Christian Ebeke
email: christian_hubert.ebeke@u-clermont1.fr

Does the dual-citizenship recognition determine the level and the utilization of international remittances? Cross-Country Evidence

Keywords: Dual-citizenship, Remittances, Investment, Developing countries JEL codes: F24 F22 E22 O2

Does the dual-citizenship recognition determine the level and the utilization of international remittances?

Cross-Country Evidence

Christian Hubert Ebeke

INTRODUCTION

International remittances represent one of the most important components of external capital flows received by developing countries. According to the World Bank (2011), remittances exceed the levels of external development assistance to developing countries. It is not only the levels of remittances that retain the attention but also their dynamic properties.

Several studies pointed out the fact that remittances are more stable than the other component of private capital flows [START_REF] Chami | Macroeconomic Consequences of Remittances[END_REF] and they are sometimes countercyclical vis-à-vis receiving country business cycle [START_REF] Sayan | Business cycles and workers' remittances: How do migrant workers respond to cyclical movements of GDP at home?[END_REF]Frankel, 2010). Recent papers indicated that remittances often react positively to exogenous shocks for instance natural disasters [START_REF] Yang | Coping with disaster: The impact of hurricanes on international financial flows, 1970-2002[END_REF][START_REF] Mohapatra | Remittances and natural disasters: ex-post response and contribution to ex-ante preparedness[END_REF][START_REF] David | How do international financial flows to developing countries respond to natural disasters?[END_REF]. It is therefore not surprising to see that the literature on remittances has highlighted their strong impact on poverty reduction and stabilization [START_REF] Adams | Do international migration and remittances reduce poverty in developing countries?[END_REF][START_REF] Bugamelli | Output growth volatility and remittances[END_REF][START_REF] Chami | Remittances: An automatic output stabilizer?[END_REF][START_REF] Combes | Remittances and household consumption instability in developing countries[END_REF].

Beside these developmental effects of remittances, the literature also has analyzed the potentially negative effects of large remittance inflows. The results indicate that large remittance-to-GDP ratios are appreciating the real exchange rate (Amuedo- [START_REF] Amuedo-Dorantes | Workers' remittances and the real exchange rate: a paradox of gifts[END_REF][START_REF] Barajas | Workers' remittances and the equilibrium real exchange rate: Theory and evidence[END_REF] and expanding domestic corruption [START_REF] Abdih | Remittances and institutions: Are remittances a curse?[END_REF]. The literature has also focused on the determinants of international remittances. Among the receiving country determinants of remittances, it appears that the level of financial development at home (a proxy for the cost of remitting money), the domestic shocks (natural disasters, civil war, and inflation) and the level and the composition of migration in terms of qualifications do matter [START_REF] Lueth | Determinants of bilateral remittance flows, The B.E[END_REF][START_REF] Freund | Remittances, transaction costs, and informality[END_REF][START_REF]The determinants of international remittances in developing countries[END_REF]. Regarding the last factors, the results of the literature are that countries which export more migrant and unskilled migrants receive on average more remittances than the others. In contrast, a high level of financial development seems to positively determine remittances by lowering transaction costs associated with remitting activities. [START_REF] Giuliano | Remittances, financial development, and growth[END_REF] also found that the level of financial development determines the positive effect of remittances on growth since remittances are more likely to be used for productive purposes in less financially developing countries.

This paper revisits the literature on the determinants and the consequences of international remittances in developing countries. It explores the effect of the recognition of the dual citizenship status on the level of remittances received by each country. More specifically, the paper examines whether countries which recognize a dual citizenship status tend to receive more remittances than others and investigates whether the recognition of the dual citizenship determines significantly the utilization of remittances to achieve investments goals in the country of origin.

At least four reasons justify this study. Firstly, this paper enters the debate on the effects of the citizenship status of the migrants on their remitting behavior. The paper by using a large sample of countries aims at measuring the between countries effect of the legislation on the dual citizenship recognition on remittance inflows. It therefore supplements the early evidence of [START_REF] Vadean | Citizenship Status and Remittances: Evidence from German Micro Data[END_REF] who found that immigrants in Germany who have a dual citizenship tend to remit more to their country of origin than others. The paper is also close to the one of [START_REF] Leblang | Harnessing the Diaspora: Dual citizenship, Migrant Remittances and Return[END_REF]. The author showed using panel data that the recognition of the dual-citizenship helps increase remittance inflows and the returns of migrants at home. In this paper, I test whether between differences among countries in term of the recognition of the dual-citizenship explains the between differences in remittance inflows during the last decade.

I therefore base my analysis on a cross-sectional approach. This can also been justified by the fact that a large part of the variation in the dual-citizenship legislations is between countries instead of within countries (only a small number of developing countries (9) changed their legislation on the dual-citizenship during the recent years, see Sejersen, 2008 2 ).

Secondly, I distinguish between the 'direct effect' of the dual-citizenship (defined as the effect of the dual-citizenship obtained after holding constant, the between country differences in migrant stocks of adults) and the 'indirect effect' of the recognition of the dualcitizenship on remittances which passes through the migration incentives. I also use the narrow definition of remittances as suggested in the International Monetary Fund Balance of Payments Yearbooks by taking only workers' remittances to record the remittance variable and ignoring compensation of employees and migrant transfers. This strategy is retained because these two components sometimes recorded as remittances by authors do not really reflect the decision taken by a migrant to remit nor the migrant transfers of wealth represent flows but clearly a stock. Moreover, I test the effect of the dual-citizenship recognition on alternative measures of the dependent variable: remittances per capita, remittances in percent of GDP or in logarithm form.

Thirdly, this paper is also different from earlier studies given that it quantifies the effect that the dual-citizenship exerts on the allocation of remittances into investment at home. The hypothesis tested here is that migrants with a dual citizenship are more likely to invest in their country of origin given they afford smaller transaction costs than others migrants getting only the nationality of the host country. Indeed, emigrants without a dual citizenship but only the citizenship of the host country may now require an entry visa for home visits, lose the right to buy or even own real property or start easily a business, and in the extreme they may experience discrimination in their erstwhile home country [START_REF] Vadean | Citizenship Status and Remittances: Evidence from German Micro Data[END_REF]. This might reduce the motivation of doing investments in assets (e.g. real estate) and business in the country of origin. Altogether, a country which recognizes a dual-citizenship (country A) to its international migrants can maximize the level of remittances sent for investment purposes compared to another country which do not recognize the dual-citizenship (country B).

Suppose that both countries A and B export the same level of migrants abroad but that the composition of migrants originating from A is a mix of natives which have kept their former nationality (M1A) and natives with the both nationalities of their erstwhile home country and the host country (M2A). For country B, the composition of migrants is a mix of natives which have kept their former nationality (M1B) and migrants who have only the nationality of the host country and have lost their former nationality (M2B). It comes that the country A maximizes the potential level of remittances which is sent for investment purposes given that both M1A and M2A can start business at home without huge transaction costs while in country B, the migrants M2B are less incited to start business due to the additional transaction costs that the lost of their former nationality triggers. 3 One of the best alternatives to this outcome is dual-citizenship, status which provides rights in both the host and the home country. Dual-citizenship gives the migrant the best capacity to act transnationally, i.e. full access to the markets of both the host and home country and mobility between the two, status that could stimulate investments in the home country and the related remittances to relatives. Altogether, one would therefore observe that countries which recognize a dual-citizenship status to their migrants receive (i) more remittances on average and (ii) more remittances devoted to investment purposes. 3 It is plausible that the migrants who have lost their former nationality can send remittances to their relatives at home and start business under their names. But this is based on the hypothesis that migrants fully trust their relatives and can engage large funds under their names. However, this hypothesis can be challenged according to papers examining the spread and the causes of the mistrust in developing countries [START_REF] Nunn | The slave trade and the origins of mistrust in Africa[END_REF].

