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A two-dimensional hydrogen atom offers a promising alternative for describingthe quantum interaction between an electron and a proton in the presence of astraight cosmic string. Reducing the hydrogen atom to two dimensions enhancesits suited to capture the cylindrical/conical symmetry associated with thecosmic string, providing a more appropriate description of the physical system.After solving Schr\"dinger's equation, we calculate the eigenenergies,probability distribution function, and expected values for the hydrogen atomwith logarithmic potential under the influence of the topological defect. Thecalculations for the 2D hydrogen atom are performed for the first time usingthe Finite Difference Method. The results are presented through graphics,tables, and diagrams to elucidate the system's physical properties. We haveverified that our calculations agree with a linear variational method result.Our model leads to an interesting analogy with excitons in a two-dimensionalmonolayer semiconductor located within a specific semiconductor region. Toelucidate this analogy, we present and discuss some interaction potentials andtheir exciton eigenstates by comparing them with the results from theliterature.

Cosmic string influence on a 2D hydrogen atom and its relationship with the Rytova-Keldysh logarithmic approximation in semiconductors