Patrick Höhener
email: patrick.hohener@univ-amu.fr

Zhi M Li

Maxime Julien

Pierrick Nun

Richard J Robins

Gérald S Remaud

Simulating stable isotope ratios in plumes of groundwater pollutants with BIOSCREEN-AT-ISO

Keywords: Reactive transport, saturated zone, contaminated sites, natural attenuation, stable isotopes

BIOSCREEN is a well-known simple tool for evaluating the transport of dissolved contaminants in groundwater, ideal for rapid screening and teaching. This work extends the BIOSCREEN model for the calculation of stable isotope ratios in contaminants. A threedimensional exact solution of the reactive transport from a patch source, accounting for fractionation by first-order decay and/or sorption, is used. The results match those from a previously published isotope model but are much simpler to obtain. Two different isotopes may be computed, and dual isotope plots can be viewed. The dual isotope assessment is a rapidly emerging new approach for identifying process mechanisms in aquifers. Furthermore, deviations of isotope ratios at specific reactive positions with respect to "bulk" ratios in the whole compound can be simulated. This model is named BIOSCREEN-AT-ISO and will be downloadable from the journal homepage.

Introduction

The BIOSCREEN model had been developed by the US EPA in 1996 [START_REF] Newell | BIOSCREEN Natural Attenuation Decision Support System, User's manual[END_REF] as a user-friendly simulation tool for the evaluation of the transport of dissolved contaminants in groundwater. Under the name of BIOSCREEN-AT, an improved version based on the exact analytical solution for reactive transport from a patch source in 3 dimensions was later published [START_REF] Karanovic | BIOSCREEN-AT: BIOSCREEN with an exact analytical solution[END_REF]) and distributed as an MS EXCEL-based spreadsheet. Within the last 15 years, considerable progress has been made in the analysis of isotope ratios in dissolved groundwater pollutants. Compound-specific isotope analysis (CSIA) of either 13 C, 2 H, or 15 N can be made using isotope ratio monitoring by Mass Spectrometry (irm-MS) (Hofstetter and Berg 2007;[START_REF] Elsner | Stable isotope fractionation to investigate natural transformation mechanisms of organic contaminants: principles, prospects and limitations[END_REF][START_REF] Thullner | Quantification of organic pollutant degradation in contaminated aquifers using compound specific stable isotope analysis -Review of recent developments[END_REF]. This method is able to realize multi-element analyses using a small amount of sample and to determine isotope ratios of mixtures components using Gas Chromatography (GC) or High Performance Liquid Chromatography (HPLC) coupling. The main inconvenient of this method is that it only allows determining the average over the whole molecule isotopic composition, missing the intramolecular distribution of heavy isotopes in the studied compounds. In this context, different methods have been developed in order to perform Position-Specific Isotope Analysis (PSIA). The isotope ratio measurement by 13 C Nuclear Magnetic Resonance Spectrometry (irm-13 C NMR) is a recently developed technique capable of determining the isotopic composition of each carbon position of a large panel of molecules [START_REF] Caytan | Precise and accurate quantitative C-13 NMR with reduced experimental time[END_REF]. In a previous study, irm-13 C NMR has recently proven its interest in the determination of origin of contaminants [START_REF] Julien | Enhanced forensic discrimination of pollutants by position-specific isotope analysis using isotope ratio monitoring by 13 C nuclear magnetic resonance spectrometry[END_REF]) and the study of their remediation (Julien et al. 2015a+b).

Changes in isotope ratios during reactive transport are indicative of reactive processes: bondbreaking processes can cause a large isotope fractionation at the position of the initial bond cleavage, often leading to an enrichment of the remaining non-degraded pollutant.. Smaller secondary isotope fractionation at t sites adjacent to a reactive position can also occur. Also, when the transformation products of pollutants are also components of the primary pollutants in the contamination source, isotope ratio can be used to differentiate the origins of chemicals and provide actual description of reactive processes. Thus, US EPA recommended the use of CSIA to access biodegradation processes and to identify the source of organic groundwater contaminants [START_REF] Hunkeler | A Guide for Assessing Biodegradation and Source Identification of Organic Ground Water Contaminants using Compound Specific Isotope Analysis[END_REF]. CSIA has been largely applied to study natural attenuation processes in contaminated field investigation like discussed in three critical review articles (Meckenstock et al. 2004;[START_REF] Schmidt | Compound-specific stable isotope analysis of organic contaminants in natural environments: a critical review of the state of the art, prospects, and future challenges[END_REF][START_REF] Elsner | Stable isotope fractionation to investigate natural transformation mechanisms of organic contaminants: principles, prospects and limitations[END_REF]). To our knowledge, PSIA has not been applied in field investigation but it represents a promoting trend of isotope fractionation pattern for the study of natural attenuation processes in contaminated groundwater.

Equilibrium sorption can also create small to intermediate isotope fractionation at certain positions of molecules, but unfortunately the effect also causes in most cases an enrichment of the remaining pollutant [START_REF] Kopinke | Carbon isotope fractionation of organic contaminants due to retardation on humic substances: Implications for natural attenuation studies in aquifers[END_REF][START_REF] Höhener | Stable carbon and hydrogen isotope fractionation of dissolved organic ground water pollutants by equilibrium sorption[END_REF], which could be wrongly interpreted as an effect of degradation. In contrast, the physical dilution of compounds should a priori not change isotope ratios [START_REF] Elsner | Stable isotope fractionation to investigate natural transformation mechanisms of organic contaminants: principles, prospects and limitations[END_REF]. In combination, these isotope fractionations can impact on the bulk (average over the whole molecule) isotope ratios observed, but position specific fractionations are inevitably diluted out when only CSIA is exploited. PSIA, in contrast, gives access to the individual values.

Because of the high information content of isotope ratios in contaminants in groundwater plumes, they are often measured to obtain a better understanding of natural attenuation processes.

Moreover, fractionation factors for a large number of bond-breaking reactions and for equilibrium sorption are available [START_REF] Aelion | Environmental Isotopes in Biodegradation and Bioremediation[END_REF][START_REF] Höhener | Stable carbon and hydrogen isotope fractionation of dissolved organic ground water pollutants by equilibrium sorption[END_REF].