Fourthly, this paper uses several informations on the international recognition of the dual-citizenship. Indeed, as it has been pointed out by [START_REF] Blatter | Acceptance of dual citizenship: Empirical data and political contexts, Working Paper Series[END_REF], given the very different definitions and scopes of the studies, it is not surprising that the results of the various studies are not coherent. Indeed, there are four studies with a large number of countries and a global perspective (US Office of Personnel Management, 2001[START_REF] Renshon | The 50% American. Immigration and National Identity in an Age of Terror[END_REF][START_REF] Boll | Multiple Nationality and International Law[END_REF][START_REF] Sejersen | I Vow to Thee My Countries" The Expansion of Dual Citizenship in the 21st Century[END_REF] and these studies are very diverse in respect to the definitions, the number of countries included and the quality of information gathered. Some studies took into account the issue of the recognition of the dual-citizenship with treaty nations while those did not take into account these arrangements. Other studies recognize that some countries could 'tolerate' dual-citizenship (a form of de facto recognition) and all these discrepancies characterizing the four studies lead to different results regarding the spread of the dual-citizenship phenomenon around the world.

One advantage of the work of [START_REF] Blatter | Acceptance of dual citizenship: Empirical data and political contexts, Working Paper Series[END_REF] is to aggregate and merge the findings of the previous four studies and to finally propose a new classification of countries.

They distinguish between the 'full' recognition (if the majority of studies concludes on the same way) and the 'no' recognition. Between the two parts of the spectrum they allow for some categories such as 'tolerance and acceptance with treaty nations' and countries with a 'very limited' acceptance. In this paper, I take advantage of this recent codification. I present results obtained with the 'full' recognition and with the broad definition (merging the 'full', the 'tolerance' and the 'very limited acceptance' dimensions into a single dummy variable).

The remainder of the paper is as follows. I start with the description of the empirical strategy devoted to quantify the effect of the dual-citizenship recognition on remittance inflows. The paper uses a large cross-section of developing countries (103) observed over the period 2000-2008. This period is chosen for at least two reasons. Firstly, by retaining the most recent decade I use the most reliable data on remittances at the cross-country level. Secondly, using the period 2000-2008 reduces the self-selection bias that could be associated with the decision to modify the legislation on the dual-citizenship. Given that almost all the countries included in the sample have not changed their legislation during this period, this ensures that the effect of the dual-citizenship on remittances is not affected by the self-selection bias during this period. However, it remains 9 countries which have changed their legislation in the mid of 2000s and these countries could bias the results. For robustness purposes, I run also regressions by excluding these 'switchers' from the sample.

Moreover, since the existing datasets on the dual-citizenship look at the dual citizenship regulations at the beginning of the 21 st Century, this study would therefore provide evidence on the impact of earliest recognitions of the dual-citizenship and thus neglect the effect of the very recent innovations (recognitions occurred in 2008) for example. This can be justified by the fact that the innovations in the legislation act with a delay (due to effective implementation and enforcement costs).

The results indicate that countries which recognize a dual-citizenship receive on average more remittances than others and this effect is the combination of an indirect effect (the dual-citizenship recognition increases incentive to migrate abroad and finally increase the stock of emigrants and remittances) and a direct effect obtained once I control for the level of migration stock. This effect captures the incentives to remit more due to the economic advantages of the dual-citizenship described above.

Section 3 examines whether the positive elasticity of the private investment with respect to remittance inflows is conditional upon the recognition of the dual-citizenship law.

The results obtained do not reject the hypothesis that the dual-citizenship matters. Section 4 concludes on the policy implications of these results.

QUANTIFYING THE EFFECT OF THE DUAL-CITIZENSHIP RECOGNITION ON THE LEVEL OF REMITTANCE INFLOWS

a) The econometric models I take advantage of the literature on the macroeconomic determinants of remittances in developing countries [START_REF] Freund | Remittances, transaction costs, and informality[END_REF][START_REF]The determinants of international remittances in developing countries[END_REF] to quantify the effect of the dual-citizenship. The following equation is specified:

i j i i i i M D X R ε α δ θ β + + + + ′ = 1 1 (1)
where R is either remittances per capita in $ US, remittance-to-GDP ratio, the log level of remittances per capita in $ US, and the logarithm of remittances in percentage of GDP. D is a dummy variable taking 1 if the country recognizes the dual-citizenship and 0 otherwise, 4 M is the log of migration of workers abroad (individuals born in their country of origin, aged 25+ and living and working abroad), X is the matrix of control variables (GDP per capita in the receiving country, GDP per capita growth in the host country, 5 age dependency ratio and financial development6 ), j α the vector of regional dummies and ε is the idiosyncratic error term. According to the literature, the level of income would be negatively correlated with remittances while the proportion of inactive people, the stock of migrants abroad, the economic conjuncture in the host countries and financial development 4 I use alternative measures for this dummy. I use the work of [START_REF] Blatter | Acceptance of dual citizenship: Empirical data and political contexts, Working Paper Series[END_REF] by creating two dummies for the 'full acceptance' and the 'broad acceptance', respectively. 5 The GDP per capita growth in the host countries is computed as the weighted sum of the GDP per capita growth in each migrant destination countries with weights being the share that each destination country represents in the emigration of individuals originating from the country observed. Data on the bilateral migration shares are drawn from the World Bank website, Migration and Remittance Prospects.

(proxy for transaction costs in the remittance market) are positively correlated with remittances. i is the indicator of the country. 1 θ in equation ( 1) identifies the effect on remittances of the dual citizenship status. Put differently, it measures the average differences in remittances in countries recognizing a dual citizenship versus countries that did not. This is the direct effect of the dual-citizenship since I already control for the level of migrant stock abroad. By doing so I ensure that the effect observed for the dual-citizenship is not driven by its positive effect on migration incentives.

Because the dual-citizenship could also determine the level of remittance inflows through its indirect effect on migration (people living in countries which recognize a dualcitizenship have more incentives to migrate abroad for work and to get a second nationality for themselves and for their children or spouses), the equation 2 is specified to quantify the 'total effect' of the dual-citizenship on remittances. The indirect effect which passes through the increase of the stock of migrant abroad will be derived as the difference between the 'total effect' and the direct effect 1 θ . The equation measuring the 'total effect' of the dual citizenship is the following:

i j i i i i Mr D X R ε α δ θ β + + + + ′ = 2 2 (2)
with Mr being the component of the migrant stock variable not explained by the dualcitizenship dummy. Extracting from M the potential effect of the dual-citizenship ensures that 2 θ now measures the 'total effect' of D. It is worth noting that this transformation affects only the estimated coefficient ( 2 θ ) on the dual-citizenship variable. [START_REF] Pagan | Econometric issues in the analysis of regressions with generated regressors[END_REF] shows that the two-step procedure, of estimating (2) gives asymptotically efficient estimates and the correct values for the standard errors.

In cases where the 'transmission' variable (the stock of international migrants) has a positive effect on remittance inflows, and the dual-citizenship dummy has a positive effect on migration, this method will provide for a larger coefficient on the dual citizenship in equation ( 2). If it transpires that the dual-citizenship is not a determinant of migration, there is no effect on the coefficient of the dual-citizenship. The indirect effect of the dual-citizenship which passes through the volume of migrants is given by 1 2 θ θ -.

b) Data

The remittance variable records the money sent by migrants residing in host countries for at least one year. Data in $ US come from the IMF Balance of Payments Yearbook and are divided by each country total population. Information on the dual-citizenship laws were drawn from [START_REF] Blatter | Acceptance of dual citizenship: Empirical data and political contexts, Working Paper Series[END_REF]. All the other remaining control variables were drawn from the World Bank Tables. Data are again averaged for the period 2000-2008 for each country.

The list of countries (with the information on the dual-citizenship recognition) and the descriptive statistics of all the variables used in the paper are presented in appendix.

c) Dual-citizenship and the level of remittances: Econometric results

(i) The direct effect of the recognition of the dual-citizenship

The results of estimations are presented in Table 1 andTable 2. The method of estimation is the OLS estimator with a full set of regional dummies. In Table 1, I present the results obtained using the narrower definition of the dual-citizenship recognition (the 'full' status corresponds to countries in which all the available informations presented in the four studies mentioned in the introduction of this paper converge toward the same conclusion).

