extracted the connectivity coefficients in significant clusters (previous analysis) for each participant and computed partial correlations between connectivity coefficients and SDQ scores, adjusting for child age and sex.

Although mediation analysis is the optimal statistical approach to examine the presumed three-way relation between SES, brain connectivity, and child adjustment, the current study is under-powered to properly test for mediation. Accordingly, these preliminary correlational analyses are exploratory and presented in the spirit of encouraging larger-scale hypothesis-driven investigations.

Results

Predicting functional connectivity from parental education

ROI-to-Voxels multiple regression analyses were performed to examine whether parental education predicted rsFC of the bilateral hippocampi and amygdalae at 10 years (Table 3; Fig. 1 and2). The analyses revealed that after accounting for child age and sex, higher parental education was associated with significantly stronger positive connectivity between the bilateral hippocampi and the dorsal anterior cingulate cortex (dACC, also known as the mid-cingulate cortex [MCC]) and between the right amygdala and the dACC. Higher parental education was also associated with significantly stronger negative connectivity between the left amygdala and bilateral angular gyri. The link between parental education and the connectivity of the right hippocampus and the right amygdala with the dACC remained significant when income-to-needs ratio was covaried, whereas the connectivity of the left hippocampus with the dACC and of the left amygdala with the angular gyrus did not.

To test whether the results may have been driven by the three children of parents with lower education (<13 years of education), we conducted a secondary regression analysis between connectivity coefficients extracted in each significant cluster and parental education, adjusting for child age and sex, excluding the data from these three children (Table S1,Fig S1). Higher parental education remained significantly associated with stronger positive connectivity between the bilateral hippocampi and the dorsal anterior cingulate cortex (all βs > .49, ps < .02). Higher parental education also tended to be associated with stronger positive connectivity between the right amygdala and the dACC (β = .39, p = .06; this marginal finding can likely be explained by the decrease in statistical power ensuing from the smaller sample size for these analyses). Overall, these secondary analyses suggest that the findings apply broadly and are not due to a few unusual cases.

On an exploratory basis, we examined the unique contributions of maternal and paternal education to brain connectivity using a secondary ROI-to-Voxels multiple regression with maternal and paternal education entered separately. The connectivity between the right hippocampus and the dorsal anterior cingulate cortex was predicted by maternal education beyond paternal education (Figure S2). There was no significant effect of paternal education beyond maternal education.

Predicting functional connectivity from income-to-needs ratio

ROI-to-Voxels regression analyses were performed to examine whether income-to-needs ratio predicted rsFC of the right and left amygdala and hippocampus at 10 years. The analyses revealed no significant results.

Associations between functional connectivity and child socioemotional adjustment

Partial correlations (adjusted for child age and sex) between functional connectivity coefficients and child socioemotional adjustment indices are displayed in Table 4 (see also Fig. 3). Overall, the results suggest that children with stronger positive connectivity of bilateral hippocampi and the right amygdala with the dACC, and stronger negative connectivity of the left amygdala with the right angular gyrus, displayed higher prosocial behavior.

Discussion/Conclusion

The goal of this report was to investigate whether SES in infancy, namely parental education and family income, predicted rsFC of bilateral hippocampi and amygdalae in late childhood, and whether this connectivity was associated with child socioemotional functioning in a middle-class sample. Although the findings of this small-scale preliminary study require replication in larger samples to be generalized, we observed that higher parental education during infancy predicted stronger rsFC of the hippocampus and the amygdala with the dACC and the angular gyrus in late childhood, which in turn was associated with higher prosocial behavior. No significant findings emerged with family income. These findings are the first to provide evidence of an association between parental education and subcortical-cortical rsFC, and contribute to the emerging literature suggesting that childhood SES is associated with the development of the neural substrates of social abilities in childhood [7].

Higher parental education during infancy predicted stronger positive connectivity of the bilateral hippocampi and the right amygdala with the dACC (also known as the mid-cingulate cortex [MCC]), and stronger negative connectivity between the left amygdala and bilateral angular gyrus at 10 years of age. In adults, the dACC has positive connectivity with the hippocampus and the right amygdala [59-61], and the angular gyrus (inferior parietal lobe) is negatively coupled with the amygdala [29,61]. Subcortical-cortical connectivity undergoes protracted maturation throughout childhood and adolescence characterized by changes in both the valence and the strength of connectivity [28,29]. Specifically, studies report age-related increases in rsFC strength (which becomes more positive with age) between the hippocampus and the dACC, and between the amygdala and the dACC, and reversal in rsFC valence (from positive to negative) between the amygdala and the inferior parietal cortex from childhood to adulthood [28,29]. suggest that changes in amygdala-cortical connectivity are likely to occur at the transition between childhood and adolescence (around age 10 years, age at which MRI acquisition was performed in the present study). Based on adult patterns of connectivity and these developmental trends, our results, although preliminary, appear to indicate that rsFC is more mature in children whose parents have higher levels of education.

The enhanced connectivity of the right hippocampus and the right amygdala with the dACC appears to be robustly associated with parental education, in that these links hold above and beyond family income. In contrast, the connectivity of the left hippocampus with the dACC, and of the left amygdala with the angular gyrus, appears to be associated with the variance shared by parental education and family income, rather than to variance unique to each. Consistent with the multidimensional nature of SES that reflects both economic and social facets of the familial environment and previous findings suggesting that each may bear differently on child development [1,2,14,41,42], the present findings highlight that parental education may have both unique effects on neural development and combined effects with income.

Stronger parental education-related connectivity (i.e., hippocampus-dACC, right amygdala-dACC and left amygdala-right angular connectivity) was associated with higher child prosocial behavior. Prosocial behaviors include voluntary actions that respond to another's need or improve their welfare (i.e., helping, consoling, sharing) and are mutually dependent on socio-cognitive abilities such as empathy and mentalizing [62][63][64][65]. The amygdala, hippocampus, dACC and angular gyrus are interconnected brain regions involved in empathy and mentalizing [64][65][66][67][68]. In older adolescents and young adults, SES has been associated with the neural substrates of socio-cognitive abilities. However, in children, while higher SES is associated with higher performance on mentalizing tasks [69][70][71], no study to date has explored the neural correlates of this association [7]. Thus, the current findings are the first to suggest that higher parental education may promote the development of the neural substrates of child social abilities, with higher parental education predicting stronger rsFC associated with prosocial behavior in 10 year-old children.

Children growing up in higher-SES families are more likely to be exposed to positive early experiences promoting their development and brain maturation [5,6]. More educated parents generally provide richer linguistic, cognitive and social stimulation [5]. Although speculative, it is possible that such stimulation promotes the development of the neural substrates underlying child social abilities through experience-dependent processes that regulate synaptogenesis and synaptic pruning (i.e., micro-mechanisms that underlie the maturation of brain connectivity). Through experiencedependent processes, experience determines the strength and effectiveness of synaptic connections based on their recruitment [24,72]. Parents who have higher levels of education may be better equipped (e.g., access to and tendency to look for educational resources and information) to expose their children to a variety of cognitively and socially stimulating situations. Such situations may increase child prosocial behaviors resulting in frequent coactivation of brain regions involved in prosociality, and in turn, strengthening the connectivity of these regions. Thus, the links between brain connectivity and child prosociality observed here may be bidirectional. Barch and colleagues [39], who reported that higher family income during childhood predicted stronger rsFC of the amygdala and hippocampus, no association between family income and rsFC was found in the present study. This discrepancy may come from the different sample sizes and populations. Barch and colleagues' sample was much larger than the current one (N = 105 vs. 28) and thus provided better-powered analyses. Further, there was more income variability in their sample: about 21% of their families were below the poverty line (as defined by an income-to-needs ratio < 1; income-to-needs ratio range = 0 -3.93) compared to 11% in the current sample (incometo-needs ratio range = 0.7 -3.5). These parameters may have limited our ability to detect family income effects. Age effects may also explain the difference in the findings; Barch and colleagues assessed family income when children were between 3 and 6 years old (vs. 7 months in the present study). Another explanation for the absence of association between family income and rsFC in the present study is that family income has been shown to be volatile and can change substantially from early to late childhood [41]. Consistent with this, a recent study reported that mean 7-year family income during adolescence was not associated with brain rsFC, but change in income across adolescence predicted rsFC [38].