Analytical or numerical models for compound-specific isotope ratios in aquifer pollutants have been developed. [START_REF] Höhener | Multidimensional analytical models for isotope ratios in ground water pollutant plumes of organic contaminants undergoing different biodegradation kinetics[END_REF] showed with analytical models on MAPLE worksheets that only the exact analytical solution in BIOSCREEN-AT gives correct isotope ratios at lateral plume margins. However, all current isotope models are quite demanding in operational skills and partly also in CPU time and therefore have not been widely exploited.

The present work here combines the know-how of isotope evolution from the more complex models with the well-known and user-friendly model BIOSCREEN-AT to propose a simple tool predicting isotope ratios in groundwater as a function of time and space. The tool should compute two different isotopes in each compound (e.g. 13 C and 2 H) in order to create the so-called dualisotope plots [START_REF] Vogt | Multi-element isotope fractionation concepts to characterize the biodegradation of hydrocarbonsfrom enzymes to the environment[END_REF]). These plots are very sensitive to different reaction mechanisms.

The overall goal is that a user can rapidly deduce whether their combined data on concentration and isotope ratios prove unambiguously the degradation and/or sorption of the target pollutant in the studied aquifer.

Analytical solution and its implementation

The schematic representation of the pollution scenario in a homogeneous 3-dimensional aquifer is shown in Figure 1. The aquifer is semi-infinite in the x direction, and infinite in the y and z directions. Reactive transport is modeled with the advection-dispersion equation given for the xy-z space as in equation ( 1), under the following assumptions: groundwater flow is steady and uni-directional along the x-axis; material properties are homogeneous; partitioning between dissolved and sorbed phases is instantaneous and reversible; neither gas phases nor volatilization from groundwater is modeled; the solute undergoes degradation following first-order kinetics, but only in the dissolved phase (which is realistic for microorganisms which degrade dissolved pollutants at low environmental concentrations).

C x C v z C v y C v x C v t C R z y x f                    2 2 2 2 2 2 (1)
In Eq. ( 1), v is the unretarded groundwater flow velocity (m yr -1

), 

x is longitudinal dispersivity (m) ,  y , and  z are the transversal dispersivities in the y (horizontal), and z (vertical) direction (m), is a first-order degradation rate (yr -1

), and R f is the retardation factor (see eq. ( 4)). We use the exact solution of eq. ( 1) from [START_REF] Cleary | Analytical models for ground water pollution and hydrology[END_REF] which is equation ( 2):

(2) with and where i stands for a specific isotope,  is a first-order decay rate of the concentration in the source, v' = v/R f,i , and C 0,i is the (constant) concentration of isotope i in the source (mol L -1

).

The use of the exact solution fixes problems associated with the first BIOSCREEN model which was based on Domenico's analytical solution. This issue was broadly discussed in several publications [START_REF] Guyonnet | Dimensionless analysis of two analytical solutions for 3-D solute transport in groundwater[END_REF][START_REF] Srinivasan | Domenico solution -Is it valid?[END_REF][START_REF] West | On the use and error of approximation in the Domenico (1987) solution[END_REF].

For isotope modeling, we use the isotope approach [START_REF] Hunkeler | Modeling chlorine isotope trends during sequential transformation of chlorinated ethenes[END_REF]

 =  (3a) ) , , , , ( , 0 ) , , , ( t z y x i i t z y x i C C                     d v Z z erfc v Z z erfc v Y y erfc v Y y erfc v x R v x t v x y y y y x i f i t x t z y x i                                                                                        ' 2 ' 2 ' 2 5 . 0 ' 2 5 . 0 ' 4 exp 1 ' exp ' 8 2 , 0 5 . 1 ) , , , , ( h  =  react (3a) OC OC b f l K f n R    1 (4a) sorption OC OC b f h K f n R     1 (4b)   0 0 0 1 / R C C l   (5a)   0 0 0 0 1 / R R C C h   (5b)
Here,  b is the soil bulk density (kg L -1 ), n is effective porosity, K OC is the partitioning coefficient of the contaminant between organic carbon and water (L kg -1 ), f OC is the unitless fraction of organic carbon of the aquifer solids, and R 0 is the initial (constant) isotope ratio of the contaminant in the source. Equations 3 and 4 create different transport behavior of light and heavy isotopes, which finally lead to changes in the isotope ratios.

These isotope ratios in delta notation (in ‰) are finally obtained by equation ( 6):

1000 1 ) , , , ( ) , , , ( ) , , , (  
                      standard t z y x l t z y x h t z y x R C C   (6)
where R standard is the isotope ratio of the international standard for the element of interest.

For the purpose of the assessment of degradation, equation ( 7), which computes the percent of degradation B (%) compared to overall concentration decrease in the contaminant, was incorporated into the model:

100 1000 1000 1 (%) / 1000                      reaction source B    (7)
B (%) is mainly caused by biotic reaction, but at some sites also abiotic reactions were found to fractionate isotopes. The equation 7 is only valid when the change in isotope ratios is uniquely caused by reaction. In cases where sorption fractionates isotopes, the B (%) will be wrong. This is illustrated in the spreadsheet because the model gives also the true B (%) obtained from modeling a sorption-affected tracer and equation ( 8). Here, C tracer is a simulated concentration of a sorbing tracer affected by R f . The equations were implemented on a spreadsheet (MS EXCEL, version 2010). The integration of the  factors of equation ( 1) is made in 100 steps of d. In order to test the exactness of the approach implemented in EXCEL, the results were compared to numerical integrations made by MAPLE (version 13, Waterloo Maple Inc, Waterloo, Canada) using the Maple worksheet from Höhener and Atteia (2010) with the exact solution of [START_REF] Wexler | Analytical solutions for one-, two, and three-dimensional solute transport in ground water systems with uniform flow[END_REF].

Example calculations

The presented example here is for methyl tert-butyl ether (MTBE). Fractionation factors of 13 C and 2 H during aerobic degradation were chosen as similar to those found in laboratory experiments [START_REF] Rosell | Variations in C-13/C-12 and D/H enrichment factors of aerobic bacterial fuel oxygenate degradation[END_REF]). MTBE has an intra-molecular variation in 13 C isotope ratio, with the methyl group (in the NMR spectrum position 2 according to the chemical shift) being most negative [START_REF] Julien | Enhanced forensic discrimination of pollutants by position-specific isotope analysis using isotope ratio monitoring by 13 C nuclear magnetic resonance spectrometry[END_REF]. Initial enzymatic attack will occur at this position, leading to the largest isotope fractionation at this C atom. Two scenarios were modeled (see Table 1 for model parameters). Scenario 1 presents an old plume in groundwater with low flow velocity, without any sorption, whereas scenario 2 is a young plume in a faster flowing groundwater where sorption occurs and sorption fractionates the pollutant in addition to the fractionation by degradation. Both scenarios were modeled with the present model (BIOSCREEN-AT-ISO) and the MAPLE model [START_REF] Höhener | Multidimensional analytical models for isotope ratios in ground water pollutant plumes of organic contaminants undergoing different biodegradation kinetics[END_REF]. The results are shown in Figures 2 and3.