The first four columns of the Table 1 indicate that countries which have allowed the dualcitizenship status receive on average more remittances: 43 $US per capita more remittances a year (column1), an additional 1.7% percentage point of GDP of remittances (column 2) and around 90% more remittances than the countries which did not recognize the dual-citizenship (columns 3 and 4). Interesting and expected results are also observed regarding the control variables. Indeed, the estimation results suggest that remittances are more likely to flow in countries which export a large number of migrants, in countries with high demographic dependency ratios and finally in countries with lower transaction costs associated with remittances. Moreover, in some specifications, it appears that a better economic conjuncture in the host countries affects positively remittances.

[Table 1 about here]

In the last four columns in Table 1, I exclude from the sample the countries which changed their legislation during the decade. Although these switchers had been taken in the analysis as countries allowing the dual-citizenship status, the results obtained before can be biased because these countries recognized the dual-citizenship belatedly compared to the rest of the sample for which the recognition was already enabled or not before the beginning of the period of analysis (year 2000). This corresponds to around 7 countries and excluding them does not really affect the previous results. Indeed, the results obtained in columns 5-8 are qualitatively and quantitatively similar to the previous ones.

In Table 2, I repeat the same exercises as previously but I use the second dummy of dual-citizenship (the broad codification which takes into account countries which tolerate or with a limited acceptance). The impacts quantified are a higher compared to those obtained in Table 1. Indeed, the results indicate that when I take into account the de facto recognitions, the effect of the dual-citizenship is stronger. Countries which fully recognize the dualcitizenship or tolerate it receive 60 $US per capita more remittances or 3 percentage points of GDP more remittances than others. In terms of relative gap, these countries receive on average the double of the amount of remittances received by the others countries (columns 3 and 4). Even when I exclude from the sample the switchers, the results remain broadly unchanged.

[Table 2 about here] (ii) The indirect and total effects of the recognition of the dual-citizenship

To measure the total effect of the dual-citizenship and derive the indirect effect, the migration variable is replaced by its residual component obtained after controlling for the effect of the dual-citizenship dummy on migration. The results are presented in Table 3 andTable 4.

In Table 3, I present the results obtained using the narrower measure (the 'full recognition'). It appears that the coefficients associated with the dual-citizenship dummy have increased compared to its level observed in Table 1. As expected, it is worth noting that the coefficients and the standard-errors of all the remaining variables included in the model did not change. The shift of the coefficient associated with the dual-citizenship dummy suggests that its impact on remittances is the combination of one 'direct effect' and one 'indirect effect' through the migration channel. Knowing that the coefficient estimated in Table 3 measures the 'total effect' of the dual-citizenship on remittances, the 'indirect effect' is therefore computed as the difference between the 'total effect' and the 'direct effect'. This 'indirect effect' represents around one-quarter of the 'total effect'. These results are not affected by the exclusion of the switchers from the sample (columns 5-8).

[Table 3 about here]

In Table 4, I replace the 'full recognition' dummy by the 'broad recognition' dummy.

The results highlight an increase in the magnitude of the coefficients associated with the dual-citizenship dummy. This result suggests that even with this definition of the dual-citizenship, the hypothesis that the effect is the combination of two direct and indirect effects is not rejected. The indirect effect is obtained by taking the difference between the coefficients of Table 4 and the coefficients of Table 2. It appears that the indirect effect corresponds to around one-fifth of the total effect, the remaining effect being due to the direct effect. This result is also robust to the exclusion of switchers from the sample.

[Table 4 about here]

Altogether, the effect on remittances of the recognition of the dual-citizenship is primarily due to what I called the 'incentive effect' to remit more since it is easier for migrants with dual-citizenship to start business at home and monitor it by travelling frequently. In the next section, I test empirically this proposition that remittances are more likely to be channeled into investment in countries which have recognized a dual-citizenship for their migrants.

DUAL-CITIZENSHIP, REMITTANCES AND PRIVATE INVESTMENT IN

RECEIVING COUNTRIES

This section tests empirically whether remittances are more likely to foster the private investment ratio in receiving countries which have recognized officially a dual-citizenship status. In the previous section, I addressed the issue of the relationship between the citizenship law and the level of remittances. I turn now on the relationship between the dual citizenship law and the cross-country differences in terms of invested remittances.

a) The econometric model

The following model describes the effect of the dual-citizenship on the elasticity of the private investment ratio with respect to the remittance ratio:

( ) i j i i i i i i D D R R X I ε α θ θ θ β + + + × + + ′ = 5 4 3 (3)
where I represents the private investment-to-GDP ratio, R the log of the remittance-to-GDP ratio, D is the dual citizenship dummy, and X is the few set of control variables generally introduced in the private investment equations at the macroeconomic level [START_REF] Serven | Real-exchange-rate uncertainty and private investment in LDCs[END_REF]. I therefore control for the inflation rate, the public investment-to-GDP ratio, the private credit-to-GDP ratio, the level of public debt service-to-GDP ratio, the level of trade openness, and for the political stability index.

In equation ( 3), the coefficient of interest is 4 θ which measures the differential elasticity of private investment with respect to remittances in countries which recognize a dual citizenship (D = 1) versus countries that did not (D = 0). 0 4 > θ suggests that remittances increase the most private investment when the migrants can enjoy a dual-citizenship. Put differently, this means that remittances are more devoted to investment purposes in countries which recognize the dual-citizenship status for the natives. Finally, 3 θ measures the effect of remittances on investment in countries which do not recognize the dual-citizenship status for their migrants (D = 0).

b) Identification strategy

The main shortcoming that characterizes the estimation of the equation ( 3) is neglecting the plausibility that remittance inflows and the private investment are simultaneously determined. I therefore resort to the instrumental variables approach in which I allow for exogenous sources of variations in remittances. I use as instruments for remittances the remittance-to-GDP ratio for each country in 1995 and the GDP per capita growth rate in host country.

My identification assumption is that the level of remittances in 1995 and the economic conjuncture in the host country are exogenous and determine the private investment ratio only through remittances after controlling for my set of explanatory variables. For instance since I already control for the trade openness, I reduce the possibility that the economic conjuncture in the host countries could affect the private investment in receiving economic through other channels of the globalization. Finally, the validity of the instrumental variables approach will be assessed through the Hansen overidentification test and through the first-stage F-test statistics.

The second endogenous variable is the interactive term of remittances crossed with the dual-citizenship dummy. This variable is instrumented by the instruments of remittances crossed with the dual-citizenship dummy.

c) Data

The private and public investment variables are drawn from the IMF-World Economic Outlook database. The quality of institutions measured by the indicator of political stability comes from the World Bank Governance Indicators dataset. All the other explanatory variables are drawn from the World Bank Tables.

d) Dual citizenship, remittances and private investment: Econometric results

The results of the estimations are presented in Table 5 andTable 6. In Table 5, I present the results obtained using the 'full recognition' variable while in Table 6 I use the 'broad recognition' dummy.

In Table 5, I begin first by presenting the results obtained using the OLS with regional dummies method (column 1). The result indicates that the differential impact of remittances on the private investment between countries recognizing a dual-citizenship and the other countries is positive and highly significant (the coefficient stands at around 0.4). The coefficient of the additive term of remittances which identifies the impact of remittances in countries which do not recognize the dual-citizenship status, is not statistically different from 0. Finally, the sum of the coefficients of the additive term and the multiplicative term of remittances leads to a globally positive impact of remittances in countries recognizing the dual-citizenship status.

[Table 5 around here]

In column 3, the causal effect on remittances is measured by performing a two-stage least squares estimations in order to reduce the endogeneity of remittances. Column 2 presents the results of the first-stage regression of the instrumentation of remittances. 7 The diagnostic statistics associated with the first-stage regression do not reject the hypotheses that the instruments are jointly strong and not correlated with the residuals of the second-stage (F-stat and Hansen OID p-value). The results obtained from the second-stage regression confirm the hypothesis that remittance inflows are more likely to be positively correlated with the private investment ratio in countries which recognize the dual-citizenship. Indeed the coefficient of the interactive is positive and statistically significant. Moreover, the sum of the two coefficients associated with remittances (which identify the effect of remittances on investment in countries which recognize the dual-citizenship) is positive and statistically significant. 8In column 4, even after excluding the switchers (countries which have changed their legislation in the mid of 2000s), the result that the effect of remittances on investment is more 7 It is worth noting that the variable remittances*dual-citizenship has been also instrumented by the same set of instruments introduced linearly and non-linearly. The results are not shown but are available upon request. However, I present at the bottom of Table 5, the F-test of instrumentation of the multiplicative variable. positive and significant in countries which recognize the dual-citizenship status is not rejected by the data.