Contrary to

The current results must be interpreted in the context of some limitations. The links between SES and child brain connectivity and socioemotional functioning may be underestimated because of the limited statistical power, as well as the conservative use of covariates and severe p-value thresholds and statistical corrections. Hence, these results should be considered preliminary and examined in larger samples to test their generalizability, notably in samples including parents with very low education levels (all parents in the current sample had at least a high school degree). Although the short 5-minute rs-fMRI sequence used here was chosen to reduce the risk of motion artefacts in this pediatric population and is sufficient to capture connectivity between the hippocampus and the amygdala with the rest of the brain [45,46], the present results need to be replicated with longer rs-fMRI sequences. Although longitudinal, the non-experimental design precludes causal inference and notably does not allow us to rule out that child brain connectivity may be a genetic inheritance from parents. However, a recent study reported that connectivity between the amygdala and the prefrontal cortex is largely influenced by environmental factors [73], suggesting that brain connectivity may be subject to only modest genetic influences. Finally, the lack of direct assessment of stimulating environment input (e.g., linguistic, cognitive and social stimulation) prevented us from testing the hypothesis that richer stimulation, presumably provided by more educated parents, may promote the development of the neural substrates underlying child social abilities. Future studies should assess such aspects of the familial environment to better understand the interactive mechanisms through which parental education may promote child brain development.

Conclusion

This report suggests that higher parental education during infancy predicts more mature subcorticalcortical rsFC in late childhood, which in turn is associated with higher concurrent prosocial behavior in children. These findings are congruent with a growing literature suggesting that childhood SES is associated with the development of the neural substrates of child social abilities [7]. In addition, they re-emphasize the idea that small variations of SES within the normative range meaningfully relate to child neural and socioemotional functioning. Nevertheless, child brain development is influenced by many environmental factors inside and outside the family environment (e.g., siblings, daycare and school experience, peers, etc.), and the present findings should in no way be taken to imply that only higher parental education may promote better brain development. While the present results need be replicated in larger and more diverse samples, this study provides rare evidence that parental education may contribute to the development of rsFC between brain regions important for social abilities in typically developing children.

Appendix 10. Figure Legends

Fig. 1 .

 1 Fig. 1. Parental education during infancy predicts stronger positive functional connectivity between the left and right hippocampi and the right amygdala seeds with the dorsal anterior cingulate cortex (A, B, D), and stronger negative connectivity between the left amygdala and bilateral angular gyrus (C).

Fig. 2 .

 2 Fig. 2. Association between parental education and functional connectivity of the left hippocampus (A), right hippocampus (B), left amygdala (C) and right amygdala (D). Scatter plots are presented for descriptive purposes to illustrate the direction of the association. Abbreviations: dACC, dorsal anterior cingulate cortex; Lamyg, left amygdala; Lang, left angular gyrus; Lhipp, left hippocampus; Ramyg, right amygdala; Rang, right angular gyrus; Rhipp, right hippocampus.

Fig. 3 .

 3 Fig. 3. Scatter plots of partial correlations between functional connectivity and child social-emotional adjustment. Only significant associations are represented. Abbreviations: dACC, dorsal anterior cingulate cortex; Lamyg, left amygdala; Lhipp, left hippocampus; Rang, right angular gyrus; Rhipp, right hippocampus.

Fig. S1 .

 S1 Fig.S1. Association between connectivity coefficients and parental education, adjusting for child age and sex, excluding the data from the three children of parents with lower education (<13 years of education). Scatter plots are presented for descriptive purposes to illustrate the direction of the association. Abbreviations: dACC, dorsal anterior cingulate cortex; Lamyg, left amygdala; Lang, left angular gyrus; Lhipp, left hippocampus; Ramyg, right amygdala; Rang, right angular gyrus; Rhipp, right hippocampus.

Fig. S2 .

 S2 Fig. S2. Maternal education predicts stronger functional connectivity of the right hippocampus with the left superior frontal gyrus (xyz = -14 3 74, k = 432, F = 50.46, p-FWE < .01), the left inferior OPER frontal gyrus (xyz = -48 10 6, k = 325, F = 33.84, p-FEW = .02) and the dorsal anterior cingulate cortex (xyz = 3 8 48, k = 432, F = 31.38, p-FWE < .01).

Figure 3 Figure S2

 3S2 Figure 3

Funding Sources

This work was supported by grants awarded to A. Bernier by the Canadian Institutes of Health Research [MOP-119390], the Social Sciences and Humanities Research Council of Canada [410-2010-1366], and the Fonds de Recherche du Québec -Société et Culture [2012-RP-144923].

None

Supplementary Material

Secondary analyses are presented in supplementary Table S1 andFigures S1 andS2.

Statements

Acknowledgement

The authors wish to acknowledge Natasha Ballen, Marie-Ève Bélanger, Stéphanie Bordeleau, Andrée-Anne Bouvette-Turcot, Catherine Cimon-Paquet, Isabelle Demers, Marie Deschênes, Christine Gagné, Sarah Hertz, Véronique Jarry-Boileau, Jessica Laranjo, Élodie Larose-Grégoire, Nadine Marzougui, Célia Matte-Gagné, Rachel Perrier, Émilie Rochette, Marie-Soleil Sirois, Émilie Tétreault, and Natasha Whipple for help with data collection. The authors want to express special gratitude to the participating families of the Grandir Ensemble project who generously opened their homes to us. The authors thank Andre van der Kouwe from the Massachusetts General Hospital for the use of the MPRAGE 4-echo sequence.

Statement of Ethics

The present study has been approved by the local human research ethics committee and all families have provided written informed consent for participation.

Disclosure Statement

The authors have no conflicts of interest to declare.

Author Contributions


Vitali Vougalter
email: vitali@math.toronto.edu

Vitaly Volpert
email: volpert@math.univ-lyon1.fr

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 with the methods of spectral theory and scattering theory for Schrödinger type operators. One of the main results of the work concerns solvability conditions for the equation -∆u + V (x)u -au = f where a ≥ 0. They are formulated in terms of orthogonality of the function f to the solutions of the homogeneous adjoint equation.

Introduction

Linear elliptic problems in bounded domains with a sufficiently smooth boundary satisfy the Fredholm property if and only if the ellipticity condition, proper ellipticity and the Lopatinskii conditions are satisfied. Fredholm property implies the solvability conditions: the nonhomogeneous operator equation Lu = f is solvable if and only if the right-hand side f is orthogonal to all solutions of the homogeneous adjoint problem L * v = 0. The orthogonality is understood in the sense of duality in the corresponding spaces.