Figures 2a and3a show that concentrations along the plume centerline both match exactly the concentrations modeled by the MAPLE worksheet, indicating that the integration of time in the EXCEL model in 100 steps is sufficiently accurate compared to an independent numerical integration both for short and long times (2 and 50 years). The other subsets in Figures 2 and3 show that both models also yield identical results for isotope ratios of 13 C and of 2 H. The modeled curves along the plume centerline indicate that the isotope data enable a clear distinction between scenario 1, wherein only degradation fractionates, and scenario 2, wherein sorption also contributes to fractionation. The increases of the ratios are enhanced in scenario 2, especially at the forerunning plume front. The explanation for this is that only at the fore-running front of young plumes, there are sorption sites unoccupied by the contaminant, and therefore fractionation can occur when the contaminant sorbs to these sites. Near the source, or everywhere in old plumes, sorption sites are already occupied by contaminant, and no fractionation occurs anymore. This had previously been predicted [START_REF] Kopinke | Carbon isotope fractionation of organic contaminants due to retardation on humic substances: Implications for natural attenuation studies in aquifers[END_REF]. Figures 2c and3c show that the slope  in the dual isotope plots  2 H vs  13 C does not lie exactly on the approximation   H/C15 in both cases, and that the present model is a helpful tool to investigate why and how such slopes change as a function of two fractionating processes and of dispersion. It had been shown previously that dispersion must be taken into account for the interpretation of field isotope data, even when dispersion itself does not create fractionation [START_REF] Abe | Does the Rayleigh equation apply to evaluate field isotope data in contaminant hydrogeology?[END_REF]. Finally, Figures 2b and3b show in addition that, for the carbon isotope ratios, a position-specific isotope measurement of  13 C would enhance even more the discriminatory power of the isotope approach, since the  13 C in the reactive position 2 increases by up to 10‰, whereas the bulk  13 C increases only by 2‰. Once progress is made in the purification of samples in order to perform PSIA by NMR (Julien et al. 2015) in real-world groundwater, our model is operational for data interpretation.

Conclusions

To sum up, the BIOSCREEN-AT-ISO model with stable isotopes is validated in this work and can serve in future as a tool for isotope geochemists for the assessment of natural attenuation of dissolved groundwater pollutants during reactive transport. The BIOSCREEN-AT format was chosen because it gained popularity in the community of groundwater remediation and proved to be useful for rapid assessment and teaching. The study of field processes using compoundspecific isotope analysis is recommended by US EPA which published guidelines for isotope data interpretations [START_REF] Hunkeler | A Guide for Assessing Biodegradation and Source Identification of Organic Ground Water Contaminants using Compound Specific Isotope Analysis[END_REF]. Field data of concentrations and isotope ratios are easily assessed with the model, and the contribution of biodegradation to natural attenuation can be quantified at any point in the aquifer. The model predicts dual isotope evolutions in space and time and reinforces interpretations of degradation mechanisms [START_REF] Vogt | Multi-element isotope fractionation concepts to characterize the biodegradation of hydrocarbonsfrom enzymes to the environment[END_REF]. The model is an ideal complement to more sophisticated numerical models: this analytical model is free of numerical dispersion and can be used to validate results from numerical codes for homogeneous cases. More sophisticated numerical approaches would need very time-consuming tailor-made modeling and maybe development of codes, while the use of a spreadsheet model like BIOSCREE-AT-ISO can simulate in some hours a three-dimensional field case and can check whether an investment in more complex models is worthwhile. The limitations of this model compared to numerical approaches are: 1) it is only valid only in homogeneous systems; 2) only for constant or experimentally decaying sources; 3) only for linear sorption isotherms; and 4) only for stable isotopes of C, H, N, and O, but not for Cl which behaves differently [START_REF] Hunkeler | Modeling chlorine isotope trends during sequential transformation of chlorinated ethenes[END_REF]. The BIOSCREEN-AT-ISO is a Microsoft EXCEL spreadsheet compatible with versions 2010 or later and can be downloaded free of charge from the Journal website.

 C react.-pos1/2/3  C sorption  C sorp.-pos1/2/3 (‰) (‰) (‰) (‰)
-2 0/-10/0 na na -2 0/-10/0 -0.3 0/0/-0.5 Hydrogen enrichment factor for degradation for sorption

  H react.-bulk  H sorption (‰) (‰) -30 na -30 -5 Initial isotope ratio  13 C 0 bulk  13 C 0 pos1/2/3  2 H 0 (‰) (‰) (

  l

FIGURE CAPTIONS Fig. 1 :

 CAPTIONS1 FIGURE CAPTIONS

Fig. 2 :

 2 Fig. 2: Results from BIOSCREEN-AT-ISO (this work) compared to results from MAPLE models for the scenario 1: a) concentrations, b) isotope ratios, and c) dual isotope evolution, with linear regression of slope .

Fig. 3 :

 3 Fig. 3: Results from BIOSCREEN-AT-ISO (this work) compared to results from MAPLE models for the scenario 2: a) concentrations, b) isotope ratios, and c) dual isotope evolution, with linear regression of slope .

  Figure 1:

  Fig. 2a

  Fig. 2b

  Fig. 3a

  Fig. 3b

Table 1 :

 1 Aquifer and compound properties used for modeling isotope fractionation in the groundwater plumes with equation (1).

	Aquifer		Notation	Unit	Value	Value
	Parameters					Scenario 1	Scenario 2
	Groundwater		v	m yr -1	180	365
	flow velocity
	Longitudinal		 x	m	10	10
	dispersivity
	Transversal		 y	m	1	1
	dispersivities Ratio of solids to		 z r sw	m kg L -1	0.1 5	0.1 5
	water
	Organic carbon content		f OC	kg -1 kg OC	0	0.05
	Source width and		Y	m	20	20
	depth		Z	m	infinite	infinite
	Compound		MTBE
	parameters
	Concentration of		C 0	m	1	1
	MTBE
	Org. carbon-water		K OC	L kg OC	-1	12	12
	part. coefficient
	Retardation factor		R f	(-)	1	5.8
	Carbon		
	enrichment factor	 C	react.-bulk
	for degradation
	for sorption

Acknowledgments

This work was funded by the French National Research Agency ANR, project ISOTO-POL funded by the program CESA (N° 009 01). M. Julien thanks the ANR for funding his PhD bursary through this project. We thank three unknown reviewers for their helpful comments.

