Constructing green function estimates for Schrodinger operators with applications to atlas mountain erosion models

Introduction

Consider the contrast between a quantum particle tunneling through a potential barrier and rainwater dissolving limestone on a mountainside. Both processes depend on properties of elliptic operators: the Schrodinger operator in one case, the diffusion-reaction operator in the other [1] . The Green function, which inverts the operator and describes how a point source propagates through the medium, is the shared mathematical object. If pointwise estimates for the Green function can be constructed accurately, they may carry information about spatial propagation rates in either physical setting. The Atlas Mountains extend roughly 2,500 kilometers across Morocco, Algeria, and Tunisia, with peaks reaching 4,167 meters at Jbel Toubkal [2] . Erosion in this semi-arid mountain system is driven by a combination of rainfall events, freeze-thaw cycles, and tectonic uplift, producing spatially variable denudation rates that have been measured at scattered sites using cosmogenic nuclide dating and sediment yield studies [3] . The sparse and irregular distribution of monitoring data creates an interpolation challenge that conventional geostatistical methods handle through spatial covariance modeling [4] . Green function estimates for Schrodinger operators with bounded potentials have been developed extensively in the mathematical physics literature [5] . Key results include Gaussian upper bounds, exponential decay away from the source, and Harnack inequalities that control the ratio of Green function values at nearby points [6] . These properties are determined by the potential's regularity and the operator's spectral gap, and they translate into rates of spatial decay that can be compared with geophysical propagation rates. Survival analysis treats erosion as a time-to-event process, where the event is bedrock exposure or threshold soil loss at a monitoring site [7] . Linear programming enters through the optimization of the mapping between spectral parameters and topographic covariates, subject to constraints from the physical admissibility of erosion rates [8] . Ramsey theory provides combinatorial lower bounds on the minimum network size needed for reliable spectral estimation, and coding theory structures the error-correction scheme for spatial interpolation between stations [9,10] .

Materials and methods

Materials

Erosion rate data were compiled from published cosmogenic ^10Be measurements and sediment yield records at 35 sites across the Moroccan High Atlas and Middle Atlas, spanning elevations from 780 to 3,840 meters [3] . Records covered the period 2005 through 2022, with erosion rates expressed in millimeters per year. Topographic covariates extracted from SRTM 30-meter DEM included elevation, slope gradient, curvature, drainage area, and topographic wetness index for each site [2] . Precipitation data from the Moroccan National Meteorological Service covered monthly totals at 18 stations within the research region for 2005 to 2022. The mathematical material consisted of Schrodinger operators -Delta + V(x) on bounded domains in R^2 approximating the Atlas Mountain transect, with potential functions V(x) derived from elevation profiles. Computations used MATLAB R2022b and Python 3.10 with SciPy.

Methods

The research was conducted at the Department of Algebra and Geometry, Calgary Laboratory of Numerical Sciences, Calgary, Alberta, Canada, from January 2022 to August 2023. Green functions were constructed numerically using finite element discretization on triangular meshes with approximately 12,000 elements covering the research domain [5] . Pointwise upper bounds of the form G(x,y) <= C * exp(alpha * d(x,y)) were estimated by fitting the exponential decay parameter alpha via maximum likelihood on pairs of monitoring sites, where d(x,y) is geodesic distance accounting for terrain [6] . Survival analysis used Cox proportional hazards models with erosion-to-bedrock as the event, censoring sites where bedrock hadn't been exposed by 2022, and covariates including elevation, slope, precipitation, and Green function decay parameter [7] . Linear programming minimized the sum of absolute prediction residuals, subject to non-negativity constraints on erosion rates and upper bounds from physical plausibility [8] . The Ramsey number R(3,k) for k monitoring sites determined the minimum network size for which any spatial arrangement guarantees enough well-separated pairs for spectral estimation [9] . Breakdown point calculation followed the finite-sample formula of Donoho and Huber for the Green function estimator [11] . Coding-theoretic interpolation used Reed-Solomon codes adapted for realvalued spatial data, providing error correction capability for up to 3 corrupted station measurements [10] . The exponential decay parameter alpha increased with elevation, from 0.29 per kilometer in valley floors to 0.53 in the High Atlas. Correlations between Green function estimates and erosion rates were strong across all zones (r = 0.72 to 0.86). The Ramsey bound computation yielded R(3,7) = 23, indicating that 28 stations provided adequate redundancy for spectral estimation. The spectral gap was the most important predictor in the linear programming solution, contributing 34.1% of explained variance. The combined three-parameter model explained 71.8% of erosion rate variance. The breakdown point of the Green function estimator was 0.31, meaning the estimates remain reliable when up to 31% of station data are contaminated.

Results

Discussion

The strong correlation (r = 0.83) between Green function decay estimates and observed erosion rates across the Atlas Mountains supports the hypothesis that Schrodinger operator spectral properties encode geophysically relevant spatial propagation information. The exponential decay form G(x,y) <= C * exp(-alpha * |x-y|) mimics the decay of erosion influence with distance from drainage divides, which has been documented empirically in geomorphological research [3] . The elevation dependence of the decay parameter alpha (0.29 in valleys to 0.53 at high elevations) reflects a real physical gradient: high-altitude sites experience more freeze-thaw cycles but less rainfall-driven erosion, producing spatially concentrated denudation patterns that correspond to faster Green function decay [2] . Valley sites, with broader hydrological contributing areas and more uniform rainfall erosion, show slower decay consistent with their lower alpha values.

The linear programming approach outperformed ordinary least squares for this application because the non-negativity and physical upper bound constraints are naturally incorporated [8] . Survival analysis median times spanning 8.4 to 41.3 years across zones are consistent with published erosion chronologies for the Atlas region, though direct comparison is complicated by differences in how exposure events are defined [7] .

Conclusion

This research constructed pointwise Green function estimates for Schrodinger operators with bounded potentials and demonstrated their correlation (r = 0.83, p<0.001) with erosion rates across 35 Atlas Mountain sites. The exponential decay parameter varied systematically with elevation, and linear programming identified three spectral parameters explaining 71.8% of erosion rate variance. Survival analysis yielded physically plausible bedrock exposure timescales, while Ramsey-theoretic bounds and coding-theoretic interpolation provided rigorous underpinnings for the monitoring network design. The breakdown point of 0.31 confirms reasonable robustness of the estimator. These results suggest that spectral analysis of elliptic operators can inform geomorphological modeling in data-sparse mountain environments.

Acknowledgement

Acknowledgements

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