Reduction to a Fredholm integral equation and numerical solution of the inverse Cauchy problem for the Schrödinger-Pauli equation

This paper studies the inverse Cauchy problem for the two-dimensional Schrödinger-Pauli equation, which models a spin- <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:math> {1 2} quantum particle in a magnetic field. The problem involves reconstructing an inaccessible boundary condition from overdetermined data, a severely ill-posed inverse problem. We develop a numerical method combining Lavrentiev regularization with Haar wavelet discretization, yielding a regularized Fredholm equation solved via an efficient collocation scheme with explicit matrix entries. Numerical results demonstrate first-order convergence and robustness to noise up to 10%. Notably, multi-frequency solutions exhibit enhanced noise stability compared to single-mode cases. The method provides a stable, efficient framework for boundary reconstruction in quantum systems with partial data.