Supplementary Material

The following supplementary material is available for this article: EXCEL Spreadsheets containing the model and the example calculations of scenarios 1+ 2.


Benjamin Landon

Annalisa Panati

Jane Panangaden

Justine Zwicker

Reflection probabilities of one-dimensional Schrödinger operators and scattering theory

The dynamic reflection probability of [DS] and the spectral reflection probability of [GNP, GS4] for a one-dimensional Schrödinger operator H " ´∆ `V are characterized in terms of the scattering theory of the pair pH, H 8 q where H 8 is the operator obtained by decoupling the left and right half-lines R ď0 and R ě0 . An immediate consequence is that these reflection probabilities are in fact the same, thus providing a short and transparent proof of the main result of [BRS]. This approach is inspired by recent developments in non-equilibrium statistical mechanics of the electronic black box model and follows a strategy parallel to [JLPa].

Introduction

One-dimensional Schrödinger operators H " ´∆ `V describe a quantum particle on R under the influence of a potential V . The transmission and reflection properties of such operators concern the behaviour of wave packets originating on, say, the far left of the system and describe whether the packet is transmitted through the system to the right, or reflected back to the left.

The dynamical characterization of transmission and reflection, originating in [DS], uses the solution of the time dependent Schrödinger equation to define states asymptotically concentrated on the left or right in the distant future or past. The notions of reflection and transmission coefficients then enter through the decomposition of a state concentrated asymptotically on the left in the distant past into the superposition of two states concentrated on the left and right in the distant future.

The spectral characterization of transmission and reflection is given in terms of two special solutions of the time independent Schrödinger equation. The reflected and transmitted waves are obtained by expanding one solution as a superposition of the other and its conjugate (for the energies under consideration, each solution is not the multiple of a real-valued solution); in the mathematical literature, this approach originates in [START_REF] Gesztesy | Uniqueness theorems in inverse spectral theory for onedimensional Schrdinger operators[END_REF][START_REF] Gesztesy | Inverse spectral analysis with partial information on the potential, I. the case of an AC Component[END_REF].

Our contribution is to characterize both of these notions of reflection and transmission in terms of the scattering theory of the pair pH, H 8 q where H 8 is the direct sum of the restrictions of H to the left and right half-lines, with Dirichlet boundary conditions. The scattering theory of this pair provides yet another notion of reflection and transmission through the elements of the scattering matrix. In fact, it will be immediate that all three notions coincide thus providing a short and transparent proof of the main result of [BRS].

The consideration of the decoupled operator H 8 appears somewhat unnatural here. This approach originates in studies of the nonequilibrium statistical mechanics of the electronic black box model.

There, the relevent transport phenomena are computed using an associated one-dimensional Jacobi matrix acting on 2 pZq, where the left and right half-spaces correspond to two semi-infinite thermal reservoirs which are initially uncoupled and at thermal equilibrium. The study of the scattering theory of the decoupled and coupled operators thus enters quite naturally. We refer the interested reader to [JLPi] for a complete discussion.

This approach to reflection and transmission was first developed for Jacobi matrices [JLPa] and later extended to the five-diagonal unitary CMV matrices in [CLP]. In all of these cases, the question of whether the dynamical and spectral definitions of reflection and transmission coincide was already answered in [BRS].

The remainder of the paper is organized as follows. In the next section we introduce the different notions of reflection and transmission and state our main results. In the third and final section we provide an elementary derivation of the scattering matrix for the pair pH, H 8 q. The formula for the scattering matrix is known and can be deduced from, e.g., [START_REF] Behrndt | Scattering matrices and Weyl functions[END_REF][START_REF] Behrndt | Scattering theory for open quantum systems with finite rank coupling[END_REF] or from [AP] (see also [AK]). Acknowledgements: The authors thank V. Jakšić for useful and enlightening discussions. The work of B.L. is partly supported by NSERC. The research of A.P. was partly supported by ANR (grant 12-JS01-0008-01).

Main results

Preliminaries

We consider Schrödinger operators H :" ´∆ `V (2.1) on R. We assume that V `P L 1 loc pRq and its negative part V ´obeys Qp´∆q Ď QpV ´q and xϕ, V ´ϕy ď αxϕ, ´∆ϕy `βxϕ, ϕy

for some α ă 1 and all ϕ P Qp´∆q. We assume that V is in the limit point case at ˘8. Under these assumptions H defines a self-adjoint operator bounded from below. For our purposes it will be no loss of generality to assume H ě 0.

(2.3)

We define now H l{r :" ´∆8,l{r `V (2.4) to be the restriction of H to the left and right half spaces L 2 pR l{r , dxq where we denote R l " p´8, 0s and R r " r0, 8q. Here, ´∆8,l{r is the Laplacian with Dirichlet boundary conditions on the left/right half-space.

We summarize now some of the elements of the Weyl-Titschmarsh theory (our notation is based on Appendix A of [START_REF] Gesztesy | Uniqueness theorems in inverse spectral theory for onedimensional Schrdinger operators[END_REF]; complete proofs may be found in the references there, or in [T]). For every z P CzR there are unique solutions u l{r pzq of the ODE In fact, it follows from [GS1, GS3, T] that ρ l{r is the spectral measure for H l{r and δ 1 0 . Here, δ 1 0 P H ´2pH l{r q where H s pH l{r q is the scale of spaces associated to H l{r and the associated spectral measure is defined as in [S]. The vector δ 1 0 is the continuous linear functional on H 2 pH l{r q defined by xδ 1 0 , φy " φ 1 p0q on test functions φ P C 8 0 pRq Ă H 2 pH l{r q . The associated spectral measure is defined as in [

Hu " zu, up0q "

Reflection probabilities

We discuss first the spectral reflection probability of [START_REF] Gesztesy | One-dimensional scattering theory for quantum systems with nontrivial spatial asymptotics[END_REF][START_REF] Gesztesy | Inverse spectral analysis with partial information on the potential, I. the case of an AC Component[END_REF]. For any λ P R s.t. m l{r pλ `i0q exists, we define the two solutions of Hψ l{r " λψ l{r (2.12) by requiring

ψ l{r pλ, 0q " 1, ψ 1 l pλ, 0q " ´ml pλ `i0q, ψ 1 r pλ, 0q " m r pλ `i0q.
(2.13) Let S l{r be an essential support of the a.c. spectrum for H l{r . For a.e. λ P S l we have that ψ l pλq is not a multiple of a real solution and is therefore linearly independent of ψl pλq. For such λ we can expand ψ r pλq " Apλq ψl pλq `Bpλqψ l pλq.