In the case of unbounded domains, one more condition should be imposed in order to preserve the Fredholm property. This condition can be formulated in terms of limiting operators and requires that all limiting operators should be invertible or that the only bounded solution of limiting problems is trivial [START_REF] Volpert | Fredholm property of elliptic operators in unbounded domains[END_REF]. Limiting operators are the operators with limiting values of the coefficients at infinity, if such limiting values exist. Otherwise, limiting coefficients are determined by means of sequences of shifted coefficients and locally convergent subsequences. If we consider for example the operator Lu = -∆uau in R n , where a is a positive constant, then its only limiting operator is the same operator L. Since the limiting equation Lu = 0 has a nonzero bounded solution, then the operator L, considered in Sobolev or in 1 Hölder spaces does not satisfy the Fredholm property. Therefore the solvability conditions are not applicable. However, the particular form of the equation -∆uau = f in R n allows us to apply the Fourier transform and to find its solution. It can be easily verified that it has a solution u ∈ L 2 (R n ) if and only if f (ξ)/(ξ 2a) ∈ L 2 (R n ), where "hat" denotes the Fourier transform. In the other words, the solvability conditions are given by the equality R n e iξx f (x)dx = 0 for any ξ ∈ R n such that |ξ| 2 = a. This means that formally we obtain solvability conditions similar to those for Fredholm operators: the right-hand side is orthogonal to all solutions of the homogeneous formally adjoint problem.

In this example, we are able to obtain solvability conditions due to the fact that the operator has constant coefficients and we can apply the Fourier transform. In general, the question about solvability conditions for non Fredholm operators is open and represents one of the major challenges in the theory of elliptic problems. Some classes of reaction-diffusion operators without Fredholm property can be studied by the introduction of weighted spaces [START_REF] Volpert | Fredholm property of elliptic operators in unbounded domains[END_REF] or reducing them to integro-differential operators [START_REF] Ducrot | Systemes de réaction-diffusion sans propriété de Fredholm[END_REF], [START_REF] Ducrot | Reaction-diffusion problems with non Fredholm operators[END_REF]. Other types of solvability conditions, different from the usual orthogonality conditions, are obtained for some second order operators on the real axis or in cylinders [START_REF] Kryzhevich | Different types of solvability conditions for differential operators[END_REF]. Some elliptic problems in R 2 are studied in [START_REF] Volpert | Solvability conditions for elliptic problems with non-Fredholm operators[END_REF] where the solvability conditions are obtained with the help of space decomposition of the operators.

A special class of elliptic operators in R n , A = A ∞ + A 0 , where A ∞ is a homogeneous operator with constant coefficients and A 0 is an operator with rapidly decaying coefficients is studied in specially chosen spaces with a polynomial weight. The finiteness of the kernel is proved in [W1], [NW], their Fredholm property in [W2], [L], [LM] in the case of weighted Sobolev spaces and in [B], [BP] for weighted Hölder spaces. The Fredholm property and the index of such operators are determined by their principal part A ∞ . The operator A 0 does not change them due to the rapid decay of the coefficients. Laplace operator in exterior domains is studied in [AB].

In this work we consider two classes of non Fredholm operators and establish the solvability conditions for the equations involving them. The methods cited above are not applicable here and we develop some new approaches. In the first case we study the operator

H a on L 2 (R 3 ), such that H a u = -∆u + V (x)u -au
where a ≥ 0 is a parameter, the potential V (x) decays to zero as x → ∞. We investigate the conditions on the function f ∈ L 2 (R 3 ) under which the equations

H a u = f (1.1)
and

H 0 u = f, (1.2)
the second one is the limiting case of the first one as a → 0, have the unique solution in L 2 (R 3 ). Since the potential equals zero at infinity, the operator H a has a unique limiting operator Lu = -∆uau, which is the same as discussed above. The limiting problem Lu = 0 has nonzero bounded solutions. Therefore, the operator H a , a ≥ 0 does not satisfy the Fredholm property, and the solvability of equations (1.1) and (1.2) is not known. The coefficients of the operators are not constant any more and we cannot simply apply the Fourier transform as in the example above. We will use the spectral decomposition of selfadjoint operators. We note that in the case where a = 0 and the potential is rapidly decaying at infinity, the operator H 0 belong to the class of operators A ∞ + A 0 discussed above. The results of this work differ from the results in the cited papers. We do not work in the weighted spaces and obtain solvability conditions without proving the Fredholm property which may not hold. However, more important difference is that we consider also the case a > 0. It is essentially different and the previous methods are not applicable. To the best of our knowledge, solvability conditions for equation (1.1) with a > 0 and n ≥ 2 were not obtained before. The solvability conditions are formulated in terms of the orthogonality of the righthand side f to all solutions of the homogeneous adjoint equation H a v = 0 (the operator is self-adjoint).

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 the 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 and 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 )
with the constant c HLS given on p.98 of [START_REF] Lieb | Loss: "Analysis[END_REF]. In our notations (f

1 (x), f 2 (x)) L 2 (R 3 ) := R 3 f 1 (x) f2 (x)
dx and for a vector function A(x) = (A 1 (x), A 2 (x), A 3 (x)) 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 the same notation will be used even if the functions above are not square integrable, like the so called perturbed plane waves ϕ k (x) which are normalized to a delta function (see the equation (2.1) in the Section 2). We make the following technical assumption.

Assumption 1.1. The potential function V (x) : R 3 → R is continuous and satisfies the bound |V (x)| ≤ C 1 + |x| 3.5+ε with some ε > 0 and x ∈ 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 and √ c HLS V L 3 2 (R 3 ) < 4π
The function f (x) ∈ L 2 (R 3 ) and |x|f (x) ∈ L 1 (R 3 ).

Here and further down C stands for a finite positive constant. Since under our assumptions on the potential the essential spectrum σ ess (H a ) of the Schrödinger type operator H a = H 0a fills the interval [-a, ∞) (see e.g. [JMST] ), the Fredholm alternative theorem fails to work in this case. The problem can be easily handled by the method of the Fourier transform in the absence of the potential term V (x). We show that this method can be generalized in the presence of a shallow, short-range V (x) by means of replacing the Fourier harmonics by the functions ϕ k (x), k ∈ R 3 of the continuous spectrum of the operator H 0 , which are the solutions of the Lippmann-Schwinger equation (see (2.1) in Section 2 and the explicit formula (2.2)).

While the wave vector k attains all the possible values in R 3 , the function ϕ 0 (x) corresponds to k = 0 in the formulas (2.1) and (2.2). 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 | stands for its Lebesgue measure. Our first main result is as follows.

Theorem 1. Let the Assumption 1.1 hold. Then a) The problem (1.1) admits a unique solution u ∈ L 2 (R 3 ) if and only if

(f (x), ϕ k (x)) L 2 (R 3 ) = 0 f or k ∈ S 3 √ a a.e. b) The problem (1.2) has a unique solution u ∈ L 2 (R 3 ) if and only if (f (x), ϕ 0 (x)) L 2 (R 3 ) = 0
In the second part of the article we consider the operator L = -∆ x -∆ y + V(y) on L 2 (R n+m ) with the Laplacian operators ∆ x and ∆ y in x = (x 1 , x 2 , ..., x n ) ∈ R n , y = (y 1 , y 2 , ..., y m ) ∈ R m and prove the necessary and sufficient conditions for the solvability in L 2 (R n+m ) of the inhomogeneous problem Lu = g(x, y),

(1.3) where g(x, y) ∈ L 2 (R n+m ). We assume the following.

Assumption 1.2. The function V(y) : R m → R is continuous and lim y→∞ V(y) = V + > 0.

Thus for the operator h := -∆ y + V(y) 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 ϕ k j , such that

hϕ k j = e j ϕ k j , 1 ≤ k ≤ m j , (ϕ k i , ϕ l j ) L 2 (R m ) = δ i,j δ k,l
, where m j stands for the eigenvalue multiplicity, which is finite since the essential spectrum starts only at V + and δ i,j for the Kronecker symbol. We make the following key assumption on the discrete spectrum of the operator h relevant to the problem (1.3).