In Table 6, I replace the 'full recognition' dummy by the 'broad recognition' dummy and run the same estimations. Whatever the method used (OLS or IV, including or excluding the switchers), the results remain qualitatively unchanged. Countries which enable the dualcitizenship status for their natives receive more remittances channeled into investment purposes compared to the others remittance-dependent economies.

[Table 6 around here]

CONCLUDING REMARKS

This paper examined the role of the recognition of the dual-citizenship on both the level of remittance inflows and the utilization of remittances in receiving economies of the developing world. It mobilizes a large cross-section of countries (103) observed over the period 2000-2009 and finds that countries which recognize a dual-citizenship status receive x% more remittances than others. Moreover, the results also highlighted that remittances are strongly channeled into investment in these countries compared to the others.

The policy implications of these results are two-folds. First, the issue of the dualcitizenship recognition would be taken into account among all the strategies designed domestically and internationally in favor of sustaining remittances. Second, the recognition of the dual citizenship also enters the debate on the ways to increase the benefits of remittance inflows by encouraging their reinvestment. The paper shows that countries which have already allowed getting a dual-citizenship perform better in terms of more investmentmotivated remittances. 18.23 21.32 Hansen OID:p-value 0.866 0.586 Notes: Robust standard errors in parentheses. 'Broad recognition' of the dual-citizenship takes countries with a 'fully recognition' and countries which tolerate or accept a dual-citizenship with treaty nations and finally countries with a limited acceptance (see [START_REF] Blatter | Acceptance of dual citizenship: Empirical data and political contexts, Working Paper Series[END_REF] for more details). Regional dummies are included in all equations. Data are averaged for the period 2000-2009. The dependent variable is the private investment-to-GDP ratio. *** p<0.01, ** p<0.05, * p<0.1.

Table 1 : Direct effect of the dual-citizenship law ('full recognition' drawn from Blatter et al., 2009) on remittance inflows.

 1

	Excluding 'switchers'

Table 2 : Direct effect of the dual-citizenship law ('Broad recognition' drawn from Blatter et al., 2009) on remittance inflows.

 2 Robust standard errors in parentheses. 'Broad recognition' of the dual-citizenship records countries with a 'fully recognition' and countries which tolerate or accept a dual-citizenship with treaty nations and finally countries with a limited acceptance (see[START_REF] Blatter | Acceptance of dual citizenship: Empirical data and political contexts, Working Paper Series[END_REF] for more details). Regional dummies are included in all equations. Data are averaged for the period 2000-2009. Rpc: Remittances in per capita $ US, RY: remittance in percentage of GDP. *** p<0.01, ** p<0.05, * p<0.1.

	Excluding 'switchers'

Table 3 : Total effect of the dual-citizenship law ('full recognition' drawn from Blatter et al., 2009) on remittance inflows.

 3 Robust standard errors in parentheses. Regional dummies are included in all equations. Data are averaged for the period 2000-2009. Rpc: Remittances in per capita $ US, RY: remittance in percentage of GDP.

	Excluding 'switchers'

Table 4 : Total effect of the dual-citizenship law ('Broad recognition' drawn from Blatter et al., 2009) on remittance inflows.

 4 Robust standard errors in parentheses. 'Broad recognition' of the dual-citizenship records countries with a 'fully recognition' and countries which tolerate or accept a dual-citizenship with treaty nations and finally countries with a limited acceptance (see[START_REF] Blatter | Acceptance of dual citizenship: Empirical data and political contexts, Working Paper Series[END_REF] for more details). Regional dummies are included in all equations. Data are averaged for the period 2000-2009. Rpc: Remittances in per capita $ US, RY: remittance in percentage of GDP. *** p<0.01, ** p<0.05, * p<0.1.

	Excluding 'switchers'

Table 5 : Dual-citizenship ('Full recognition' drawn Blatter et al., 2009), remittances and private investment.

 5 Robust standard errors in parentheses. Regional dummies are included in all equations. Data are averaged for the period 2000-2009. The dependent variable is the private investment-to-GDP ratio. *** p<0.01, ** p<0.05, * p<0.1.

	OLS	Remittance First-stage	IV	IV-excluding switchers
	(1)	(2)	(3)	(4)

Table 6 : Dual-citizenship ('Broad recognition' drawn Blatter et al., 2009), remittances and private investment.

 6

	OLS	Remittance First-stage	IV	IV-excluding switchers
	(1)	(2)	(3)	(4)

According to[START_REF] Sejersen | I Vow to Thee My Countries" The Expansion of Dual Citizenship in the 21st Century[END_REF] since the year 2000 the developing countries that changed their legislation in favor of the recognition of the dual-citizenship are the following: Afghanistan (2004),India (2004), Philippines (2003), Russia (2001), Belarus (2002), Lithuania (2002), Moldova (2003), Armenia (2004), and Hungary (2004).

The financial development variable and the dependency demographic ratio are expressed in log when the dependent variable is expressed in log.

The F-statistic test of the joint significance of these coefficients rejects the null hypothesis that the coefficients are jointly equal to zero. Results are available upon request.

APPENDIX


François Rodier

Some More Functions That Are Not APN Infinitely Often. The Case of Kasami exponents

We prove a necessary condition for some polynomials of Kasami degree to be APN over F q n for large n.

Introduction

The vector Boolean functions are used in cryptography to construct block ciphers and an important criterion on these functions is high resistance to differential cryptanalysis.

Let q = 2 n for some positive integer n. A function f : F q -→ F q is said to be almost perfect nonlinear (APN) on F q if the number of solutions in F q of the equation f (x + a) + f (x) = b is at most 2, for all a, b ∈ F q , a = 0. Because F q has characteristic 2, the number of solutions to the above equation must be an even number, for any function f on F q . This kind of function has a good resistance to differential cryptanalysis as was proved by Nyberg in [START_REF] Nyberg | Differentially uniform mappings for cryptography[END_REF]. So far, the study of APN functions has focused on power functions. Recently it was generalized to polynomials (cf. [START_REF] Aubry | A Few More Functions That Are Not APN Infinitely Often, Finite Fields: Theory and applications[END_REF]).

There are many classes of function for which it can be shown that each function is APN for at most a finite number of extensions. So we fixe a finite field F q and a function f : F q → F q given by a polynomial in F q [x] and we set the question of whether this function can be APN for an infinite number of extensions of F q .

In this approach, Hernando and McGuire [START_REF] Hernando | Proof of a conjecture on the sequence of exceptional numbers, classifying cyclic codes and APN functions[END_REF] showed a result on the classification of APN monomials which has been conjectured for 40 years: the only exponents such that the monomial x d are APN over infinitely many extension of F 2 are of the form 2 i + 1 or 4 i -2 i + 1. One calls these exponents exceptional exponents. Then it is natural to formulate for polynomial functions the following conjecture.

Conjecture 1.1 (Aubry, McGuire and Rodier) A polynomial on F q can be APN for an infinity of extensions of F q only if it is CCZ equivalent (as was defined by Carlet, Charpin and Zinoviev in [START_REF] Carlet | Codes, bent functions and permutations suitable for DES-like cryptosystems[END_REF]) to a monomial x t where t is an exceptional exponent.

Some cases for f of small degree have been proved by the author [START_REF] Rodier | Bornes sur le degré des polynômes presque parfaitement non-linéaires[END_REF]. We showed there that for some polynomial functions f which are APN on F m 2 , the number m is bounded by an expression depending on the degree of f .