(2.14)

Definition 2.1. For a.e. λ P S l the spectral reflection coefficient (relative to incidence from the left) is defined by

R psq l pλq :" Bpλq Apλq .
(2.15)

and the spectral reflection probability is defined by

|R psq l pλq| 2 . For λ R S l set R psq l pλq " 1. If R psq l
pλq " 0 a.e. on a Borel set e then H is said to be spectrally reflectionless on e.

Using (2.13) one obtains for λ P S l , R psq l pλq " ´mr pλ `i0q `m l pλ ´i0q m l pλ `i0q `mr pλ `i0q .

(2.16)

We now discuss the dynamical reflection probability of [DS,BRS]. Let j x ď 1 be a smooth function that is 0 on p´8, xs and 1 for rx `1, 8q. The strong limits P ȓ :" s ´lim tÑ˘8 e itH j x e ´itH P ac pHq, P l :" s ´lim tÑ˘8 e itH p1 ´jx qe ´itH P ac pHq, (2.17) exist and do not depend on the choice of x. The operator P l{r is the projection onto the space H l{r where H l :"

" ϕ P H ac pHq : @x, lim tÑ˘8 ||j x e ´itH ϕ|| " 0 * (2.18)
is the set of states concentrated asymptotically on the left in the distant future/past, and H ȓ is given by a similar definition with j x Ñ 1 ´jx . Davies and Simon proved the following:

Theorem 2.2 (Theorem 3.3 of [DS]). Under our hypotheses on V we have the following decomposition for the absolutely continuous subspace of H.

H ac pHq " H l ' H r " H ĺ ' H ŕ . (2.19)
Note by time reversal invariance, H l{r " Ě H r{l . We will see later that the restriction of H to H l{r has simple spectrum. As each of the spectral projections commutes with H, the operator P l P ĺ P l acts as an operator of multiplication by a function we denote by |R pdq l pλq| 2 . This function is defined for a.e. λ in an essential support of the a.c. spectrum of the restriction of H to H l . We will see later that this is precisely S l . Following [BRS] we have the following definition. Remark. In [BRS] H is said to be dynamically reflectionless on a Borel e Ď R if P e pHqP l " P e pHqP ŕ (where P e pHq is the spectral projection onto e) and e is contained up to a set of measure 0 in an essential support of the a.c. spectrum of H. We will see later that |R pdq l pλq| 2 ‰ 1 only on S l X S r and on this set the operator P r P ŕ P r acts by multiplication by the same function |R pdq l pλq| 2 . It follows that this definition coincides with ours. 1As we will see, the link between the dynamical and spectral notions of reflection is provided by scattering theory. To be more precise, we will prove:

Theorem 2.4. The spectral and dynamic reflection probabilities are identical,

|R psq l pλq| 2 " |R pdq l pλq| 2 .
(2.20)

Moreover, they are given by any diagonal element of the scattering matrix of the pair pH, H 8 q, which will be defined below. Consequently, H is dynamically reflectionless on a Borel set e iff it is spectrally reflectionless on e.

Scattering theory of decoupled Schrödinger operators

In this section we review the elements of scattering theory we need and provide a proof of Theorem 2.4. Recall our definition of H 8 as the direct sum of the restriction of H to the right and left half-lines.

For the difference of the resolvents we have the formula [S] pH ´zq ´1 ´pH 8 ´zq ´1 " G 00 pzq ´1|pH ´zq ´1δ 0 yxpH ´zq ´1δ 0 | (2.21) which holds in particular for z " ´1. Recall that δ 0 P H ´1pH q and so pH ´zq ´1δ 0 is a bonafide element of H. From (2.21) we see immediately that the wave operators [RS] w ˘:" s ´lim tÑ˘8

e itH e ´itH8 P ac pH 8 q (2.22) exist and are complete (complete meaning that Ran w ˘" H ac pHq). The scattering matrix s :" w ˚w´( 2.23) is a unitary operator taking H ac pH 8 q to itself. Moreover, it commutes with H 8 and hence acts as multiplication by a 2 ˆ2 matrix which we denote by spλq on H ac pH 8 q " H ac pH l q ' H ac pH r q -L 2 pR, dρ l,ac q ' L 2 pR, dρ r,ac q.

(2.24)

Let us elucidate the above identification. First, the spectral measure for δ 1 0 and H l{r is defined as in [S]. Secondly, δ 1 0 is cyclic for H l{r in the sense that the linear span of pH l{r ´zq ´1δ 1 0 : z P CzR

( (2.25)
is dense in L 2 pR ď0 , dxq{L 2 pR ě0 , dxq [GS1, [START_REF] Gesztesy | Uniqueness theorems in inverse spectral theory for onedimensional Schrdinger operators[END_REF][START_REF] Teschl | Mathematical methods in quantum mechanics[END_REF]. From this the identification in (2.24) follows.

We now give a formula for the scattering matrix. This formula is known and can be deduced as a special case from general results on extension theory of symmetric operators in [START_REF] Behrndt | Scattering matrices and Weyl functions[END_REF] (see also Theorem 4.5 of [START_REF] Behrndt | Scattering theory for open quantum systems with finite rank coupling[END_REF] as well as related results of [AK, AP]). We include an elementary derivation in Section 3 for completeness. Before we state the formula we apply a unitary transformation taking the space on the RHS of (2.24) to p H :" L 2 pR, η l dλq ' L 2 pR, η r dλq (2.26)

where η l{r is the characteristic function of S l{r (the transformation is given by multiplication by the square root of the appropriate Radon-Nikodyn derivative). The formula we give is for the scattering matrix acting on this transformed space. This transformation is essentially notational and is standard in scattering theory; with this convention, the scattering matrix is a unitary 2 ˆ2 matrix for every λ.