Assumption 1.3. The eigenvalues e j < 0 for all 1 ≤ j ≤ N -1 and e N = 0. Thus under our assumptions the operator L is not Fredholm. Zero is the bottom of the essential spectrum of the operator -∆ x and h has the square integrable zero modes. Moreover, the operator h has the negative eigenvalues e j , j = 1, ..., N -1 and -∆ x the Fourier harmonics e ipx (2π)

n 2 , such that p ∈ S n √ -e j
. However, the equation (1.3) can be solved on the proper subspace and the orthogonality conditions will strongly depend on the dimensions of the problem.

Let us introduce the following subspace weighted in the first variable for the right side of the equation (1.3).

L 2 α, x = {g(x, y) : g(x, y) ∈ L 2 (R n+m ) and |x| α 2 g(x, y) ∈ L 2 (R n+m )}, α > 0 (1.4)
Our second main result is as follows.

Theorem 2. Let the Assumptions 1.2 and 1.3 hold. Then for the equation ( 1.3):

I) When n = 1 and g(x, y) ∈ L 2 α,
x for some α > 5 there exists a unique solution u ∈ L 2 (R 1+m ) if and only if:

(g(x, y), ϕ k N (y)) L 2 (R 1+m ) = 0, (g(x, y), xϕ k N (y)) L 2 (R 1+m ) = 0, 1 ≤ k ≤ m N and (g(x, y), e ±i √ -e j x √ 2π ϕ k j (y)) L 2 (R 1+m ) = 0, 1 ≤ j ≤ N -1, 1 ≤ k ≤ m j II) When n = 2 such that x = (x 1 , x 2 ) ∈ R 2 and g(x, y) ∈ L 2 α,
x for some α > 6 there exists a unique solution u ∈ L 2 (R 2+m ) if and only if:

(g(x, y), ϕ k N (y)) L 2 (R 2+m ) = 0, (g(x, y), x i ϕ k N (y)) L 2 (R 2+m ) = 0, i = 1, 2, 1 ≤ k ≤ m N and (g(x, y), e ipx 2π ϕ k j (y)) L 2 (R 2+m ) = 0, a.e. p ∈ S 2 √ -e j , 1 ≤ j ≤ N -1, 1 ≤ k ≤ m j III) When n = 3, 4 and g(x, y) ∈ L 2 α,
x for some α > n + 2 there exists a unique solution u ∈ L 2 (R n+m ) if and only if:

(g(x, y), ϕ k N (y)) L 2 (R n+m ) = 0, 1 ≤ k ≤ m N and (g(x, y), e ipx (2π) n 2 ϕ k j (y)) L 2 (R n+m ) = 0, a.e. p ∈ S n √ -e j , 1 ≤ j ≤ N -1, 1 ≤ k ≤ m j IV) When n ≥ 5 and g(x, y) ∈ L 2 α,
x for some α > n + 2 there exists a unique solution u ∈ L 2 (R n+m ) if and only if:

(g(x, y), e ipx (2π) n 2 ϕ k j (y)) L 2 (R n+m ) = 0, a.e. p ∈ S n √ -e j , 1 ≤ j ≤ N -1, 1 ≤ k ≤ m j
Proving solvability conditions for linear elliptic problems with non-Fredholm operators plays the crucial role in various applications including those to travelling wave solutions of reaction-diffusion systems (see [START_REF] Volpert | Solvability conditions for elliptic problems with non-Fredholm operators[END_REF]). Let us first establish several important properties for the functions of the spectrum of the Schrödinder operator in the left side of the equation (1.1) and for the related quantities.

Spectral properties of the operator H 0 and proof of Theorem 1

The functions of the continuous spectrum satisfy 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 (2.1)
and the orthogonality relations

(ϕ k (x), ϕ q (x)) L 2 (R 3 ) = δ(k -q), k, q ∈ R 3 . We 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 )
Let us show that the norm of the operator

Q : L ∞ (R 3 ) → L ∞ (R 3 ) denoted as Q ∞ is small
when the potential V (x) satisfies our assumptions. We prove the following lemma.

Lemma 2.1. Let the Assumption 1.1 hold. Then ||Q|| ∞ < 1.

Proof.

Clearly

Q ∞ ≤ sup x∈R 3 1 4π R 3 |V (y)| |x -y| dy
The expression involved in the right side of the inequality above can be written as

1 4π (χ {|x|≤R} 1 |x| ) * |V (x)| + 1 4π (χ {|x|>R} 1 |x| ) * |V (x)|
with some R > 0 and χ denoting the characteristic function of the correspondent set. This can be estimated above using the Young's inequality as

1 4π V L ∞ (R 3 ) R 0 4πrdr + 1 4π χ {|x|>R} 1 |x| L 4 (R 3 ) V L 4 3 (R 3 ) = = 1 2 V L ∞ (R 3 ) R 2 + 1 (4π) 3 4 V L 4 3 (R 3 ) R -1 4
We optimize the right side of the equality above over R. The minimum occurs when

R = V L ∞ (R 3 ) (4π) 3 4 4 V L 4 3 (R 3 ) -4 9 , such that Q ∞ ≤ 4 1 9 9 8 (4π) -2 3 V 1 9 L ∞ (R 3 ) V 8 9 L 4 3 (R 3 )
, which is k-independent. The Assumption 1.1 yields the statement of the Lemma. Note that V ∈ L Corollary 2.2. Let the Assumption 1.1 hold. Then the functions of the continuous spectrum of the operator

H 0 are ϕ k (x) ∈ L ∞ (R 3 ) for all k ∈ R 3 , such that ϕ k (x) L ∞ (R 3 ) ≤ 1 1 -Q ∞ 1 (2π) 3 2 , k ∈ R 3
Proof. By means of the Lippmann-Schwinger equation (2.1) and the fact that Q ∞ < 1 the functions can be expressed as

ϕ k (x) = (I -Q) -1 e ikx (2π) 3 2 , k ∈ R 3 (2.2)
The Lemma 2.1 yields the bound on the operator norm (

I -Q) -1 ∞ ≤ 1 1 -Q ∞ .
The following elementary lemma shows that in our problem the operator H 0 possesses the spectrum analogous to the one of the minus Laplacian and therefore only the functions ϕ k (x), k ∈ R 3 are needed to be taken into consideration.

Lemma 2.3. Let the Assumption 1.1 be true. Then the operator H 0 is unitarily equivalent to -∆ on L 2 (R 3 ).

Proof. By means of the Hardy-Littlewood-Sobolev inequality (see e.g. p.98 [START_REF] Lieb | Loss: "Analysis[END_REF]) and the Assumption 1.1 we have

R 3 R 3 |V (x)||V (y)| |x -y| 2 dxdy ≤ c HLS V 2 L 3 2 (R 3 ) < (4π) 2
The left side of the inequality above is usually referred to as the Rollnik norm (see e.g. [S71]) and the upper bound we obtained on it is the sufficient condition for the operator

H 0 = -∆ + V (x) on L 2 (R 3
) to be self-adjoint and unitarily equivalent to -∆ via the wave operators (see e.g. [K65], also [START_REF] Rodnianski | Time decay for solutions of Schrödinger equations with rough and time-dependent potentials[END_REF]) given by

Ω ± := s -lim t→∓∞ e it(-∆+V ) e it∆
where the limit is 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).

By means of the spectral theorem for the self-adjoint operator H 0 any function ψ(x) ∈ L 2 (R 3 ) can be expanded through the functions ϕ k (x), k ∈ R 3 forming the complete system in L 2 (R 3 ). The generalized Fourier transform with respect to these functions is being denoted as ψ(k

) := (ψ(x), ϕ k (x)) L 2 (R 3 ) , k ∈ R 3 (2.3)
We prove the following technical estimate concerning the above mentioned generalized Fourier transform for the right side of the equations (1.1) and (1.2).