We used it for a method already used by Janwa who showed, with the help of Weil bounds, that certain cyclic codes could not correct two errors [START_REF] Janwa | Hyperplane sections of Fermat varieties in P 3 in char. 2 and some applications to cyclic codes, Applied Algebra[END_REF]. Canteaut showed by the same method that some power functions were not APN for a too large value of the exponent [START_REF] Canteaut | Differential cryptanalysis of Feistel ciphers and differentially δ-uniform mappings[END_REF]. We were able to generalize this result to all polynomials by applying Lang-Weil's results. Some cases of this conjecture have been studied already, in particular the case of Gold degree. We recall them in section 3. In this paper, we will study polynomials of Kasami degree. The proofs happen to be somehow the same as in Gold degree, with a few changes anyway.

Preliminaries

We define

φ(x, y, z) = f (x) + f (y) + f (z) + f (x + y + z) (x + y)(x + z)(y + z)
which is a polynomial in F q [x, y, z]. This polynomial defines a surface X in the three dimensional affine space A 3 . If X is absolutely irreducible (or has an absolutely irreducible component defined over F q ) then f is not APN on F q n for all n sufficiently large. As shown in [START_REF] Rodier | Bornes sur le degré des polynômes presque parfaitement non-linéaires[END_REF], this follows from the Lang-Weil bound for surfaces, which guarantees many F q n -rational points on the surface for all n sufficiently large.

We call φ j (x, y, z) the φ function associated to the monomial x j . The function φ j (x, y, z) is is homogeneous of degree j -3.

We recall a result due to Janwa, Wilson, [6, Theorem 5] about Kasami exponents.

Theorem 2.1 If f (x) = x 2 2k -2 k +1 then φ(x, y, z) = α∈F 2 k -F 2 p α (x, y, z) (1)
where for each α, p α (x, y, z) is an absolutely irreducible polynomial of degree

2 k + 1 on F 2 k such that p α (x, 0, 1) = (x -α) 2 k +1 .
3 Some Functions That Are Not APN Infinitely Often

The best known examples of APN functions are the Gold functions x 2 k +1 and the Kasami-Welch functions x 4 k -2 k +1 . These functions are defined over F 2 , and are APN on any field F 2 m where gcd(k, m) = 1. For other odd degree polynomial functions, we can state a general result.

Theorem 3.1 (Aubry, McGuire and Rodier, [START_REF] Aubry | A Few More Functions That Are Not APN Infinitely Often, Finite Fields: Theory and applications[END_REF]) If the degree of the polynomial function f is odd and not a Gold or a Kasami-Welch number then f is not APN over F q n for all n sufficiently large.

In the even degree case, we can state the result when half of the degree is odd, with an extra minor condition. Theorem 3.2 (Aubry, McGuire and Rodier, [START_REF] Aubry | A Few More Functions That Are Not APN Infinitely Often, Finite Fields: Theory and applications[END_REF]) If the degree of the polynomial function f is 2e with e odd, and if f contains a term of odd degree, then f is not APN over F q n for all n sufficiently large.

In [START_REF] Rodier | Functions of degree 4e that are not APN Infinitely Often[END_REF] we have some results for the case of polynomials of degree 4e where e is odd. Theorem 3.3 If the degree of the polynomial function f is even such that deg(f ) = 4e with e ≡ 3 (mod 4), and if the polynomials of the form

(x + y)(y + z)(z + x) + P with P (x, y, z) = c 1 (x 2 + y 2 + z 2 ) + c 4 (xy + xz + zy) + b 1 (x + y + z) + d (2) for c 1 , c 4 , b 1 , d ∈ F q 3 , do not divide φ then f is not APN over F q n for n large.
We have more precise results for polynomials of degree 12. Theorem 3.4 If the degree of the polynomial f defined over F q is 12, then either f is not APN over F q n for large n or f is CCZ equivalent to the Gold function x 3 . In this case f is of the form

L(x 3 ) + L 1 or (L(x)) 3 + L 1
where L is a linearized polynomial

x 4 + x 2 (c 1+q + c 1+q 2 + c q+q 2 ) + xc 1+q+q 2 ,
c is an element of F q 3 such that c+c q +c q 2 = 0 and L 1 is a q-affine polynomial of degree at most 8 (that is a polynomial whose monomials are of degree 0 or a power of 2).

We have some results on the polynomials of Gold degree d = 2 k + 1.

Theorem 3.5 (Aubry, McGuire and Rodier

, [1]) Suppose f (x) = x d + g(x) where deg(g) ≤ 2 k-1 + 1 . Let g(x) = 2 k-1 +1
j=0 a j x j . Suppose moreover that there exists a nonzero coefficient a j of g such that φ j (x, y, z) is absolutely irreducible (where φ i (x, y, z) denote the polynomial φ(x, y, z) associated to x i ). Then f is not APN over F q n for all n sufficiently large.

Polynomials of Kasami Degree

Suppose the degree of f is a Kasami number d = 2 2k -2 k + 1. Set d to be this value for this section. Then the degree of φ is d -3 = 2 2k -2 k -2. We will prove the absolute irreducibility for a certain type of f .

Theorem 4.1 Suppose f (x) = x d + g(x) where deg(g) ≤ 2 2k-1 -2 k-1 + 1 . Let g(x) = 2 k-1 +1
j=0 a j x j . Suppose moreover that there exists a nonzero coefficient a j of g such that φ j (x, y, z) is absolutely irreducible. Then φ(x, y, z) is absolutely irreducible.

Proof: Suppose φ(x, y, z) = P (x, y, z)Q(x, y, z) with deg P ≥ deg Q. Write each polynomial as a sum of homogeneous parts:

d j=3 a j φ j (x, y, z) = (P s + P s-1 + • • • + P 0 )(Q t + Q t-1 + • • • + Q 0 ) (3)
where P j , Q j are homogeneous of degree j. Then from the Theorem (2.1) we get

P s Q t = α∈F 2 k -F 2 p α (x, y, z).
In particular this implies that P s and Q t are relatively prime as the product is made of distinct irreducible factors.

The homogeneous terms of degree less than d-3 and greater than 2 2k-1 -2 k-1 are 0, by the assumed bound on the degree of g. Equating terms of degree s + t -1 in the equation (3) gives P s Q t-1 + P s-1 Q t = 0. Hence P s divides P s-1 Q t which implies P s divides P s-1 because gcd(P s , Q t ) = 1, and we conclude P s-1 = 0 as deg P s-1 < deg P s . Then we also get Q t-1 = 0. Similarly, P s-2 = 0 = Q t-2 , P s-3 = 0 = Q t-3 , and so on until we get the equation

P s Q 0 + P s-t Q t = 0
since we suppose that s ≥ t. This equation implies P s divides P s-t Q t , which implies P s divides P s-t , which implies P s-t = 0. Since P s = 0 we must have Q 0 = 0.

We now have shown that Q = Q t is homogeneous. In particular, this means that φ j (x, y, z) is divisible by p α (x, y, z) for some α ∈ F 2 k -F 2 and for all j such that a j = 0. We are done if there exists such a j with φ j (x, y, z) irreducible. Since φ j (x, y, z) is defined over F 2 it implies that p α (x, y, z) also, which is a contradiction with the fact that α is not in F 2 .

⊔ ⊓

Remark:

The hypothesis that there should exist a j with φ j (x, y, z) is absolutely irreducible is not a strong hypothesis. This is true in many cases (see remarks in [START_REF] Aubry | A Few More Functions That Are Not APN Infinitely Often, Finite Fields: Theory and applications[END_REF]). However, some hypothesis is needed, because the theorem is false without it. One counterexample is with g(x) = x 13 and k ≥ 4 and even.

Corollary 4.1 Suppose f (x) = x d + g(x) where g is a polynomial in F q [x] such that deg(g) ≤ 2 2k-1 -2 k-1 + 1 . Let g(x) = 2 k-1 +1 j=0
a j x j . Suppose moreover that there exists a nonzero coefficient a j of g such that φ j (x, y, z) is absolutely irreducible. Then the polynomial f is APN for only finitely many extensions of F q .