It is clear that the value of the diagonal elements of the scattering matrix are unaffected.

Theorem 2.5. The scattering matrix for the pair pH, H 8 q acting on p H is given by

s ab pλq " δ ab `2iG 00 pλ `i0q a Imrm a pλ `i0qsImrm b pλ `i0qs (2.27)
for a, b P tl, ru.

Note that by unitarity of the scattering matrix we have |s ll pλq| 2 " |s rr pλq| 2 and by the above formula both are identically 1 except for λ P S l X S r .

¿From the point of view of scattering theory it is natural to introduce |s ll pλq| 2 as the reflection probability and introduce the definition of reflectionless operators as in the Jacobi and CMV cases. The following proof shows that all of these definitions coincide. Proof of Theorem 2.4. From the formula [START_REF] Gesztesy | Uniqueness theorems in inverse spectral theory for onedimensional Schrdinger operators[END_REF] G 00 pzq " ´1 m l pzq `mr pzq

(2.28)

we see |R psq l pλq| 2 " |s ll pλq| 2 .

(2.29)

Next we claim that we can replace the smooth cut-off function j a in the definition of the asymptotic projections. Let χ 0 be the characteristic function of r0, 8q. We claim that s ´lim tÑ8 e itH pj x ´χ0 qe ´itH P ac pHq " 0.

(2.30)

It suffices to prove that ||pj x ´χ0 qe ´itH P ac pHqϕ|| Ñ 0 as t Ñ 8 for ϕ P DpHq. Let ψ " pH `1qϕ and C " pj x ´χ0 qpH `1q ´1. Then, ||pj x ´χ0 qe ´itH P ac pHqϕ|| " ||Ce ´itH P ac ψ||.

(2.31) By Theorem 3.2 of [DS], C is compact (trace-class even), hence by a well-known fact the RHS then goes to 0 as t Ñ 8 (see Lemma 2 on page 24 of [RS]).

We may therefore compute P l{r with χ pl{rq 0 instead of j a and 1 ´ja , where χ pl{rq 0

is the characteristic function of p´8, 0s and r0, 8q respectively. Since H 8 commutes with χ pl{rq 0

we immediately obtain that P l{r " w ˘χl{r 0 w ˚.

(2.32)

As a consequence, w ˘is a unitary map from H ac pH l{r q to H l{r intertwining H 8 and H. From this we see that the restriction of H to H l{r has simple spectrum and that S l{r is an essential support of the a.c. spectrum for this restriction.

The formula (2.32) and the definition of the dynamical reflection probability yields

|R pdq l pλq| 2 " |s ll pλq| 2 (2.33)
for a.e. λ P S l . Since the RHS is identically 1 outside of S l we see that the above equality holds a.e. A similar calculation yields that P r P ŕ P r acts by multiplication by the same function (recall

|s ll | " |s rr |).
Hence we see that |R pdq l pλq| 2 ‰ 1 only on S l XS r justifying the remark following Definition 2.3. We conclude Theorem 2.4.

We would like to give a final comment on the decomposition of a ψ P H ĺ as ψ " P l ψ `P r ψ. From the above proof we see that w ˘δ1 0 is cyclic for the restriction of H to H l{r (where we are implicitly thinking of δ 1 0 as an element of H ´2pH l{r q). After applying a unitary map (again, multiplication by the square root of the appropriate Radon-Nikodyn derivative) to associated spectral representation, we get a representation of (the restriction of) H as multiplication by λ on L 2 pR, η l{r pλqdλq. In this representation, P l{r ϕpλq " s l,l{r pλqϕpλq. This motivates our choice of notation for the dynamical reflection probability as |R pdq l pλq| 2 , as we could define R pdq l pλq as the coefficient of P l ϕ which is now seen to be s ll pλq.

Derivation of scattering matrix

In this section we give an elementary derivation of Theorem 2.5. By the invariance principle [RS] the wave operators and scattering matrix of the pair pH, H 8 q is the same as that for the pair p´pH 1q ´1, ´pH 8 `1q ´1q so we compute the latter. For notational convenience we denote R " ´pH `1q where in the last line we use the fundamental theorem of calculus and then replace the integral lim uÑ8 ş u ´u by an Abel sum, which is allowed as the integrand is bounded and we know the limit exists. By (2.21) and the fact that G 00 pzq ´1pH ´zq ´1δ 0 " pH l ´zq ´1δ 1 0 ' pH r ´zq ´1δ 1 0 (3.3) (this can be derived from the formulas in Appendix A of [START_REF] Gesztesy | Uniqueness theorems in inverse spectral theory for onedimensional Schrdinger operators[END_REF] in the same way that the half-line case is done in [START_REF] Gesztesy | Rank one perturbations at infinite coupling[END_REF] -it is ultimately a simple consequence of the Wronskian formula for the Green's function) we see that the integrand in the last line of (3.2) is given by ie ´ |u| xf l , e iuR8 pR 8 ´Rqe ´iuR w ´gl y " ´ie ´ |u| xf l , e iuR8 pH l `1q ´1δ 1 0 yxRδ 0 , w ´e´iuR8 g l y.

(3.4)

We evaluate the latter inner product following an argument similar to [JKP]. Let h l " e ´iuR8 g l . We have, xRδ 0 , w ´hl y " lim tÑ8 xRδ 0 , e ´itR e itR8 h l y " xRδ 0 , h l y ´lim

1 Ó0 ż 8 0 e ´ 1 t ixRδ 0 , e ´itR pR ´R8 qe itR8 h l ydt (3.5)
where we have used the FTC and replaced the integral lim tÑ8 ş t 0 by its Abel sum. Using (2.21) and (3.3) we see that the integral in the last line of (3.5) equals ż 8 0 e ´ 1 t ixRδ 0 , e ´itR pR ´R8 qe itR8 h l ydt "

ż 8 0 ie ´ 1 t xδ 0 , Re ´itR Rδ 0 y "ż R e ´itpλ`1q ´1 h l pλq dρ l,ac pλq λ `1  dt. (3.6)
Everything here is absolutely integrable so we use Fubini and evaluate the t integral yielding