Lemma 2.4. Let the Assumption 1.1 hold. Then

∇ k f (k) ∈ L ∞ (R 3 ) Proof. Obviously ∇ k f (k) = (f (x), ∇ k ϕ k (x)) L 2 (R 3
) . From the Lippmann-Schwinger equation (2.1) we easily obtain

∇ k ϕ k = e ikx (2π) 3 2 ix + (I -Q) -1 Q e ikx (2π) 3 2 ix + (I -Q) -1 (∇ k Q)(I -Q) -1 e ikx (2π) 3 2 , (2.4)
where

∇ k Q : L ∞ (R 3 ) → L ∞ (R 3 ; C 3
) stands for the operator with the integral kernel

∇ k Q(x, y, k) = - i 4π e i|k||x-y| k |k| V (y)
An elementary computation shows that its norm

∇ k Q ∞ ≤ 1 4π V L 1 (R 3 ) < ∞
due to the rate of decay of the potential V (x) given explicitly by the Assumption 1.1. It is clear from the identity (2.4) that we need to show the boundedness in the k-space of the three terms. The first one is

T 1 (k) := (f (x), e ikx (2π) 3 2 ix) L 2 (R 3 ) , such that |T 1 (k)| ≤ 1 (2π) 3 2
xf L 1 (R 3 ) < +∞ by the Assumption 1.1. The second term to be estimated is

T 2 (k) := (f (x), (I -Q) -1 Q e ikx (2π) 3 2 ix) L 2 (R 3 ) Thus |T 2 (k)| ≤ 1 (2π) 3 2 f L 1 (R 3 ) 1 1 -Q ∞ Qe ikx x L ∞ (R 3 ) . Note that f (x) ∈ L 1 (R 3
) by means of the Assumption 1.1 and Fact 1 of the Appendix. Using the definition of the operator Q along with the Young's inequality we have the upper bound

|Qe ikx x| ≤ 1 4π R 3 |V (y)||y| |x -y| dy = 1 4π {(χ {|x|≤1} 1 |x| ) * |V (x)||x| + (χ {|x|>1} 1 |x| ) * |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
) due to the explicit rate of decay of the potential V (x) stated in the Assumption 1.1. Therefore, T 2 (k) ∈ L ∞ (R 3 ). We complete the proof of the lemma with the estimate on the remaining term

T 3 (k) := (f (x), (I -Q) -1 (∇ k Q)(I -Q) -1 e ikx (2π) 3 2 ) L 2 (R 3 ) ,
such that we easily arrive at the k-independent upper bound

|T 3 (k)| ≤ 1 4π(2π) 3 2 f L 1 (R 3 ) 1 (1 -Q ∞ ) 2 V L 1 (R 3 ) < ∞
Armed with the auxiliary lemmas established above we proceed to prove the first theorem.

Proof of Theorem 1. First of all if the equation (1.1) admits two solutions

u 1 (x), u 2 (x) ∈ L 2 (R 3 ) their difference v(x) := u 1 (x) -u 2 (x) would satisfy the homogeneous problem H a v = 0.
Since the operator H a possesses no nontrivial square integrable zero modes, v(x) will vanish a.e. The analogous argument holds for the solutions of the equation (1.2). From the equation (1.1) by applying the transform (2.3) we obtain

ũ(k) = f (k) k 2 -a , k ∈ R 3 ,
which is convenient to write as the sum of the singular and the nonsingular parts

ũ(k) = f (k) k 2 -a χ Aσ + f (k) k 2 -a χ A c σ , (2.5)
where χ Aσ is the characteristic function of the spherical layer

A σ := {k ∈ R 3 : √ a -σ ≤ |k| ≤ √ a + σ} , 0 < σ < √ a
and χ A c σ of the layer's complement in the three-dimensional k-space. For the second term in the right side of the identity (2.5)

f (k) k 2 -a χ A c σ ≤ | f (k)| √ aσ ∈ L 2 (R 3 )
To estimate the remaining term we will make use of the identity

f (k) = f ( √ a, ω) + |k| √ a ∂ f (|s|, ω) ∂|s| d|s|
Here and further ω stands for the angle variables on the sphere and dω will denote integration with respect to these variables. Thus we can split the first term in the right side of (2.5) as ũ1 (k) + ũ2 (k), where

ũ1 (k) = |k| √ a ∂ f (|s|,ω) ∂|s| d|s| k 2 -a χ Aσ , ũ2 (k) = f ( √ a, ω) k 2 -a χ Aσ (2.6)
Clearly, we have the bound

|ũ 1 (k)| ≤ ∇ k f (k) L ∞ (R 3 ) |k| + √ a χ Aσ ∈ L 2 (R 3 )
by means of Lemma 2.4. We complete the proof of the Part a) of the theorem by estimating the norm

ũ2 (k) 2 L 2 (R 3 ) = √ a+σ √ a-σ d|k| |k| 2 (|k| - √ a) 2 (|k| + √ a) 2 S 3 dω| f( √ a, ω)| 2 < ∞ if and only if (f (x), ϕ k (x)) L 2 (R 3 ) = 0 for k a.
e. on the sphere S 3 √ a . Then we turn our attention to the equation (1.2) by applying to it the generalized Fourier transform with respect to the eigenfunctions of the continuous spectrum of the operator H 0 , which yields

ũ(k) = f (k) k 2 = f (k) k 2 χ {|k|≤1} + f (k) k 2 χ {|k|>1} Clearly f (k) k 2 χ {|k|>1} ≤ | f (k)| ∈ L 2 (R 3
). We use the formula

f (k) = f(0) + |k| 0 ∂ f (|s|, ω) ∂|s| d|s| with f (0) = (f (x), ϕ 0 (x)) L 2 (R 3
) and ϕ 0 (x) is given by (2.2) with k = 0. Hence

|k| 0 ∂ f (|s|,ω) ∂|s| d|s| k 2 χ {|k|≤1} ≤ ∇ k f (k) L ∞ (R 3 ) χ {|k|≤1} |k| ∈ L 2 (R 3 )
via Lemma 2.4. Therefore it remains to estimate the norm f (0)

k 2 χ {|k|≤1} 2 L 2 (R 3 ) = 4π 1 0 d|k| | f(0)| 2 |k| 2 < ∞
if and only if (f (x), ϕ 0 (x)) L 2 (R 3 ) = 0 which completes the proof of the theorem.

Note that if we let the potential function V (x) in the statement of the Theorem 1 vanish, we obtain precisely the usual orthogonality conditions in terms of the Fourier harmonics.

In the following chapter we prove the Theorem 2. As distinct from the first example the dimensions of the problem are not fixed anymore and we show how robust the dependence of the solvability conditions on these dimensions can be.

Spectral properties of the operator L and proof of Theorem 2

Let P ± and P 0 be the orthogonal projections onto the positive, negative and zero subspaces of the operator h. Applying these operators to both sides of the equation (1.3) via the spectral theorem we relate the problem to the equivalent system of the three following equations.

L + u + = g + , (3.1) L -u -= g -, (3.2) and L 0 u 0 = g 0 , (3.3)
where the operators L ± = P ± LP ± and L 0 = P 0 LP 0 act on the functions u ± = P ± u and u 0 = P 0 u respectively and the right sides of the equations above are g ± = P ± g and g 0 = P 0 g. Without loss of generality we can assume that

g 0 (x, y) = v 0 (x)ϕ 1 N (y) , (3.4) where v 0 (x) = (g 0 , ϕ 1 N ) L 2 (R m ) = (g, ϕ 1 N ) L 2 (R m ) .
Let us first turn our attention to the equation (3.1). We have the following lemma.