On the Boundary of the First Case

If we jump one degree more we need other arguments to prove irreducibility. Theorem 4.2 Let q = 2 n . Suppose f (x) = x d + g(x) where g(x) ∈ F q [x] and deg(g) = 2 2k-1 -2 k-1 + 2. Let k ≥ 3 be odd and relatively prime to n. If g(x) does not have the form ax 2 2k-1 -2 k-1 +2 + a 2 x 3 then φ is absolutely irreducible, while if g(x) does have the form ax 2 2k-1 -2 k-1 +2 +a 2 x 3 then either φ is irreducible or φ splits into two absolutely irreducible factors which are both defined over F q .

Proof: Suppose φ(x, y, z) = P (x, y, z)Q(x, y, z) with deg P ≥ deg Q and let

g(x) = 2 2k-1 -2 k-1 +2 j=0 a j x j .
Write each polynomial as a sum of homogeneous parts:

d j=3 a j φ j (x, y, z) = (P s + P s-1 + • • • + P 0 )(Q t + Q t-1 + • • • + Q 0 ). Then P s Q t = α∈F 2 k -F 2 p α (x, y, z).
In particular this means P s and Q t are relatively prime as in the previous theorem.

Since s ≥ t, we have s ≥ 2 2k-1 -2 k-1 -1. Comparing each degree gives

P s-1 = 0 = Q t-1 , P s-2 = 0 = Q t-2
, and so on until we get the equation of degree s + 1

P s Q 1 + P s-t+1 Q t = 0 which implies P s-t+1 = 0 = Q 1 . If s = t then s ≥ 2 2k-1 -2 k-1 .
Note then that a s+3 φ s+3 = 0. The equation of degree s is

P s Q 0 + P s-t Q t = a s+3 φ s+3 = 0.
This means that P s-t = 0, so Q 0 = 0. We now have shown that Q = Q t is homogeneous. In particular, this means that φ(x, y, z) is divisible by p α (x, y, z) for some α ∈ F 2 k -F 2 , which is impossible, as we will show. Indeed, since the leading coefficient of g is not 0, the polynomial φ 2 2k-1 -2 k-1 +2 occurs in φ; as

φ 2 2k-1 -2 k-1 +2 = φ 2 2 2k-2 -2 k-2 +1 (x + y)(y + z)(z + x), (4)
this polynomial is prime to φ, because if p α (x, y, z) occurs in the polynomials

φ 2 2k-1 -2 k-1 +2 , then it will occur in φ 2 2k-2 -2 k-2 +1 . If that is the case, the polynomial p α (x, 0, 1) = (x-α) 2 k +1 would divide φ 2 2k-2 -2 k-2 +1 (x, 0, 1). One has (x + y)(y + z)(z + x)φ 2 2k-2 -2 k-2 +1 (x, y, z) = x 2 2k-2 -2 k-2 +1 + y 2 2k-2 -2 k-2 +1 + z 2 2k-2 -2 k-2 +1 + (x + y + z) 2 2k-2 -2 k-2 +1 hence x(x + 1)φ 2 2k-2 -2 k-2 +1 (x, 0, 1) = x 2 2k-2 -2 k-2 +1 + 1 + (x + 1) 2 2k-2 -2 k-2 +1 .
Let s = xα. We have, for some polynomial R:

(s + α)(s + α + 1)s 2 k +1 = (s + α) 2 2k-2 -2 k-2 +1 + 1 + (s + α + 1) 2 2k-2 -2 k-2 +1 = α 2 2k-2 -2 k-2 +1 + sα 2 2k-2 -2 k-2 + s 2 k-2 α 2 2k-2 -2 k-1 +1 + 1 + +(α + 1) 2 2k-2 -2 k-2 +1 + s(α + 1) 2 2k-2 -2 k-2 + s 2 k-2 (α + 1) 2 2k-2 -2 k-1 +1 + s 2 k-2 +1 R(s).
As

α 2 k -1 = 1 we have α 2 2k-2 -2 k-2 = α 2 k-2 (2 k -1) = 1. So (s + α)(s + α + 1)s 2 k +1 = α + s + s 2 k-2 α 1-2 k-2 + 1 + (α + 1) + s + s 2 k-2 (α + 1) 1-2 k-2 + s 2 k-2 +1 R(s) = s 2 k-2 (α 1-2 k-2 + (α + 1) 1-2 k-2 ) + s 2 k-2 +1 R(s)
which is a contradiction. Suppose next that s = t = 2 2k-1 -2 k-1 -1 in which case the degree s equation is

P s Q 0 + P 0 Q s = a s+3 φ s+3 . If Q 0 = 0, then φ(x, y, z) = d j=3 a j φ j (x, y, z) = (P s + P 0 )Q t which implies that φ(x, y, z) = a d φ d (x, y, z)+a 2 2k-1 -2 k-1 +2 φ 2 2k-1 -2 k-1 +2 (x, y, z) = P s Q t +P 0 Q t
and P 0 = 0, since g = 0. So one has φ 2 2k-1 -2 k-1 +2 divides φ d (x, y, z) which is impossible by [START_REF] Carlet | Codes, bent functions and permutations suitable for DES-like cryptosystems[END_REF].

We may assume then that P 0 = Q 0 . Then we have φ(x, y, z)

= (P s + P 0 )(Q s + Q 0 ) = P s Q s + P 0 (P s + Q s ) + P 2 0 . (5)
Note that this implies a j = 0 for all j except j = 3 and j = s + 3. This means

f (x) = x d + a s+3 x s+3 + a 3 x 3 .
So if f (x) does not have this form, this shows that φ is absolutely irreducible. If on the contrary φ splits as (P s + P 0 )(Q s + Q 0 ), the factors P s + P 0 and Q s + Q 0 are irreducible, as can be shown by using the same argument.

Assume from now on that f (x) = x d + a s+3 x s+3 + a 3 x 3 and that (5) holds. Then a 3 = P 2 0 , so clearly P 0 = √ a 3 is defined over F q . We claim that P s and Q s are actually defined over F 2 . We know from (1) that P s Q s is defined over F 2 . Also P 0 (P s + Q s ) = a s+3 φ s+3 , so P s + Q s = (a s+3 / √ a 3 )φ s+3 . On the one hand, P s + Q s is defined over F 2 k by Theorem 2.1. On the other hand, since φ s+3 is defined over F 2 we may say that P s + Q s is defined over F q . Because (k, n) = 1 we may conclude that P s + Q s is defined over F 2 . Note that the leading coefficient of P s + Q s is 1, so a 2 s+3 = a 3 . Whence if this condition is not true, then φ is absolutely irreducible.

Let σ denote the Galois automorphism x → x 2 . Then P s Q s = σ(P s Q s ) = σ(P s )σ(Q s ), and P s + Q s = σ(P s + Q s ) = σ(P s ) + σ(Q s ). This means σ either fixes both P s and Q s , in which case we are done, or else σ interchanges them. In the latter case, σ 2 fixes both P s and Q s , so they are defined over F 4 . Because they are certainly defined over F 2 k by Theorem 2.1, and k is odd, they are defined over

F 2 k ∩ F 4 = F 2 .
Finally, we have now shown that X either is irreducible, or splits into two absolutely irreducible factors defined over F q .

⊔ ⊓

Remark: For k = 3, the polynomial φ corresponding to f (x) = x 57 + ax 30 + a 2 x 3 where a ∈ F q is irreducible. Indeed if it were not, we would have P 27 and Q 27 defined over F 2 , so by Theorem 2.1 we would have P 27 = p β (x, y, z)p β 2 (x, y, z)p β 4 (x, y, z) and Q 27 = p β 3 (x, y, z)p β 5 (x, y, z)p β 6 (x, y, z)

for some β ∈ F 8 -F 2 . So, up to inversion, we would check that P 27 (x, 0, 1) = (1+x+x 3 ) 9 and Q 27 (x, 0, 1) = (1+x 2 +x 3 ) 9 , hence P 27 (x, 0, 1)+Q 27 (x, 0, 1) = (1 + x + x 3 ) 9 + (1 + x 2 + x 3 ) 9 , and one can check that this is not equal to φ 30 (x, 0, 1) as it should be.