ż R h l pλqi ż 8 0 xδ 0 , Re ´itpR`pλ`1q ´1´i 1 q Rδ 0 ydt dρ r,ac pλq λ `1 " ż R h l xδ 0 , RpR `pλ `1q ´1 ´i 1 q ´1Rδ 0 y dρ l,ac pλq λ `1 . (3.7)
We now concern ourselves with the limit 1 Ó 0. First note that lim 1 Ó0

xδ 0 , RpR `pλ `1q ´1 ´i q ´1Rδ 0 y " ´G00 p´1q `lim

1 Ó0
xδ 0 , pH ´λ ´i pH `1qpλ `1qq ´1δ 0 y (3.8) whenever the right hand limit exists. In a moment we will prove that lim

1 Ó0 xδ 0 , pH ´λ ´i 1 pH `1qpλ `1qq ´1δ 0 y " lim 1 Ó0
xδ 0 , pH ´λ ´i 1 q ´1δ 0 y " G 00 pλ `i0q (3.9) whenever the limit on the RHS exists and is finite. Suppose for the remainder of the proof that g l is nonzero only on a set on which the RHS of (3.9) converges to G 00 pλ `i0q uniformly. By the DCT we see that we can pass the limit 1 Ó 0 through the integral and we obtain xRδ 0 , w ´e´iuR8 g l y " ´żR e iupλ`1q ´1 G 00 pλ `i0qg l pλq dρ l,ac pλq λ `1 .

(3.10) Plugging this into (3.4) and using Fubini we see that xf l , ps ´1qg l y " lim To complete the computation we use the fact that lim Ó0 ż R e iuppλ`1q ´1´pλ 1 `1q ´1q´ |u| du Ñ δppλ `1q ´1 ´pλ 1 `1q ´1q " pλ `1q 2 δpλ ´λ1 q.

(3.12)

To be more precise, since gpλqG 00 pλ `i0q dρ l,ac dλ pλqpλ `1q ´1 is in L 1 pR, dxq it is not too hard to see that ż g l pλqG 00 pλ`i0q ż R e iuppλ`1q ´1´pλ 1 `1q ´1q´ |u| du dρ l,ac pλq λ `1 Ñ g l pλ 1 qG 00 pλ 1 `i0q dρ l,ac dλ pλ 1 qpλ 1 `1q (3.13) strongly in L 1 provided that we also take g l to have compact support. If we then chose f l so that f l pλ 1 q dρ l,ac dλ pλ 1 q is bounded with f l also of compact support then we can pass the above limit through the λ 1 integration and obtain xf l , ps ´1qg l y " ´i ż R ˆdρ l,ac dλ pλq ˙2 fl pλqg l pλqG 00 pλ `i0qdλ.

(3.14)

We have therefore obtained the formula for the special class of f l and g l described in the proof. The extension to all of L 2 is a density argument which we omit. It remains to obtain (3.9). With µ 0 the spectral measure for pH, δ 0 q we have |xδ 0 , pH ´λ ´i pH `1qpλ `1qq ´1δ 0 y ´xδ 0 , pH ´λ ´i pλ `1q 2 q ´1δ 0 y| ď ż pλ ´λ1 qpλ `1q |pλ 1 ´λ ´i pλ `1q 2 qpλ 1 ´λ ´i pλ 1 `1qpλ `1qq| dµ 0 pλ 1 q ď ż pλ `1q |λ 1 ´λ ´i pλ `1q

  Definition 2.3. The dynamical reflection probability is defined for a.e. λ P S l by |R pdq l pλq| 2 and by 1 for λ R S l . If |R pdq l | 2 vanishes a.e. on a Borel set e then we say that H is dynamically reflectionless on e.

  1q ´1´pλ 1 `1q ´1´u| | du  dρ l,ac pλq λ `1 dρ l,ac pλ 1 q λ 1 `1 (3.11)

  |λ 1 ´λ ´i pλ `1q 2 | dµ 0 pλ 1 q ` C δ (3.15)for any δ ą 0. The last integral is bounded byC λ a 2 `δ2 ImrG 00 pλ `i pλ `1q 2 qs (3.16)and the claim follows.

  S] (see Section 2.3 for further details). Being Herglotz functions, the boundary values Imrm l{r pλ ˘i0qs :" lim Nikodyn derivative of the a.c. part of ρ l{r . Lastly, we define the Green's function by G xy pzq " xδ x , pH ´zq ´1δ y y.(2.11)

	Ó0	Imrm l{r pλ ˘i qs	(2.9)
	exist and are finite for Lebesgue a.e. λ. Moreover,
	Imrm l{r pλ `i0qs "	dρ l{r,ac dλ	pλq	(2.10)
	is the Radon-

  We compute only s ll pλq. The other elements are the same. For f l and g l P H ac pH l q we have xf l , ps ´1qg l y " xf l , pw ˚´w ˚qw ´gl y

			´1,	R 8 " ´pH 8 `1q ´1.	(3.1)
	" lim uÑ8	xf l , pe iuR8 e ´iuR ´e´iuR8 e iuR qw ´gl y
	ż
	" lim Ó0	R	ie ´ |u| xf l , e iuR8 pR 8 ´Rqe ´iuR w ´gl ydu	(3.2)

These observations have reduced the question of the equivalence of our definition and that of[BRS] to the fact that for two orthogonal projections we have P `Q " 1 iff P QP " 0 and p1 ´P qp1 ´Qqp1 ´P q " 0.


Madelyn Yiseth

Rojas Castro

Marina Travanca
email: marina.travanca@u-bordeaux.fr

Marta Avalos Fernandez
email: marta.avalos-fernandez@u-bordeaux.fr

David Valentin Conesa
email: david.v.conesa@uv.es

Ludivine Orriols
email: ludivine.orriols@u-bordeaux.fr

Emmanuel Lagarde
email: emmanuel.lagarde@u-bordeaux.fr

MAVIE-Lab Sports: a mHealth for Injury Prevention and Risk Management in Sport PhD Student Track

Keywords: • Mathematics of computing → Bayesian networks, • Humancentered computing → Personal digital assistants, App, eHealth, Injury, HLIs, Prediction

Smart-phones technology and the development of mHealth (Mobile Health) applications offer an opportunity to design intervention tools to influence health behavior changes. The MAVIE-Lab is a mHealth application including a DSS (Desicion Support System) to assist in the personalized evaluation of HLIs (Home, Leisure and Sport Injuries) risk and to promote the adoption of prevention measures. MAVIE-Lab Sports will be the first module of the mobile application.