Lemma 3.1 The equation (3.1) possesses a solution u

+ ∈ L 2 (R n+m ), n ∈ N, m ∈ N.
Proof. By means of the orthogonal decomposition of the right side of the equation (1.3) g = g + + g 0 + g -we have the estimate

g + L 2 (R n+m ) ≤ g L 2 (R n+m )
The lower bound in the sense of the quadratic forms

L + ≥ P + hP + ≥ e N +1 > 0 ,
where e N +1 is either the bottom of the essential spectrum V + of the operator h or its lowest positive eigenvalue, whichever is smaller. Thus L + is the self-adjoint operator on the product of spaces L 2 (R n ) and the range Ran(P + ) such that the bottom of its spectrum is located above zero. Therefore it is invertible and the norm of the inverse L -1

+ : L 2 (R n ) ⊗ Ran(P + ) → L 2 (R n+m ) is bounded above by 1 e N +1
. Thus the equation (3.1) has the solution u + = L -1 + g + and its norm

u + L 2 (R n+m ) ≤ 1 e N +1 g L 2 (R n+m ) < ∞
Let us turn our attention to the analysis of the solvability conditions for the equation

(3.3) which is equivalent to (-∆ x )u 0 = g 0 (3.5)
The solution of this Poisson equation can be expressed as

u 0 = g 0 p 2 χ 1 + g 0 p 2 χ 1 c , (3.6)
where χ 1 stands for the characteristic function of the unit ball in the Fourier space centered at the origin and χ 1 c for the characteristic function of its complement. Here and below the hat symbol stands for the Fourier transform in the first variable, such that

ψ(p) := 1 (2π) n 2 R n ψ(x)e -ipx dx
The second term in the right side of (3.6) is square integrable for all dimensions n, m ∈ N since g 0 ∈ L 2 (R n+m ) and 1 p 2 is bounded away from the origin. Thus it remains to analyze the first term. We have the following lemma when the dimension n = 1. Lemma 3.2 Let the assumptions of the Theorem 2 hold. Then the equation (3.5) possesses a solution u 0 ∈ L 2 (R 1+m ), m ∈ N if and only if

(g(x, y), ϕ k N (y)) L 2 (R 1+m ) = 0, (g(x, y), ϕ k N (y)x) L 2 (R 1+m ) = 0, 1 ≤ k ≤ m N
Proof. We will make use of the following representation

g 0 (p, y) = g 0 (0, y) + ∂ ∂p g 0 (0, y)p + p 0 s 0 ∂ 2 ∂q 2 g 0 (q, y)dq ds , where ∂ 2 ∂p 2 g 0 (p, y) = - 1 √ 2π +∞ -∞ g 0 (x, y)e -ipx x 2 dx
Hence the first term in the right side of (3.6) in our case equals to

g 0 (0, y) p 2 χ 1 + ∂ ∂p g 0 (0, y) χ 1 p + p 0 s 0 ∂ 2 ∂q 2 g 0 (q, y)dq ds χ 1 p 2 (3.7)
Clearly we have the upper bound

∂ 2 ∂q 2 g 0 (q, y) ≤ 1 √ 2π +∞ -∞ |g 0 (x, y)|x 2 dx
By means of the Schwarz inequality and (3.4) we have the estimate valid in a space of arbitrary dimensions

|g 0 (x, y)| ≤ R m |g(x, z)| 2 dz|ϕ 1 N (y)| , x ∈ R n , y ∈ R m , n, m ≥ 1 (3.8) which yields ∂ 2 ∂q 2 g 0 (q, y) ≤ 1 √ 2π +∞ -∞ dx x 2 1 + |x| α 1 + |x| α R m |g(x, s)| 2 ds|ϕ 1 N (y)|
with α > 5 such that g(x, y) ∈ L 2 α, x . The Schwarz inequality yields the upper bound

1 √ 2π +∞ -∞ dx x 4 1 + |x| α g 2 L 2 (R 1+m ) + |x| α 2 g 2 L 2 (R 1+m ) |ϕ 1 N (y)| = C|ϕ 1 N (y)|
Therefore, for the last term in (3.7) we obtain p 0 s 0 ∂ 2 ∂q 2 g 0 (q, y)dq ds

χ 1 p 2 ≤ C 2 |ϕ 1 N (y)|χ 1 ∈ L 2 (R 1+m )
Due to the behavior in the Fourier space of the first two terms, the expression (3.7) belongs to L 2 (R 1+m ) if and only if

g 0 (0, y) = 0, ∂ ∂p g 0 (0, y) = 0 a.e. ,
which is equivalent to

(g(x, y), ϕ k N (y)) L 2 (R 1+m ) = 0, (g(x, y), ϕ k N (y)x) L 2 (R 1+m ) = 0, 1 ≤ k ≤ m N
When the dimension n = 2 we come up with the following analogous statement.

Lemma 3.3 Let the assumptions of the Theorem 2 hold. Then the equation (3.5) possesses a solution u 0 ∈ L 2 (R 2+m ), m ∈ N if and only if

(g(x, y), ϕ k N (y)) L 2 (R 2+m ) = 0, (g(x, y), ϕ k N (y)x i ) L 2 (R 2+m ) = 0, i = 1, 2, 1 ≤ k ≤ m N Proof.
Let us use the expansion analogous to the one we had for proving the previous lemma.

g 0 (p, y) = g 0 (0, y) + ∂ ∂|p| g 0 (0, θ p , y)|p| + |p| 0 s 0 ∂ 2 ∂|q| 2 g 0 (|q|, θ p , y)d|q| ds , with g 0 (|p|, θ p , y) = 1 2π R 2 g 0 (x, y)e -i|p||x|cosθ dx
and the angle between the p = (|p|, θ p ) and x = (|x|, θ x ) vectors on the plane is θ = θ pθ x .

Therefore the first term in the right side of (3.6) when n = 2 equals to

g 0 (0, y) p 2 χ 1 + ∂ ∂|p| g 0 (0, θ p , y) χ 1 |p| + |p| 0 s 0 ∂ 2 ∂|q| 2 g 0 (|q|, θ p , y)d|q| ds χ 1 p 2 (3.9) Obviously ∂ 2 ∂|q| 2 g 0 ≤ 1 2π R 2 |g 0 (x, y)||x| 2 dx
Using the estimate (3.8) we arrive at the upper bound for the right side of this inequality

1 2π |ϕ 1 N (y)| R 2 dx |x| 2 1 + |x| α 1 + |x| α R m |g(x, z)| 2 dz with α > 6 such that g(x, y) ∈ L 2 α, x
. By means of the Schwarz inequality we estimate this from above as

1 √ 2π |ϕ 1 N (y)| ∞ 0 d|x| |x| 5 1 + |x| α g 2 L 2 (R 2+m ) + |x| α 2 g 2 L 2 (R 2+m ) = C|ϕ 1 N (y)|
Therefore, for the last term in (3.9) we arrive at

χ 1 p 2 |p| 0 s 0 ∂ 2 ∂|q| 2 g 0 (|q|, θ p , y)d|q| ds ≤ C 2 χ 1 |ϕ 1 N (y)| ∈ L 2 (R 2+m )
A simple computation using the Fourier transform yields

∂ ∂|p| g 0 (0, θ p , y) = - i 2π R 2 g 0 (x, y)|x|cosθdx = Q 1 (y)cosθ p + Q 2 (y)sinθ p , where Q 1 (y) := - i 2π R 2 g 0 (x, y)x 1 dx , Q 2 (y) := - i 2π R 2 g 0 (x, y)x 2 dx and x = (x 1 , x 2 ) ∈ R 2 .
Computing the square of the L 2 (R 2+m ) norm of the first two terms of (3.9) we arrive at

2π 1 0 d|p| |p| 3 R m dy| g 0 (0, y)| 2 + π 1 0 d|p| |p| R m (|Q 1 (y)| 2 + |Q 2 (y)| 2 )dy ,
which is finite if and only if the quantities g 0 (0, y), Q 1 (y) and Q 2 (y) vanish a.e. This is equivalent to the orthogonality conditions

(g(x, y), ϕ k N (y)) L 2 (R 2+m ) = 0, (g(x, y), x 1 ϕ k N (y)) L 2 (R 2+m ) = 0, (g(x, y), x 2 ϕ k N (y)) L 2 (R 2+m ) = 0, with 1 ≤ k ≤ m N .
Let us investigate how the situation with solvability conditions differs in dimensions n = 3, 4.