Luca Pezzé

Laurent Sanchez-Palencia

Localized and Extended States in a Disordered Trap

Keywords: numbers: 03.75.-b, 03.75.Ss, 72.15.Rn

We study Anderson localization in a disordered potential combined with an inhomogeneous trap. We show that the spectrum displays both localized and extended states, which coexist at intermediate energies. In the region of coexistence, we find that the extended states result from confinement by the trap and are weakly affected by the disorder. Conversely, the localized states correspond to eigenstates of the disordered potential, which are only affected by the trap via an inhomogeneous energy shift. These results are relevant to disordered quantum gases and we propose a realistic scheme to observe the coexistence of localized and extended states in these systems.

Disorder underlies many fields in physics, such as electronics, superfluid helium and optics [START_REF] Mott | Metal-Insulator Transitions[END_REF][START_REF] Reppy | [END_REF][START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF]. It poses challenging questions, regarding quantum transport [START_REF] Lagendijk | [END_REF] and the interplay of disorder and interactions [5]. In this respect, ultracold gases offer exceptionally well controlled simulators for condensed-matter physics [6] and are particularly promising for disordered systems [7]. They recently allowed for the direct observation of one-dimensional (1D) Anderson localization of matter waves [8][9][10][11]. It should be noticed however that ultracold gases do not only mimic standard models of condensedmatter physics, but also raise new issues which require special analysis in its own right. For instance, they are most often confined in spatial traps, which has significant consequences. On the one hand, retrieving information about bulk properties requires specific algorithms [12]. On the other hand, trapping induces novel effects, such as the existence of Bose-Einstein condensates in low dimensions [13], and suppression of quantum tunneling in periodic lattices [14].

Consider Anderson localization [15]. In homogeneous disorder, linear waves can localize owing to coherent multiple scattering, with properties depending on the system dimension and the disorder strength [START_REF] Mott | Metal-Insulator Transitions[END_REF]. A paradigm of Anderson localization is that localized and extended states generally do not coexist in energy. This relies on Mott's reductio ad absurdum [START_REF] Mott | Metal-Insulator Transitions[END_REF]: Should there exist a localized state and an extended state with infinitely close energies for a given configuration of disorder, an infinitesimal change of the configuration would hybridize them, forming two extended states. Hence, for a given energy, almost all states should be either localized or extended. Exceptions only appear for peculiar models of disorder with strong local symmetries [16]. Then, a question arises: Can inhomogeneous trapping modify this picture so that localized and extended states coexist in energy?

In this Letter, we study localization in a disordered potential combined with an inhomogeneous trap. The central result of this work is the coexistence, at intermediate energies, of two classes of eigenstates. The first class corresponds to states which spread over the full (energydependent) classically allowed region of the bare trap, and which we thus call "extended". The second class corresponds to states of width much smaller than the trap size, which are localized by the disorder, and which we thus call "localized". We give numerical evidence of the coexistence of extended and localized states for different kinds of traps. We show that while the extended states are confined by the trap and weakly affected by the disorder, the localized states correspond to eigenstates of the disordered potential, which are only affected by the trap via an inhomogeneous energy shift. Finally, we propose an experimentally-feasible scheme using energy-selective time-of-flight (TOF) techniques to observe this coexistence with ultracold Fermi gases.

Let us consider a d-dimensional gas of noninteracting particles of mass m, confined into a spatial trap V T (r) and subjected to a homogeneous disordered potential V (r) of zero average, amplitude V R and correlation length σ R . Hereafter, we use "red-detuned" speckle potentials (V R < 0), which are relevant to quantum gases [7,[START_REF]We obtained the same qualitative behavior as reported in the Letter using other models of disorder, such as "bluedetuned" speckle potentials and Gaussian impurities[END_REF]. For the trap, we take V T (r) = ( 2 /2ma 2 )|r/a| α , being a the trap length scale. For instance, α = ∞ and a = L/2 for a homogeneous box of length L, while α = 2 and a = /mω for a harmonic trap of angular frequency ω. We numerically compute the eigenstates |ψ n and eigenenergies E n of the Hamiltonian

Ĥ = -2 ∇ 2 /2m + V (r) + V T (r). (1)
The eigenstates are characterized by their center of mass, r n ≡ ψ n |r|ψ n , and spatial extension (rms size), ∆r n ≡ ψ n |r 2 |ψ n -r 2 n 1/2 . The quantity ∆r n quantifies localization: the smaller, the more localized. Numerical results for the 1D (d=1) case are reported in Fig. 1. In infinite, homogeneous disorder (α = ∞, L = ∞), all states |ψ n are localized, uniformly distributed in space, and, for most models of disorder, their extension ∆z n monotonically increases with the energy [START_REF] Lifshits | Introduction to the Theory of Disordered Systems[END_REF]. As Figs. 1(a),(f) show, using a finite flat box (L < ∞) only induces a trivial finite-size effect: For low- enough energy E, we find ∆z n ≪ L and the states are not significantly affected by the finite size of the box. For larger energies however, boundary effects come into the picture. The states are centered close to the box center and their extension saturates to the value obtained for a plane wave, i.e. ∆z 0 = L/2 √ 3. A central outcome of these results is that the curve giving ∆z n versus E displays a single branch. In particular, there is no energy window where localized and extended states coexist. This finding holds independently of the finite box size and is in agreement with Mott's argument [START_REF] Mott | Metal-Insulator Transitions[END_REF].

For inhomogeneous traps (α < ∞), we find a completely different behavior. The curves giving ∆z n and z n versus E now display two clearly separated branches [see Figs. 1(b)-(e) and (g)-(j)]. For low energy, the states are strongly localized and, for E > 0, they are roughly uniformly distributed in a region bounded by the (energydependent) classical turning points, z cl (E), defined as the solutions of V T (z cl ) = E. For higher energy, the extension of the states corresponding to the upper branch in Figs. 1(b)-(e) grows and eventually saturates to that of the eigenstates of the nondisordered trap, ∆z 0 (E). The centers of mass of these states approach the trap center and form the horizontal branch in Figs. 1(g)-(j). This branch corresponds to extended states. It is easily interpreted in terms of finite-size effects, similarly as for a finite flat box. The lower branch in Figs. 1(b)-(e) is more surprising. It identifies strongly localized states of relatively large energy. It has no equivalent in the flat box and cannot be interpreted as a finite-size effect. The corresponding states are located close to the classi-cal turning points z cl (E) and generate the outer branches in Figs. 1(g)-(j). As Fig. 1 shows, this holds for all inhomogeneous traps. When the trap power α increases, the branch of extended states gets denser at the expense of that of localized states, and completely vanishes for α = ∞ (homogeneous box).

The coexistence of localized and extended states in the same energy window for disordered traps is confirmed on more quantitative grounds in the last row of Fig. 1. It shows the full density of states (solid black line), as well as the density of localized (ρ < , solid red line) and extended (ρ > , dashed blue line) states [START_REF]We use the condition ∆z < 10σR to identify localized states[END_REF]. The different nature of the localized and extended states is even more striking when one studies the wavefunctions. Let us focus for instance on the harmonic trap (α = 2) and on a narrow slice of the spectrum around E ∼ 4|V R |, where ρ < /ρ ≃ 14% of the states are localized [20]. Figure 2(a) shows the spatial density |ψ n (z)| 2 of all states found for a single realization of the disorder. We can clearly distinguish localized (thick red lines) and extended (thin blue lines) states, which shows that they coexist in the same energy window for each realization of the disorder. The localized states are very narrow and present no node (e.g. states A and E) or a few nodes (e.g. states C and H). They may be identified as bound states of the local deep wells of the disordered potential, similarly as the lowest-energy states creating the Lifshits tail in bare disorder [START_REF] Lifshits | Introduction to the Theory of Disordered Systems[END_REF]. To confirm this, let us decompose the eigenstates |ψ n of the disordered trap onto the basis of the eigenstates |χ p of the bare disordered potential [i.e. Hamiltonian (1) with V T ≡ 0], associated to the eigenenergies ǫ p . For a local- [START_REF]For instance, we find En ≃ 3.97[END_REF]. This explains that the localized states are located close to the classical turning points, as observed in Figs. 1 and2(a).