The purpose of this PhD project is to improve a particular module of MAVIE-Lab, devoted to sports (MAVIE-Lab Sports), in different aspects: statistical modeling, design and ergonomics. It also aims to evaluate system usability, acceptability, safety and efficacy. The development structure proposed and executed in this thesis will be replicated for the development of future modules for different types of HLIs. This document develops the argument, objectives and advances in the development of the MAVIE-Lab Sports and the future work.

INTRODUCTION

Injuries are a main concern in Public Health, leading cause of mortality, leaving further major consequences on the quality of life of the population. Each year in France, HLIs cause more than 11 million victims and 20,000 deaths [START_REF] Jean-Baptiste | Les accidents en France: évolution et factores associés[END_REF]. In Horizon 2020 call for Research and Innovation actions, the European Commission promoted the development of DSSs to empower individuals to self-management their health condition [START_REF]Horizon 2020 Work Programme 2014[END_REF].

This context offers a novel opportunity to develop the MAVIE-Lab, an innovative mHealth for primary prevention of HLIs.

The MAVIE-Lab has been developed under the MAVIE project framework (http://www.observatoire-mavie.com/), a large webbased cohort study with the objective of prospectively collecting data related to HLIs in France. Currently, over three years of recruitment, 27 000 volunteers have already been enrolled in the cohort.

A first version of MAVIE-Lab Sport is based on this available MAVIE volunteers information. This prototype is available online, and includes: running, hiking, road cycling, downhill skiing and basketball (https://ssl3.isped.u-bordeaux2.fr/MAVIE-OBS/Appli/#/). However, the statistical models currently included in the App are non-optimal model predicting personal risk. Besides, it did not achieve a design, structure and popularity for a potential success as intervention tool.

This thesis aims to maximize the potential of MAVIE-Lab Sports. The first objective is to select and to implement an appropriate methodology to develop a personalized prediction risk model using MAVIE cohort data together with experts information. In addition, the thesis also aims to improve the App scientific value, design and ergonomics and to evaluate it among MAVIE volunteers.

PhD Reports DH'18, April 23-26, 2018, Lyon, France DSSs are intended to induce beneficial behavior changes, specifically, the use of a DSS with the possibility of virtually experiencing solutions is hypothesized to enhance perceived susceptibility, perceived vulnerability, perceived behavioral control of self-efficacy [START_REF] David | Application of Behavior-Change Theories and Methods to Injury Prevention[END_REF][START_REF] Chou Wen Ying Sylvia | Web 2.0 for Health Promotion: Reviewing the Current Evidence[END_REF]. Our assumptions are that good personalized advice will call the attention of the users. The exposition to this DSS could induce change in risk behaviors and the adoption of safety tools.

METHODOLOGICAL PROPOSAL 2.1 MAVIE-Lab Sports Development

2.1.1 Data Sources. The MAVIE cohort is composed of approximately 6,000 practitioners over 15 years old who have authorized the use of their data. MAVIE survey includes detailed information: demographic, life-style, health data, environment and sport practice. In addition, there are specific variables about training, coaching, protective clothing, equipment, among others. Moreover, there are HLIs injuries follow-up, together with information about their causes, consequences and severity.

2.1.2 Modeling. We suggest implementing Bayesian Networks models (BN) using re-sampling techniques [START_REF] Randa | Bayes classifiers for imbalanced traffic accidents datasets[END_REF]. BN combine quantitative and qualitative modeling, probabilistic conditional dependencies and causal diagrams reasoning DAGs (Directed acyclic Graph). It allows us to combine previous qualitative causal relationships between variables (risk and protection injury factors), quantitative previous knowledge and MAVIE data evidence.

2.1.3 Design and software. The BN model relations and risk predictions will be used to build a DSS to produce adequate personal prevention advice. This DSS system will allow the user to virtually experiment the impact of a range of preventive decisions. The functions that have been designed for this objective are:

(1) A graphical overview of each risk as compared to the average level of risk between different sports. (2) An estimation of the personal injury risk.

(3) Experience potential risk change for a set of proposed behavioral changes, protective devices, equipment or sport practice environments.

Graphic design, software programming and use of technologies for the development of Apps is carried out as an interdisciplinary work in the IETO (Injury Epidemiology, Transport, Ocupation) team following the practices proposed by HAS (The French National Authority for Health) to mHeath development [START_REF]Good Practice Guidelines on Health Apps and Smart Devices[END_REF].

2.1.4 Evaluation. The evaluation will be carried out among the volunteers of the MAVIE cohort. It will include the following components:

(1) Safety. Experts will evaluate if the advice and recommendations given by the App are suitable for users health. (2) Acceptability. A questionary will be integrated into the application asking about user perception. (3) Usability. Tracking statistics of Google Analytics which are being taken for the current version of MAVIE-Lab. (4) Efficacy. Change in prevention behaviors and reduction of injury rates will be evaluated comparing MAVIE-Lab users and non-users, before and after MAVIE-Lab implementation.

RESULTS AND ONGOING WORK

The most important work done so far relates to the development of the first version of MAVIE-Lab Sports. Nevertheless, the main thesis progress has been the statistical methodological proposal for the improvement of prediction models in the App. BN elicitation requires a complete review of scientific literature, results of similar studies and expert information to correctly quantify the different degrees of knowledge of risk and protection factors and their relationships in terms of probability. Currently, we are working in DAGs structures using literature information. On the other hand, we are working on an elicitation protocol adapted to risk injury evaluation according to recommendations of previous experts elicitation applications [START_REF] Anne | The use of expert elicitation in environmental health impact assessment: a seven step procedure[END_REF].

CONCLUSION

MAVIE-Lab is a novel idea for safety promotion and HLIs prevention. The key expected result is to achieve a validated and functional version of the MAVIE-Lab Sports that would motivate users to change their behavior regarding injury prevention. The research aims to evaluating whether the exposition to a DSS displaying personalized risk could generate behavioral changes leading to the reduction of the effective number of victims, which remains a major public health problem.

COMPETING INTEREST

The authors have declared that no competing interests exist.

ETHICAL CONSIDERATIONS

The MAVIE project has already received the approval of the CNIL Commission Nationale Informatique & Libertés (France), which will be updated to take into account the specificity of the MAVIE-Lab, in particular on the issue of the provision of personalized advice without the possibility of human intervention. MAVIE volunteers signed online the informed consent to participate in the research.