Lemma 3.4 Let the assumptions of the Theorem 2 hold. Then the equation ( 3

.5) pos- sesses a solution u 0 ∈ L 2 (R n+m ), n = 3, 4, m ∈ N if and only if (g(x, y), ϕ k N (y)) L 2 (R n+m ) = 0, n = 3, 4, 1 ≤ k ≤ m N Proof.
Let us expand as

g 0 (p, y) = g 0 (0, y) + |p| 0 ∂ ∂|s| g 0 (|s|, ω, y)d|s|
Thus by means of (3.6) we need to estimate

χ 1 p 2 [ g 0 (0, y) + |p| 0 ∂ ∂|s| g 0 (|s|, ω, y)d|s|] (3.10)
By means of the Fourier transform

∂ ∂|p| g 0 (|p|, ω, y) = -i (2π) n 2 R n g 0 (x, y)e -i|p||x|cosθ |x|cosθdx ,
where θ is the angle between p and x in R n . Using (3.8) along with the Schwarz inequality and α > n + 2 such that g(x, y) ∈ L 2 α, x we easily obtain

∂ ∂|s| g 0 ≤ 1 (2π) n 2 R n dx|x| R m |g(x, z)| 2 dz|ϕ 1 N (y)| ≤ ≤ 1 (2π) n 2 ∞ 0 |S n | |x| n+1 1 + |x| α d|x| g 2 L 2 (R n+m ) + |x| α 2 g 2 L 2 (R n+m ) |ϕ 1 N (y)| = C|ϕ 1 N (y)| ,
which implies the bound

χ 1 p 2 |p| 0 ∂ ∂|s| g 0 (|s|, ω, y)d|s| ≤ C χ 1 |p| |ϕ 1 N (y)| ∈ L 2 (R n+m ), n = 3, 4
We finalize the proof of the lemma by estimating the square of the L 2 norm of the first term in (3.10).

|S n | R m dy| g 0 (0, y)| 2 1 0 d|p||p| n-5 < ∞, n = 3, 4
if and only if g 0 (0, y) = 0 a.e., which is equivalent to

(g(x, y), ϕ k N (y)) L 2 (R n+m ) = 0, n = 3, 4, 1 ≤ k ≤ m N
Thus it remains only to establish the orthogonality conditions in dimensions five and higher in the x-variable under which the equation (3.5) admits a square integrable solution.

Lemma 3.5 Let the assumptions of the Theorem 2 hold. Then the equation (3.5) possesses a solution u 0 ∈ L 2 (R n+m ), n ≥ 5, m ∈ N.

Proof. We estimate the Fourier transform using the bound (3.8) along with the Schwarz inequality and α > n + 2 such that g(x, y) ∈ L 2 α, x .

| g 0 (p, y)| ≤ 1 (2π) n 2 R n |g 0 (x, y)|dx ≤ |ϕ 1 N (y)| (2π) n 2 R n dx R m |g(x, z)| 2 dz ≤ ≤ |ϕ 1 N (y)| (2π) n 2 R n dx 1 + |x| α R n dx(1 + |x| α ) R m |g(x, z)| 2 dz = = |ϕ 1 N (y)| (2π) n 2 ∞ 0 d|x| |x| n-1 1 + |x| α |S n | g 2 L 2 (R n+m ) + |x| α 2 g 2 L 2 (R n+m ) = = C|ϕ 1 N (y)|, n ≥ 5, m
∈ N This enables us to obtain the bound on the square of the L 2 norm of the first term in the right side of (3.6).

R n dp R m dy | g 0 | 2 |p| 4 χ 1 ≤ C 1 0 d|p||p| n-5 |S n | R m |ϕ 1 N (y)| 2 dy < ∞
which completes the proof of the lemma.

We proceed with establishing the conditions under which the equation (3.2) admits a square integrable solution. Let {P -,j } N -1 j=1 be the orthogonal projections onto the subspaces correspondent to {e j } N -1 j=1 , the negative eigenvalues of the operator h, such that

P -= N -1 j=1 P -,j , P -,j P -,k = P -,j δ j,k , 1 ≤ j, k ≤ N -1
Applying these projection operators to both sides of the equation (3.2) and using the orthogonal decompositions u -=

N -1 j=1 u -,j and g -=

N -1 j=1 g -,j with P -,j u -= u -,j and P -,j g -= g -,j we easily obtain the system of equations equivalent to (3.2).

[-∆ x -∆ y + V(y)]u -,j = g -,j , 1 ≤ j ≤ N -1 (3.11)

Without loss of generality we can assume that

g -,j (x, y) = v j (x)ϕ 1 j (y), 1 ≤ j ≤ N -1 , (3.12)
where v j (x) := (g -,j , ϕ

1 j ) L 2 (R m ) = (g, ϕ 1 j ) L 2 (R m )
. By means of the Schwarz inequality

|v j (x)| ≤ R m |g(x, z)| 2 dz, x ∈ R n (3.13)
Hence the goal is to establish the conditions under which such an equation as (3.11) possesses a square integrable solution. We make the Fourier transform in the x-variable and using the fact that the operator -∆ x does not have positive eigenvalues on L 2 (R n ) obtain the expression for a solution of (3.11) as

u -,j (p, y) = v j (p) p 2 + e j ϕ 1 j (y), 1 ≤ j ≤ N -1
We distinguish the two cases dependent upon the dimension of the problem in the first variable.

Lemma 3.6 Let the assumptions of the Theorem 2 hold. Then the equation (3.11) possesses a solution u -,j (x, y) ∈ L 2 (R 1+m ), m ∈ N if and only if

(g(x, y), e ±i √ -e j x √ 2π ϕ k j (y)) L 2 (R 1+m ) = 0, 1 ≤ k ≤ m j , 1 ≤ j ≤ N -1
Proof. We express a solution of (3.11) as the sum of its regular and singular components

u -,j (p, y) = v j (p)χ Ω c δ p 2 + e j ϕ 1 j (y) + v j (p)χ Ω δ p 2 + e j ϕ 1 j (y) , (3.14)
where the set in the Fourier space Ω

δ := [ √ -e j -δ, √ -e j + δ] ∪ [- √ -e j -δ, - √ -e j + δ] = Ω + δ ∪ Ω - δ with 0 < δ <
√ -e j and Ω c δ is its complement, χ Ω δ and χ Ω c δ are their characteristic functions. It is trivial to estimate the first term in the right side of (3.14) since we are away from the positive and negative singularities ± √ -e j . Thus

v j (p)χ Ω c δ p 2 + e j ϕ 1 j (y) ≤ C|ϕ 1 j (y)|| v j (p)|χ Ω c δ ,
which along with (3.13) enables us to estimate the square of its L 2 norm.