E n ≃ V T (z n )
Let us now decompose the states |ψ n of the disordered trap onto the basis of the eigenstates |ψ 0 p of the bare trap [i.e. Hamiltonian (1) with V = 0], associated to the eigenenergy E 0 p . For a localized state, the distribution is broad. Conversely, for an extended state, the distribution is sharp and peaks at E 0 p ≃ E n to a value equal to a fraction of unity [see Fig. 2(c)]. An extended state may thus be seen as reminiscent of an eigenstate of the bare trap, which is weakly affected by the disorder. Still, the main peak in Fig. 2(c) is smaller than unity. Only for significantly higher energy, the state |ψ n results from weak perturbation of |ψ 0 n , and | ψ 0 p |ψ n | 2 displays a main peak of the order of unity as predicted by standard perturbation theory.

Our results can now be easily interpreted. In bare disorder, the typical size ∆z of the localized states increases faster than the classically allowed region z cl ∝ E 1/α provided by the trap. For low energy, ∆z ≪ z cl so that the states are strongly localized by the disorder and weakly affected by the trap. For higher energy however, the disorder would localize the states on a scale exceeding z cl . The states are then bounded by the trap and the effect of disorder becomes small. This forms the branch of extended states in both the disordered box and traps. In addition, some strongly localized states with very low energy in the bare disorder and located around point z n are shifted by the trap to approximately the energy V T (z n ). This forms the branch of localized states only in disordered traps (α < ∞) since, in the box, a state cannot be placed at intermediate energy due to the infinitely sharp edges. Quantitatively, since the localized states in the bare disorder are uniformly distributed in space, the density of localized states can be estimated to roughly scale as ρ < ∝ (1/α) × E 1/α-1 , which is consistent with the disappearance of the branch of localized states when α grows and with its vanishing for α = ∞ (see Fig. 1). Still, it is striking that localized and extended states can coexist in the same energy window. The disordered potential combined with a smooth trap permits localized states to sit outside the classically allowed region occupied by extended states [see Fig. 2(a)]. Then, the Mott argument does not apply here because the spatial segregation can be strong enough to suppress hybridization for an infinitesimal change of the disorder configuration.

Let us now discuss a possible scheme to observe the coexistence of localized and extended states in a disordered trap. Consider a gas of noninteracting ultracold fermions prepared in a given internal state, at temperature T and chemical potential µ. A class of energies |E n -E| ∆ [see Fig. 3(a)] deep in the Fermi sea (i.e. with µ -E ≪ k B T ) can be selected by applying a spin-changing radio-frequency (rf) field of frequency ν = E/h and duration τ ∼ h/∆ (with h the Planck constant) [14,[START_REF] Greiner | [END_REF]23]. The rf field transfers the atoms of corresponding energies to an internal state insensitive to the disordered trap. The transfered atoms expand freely, which provides their momentum distribution:

D E,∆ (k) ≃ |En-E| ∆ | ψn (k)| 2 , (2)
where ψn (k) is the Fourier transform of ψ n (z) [TOF technique]. In the coexistence region, D E,∆ (k) has two significantly different contributions: For localized states, | ψn (k)| 2 is centered around k ≃ 0 with tails of width ∆k n ∼ ∆z -1 n . Conversely, for extended states, | ψn (k)| 2 is peaked at k ≃ √ 2mE/ with long tails towards small momenta. We however found that averaging over realizations of the disorder blurs the central peak associated to the localized states in D E,∆ (k). In turn, the quan- tity C E,∆ (k) ≡ D E,∆ (k) × D E,∆ (0)/D E,∆ (0) 2 displays two distinct peaks for a rf pulse of realistic durations [see Fig. 3(b)]. The central one is more pronounced for narrower pulses. Selecting either the localized states or the extended states [START_REF]We use the condition ∆z < 10σR to identify localized states[END_REF] confirms that the central peak corresponds to the localized states and the side peak to the extended states [see Inset of Fig. 3(b)].

Finally, we have performed similar calculations as above in a 2D harmonic trap. Figure 4 In conclusion, we have shown that, in a disordered inhomogeneous trap, localized and extended states can coexist in a given energy window. The localized states correspond to eigenstates of the disordered potential which are only affected by the trap via an inhomogeneous energy shift. Conversely, the extended states spread over the classically allowed region of the trap and are weakly affected by the disorder. This effect is directly relevant to presentday experiments with disordered quantum gases, which are most often created in harmonic traps [11,[24][25][26][27]. We have proposed a realistic scheme to observe it in these systems. In the future, it would be interesting to extend our results to higher dimensions and to other kinds of inhomogeneous disordered systems.

We thank T. Giamarchi and B. van Tiggelen for discussions and ANR (Contract No. ANR-08-blan-0016-01), Triangle de la Physique, LUMAT and IFRAF for support.

Figure 1 :

 1 Figure 1: Extension (a)-(e), center of mass (f)-(j) and density of states [DOS] (k)-(o) of the eigenstates versus energy in various kinds of 1D disordered traps. The plots result from accumulation of numerical data over 5000 realizations of a red-detuned speckle potential with mσ 2 R |VR|/ 2 = 0.256. The first column refers to a flat box of length L = 500σR. The curved line in (a) corresponds to an infinite system [28] and the horizontal line is ∆z 0 = L/2 √ 3. The other columns refer to inhomogeneous traps with a = 12.5σR and various trap powers α. The solid lines correspond to the nondisordered case, i.e. ∆z 0 in panels (b)-(e) and ±z cl (E) in panels (g)-(j). The last row shows the full DOS ρ(E) (solid black line), as well as the DOS restricted to localized (ρ<, solid red line) and extended (ρ>, dashed blue line) states [19]. The dot-dashed green lines are the nondisordered limits.

Figure 2 :

 2 Figure 2: Eigenstates for a single realization of a 1D disordered harmonic trap. (a) Non-normalized spatial densities, |ψn(z)| 2 , vertically displaced to their eigenenergy En. Thick red lines correspond to localized states, and thin blue lines to extended states [19]. Note that extended and localized states may occupy almost-degenerate energy levels (e.g. H and I). The states C and D are projected: (b) over the eigenstates of the disordered potential, |χp , and (c) over those of the harmonic trap, |ψ 0 p . The parameters are as in Fig. 1.

Figure 3 :

 3 Figure 3: Scheme to observe the coexistence of localized and extended states in disordered traps (solid red line). (a) Atoms occupying the eigenstates of energy E ±∆ (shaded region) are transfered to a different internal state via rf coupling. The corresponding momentum distribution is then measured by TOF. (b) Correlation function CE,∆(k) (black solid line) and momentum distribution DE,∆(k) (dashed green line, arbitrary units), for ∆ = 2 ω. Inset: CE,∆(k) of all states (solid black line), and separating localized (dashed red line) and extended (dotted blue line) states [19], for ∆ = 0.01 ω. Here E = 4|VR| and the other parameters are as in Fig. 1.

  (a) shows the centers of mass r n of the eigenstates with E n ≃ 4|V R |, the color scale giving ∆r n . Figure 4(b) shows a density plot of ∆r n versus |r n | for the same data. Again, the eigenstates clearly separate into two classes: Some states are extended (large ∆r n ) and centered nearby the trap center (small |r n |). The other states are strongly localized (small ∆r n ) and located nearby the line of classical turning points (|r n | ≃ r cl (E) = 2E/mω 2 ). Hence, the two classes of states can coexist at intermediate energies also in 2D disordered traps.

Figure 4 :

 4 Figure 4: Coexistence of localized and extended states in a 2D disordered harmonic trap for E/|VR| = 4 ± 0.0003. The figure results from accumulation of data from 2 × 10 4 realizations of the disorder, with mσ 2 R |VR|/ 2 = 0.8 and ω = 0.05|VR|/ . (a) Centers of mass rn of the eigenstates and corresponding values of ∆rn/σR in color scale. The solid black line is the line of classical turning points, r cl (E) = 2E/mω 2 ≃ 63.2σR. (b) Extension ∆rn versus distance from the trap center |rn|.