+∞

-∞ dp R m dy|ϕ 1 j (y)| 2 | v j (p)| 2 χ Ω c δ ≤ v j 2 L 2 (R) ≤ g 2 L 2 (R 1+m ) < ∞
To obtain the conditions under which the remaining term in (3.14) is square integrable we first study its behavior near its negative singularity using the formula

v j (p) = p - √ -e j d v j (s) ds ds + v j (--e j )
Thus one needs to estimate

v j (- √ -e j ) + p - √ -e j db v j (s) ds ds p 2 + e j χ Ω - δ ϕ 1 j (y) (3.15)
We derive the upper bound for the derivative using (3.13) along with the Schwarz inequality with α > 5 such that g(x, y) ∈ L 2 α, x .

d v j (p) dp ≤ 1 √ 2π ∞ -∞ dx|x||v j (x)| ≤ 1 √ 2π ∞ -∞ dx |x| 1 + |x| α 1 + |x| α R m |g(x, z)| 2 dz ≤ ≤ 1 √ 2π +∞ -∞ dx x 2 1 + |x| α ∞ -∞ dx(1 + |x| α ) R m |g(x, z)| 2 dz = = 1 √ 2π +∞ -∞ dx x 2 1 + |x| α g 2 L 2 (R 1+m ) + |x| α 2 g 2 L 2 (R 1+m ) = C < ∞
This enables us to prove the square integrability for the second term in (3.15).

p -√ -e j db v j (s)

ds ds p 2 + e j χ Ω - δ ϕ 1 j (y) ≤ C |p - √ -e j | χ Ω - δ |ϕ 1 j (y)| ≤ C 2 √ -e j -δ χ Ω - δ |ϕ 1 j (y)| ∈ L 2 (R 1+m )
Near the positive singularity we use the identity

v j (p) = p √ -e j d v j (s) ds ds + v j ( -e j )
to study the conditions of the square integrability of the term

v j ( √ -e j ) + p √ -e j db v j (s) ds ds p 2 + e j χ Ω + δ ϕ 1 j (y) (3.16)
Analogously to the situation at the negative singularity we prove the square integrability of the second term in (3.16) using the bound on the derivative involved in it. Hence p √ -e j db v j (s) This bound implies that the necessary and sufficient conditions for the existence of u -,j (x, y) ∈ L 2 (R 1+m ) solving the equation (3.11) are v j ( -e j ) = 0, v j (--e j ) = 0 which by means of the definition of the functions v j (x) is equivalent to (g(x, y), e ±i √ -e j x √ 2π ϕ k j (y)) L 2 (R 1+m ) = 0, 1 ≤ k ≤ m j , 1 ≤ j ≤ N -1

ds ds p 2 + e j χ Ω + δ ϕ 1 j (y) ≤ C |p + √ -e j | χ Ω + δ |ϕ 1 j (y)| ≤ ≤ C 2 √ -e j -δ χ Ω + δ |ϕ 1 j (y)| ∈ L 2 (R 1+m
After establishing the solvability conditions for the equation (3.11) when the situation is one dimensional in the first variable we turn our attention to the cases of dimensions two and higher. where the spherical layer in Fourier space A δ := {p ∈ R n : √ -e jδ ≤ |p| ≤ √ -e j + δ}, its complement in R n is A c δ , their characteristic functions are χ A δ and χ A c δ respectively and 0 < δ < √ -e j . Clearly for the second term in the right side of (3.17

R n | v j (p)| 2 dp R m |ϕ 1 j (y)| 2 dy = v j 2 L 2 (R n ) ≤ g 2 L 2 (R n+m ) < ∞.
Hence the first term in the right side of (3.17) will play the crucial role for establishing the solvability conditions for the equation (3.11). We will make use of the formula Let us derive the upper bound for the derivative of the Fourier transform involved in it using (3.13) along with the Schwarz inequality, α > 6 for n = 2 and α > n + 2 for n ≥ 3 such that g(x, y) ∈ L 2 α, x .

∂ v j ∂|p| ≤ 1 (2π) n 2 R n |v j (x)||x|dx ≤ 1 (2π) n 2 R n dx|x| R m |g(x, z)| 2 dz = = 1 (2π) n 2 R n dx |x| 1 + |x| α 1 + |x| α R m |g(x, z)| 2 dz ≤ ≤ 1 (2π) n 2 R n dx |x| 2 1 + |x| α R n dx(1 + |x| α ) R m |g(x, z)| 2 dz = = 1 (2π) n 2 ∞ 0 d|x||S n | |x| n+1 1 + |x| α g 2 L 2 (R n+m ) + |x| α 2 g 2 L 2 (R n+m ) = C < ∞ Therefore |p| √ -e j

)

  Thus it remains to derive the conditions under which the first term in (3.15) and the first term in (3.16) are square integrable. Estimating the square of theL 2 (R 1+m ) -δ dp | v j (-√ -e j )| 2 (p 2 + e j ) 2 + √ -e j +δ √ -e j -δ dp | v j ( √ -e j )| 2 (p 2 + e j ) 2 ,which can be bounded below by| v j (-√ -e j )| 2

Lemma 3. 7

 7 Let the assumptions of the Theorem 2 hold. Then the equation (3.11) possesses a solution u -,j (x, y)∈ L 2 (R n+m ), n ≥ 2, m ∈ N if and only if (g(x, y), e ipx (2π) n 2 ϕ k j (y)) L 2 (R n+m ) = 0, a.e. p ∈ S n √ -e j , 1 ≤ k ≤ m j , 1 ≤ j ≤ N -1Proof. It is convenient to represent a solution of (3.11) as the sum of the singular and the regular parts u -,j (p, y) = v j (p)χ A δ p 2 + e j

  ∂|s| (|s|, ω)d|s| + v j ( -e j , ω) ∂|s| (|s|, ω)d|s| + v j ( √ -e j , ω) p 2 + e j χ A δ ϕ 1 j (y)

  ∂b v j ∂|s| (|s|, ω)d|s| p 2 + e j χ A δ ϕ 1 j (y) ≤ C √ -e j χ A δ |ϕ 1 j (y)| ∈ L 2 (R n+m )

3 (R 3 ) which is guaranteed by its rate of decay given explicitly in the Assumption 1.1.

Acknowledgement The first author thanks I.M.Sigal for partial support via NSERC grant No. 7901.

and it remains to estimate from below the square of the L 2 norm of the term v j ( √ -e j , ω)

which yields the necessary and sufficient conditions of solvability of the equation

. Using the definition of the functions v j (x) we easily arrive at

Having established the orthogonality conditions in the lemmas above which guarantee the existence of square integrable solutions for our equations we conclude the proof of Theorem 2.

Proof of Theorem 2. We construct the solution of the equation (1.3) as u := u + + u 0 + N -1 j=1 u -,j , where the existence of u + ∈ L 2 (R n+m ) is guaranteed by Lemma 3.1, of u 0 ∈ L 2 (R n+m ) by Lemmas 3.2-3.5, of {u -,j } N -1 j=1 ∈ L 2 (R n+m ) by Lemmas 3.6 and 3.7. Suppose the equation (1.3) admits two solutions u 1 , u 2 ∈ L 2 (R n+m ). Then their difference w := u 1u 2 ∈ L 2 (R n+m ) solves the homogeneous problem with separation of variables Lw = 0 which admits two types of solutions. The first ones are of the form γ(x)ϕ k N (y), 1 ≤ k ≤ m N with γ(x) harmonic. The second ones are of the kind e ipx (2π)

In both cases to vanish is the only possibility for them to belong to the space L 2 (R n+m ).

Appendix

Proof. The norm f L 1 (R 3 ) is being estimated from above by means of the Schwarz inequality as